README.md

---
tags:
- sentence-transformers
- sentence-similarity
- feature-extraction
- generated_from_trainer
- dataset_size:264888
- loss:CosineSimilarityLoss
base_model: sentence-transformers/all-MiniLM-L6-v2
widget:
- source_sentence: "latex_in_original_or_summarized: K(M, n)\n\n[SEP]\n\nsummarized:\
    \ $K(M, n)$\n\n[SEP]\n\nmain_note_content: Chain complexes and  spaces.   [59],\
    \ that for  simplicial sheaf   $\\text{X}$ we denote by $C_{*}(\\mathcal{X})$\
    \ the (normalized) chain complex  $C_{*}(\\mathcal{A}$  associated to the   sheaf\
    \  abelian groups   $\\mathbb{X}$. This  defines a functor\n\n$$  C_{*}: \\Delta^{o\
    \ p} S h v_{N i s}\\left(S m_{k}\\right)  C_{*}(\\text{A} b(k))  $$$ ^f7eebc\n\
    \nwhich is well  (see $[44,59]$  instance) to have a right adjoint\n\n6.2 \\mathbb{A}^{1}$-Derived\
    \ Category   Spaces\n161\n\n$$  K: C_{*}(\\mathcal{A} b(k)) \\rightarrow \\phi^{o\
    \ p} S h v_{N i s}\\left(S   $$ \n\n\ncalled the  space \n\nFor an abelian  $M\
    \   b(k)$ and an integer $n$ we define the pointed simplicial sheaf $K(M, n)$\
    \ (see [59, page 56])   $K$ to the shifted complex $M[n]$,  the complex $M$ placed\
    \ in degree 0 . If n< 0, the space $K(M, n)$ is a point. If $n \\geq 0$ then $K(M,\
    \ n)$ has only one non-trivial  sheaf which is the  and which is canonically isomorphic\
    \ to $M$. More generally, for a chain  $C_{*}$,   $K C_{*}$ has   homotopy sheaf\
    \ 0  $n< 0$, and the $n$-th homology sheaf $H_{n}\\left(C_{*}\\right)$ for $n\
    \ \\geq 0$.\n\nIt is clear that $C_{*}: \\Delta^{o p} S h  i s}\\left(S m_{k}\\\
    right) \\rightarrow  b(k))$ sends simplicial weak equivalences to quasi-isomorphisms\
    \ and $K: C_{*}(A b(k)) \\rightarrow \\Delta^{o p} S h v_{N i s}\\left(S m_{k}\\\
    right)$ maps quasi-isomorphisms to simplicial  equivalences. If $C_{*}$  fibrant,\
    \ it follows that $K\\left(C_{*}\\right)$ is simplicially  Thus the two functors\
    \ induce a pair of adjoint functors\n\n$$  C_{*}: \\mathcal{H}{s}(k) \\rightarrow\
    \ D(\\mathcal{A} b(k))  $$ ^c4a825\n\n\n\n$$  K: D(\\mathrm{A} b(k)) \\rightarrow\
    \ \\mathcal{H}_{s}(k)  $$ \n\nAs a consequence it is clear that   is an $\\mathscr{A}^{1}$-local\
    \ complex,  space $K\\left(C_{*}\\right)$ is an $\\mathbb{A}^{1}$-local space.\
    \ Thus $C_{}: \\mathbf{H}_{s}(k) \\rightarrow   maps $\\mathcal{A}^{1}$-weak \
    \ to $\\mathrm{A}^{1}$-quasi  and induces a functor\n\n    \\rightarrow D_{\\\
    mathbb{A}^{1}}(A b(k))  $$ \n\nwhich in concrete terms, maps a space $\\operatorname{X}$\
    \ to the $\\mathbb{A}^{1}$-localization of $C_{*}(\\mathcal{X})$. We denote the\
    \ latter by $C_{*}^{A^{1}}(\\mathbb{X})$ and call it the $\\mathbb{A}^{1}$-chain\
    \  of $\\mathcal{X}$.  functor $C_{*}^{\\operatorname{A}^{1}}: \\mathfrak{H}(k)\
    \ \\rightarrow  b(k))$ admits as right adjoint the functor $K^{\\mathbb{A}^{1}}:\
    \ D_{\\mathbb{A}^{1}}(\\mathcal{A} b(k)) \\rightarrow \\mathcal{H}(k)$ induced\
    \ by $C_{*} \\mapsto K\\left(L_{\\mathbb{A}^{1}}\\left(C_{*}\\right)\\right)$.\
    \ We  that for an $\\mathbb{A}^{1}$-local complex  the space $K\\left(C_{*}\\\
    right)$ is automatically $\\mathbb{A}^{1}$-local and thus simplicially equivalent\
    \ to the space \n\n\n[SEP]\n\nprocessed_content: the pointed simplicial  where\
    \ $M$ \\in  b(k)$ and $n$ is  integer. It is defined by applying  to the  complex\
    \ $M[n]$, of the complex    degree 0 ."
  sentences:
  - "latex_in_original_or_summarized: \\gamma_1=(m_1,N_1,a_1)\n\n[SEP]\n\nsummarized:\
    \ $\\gamma_1=(m_1,N_1,a_1)$\n\n[SEP]\n\nmain_note_content: \\begin{notation}\\\
    label{Dep1}\nLet $\\gamma_1=(m_1,N_1,a_1)$,  $\\gamma_2=(m_2,N_2,a_2)$ be an ordered\
    \ pair of \n(generalized) monodromy data which  hypothesis (A). Assume that $m_1|m_2$.\n\
    Set $d:=m_2/m_1$ and $r:=\\gcd(m_1, a_1(N_1))$.  \nThen, \\eqref{Dep}  to \n$\\\
    epsilon=d(r-1)$ and $g_3=dg_1+g_2+\\epsilon$.\nIn particular, $\\epsilon=0$ if\
    \ and  if $r=1$. \n\\end{notation}\n\n\n[SEP]\n\nprocessed_content: "
  - 'latex_in_original_or_summarized: \langle u\rangle  G W(F)


    [SEP]


    summarized: $\langle u\rangle \in G W(F)$


    [SEP]


    main_note_content: Let us denote (in  characteristic) by $G W(F)$ the Grothendieck-Witt
    ring of isomorphism classes of non-degenerate symmetric bilinear forms [48]: this
    is the group completion of the commutative monoid of isomorphism classes of non-degenerate
    symmetric  forms for the direct sum.


    For $u \in F^{\times}$, we denote by $\langle u\rangle  G W(F)$ the form on  vector
    space of rank one  given by $F^{2}  F,(x,  \mapsto u x y .$ By the results of
    loc.   \langle u\rangle$ generate $G  as a group. The following Lemma is (essentially)
    [48, Lemma (1.1) Chap. IV]:



    [SEP]


    processed_content: '
  - 'latex_in_original_or_summarized: $\varepsilon_{\infty}$


    [SEP]


    summarized: $\varepsilon_{\infty}$


    [SEP]


    main_note_content: To compute the genus of $X(\kappa)$, further specialize to
    $\Gamma_{1}=\Gamma$ and $\Gamma_{2}=$ $\mathfrak{SL}_{2}(\mathbb{Z}) . Let $y_{2}=\mathrm{SL}_{2}(\mathbb{Z})
    i, y_{3}=\mathrm{SL}_{2}(\mathbb{Z}) \mu_{3}$, and $y_{\infty}=\mathfrak{SL}_{2}(\mathbb{Z})
    \infty$ be the elliptic point of period 2, the elliptic point of period 3, and
    the cusp of $X(1)=$ SL_{2}(\mathbb{Z}) \backslash \mathcal{H}^{*} .$ Let $\varepsilon_{2}$
    and $\varepsilon_{3}$ be the number of elliptic points of $\Gamma$ in $f^{-1}\left(y_{2}\right)$$
    and of^{-1}\left(y_{3}\right)$, i.e., the number of elliptic points of period
    2 and 3 in $X(\Gamma)$, and let $\varepsilon_{\infty}$ be the number of cusps
    of X(\Gamma) .$ Then recalling that $d=\operatorname{deg}(f)$ and letting $h=2$
    or $h=3$, the formula for $d$ at the beginning of the section and then the formula
    for $e_{\pi_{1}(\tau)}$ at the nonelliptic points and the elliptic points over
    $\mathrm{SL}_{2}(\mathscr{Z}) y_{h}$ show that (Exercise 3.1.3(a))


    $$ d=\sum_{x \in f^{-1}\left(y_{h}\right)} e_{x}=h \cdot\left(\left|f^{-1}\left(y_{h}\right)\right|-\varepsilon_{h}\right)+1
    \cdot \varepsilon_{h} $$


    and using these equalities twice gives

    $$ \sum_{x \in f^{-1}\left(y_{h}\right)}\left(e_{x}-1\right)=(h-1)\left(\left|f^{-1}\left(y_{h}\right)\right|-\varepsilon_{h}\right)=\frac{h-1}{h}\left(d-\varepsilon_{h}\right)
    $$


    $68 \quad 3$ Dimension Formulas


    Also.

    $$ \sum_{x \in f^{-1}\left(y_{\infty}\right)}\left(e_{x}-1\right)=d-\varepsilon_{\infty}
    $$

    Since $X(1)$ has genus 0, the Riemann-Hurwitz formula now shows



    [SEP]


    processed_content: '
- source_sentence: "latex_in_original_or_summarized: $M_\\ell(C \\to S) = M_\\ell(S)$\n\
    \n[SEP]\n\nsummarized: $M_\\ell(C \\to S) = M_\\ell(S)$\n\n[SEP]\n\nmain_note_content:\
    \ If $C \\to S$ is a relative smooth proper curve of genus $g \\geq 1$ over an\
    \ irreducible base, then the $\\ell$-torsion of  relative Jacobian of $C$    information\
    \ about the family.  Suppose $\\ell$ is invertible on $S$, and let  \\in S$ be\
    \ a geometric point.  The fundamental group $\\pi_1(S,s)$ acts\nlinearly on the\
    \ fiber $\\operatorname{Pic}^0(C)[\\ell]_{s} \\cong (\\mathbb{Z}/\\ell)^{2g}$,\
    \ \none can consider the mod-$\\ell$  representation associated to $C$:\n\n$$\\\
    rho_{C \\to S, \\ell}:\\pi_1(S,s) \\rightarrow  \\cong \\operatorname{GL}_{2g}(\\\
    mathbb{Z}/\\ell).$$ ^e59a92\n\nLet $M_\\ell(C \\to S)$, or simply $M_\\ell(S)$,\
    \ be the image\nof this representation. \nIf a primitive $\\ell$th root of  is\
    \ defined   $S$, then $\\operatorname{Pic}^0(C)[\\ell]_{s}$ is equipped\nwith\
    \ a skew-symmetric form $\\langle \\cdot,\\cdot  and $M_\\ell(C \\to S) \\subseteq\n\
    \\operatorname{Sp}(\\operatorname{Pic}^0(C)[\\ell]_s,\\langle  \\rangle) \\cong\n\
    \\operatorname{Sp}_{2g}(\\mathbb{Z}/\\ell)$. \nIf C \\to S$ is a sufficiently\
    \ general family of curves, then\n$M_\\ell(C \\to S) \\cong \\operatorname{Sp}_{2g}(\\\
    mathbb{Z}/\\ell)$ \\cite{delignemumford}.\n\nIn this  we compute  when $S$ is\
    \ an irreducible component of  moduli space of hyperelliptic or  trielliptic curves\
    \ and $C \\to S$ is the tautological curve.  The first result implies that there\
    \ is no restriction on the monodromy group in the hyperelliptic case other than\
    \ that it preserve the symplectic pairing. As  trielliptic curve is a $\\mathbb{Z}/3$-cover\
    \ of a genus zero curve,  the $\\mathbb{Z}/3$-action constrains the monodromy\
    \ group to lie in a unitary group associated to $\\mathbb{Z}[\\zeta_3]$. The second\
    \ result implies that this is the only additional restriction in the trielliptic\
    \ case.   \n\n\\paragraph{Theorem \\ref{thhe}}\n{\\it \n $\\ell$ be an odd prime,\
    \ and let $k$ be an  closed  in which $2\\ell$ is invertible.\nFor $g\\geq 1$,\
    \ $M_\\ell(\\mathcal{H}_g\\otimes k)\\cong\n\\operatorname{Sp}_{2g}(\\mathbb{Z}/\\\
    ell)$.}\n\n\\paragraph{Theorem \\ref{thtri}}\n{\\it \nLet $\\ell\\geq 5$ be prime,\
    \ and let $k$ be   closed field in which $3\\ell$ is invertible.  \n$\\mathcal{T}^{\\\
    bar\\gamma}$ be any component  the moduli space \ntrielliptic curves of genus\
    \ $g\\geq  Then\n$M_\\ell(\\mathcal{T}^{\\bar\\gamma}\\otimes k) \\cong\n\\operatorname{SG}_{(r_\\\
    gamma,s_\\gamma)}(\\mathbb{Z}/\\ell)$ (where the latter is  unitary group defined\n\
    in \\eqref{eqdefsg}).}\n\n\\medskip\n\nWe also prove that the $\\ell$-adic monodromy\
    \ group  \n$\\operatorname{Sp}_{2g}(\\mathbb{Z}_\\ell)$ in the situation of Theorem\
    \ \\ref{thhe} and is $\\operatorname{SG}_{(r_\\gamma,s_\\gamma)}(\\mathbb{Z}_\\\
    ell)$\nin the  of Theorem \\ref{thtri}.\n\nTheorem \\ref{thhe} is an unpublished\
    \ result  J.K. Yu and has already been used multiple times in  literature.\nIn\
    \ \\cite{chavdarov}, Chavdarov assumes this result  show that the numerator of\
    \ the zeta function of\nthe typical hyperelliptic curve over a finite field is\
    \ irreducible.\nKowalski also uses this result in a similar fashion \\cite{kowalskisieve}.\n\
    The first author used Theorem  to prove a conjecture of  and\nWashington on class\
    \  of quadratic function fields \n\nThere are other results in the literature\
    \ which  similar to Theorem \\ref{thhe}\nbut which are not quite strong enough\
    \ for the  above.\nA'Campo \\cite[Th.\\ 1]{acampo} computes the topological  of\
    \ $\\mathcal{H}_g \\otimes  \nOn the arithmetic side, the $\\mathbb{Q}_\\ell$,\n\
    as opposed to $\\mathbb{Z}_\\ell$, monodromy of $\\mathcal{H}_g$\nis computed\
    \ in \\cite[10.1.16]{katzsarnak}.  Combined with a theorem of\nLarsen on compatible\
    \ families of representations \\cite[3.17]{larsenmax},\nthis shows that the mod-$\\\
    ell$  group \nof $\\mathcal{H}_g$ is maximal for a set of\nprimes $\\ell$ of density\
    \ one (as opposed to for all $\\ell \\geq 3$). \n\nThere are results on $\\mathbb{Q}_\\\
    ell$-monodromy  cyclic covers of the projective\nline of arbitrary degree, e.g.,\
    \  \\cite[Sec. 7.9]{katztwisted}.  Also,\nin \\cite[5.5]{fkv}, the authors prove\
    \ that the projective representation\n$\\mathbb{P} \\rho_{C \\to S,\\ell}$  surjective\
    \ for many\nfamilies of cyclic covers  the projective line.  \nDue to a combinatorial\
    \  their theorem does not apply to $\\mathcal{H}_g$\nand applies to at most one\
    \ component of the moduli space of\ntrielliptic curves for each  see Remark \\\
    ref{Rfkv}.   \nSee also work of Zarhin, e.g., \\cite{zarhincyclic}.\n\n an application,\
    \ for all $p \\geq   show using \n  exist hyperelliptic and trielliptic curves\n\
    of every genus  signature) defined over $\\bar{\\mathbb{F}}_p$ whose Jacobians\
    \  absolutely simple.\nIn contrast with the applications above, \nthese corollaries\
    \ do not use the full strength of our results.\nRelated  can be found in \\cite{HZhu}\
    \   authors produce curves with absolutely  \nJacobians over $\\mathbb{F}_p$ under\
    \ the  $g \\leq 3$.\n\n\\paragraph{Corollary \\ref{Chypabsirr}} \n{\\it Let p\
    \ \\not = 2$  let   Then there exists a\nsmooth hyperelliptic curve of genus $g$\
    \  over $\\bar{\\mathbb{F}}_p$ whose Jacobian is\nabsolutely simple.}\n\n\\paragraph{Corollary\
    \ \\ref{Ctriabsirr}}\n{\\it Let $p \\not = 3$.   $g  3$ and   be a trielliptic\
    \ signature for $g$\n \\ref{Dtrisig}).  \nThen there exists a smooth trielliptic\
    \ curve defined over  with genus $g$ and signature $(r,s)$\nwhose Jacobian is\
    \  simple.}\n\n\\medskip \n\nOur proofs proceed by induction on the genus.\nThe\
    \ base cases for the  family\nrely on the fact that every curve of genus $g=1,2$\
    \ is hyperelliptic;\nthe claim on monodromy follows from the analogous assertion\
    \  the monodromy of $\\mathcal{M}_g$.\nThe  case  for the trielliptic family involves\
    \ a comparison with\na Shimura variety of PEL type, namely, the  modular variety.\
    \   \nAn important step is to show  the monodromy group does not change in the\
    \ base cases when  \none adds a labeling of the ramification points to the moduli\
    \ problem.\n\nThe  step is similar to the method used in \\cite{ekedahlmono} \n\
    and uses the fact that families of smooth hyperelliptic (trielliptic)\ncurves\
    \ degenerate to trees of  (trielliptic) curves of lower genus.\nThe combinatorics\
    \ of admissible degenerations require us \nto compute the monodromy exactly for\
    \ the inductive step rather than up to isomorphism.  \n\nThe inductive strategy\
    \ using admissible degeneration developed here\nshould work for other  of curves,\
    \ especially for more general\ncyclic covers of  projective   The difficulty is\
    \ in  direct\ncalculation of monodromy for the necessary base cases.\n\nWe thank\
    \ C.-L.\\ Chai, R.\\ Hain, A.J.\\ de Jong, E. Kani, and J. Kass.\n\n\n[SEP]\n\n\
    processed_content: the image of the mod-$\\ell$  representation $\\rho_{C \\to\
    \  \\ell}$ of the relative smooth   $C \\to S$ of genus $g \\geq 1$ over an irreducible\
    \ base."
  sentences:
  - "latex_in_original_or_summarized: X^{\\vee}\n\n[SEP]\n\nsummarized: \n\n[SEP]\n\
    \nmain_note_content: Let  be  principally polarized abelian scheme of\nrelative\
    \ dimension $g$ over an irreducible base.  \n\nIf $\\ell$ is a\nrational  invertible\
    \ on $S$, then the $\\ell$-torsion $X[\\ell]$ of\n$\\ell$ is an \\'etale cover\
    \ of  with geometric fiber isomorphic to\n$(\\mathbb{Z}/\\ell)^{2g}$.  \nLet $s$\
    \ be a geometric point of $S$.  The  group $\\pi_1(S,s)$ \nlinearly on the $\\\
    ell$-torsion of $X$.\n\nThis yields a representation\n\n\\rho_{X \\to S, s,\\\
    ell}: \\pi_1(S,s) \\rightarrow \\operatorname{Aut}(X[\\ell]_s) \\cong \\operatorname{GL}_{2g}(\\\
    mathbb{Z}/\\ell).$$ ^dbec50\n\nThe cover $X[\\ell] \\to S$ both determines and\
    \ is determined by  representation  \\to S, s,\\ell}$. \n\nThe image of  \\to\
    \ S,  is the mod-$\\ell$ monodromy of $X \\to S$ and we denote it by $M_\\ell(X\
    \ \\to S, s), or by $M_\\ell(S,s)$ if the choice of\nabelian scheme is clear.\n\
    \nThe isomorphism class of the\n$M_\\ell(S,s)$ is independent of the choice of\
    \ base point $s$,$ and we denote it  $M_\\ell(S)$.\n\nLet $X^{\\vee}$ be the dual\
    \ abelian scheme.  There  a  pairing $X[\\ell] \\times X^{\\vee}[\\ell] \\to \\\
    boldsymbol{\\mu}_{\\ell,S}$, where  := \\boldsymbol{\\mu}_\\ell \\times S$ is\
    \  group scheme of $\\ell\\th$  of unity.\n\n polarization  induces an isomorphism\
    \ $X \\to X^{\\vee}$, and\nthus a skew-symmetric pairing $X[\\ell] \\times X[\\\
    ell] \\to \\boldsymbol{\\mu}_{\\ell,S}$.\nBecause the polarization is defined\
    \ globally, the image of monodromy\n$M_\\ell(X \\to S, s)$ is contained in the\
    \ group of symplectic\nsimilitudes of $(X[\\ell]_s,\n\\langle  \\rangle_\\phi)$,\
    \ which is isomorphic to\n$\\operatorname{GSp}_{2g}(\\mathbb{Z}/\\ell)$.  Moreover,\
    \ if a primitive $\\ell^{{\\rm  root of\nunity  globally on $S$,  $\\pi_1(S,s)$\
    \ acts trivially on\n$\\boldsymbol{\\mu}_{\\ell,S}$ and $M_\\ell(X \\to S,s) \\\
    subseteq  \\cdot,\\cdot \\rangle_\\phi) \\cong \\operatorname{Sp}_{2g}(\\mathbb{Z}/\\\
    ell).\n\nSimilarly, the  $X[\\ell^n]  S$ defines a monodromy representation \n\
    with  in $\\operatorname{Aut}(X[\\ell^n]_s) \\cong\\operatorname{GL}_{2g}(\\mathbb{Z}/\\\
    ell^n)$. Taking\n inverse limit over all n, we obtain a continuous representation\
    \ on the Tate module of $X$, \n\n$$\\rho_{X \\to S,  s}: \\pi_1(S,s) \\rightarrow\
    \ \\varprojlim_n \\operatorname{Aut}(X[\\ell^n]_s) \\cong \\operatorname{GL}_{2g}(\\\
    mathbb{Z}_\\ell).$$\n\n^f6240a\n\nWe denote the image of this representation by\
    \ $M_{\\mathbb{Z}_\\ell}(X \\to   and its isomorphism class by $M_{\\mathbb{Z}_\\\
    ell}(X \\to S)$ or $M_{\\mathbb{Z}_\\ell}(S)$.  \n\nAgain, there is an  \nM_{\\\
    mathbb{Z}_\\ell}(X \\to S) \\subseteq   \n\nIf\n$F$ is a field,  let $F_{\\ell^\\\
    infty} = F(\\boldsymbol{\\mu}_{\\ell^\\infty}(\\bar F))$. If $S$ is an  then \n\
    \n$$M_{\\mathbb{Z}_\\ell}(X \\to S, s)/  F} \\to S \\otimes{\\bar F}, s) \\cong\
    \  ^dd1bab\n\nFinally, let $M_{\\mathbb{Q}_\\ell}(X\\to$ S, s)$ be the Zariski\
    \ closure of  \\to S, s)$ in $\\operatorname{GL}_{2g}(\\mathbb{Q}_\\ell)$.\n\n\
    Now suppose that \\psi:C \\to S$ is a relative proper semi-stable curve.\n\nLet\
    \ $\\operatorname{Pic}^0(C) := \\operatorname{Pic}^0_{C/S}$ be the neutral component\
    \ of the relative Picard  of $C$ over $S$.  Since $C/S$  semi-stable, $\\operatorname{Pic}^0(C)$\
    \ is a semiabelian scheme [[bosch_lutkebohmert_raynaud_nm_Theorem 1_page_259|\\\
    cite[9.4.1]{blr}]].  \n\nSuppose that there is  least one geometric point  such\
    \  the fiber $\\operatorname{Pic}^0(C_s)$ is an abelian variety.  (This is true[^5]\
    \ if some $C_s$ is a tree  smooth curves.)  Then there is a nonempty open subscheme\
    \ $S^*$ of $S$ such that $\\operatorname{Pic}^0(C|_{S^*})$  an abelian scheme\
    \ over $S^*$.  \n\n[^5]: cf. Abelian varieties isogenous to a Jacobian by CL Chai,\
    \ which talks about a tree of smooth curves having a Jacobian that is an abelian\
    \ variety that is actually the product of the Jacobians of  irreducible \n\nWe\
    \ define the mod-$\\ell$ and $\\mathbb{Z}_\\ell$ monodromy representations of\
    \ $C$ to be those of $\\operatorname{Pic}^0(C|_{S^*}) \\to S^*$.\n\n(Alternatively,\
    \  may  constructed as the restrictions of $R^1\\psi_*\\boldsymbol{\\mu}_{\\ell,S}$\
    \ and $R^1\\psi_*\\boldsymbol{\\mu}_{\\ell^\\infty,S}$   largest subscheme of\
    \ $S$ on which these sheaves are unramified.)\n\nThus, $M_\\ell(C \\to  s) = M_\\\
    ell(\\operatorname{Pic}^0(C|_{S^*}) \\to S^*, s)$, and we denote this again by\
    \ M_\\ell(S,s) if the curve is clear and by   the base point is suppressed. ^37a851\n\
    \nThe moduli spaces $\\overline{\\mathcal{M}}_G$ and $\\widetilde{\\mathcal{M}}_G$\
    \ are Deligne-Mumford stacks, and we employ a similar formalism for \\'etale covers\
    \ of stacks \\cite{noohi}.  \n\n $\\mathcal{S}$  a connected Deligne-Mumford \
    \ The category of Galois \\'etale covers of $\\mathcal{S}$ is a Galois category\
    \  the sense of Grothendieck, and thus there is  \\'etale fundamental\n of   More\
    \ precisely, let $s\\in \\mathcal{S}$ be a geometric\n \n\nThen there is a group\
    \ $\\pi_1(\\mathcal{S},s)$ and an equivalence of  between finite $\\pi_1(\\mathcal{S},s)$-sets$\
    \ and finite \\'etale Galois covers of $\\mathcal{S}$. \n\nIf $\\mathcal{S}$ has\
    \ a coarse moduli space $S_{\\mathrm{mod}}$, then $\\pi_1(\\mathcal{S},s)$ is\
    \ the extension of $\\pi_1(S_{\\mathrm{mod}},s)$ by a group which encodes  extra\
    \ automorphism structure on the moduli space S_{\\mathrm{mod}} [[noohi_fgas_thm\
    \ 7.11|\\cite[7.11]{noohi}]]. \n\nIf $X \\to \\mathcal{S}$ is a family of abelian\
    \ varieties, we again let $M_\\ell(X\\to  be the  of $\\pi_1(\\mathcal{S}, s)$\
    \ in    ^758472\n\nLet $\\mathcal{C}^\\gamma$ be the tautological labeled curve\
    \ over\n  By the mod-$\\ell$ or $\\mathbb{Z}_\\ell$ monodromy of\n$\\widetilde{\\\
    mathcal{M}}_G^\\gamma$ we mean  of $C^\\gamma \\to \\widetilde{\\mathcal{M}}_G^\\\
    gamma$.  [^6]\n\n[^6]: #_meta/TODO/question  that  that $C^\\gamma \\to \\widetilde{\\\
    mathcal{M}}_G^\\gamma$ gets to have  relative Picard group of its own? How does\
    \ that make sense when $\\widetilde{\\mathcal{M}}_G^\\gamma$ a is not a scheme?\n\
    \n\n[SEP]\n\nprocessed_content: the dual abelian scheme of the abelian scheme\
    \ $X/S$. There is a canonical pairing $X[\\ell] \\times X^{\\vee}[\\ell] \\to\
    \ \\boldsymbol{\\mu}_{\\ell,S}$, where $\\boldsymbol{\\mu}_{\\ell,S} := \\boldsymbol{\\\
    mu}_\\ell \\times S$ is  group scheme of $\\ell\\th$ roots of unity."
  - "latex_in_original_or_summarized: \\mathbb{Th}_f \\phi\n\n[SEP]\n\nsummarized:\
    \ $_f \n\n[SEP]\n\nmain_note_content: It  be convenient to work in  stable  category\
    \ $\\mathcal{Spt}(B)$$ of $P^1$-spectra over $B$, where $B$ is a finite type scheme\
    \ over   frequently, $B=L$, where $L$ is a field extension of $k$. \n\nThe notation\
    \   be   the morphisms. $(B)$ is a  monoidal category under  smash product $\\\
    wedge$, with  $1_B$, denoting the sphere spectrum. \n\nAny pointed simplicial\
    \ presheaf $X$ determines  corresponding $\\mathbb{P}^1$-suspension spectrum $\\\
    Sigma^{\\infty} X$. \n\nFor  $\\Sigma^{} Spec L_+  1_L$ and $\\Sigma^{\\infty}\
    \ (^1_L)^{ n}$ is a suspension   When working in $\\operatorname{Spt}(L)$, we\
    \ will identify pointed  $X$ with their  spectra $\\Sigma^{} X$, omitting the\
    \ $\\Sigma^{\\infty}$.  ^1246cf\n\nWe will use  six operations $(p^*,  p_!, p^!,\
    \ \\wedge,  given by Ayoub   developed by Ayoub, and Cisinksi-Déglise \\cite{CD-triang_cat_mixed_motives}.\
    \ There  a nice summary in \\cite[\\S  \n\nWe use  following associated notation\
    \ and constructions. \n\nWhen   \\to Y$ is smooth, $p^*$ admits a left  denoted\
    \ p_{\\sharp}, induced by  forgetful functor  \\to \\operatorname{Sm}_{Y}$ from\
    \ smooth  over $X$  smooth schemes over $Y$.  \n\nFor $p:X\\to \\operatorname{Spec}\
    \ L$ a smooth scheme over $L$, the suspension spectrum of $X$ is canonically identified\
    \ with  as an object of $\\operatorname{Spt}(L)$. \n\nFor a vector bundle $p:E\
    \ \\to X$, the Thom spectrum  Th(E)$ (or just  is canonically identified  $s^*p^!\
    \ 1_X$[^2]. \n\n Perhaps $s$  a fixed section of $p$.$\n\nLet $\\Sigma^E$ equal\
    \ $\\Sigma^E = s^* p^!: (X) \\to (X)$. Let $e:  \\to X and $d: D  Y$ be two vector\
    \ bundles over smooth  $p: X   L$ and $q:Y  \\operatorname{Spec} L$.  ^123eb1\n\
    \nGiven a map $f: Y \\to X$ and a monomorphism $\\phi: D \\hookrightarrow f^*\
    \  there is an associated natural transformation  ^0f1ba8\n\n$$_f \\phi :   q^!\
    \  p_! \\Sigma^E p^!$$\n\nof endofunctors on $(L)$ inducing the map on Thom spectra.\
    \ The    \\phi$ is defined as  composition  ^0b33ea\n\n\\begin{equation}\\operatorname{Th}_f\
    \  =  {1_{f^*E}} \\circ  .\\end{equation}$$\n\nThe natural  $\\operatorname{Th}_{1_Y}\
    \  is the composition  t^*d^!  t^* ^!e^!\\to t^* \\phi^* e^! \\cong  e^!,$$ where\
    \ $t:   D$ denotes the zero section of $D$, $s: X \\to E$ denotes the zero   $E$,\
    \ and the middle arrow is  by the exchange transformation $\\phi^! \\cong   \\\
    to 1^! \\phi^* \\cong    natural transformation $\\operatorname{Th}_f   the composition\
    \ \n\n$$\\begin{equation}\\operatorname{Th}_f 1: q_! \\Sigma^{f^* E} q^! \\cong\
    \ p_!   f^! p^! \\cong p_!^E f_! f^! p^! {\\rightarrow} p_! ^E p^!,\\end{equation}$$\n\
    \nwhere $: f_! f^! \\to 1$ denotes the counit.\n\n\n[SEP]\n\nprocessed_content: "
  - "latex_in_original_or_summarized: j_0: \\mathbb{G}_m / \\bar{k} \\subset \\mathbb{A}^1\
    \ / \n\n[SEP]\n\nsummarized: $j_0$\n\n[SEP]\n\nmain_note_content: In order to\
    \ explain the simple underlying ideas, we will admit four statements, and explain\
    \ how to deduce from them equidistribution theorems about the sums $S(M, k, \\\
    chi)$ as $\\chi$ varies.\n\n(1) If $M$ and $N$ are both perverse on $\\mathbb{G}_m\
    \ / k$ (resp. on $\\mathbb{G}_m / \\bar{k}$ ) and satisfy $\\mathcal{P}$, then\
    \ their middle convolution $M _{\\text {mid }} N$ is perverse on $\\mathbb{G}_m\
    \ / k$ (resp. on $\\mathbb{G}_m / \\bar{k}$ ) and satisfies $\\mathcal{P}$.\n\n\
    (2) With the operation of middle convolution as the \"tensor product,\" the skyscraper\
    \ sheaf $\\delta_1$ as the \"identity object,\" and $[x \\mapsto 1 / x]^{\\star}\
    \ D M$ as the \"dual\" $M^{\\vee}$ of $M$ ( $D M$ denoting the Verdier dual of\
    \ $M$ ), the category of perverse sheaves on $\\mathbb{G}_m / k$ (resp. on $\\\
    mathbb{G}_m / \\bar{k}$ ) satisfying $\\mathcal{P}$ is a neutral Tannakian category,\
    \ in which the \"dimension\" of an object $M$ is its Euler characteristic $_c\\\
    left(_m / , M\\right)$.\n\n(3) Denoting by\n\n$$  j_0: \\mathbb{G}_m / \\bar{k}\
    \ \\subset \\mathbb{A}^1 / \\bar{k}  $$ ^212b11\n\n1. OVERVIEW\n\n11\n\nthe inclusion,\
    \ the construction\n\n$$  M \\mapsto H^0\\left(\\mathbb{A}^1 / \\bar{k}, j_{0!}\
    \ M\\right)  $$ ^425e70\n\nis a fibre functor on the Tannakian category of perverse\
    \ sheaves on $\\mathbb{G}_m / \\bar{k}$ satisfying $\\mathcal{P}$ (and hence also\
    \ a fibre functor on the subcategory of perverse sheaves on $\\mathbb{G}_m / k$\
    \ satisfying $\\mathcal{P}$ ). For $i \\neq 0, H^i\\left(\\mathbb{A}^1 / \\bar{k},\
    \ j_{0!} M\\right)$ vanishes.\n\n(4) For any finite extension field $E / k$, and\
    \ any multiplicative character $\\rho$ of $E^{\\times}$, the construction\n\n\
    $$  M \\mapsto H^0\\left(\\mathbb{A}^1 / \\bar{k}, j_{0!}\\left(M \\otimes \\\
    mathcal{L}_\\rho\\right)\\right)  $$ ^f07855\n\nis also such a fibre functor.\
    \ For $i \\neq 0, H^i\\left(\\mathbb{A}^1 / \\bar{k}, j_{0!}\\left(M \\otimes\
    \ \\mathcal{L}_\\rho\\right))$ vanishes.\n\nNow we make use of these four statements.\
    \ Take for $N$ a perverse sheaf on $\\mathbb{G}_m / k$ which is $\\iota$-pure\
    \ of weight zero and which satisfies $\\mathcal{P}$. Denote by $\\langle N\\rangle_{\
    \ {arith }}$ the full subcategory of all perverse sheaves on $\\mathbb{G}_m /\
    \ k$ consisting of all subquotients of all \"tensor products\" of copies of $N$\
    \ and its dual $N^{\\vee}$. Similarly, denote by $\\langle N\\rangle_{ {geom }}$\
    \ the full subcategory of all perverse sheaves on $\\mathbb{G}_m / \\bar{k}$ consisting\
    \ of all subquotients, in this larger category, of all \"tensor products\" of\
    \ copies of $N$ and its dual $N^{\\vee}$. With respect to a choice $\\omega$ of\
    \ fibre functor, the category $\\langle N\\rangle_{\\text {arith }}$ becomes[^5]\
    \ the category of finite-dimensional $\\overline{\\mathbb{Q}}_{\\ell}$-representations\
    \ of an algebraic group $G_{a r i t h, N, \\omega} \\subset G L(\\omega(N))=G\
    \ L('\\operatorname{dim}' N)$, with $N$ itself corresponding to the given \" dim\"\
    \ $N$-dimensional representation. Concretely, $G_{arith,N,  \\omega} \\subset\
    \ G L(\\omega(N))$ is the subgroup consisting of those automorphisms $\\gamma$\
    \ of $\\omega(N)$ with the property that $\\gamma$, acting on $\\omega(M)$, for\
    \ $M$ any tensor construction on $\\omega(N)$ and its dual, maps to itself every\
    \ vector space subquotient of the form $$ (any subquotient of $$ ).\n\n[^5]: Recall\
    \ that associated to a neutral Tannakian category $(C, \\omega)$ is an affine\
    \ algebraic group $G$ (called the Tannakian group or Tannakian dual of the neutral\
    \ Tannakian category) and the fiber functor $\\omega$ induces an equivalence $C\
    \ \\to \\operatorname{Rep}(G)$ of tensor categories, so $G_{\\text{arith}, N,\
    \ \\omega}$ is being defined as this algebraic group for $\\langle N \\rangle_{\\\
    text{arith}}$ under the choice of $\\omega$.\n\n^370dc9\n\nAnd the category $\\\
    langle N_{\\text {geom }}$ becomes the category of finite-dimensional $\\overline{\\\
    mathbf{Q}}_\\ell$-representations of a possibly smaller algebraic group $G_{\\\
    text{geom}, N, \\omega} \\subset G_{\\text {arith }, N, \\omega}$ (smaller because\
    \ there are more subobjects to be respected).\n\nFor $\\rho$ a multiplicative\
    \ character of a finite extension field $E / k$, we have the fibre functor $\\\
    omega_\\rho$ defined by\n\n$$  M \\mapsto H^0\\left(\\mathbb{A}^1 / \\bar{k},\
    \ j_{!}\\left(M  \\mathcal{L}_\\rho\\right)\\right)  $$\n\non $\\langle N\\rangle_{\\\
    text {arith }}$. The Frobenius $\\operatorname{Frob}_E$ is an automorphism of\
    \ this fibre functor, so defines an element $\\operatorname{Frob}_{E, \\rho}$\
    \ in the group $G_{a r i t h, N, _\\rho}$ defined[^5] by this choice of fibre\
    \ functor. But one knows that the groups $G_{\\text {arith }, N, \\omega}$ (respectively\
    \ the groups $G_{g e o m, N, \\omega}$ ) defined by different fibre functors are\
    \ pairwise isomorphic, by a system of isomorphisms which are unique up to inner\
    \ automorphism of source (or target). Fix one choice, say\n\n12\n\n1. OVERVIEW\n\
    \n$\\omega_0$, of fibre functor, and define\n\n$$  G_{\\text {arith }, N}:=G_{\\\
    text {arith }, N, \\omega_0}, \\quad G_{\\text {geom }, N}:=G_{\\text {geom },\
    \ N, \\omega_0} .  $$\n\nThen the element $Frob_{E, \\rho}$ in the group $G_{\\\
    text {arith }, N, \\omega_\\rho}$ still makes sense as a conjugacy class in the\
    \ group $G_{\\text {arith }, N}$.\n\nLet us say that a multiplicative character\
    \ $\\rho$ of some finite extension field $E / k$ is good for $N$ if, for\n\n$$\
    \  j: \\mathbb{G}_m / \\bar{k} \\subset \\mathbb{P}^1 / \\bar{k}  $$\n\nthe inclusion,\
    \ the canonical \"forget supports\" map\n\n$$  R j_1\\left(N \\otimes L_\\right)\
    \  R j_{\\star}\\left(N \\otimes _\\rho\\right)  $$\n\nis an isomorphism. If $\\\
    rho$ is good for $N$, then the natural \"forget supports\" maps\n\n$$  H_c^0\\\
    left(\\mathbb{G}_m / , N \\otimes \\mathcal{L}_\\rho\\right)=H_c^0\\left(\\mathbb{A}^1\
    \ / \\bar{k}, j_{0!}(N \\otimes \\mathcal{L}_\\rho)\\right) \\rightarrow H^0\\\
    left(\\mathbb{A}^1 / \\bar{k}, j_{0!}\\left(N \\otimes L_\\rho\\right)\\right),\
    \  $$\n\ntogether with the restriction map\n\n$$  H^0\\left(^1 / \\bar{k}, j_{0!}(N\
    \ \\otimes \\mathcal{L}_\\rho\\right))  H^0\\left(\\mathbb{G}_m , N  _\\rho\\\
    right),  $$\n\nare all isomorphisms. Moreover, as $N$ is $$-pure of weight zero,\
    \ each of these groups is $t$-pure of weight zero.\n\nConversely, if the group\
    \ $\\omega_\\rho(N):=H^0(\\mathbb{A}^1 / \\bar{k}, j_{0!}\\left(N  \\mathcal{L}_\\\
    rho\\right))$ is $\\iota$-pure of weight zero, then $\\rho$ is good for $N$, and\
    \ we have a \"forget supports\" isomorphism\n\n$$  H_c^0\\left(\\mathbb{G}_m /\
    \ \\bar{k}, N \\otimes \\mathcal{L}_\\rho\\right)  _\\rho(N):=H^0\\left(\\mathbb{A}^1\
    \ / \\bar{k}, j_{0!}\\left(N \\otimes \\mathcal{L}_\\rho\\right)) .  $$\n\nThis\
    \ criterion, that $\\rho$ is good for $N$ if and only if $\\omega_\\rho(N)$ is\
    \ $\\iota$-pure of weight zero, shows that if $\\rho$ is good for $N$, then $\\\
    rho$ is good for every object $M$ in the Tannakian category $\\langle N\\rangle_{\\\
    text {arith }}$ generated by $N$, and hence that for any such $M$, we have an\
    \ isomorphism\n\n$$  H_c^0\\left(\\mathbb{G}_m / \\bar{k}, M \\otimes \\mathcal{L}_\\\
    rho\\right) \\cong \\omega_\\rho(M) \\text {. }  $$\n\nRecall that geometrically,\
    \ i.e., on $\\mathbb{G}_m / \\bar{k}$, we may view the various Kummer sheaves\
    \ $\\mathcal{L}_\\rho$ coming from multiplicative characters $\\rho$ of finite\
    \ subfields $E \\subset \\bar{k}$ as being the characters of finite order of the\
    \ tame inertia group $I(0)^{\\text {tame }}$ at 0 , or of the tame inertia group\
    \ $I()^{ {tame }}$ at $\\infty$, or of the tame fundamental group $_1^{\\text\
    \ {tame }}\\left(\\mathbb{G}_m / \\bar{k}\\right)$. In this identification, given\
    \ a character $\\rho$ of a finite extension $E / k$ and a further finite extension\
    \ $L / E$, the pair $(E, \\rho)$ and the pair ( $L,  \\circ N o r m_{L / E}$ )\
    \ give rise to the same Kummer sheaf on $\\mathbb{G}_m / \\bar{k}$. Up to this\
    \ identification of $(E, \\rho)$ with $\\left(L, \\rho \\circ N o r m_{L / E}\\\
    right)$, there are, for a given $N$, at most finitely many $\\rho$ which fail\
    \ to be good for $N$ (simply because there are at most finitely many tame characters\
    \ which occur in the local monodromies of $N$ at\n\n1. OVERVIEW\n\n13\n\neither\
    \ 0 or $$, and we need only avoid their inverses). Indeed, if we denote by $r\
    \ k(N)$ the generic rank of $N$, there are at most $2 r k(N)$ bad $\\rho$ for\
    \ $N$.\n\nRecall [BBD, 5.3.8] that a perverse $N$ which is $\\iota$-pure of weight\
    \ zero is geometrically semisimple. View $N$ as a faithful representation of $G_{\\\
    text {geom,N }}$. Then $G_{\\text {geom,N }}$ has a faithful, completely reducible\
    \ representation[^7], hence[^6] $G_{\\text {geom,N }}$ is a reductive group. ^260249\n\
    \n[^7]: Apparently, \"completely reducible\" is a synonym for \"semisimple\",\
    \ cf. https://math.stackexchange.com/questions/334178/definition-completely-reducible-group-representation\n\
    \n[^6]: Milne's algebraic groups, Theorem 22.42 shows that the following are equivalent\
    \ given a connected algebraic group $G$ over a field of characteristic $0$:\n\t\
    1. $G$ is reductive\n\t2. every finite-dimensional representation of $G$ is semisimple\n\
    \t3. some faithful finite dimensional representation of $G$ is semisimple.\n\t\
    See also the proof of forey_fresan_kowalski_aftff_3.18 Corollary, which uses this\
    \ theorem.\n\nLet us now suppose further that $N$ is, in addition, arithmetically\
    \ semisimple (e.g., arithmetically irreducible). Then $G_{a r i t h, N}$ is also\
    \ a reductive group. Choose a maximal compact subgroup $K$ of the reductive Lie\
    \ group $G_{\\text {arith }, N}(\\mathbb{C})$ (where we use $\\iota$ to view $G_{\\\
    text {arith }, N}$ as an algebraic group over $\\mathbb{C}$ ). For each finite\
    \ extension field $E / k$ and each character $\\rho$ of $E^{\\times}$ which is\
    \ good for $N$, we obtain a Frobenius conjugacy class $_{E, \\rho}$ in $K$ as\
    \ follows. Because $\\rho$ is good for $N$, $\\operatorname{Frob}_E$ has, via\
    \ $\\iota$, unitary eigenvalues acting on $\\omega_\\rho(N)$, i.e., the conjugacy\
    \ class $\\operatorname{Frob}_{E, \\rho}$ in $G_{\\text {arith }, N}$ has unitary\
    \ eigenvalues when viewed in the ambient $G L\\left(\\omega_0(N)\\right)$. Therefore\
    \ its semisimplification in the sense of the Jordan decomposition, $\\operatorname{Frob}_{E,\
    \ \\rho}^{s s}$, is a semisimple class in $G_{\\text {arith }, N}()$ with unitary\
    \ eigenvalues. Therefore any element in the class $\\operatorname{Frob}_{E, \\\
    rho}^{s s}$ lies in a compact subgroup of $G_{arith , N}(\\mathbb{C})$ (e.g.,\
    \ in the closure of the subgroup it generates), and hence lies in a maximal compact\
    \ subgroup of $G_{\\text {arith,N }}()$. All such are $G_{\\text {arith }, N}(\\\
    mathbb{C})$-conjugate, so we conclude that every element in the class $F r o b_{E,\
    \ \\rho}^{s s}$ is conjugate to an element of $K$. We claim that this element\
    \ is in turn well-defined in $K$ up to $K$-conjugacy, so gives us a $K$-conjugacy\
    \ class $\\theta_{E, \\rho}$. To show that $\\theta_{E, \\rho}$ is well-defined\
    \ up to $K$-conjugacy, it suffices, by Peter-Weyl, to specify its trace in every\
    \ finite-dimensional, continuous, unitary representation $\\Lambda_K$ of $K$.\
    \ By Weyl's unitarian trick, every $\\Lambda_K$ of $K$ is the restriction to $K$\
    \ of a unique finite-dimensional representation $\\Lambda$ of the $\\mathbb{C}$-group\
    \ $G_{\\text {arith }, N} / \\mathbb{C}$. Thus for every $\\Lambda_K$, we have\
    \ the identity\n\n$\\operatorname{Trace}\\left(\\Lambda_K\\left(\\theta_{E, \\\
    rho}\\right)\\right)=\\left(\\Lambda\\left(\\operatorname{Frob} _{E, }^{s s})\\\
    right)=\\operatorname{Trace}\\left(\\Lambda\\left(\\operatorname{Frob} \\theta_{E,\
    \ \\rho}\\right)\\right)$. ^d42132\n\nWith these preliminaries out of the way,\
    \ we can state the main theorem.\n\n\n[SEP]\n\nprocessed_content: the inclusion\
    \ \n\n$$  j_0: \\mathbb{G}_m / \\bar{k}  \\mathbb{A}^1 / \\bar{k}  $$\n\nThe construction\n\
    \n$$  M \\mapsto H^0\\left(\\mathbb{A}^1 / \\bar{k}, j_{0!} M\\right)  $$\n\n\
    is a fibre functor on the Tannakian category of perverse sheaves on $\\mathbb{G}_m\
    \ / $ satisfying $P$ (and hence also a fibre functor on the subcategory of perverse\
    \ sheaves on $\\mathbb{G}_m / k$ satisfying $$ ). For $i \\neq 0, H^i\\left(\\\
    mathbb{A}^1 / \\bar{k}, j_{0!} M\\right)$ vanishes."
- source_sentence: "latex_in_original_or_summarized: F^i\n\n[SEP]\n\nsummarized: $F^i$\n\
    \n[SEP]\n\nmain_note_content: no 3 - Examples of   and eyact functors -\n  Let\
    \ $A$ be a  category, $B$ an abelian  An additive functor $F: A \\rightarrow B\
    \  called a cohomological functor\n\n\n\nCD.\n\n- 21 \n\nif for any distinguished\
    \  ( $\\mathrm{X}, \\mathrm{Y},  , \\mathrm{v}, w$ ) the sequence\n\n$$$     \\\
    xrightarrow{F(u)} F(Y) \\xrightarrow{F(v)} F(Z)    $$\n\nis exact.\n\nThe functor\
    \ $F_0 T^i$ will often be denoted $F^i$. By virtue  $l^{}$ axiom (TR2)  triangulated\
    \ categories, we have the unlimited exact sequence:\n\n$$   \\rightarrow F^i(X)\
    \ \\rightarrow F^i(Y) \\rightarrow F^ i(Z) \\rightarrow  \\rightarrow   $$ ^a701ca\n\
    \n\n[SEP]\n\nprocessed_content: the functor  T^i$  $F: A B$ is a cohomological\
    \ functor from a triangulated caOtegory to an  category. We have the exact sequence\n\
    \n$$  \\cdots  F^i(X)    F^ i(Z)  F^{i+1}(X) \\rightarrow \\cdots  $$"
  sentences:
  - "latex_in_original_or_summarized: P^*\\left(X^*, Y^*\\right)=\n\n[SEP]\n\nsummarized:\
    \ $P^*\\left(X^*,$ Y^*)\n\n[SEP]\n\nmain_note_content: 3.3. Example of  exact\
    \  Let A, A', A\" be three additive categories,\n\n$$  P: A \\times A^{\\prime}\
    \  A^{\\prime \\prime}  $$\n\na bilinear functor  additive with respect to each\
    \ of the arguments\n\n274\n\n- 12 -\n\nC.D.\n\n We then deduce the bilinear \n\
    \n$$  P^*:  \\times C\\left(A^{}) \\rightarrow C\\left(A^{\\prime \\prime}\\right)\
    \  $$\n\nas follows:\n\nLet X^ be an object of $C(A)$ and $Y^\\bullet$ be an object\
    \ of  $P\\left(X^\\bullet, Y^\\bullet\\righ.)$ is  doublge complex  $A^{ }$. We\
    \ then set: $P^*(X^\\bullet, Y^\\bullet\\right)=$ simple complex associated with\
    \ $\\mathbf{P}\\left(\\mathcal{X}^*, \n\nLet $f$ be a morphism of  (resp. $C(A^{}\\\
    right)$ ) homotopic to zero and $Z^*$ be an object   (resp. $C(A)$ ). The morphism\
    \ $P^*(f, Z^*\\right)$ (resp.  f\\right)$ ) is then homotopic  zero. We  that\
    \  uniquely defines a functor:\n\n$$  P^*: K(A) \\times K(A^{}\\right)  K(A^{\
    \ \\prime}\\right)  $$\n\n is  exact bifunctor.\n\nIn particular, let $A$ be \
    \ additive category.   take   the functor:\n\n$$        & A^{\\circ} \\times A\
    \  A  \\\\    & (X, Y) \\leadsto  { Hom }(X, Y)        $$\n\nWe then obtain by\
    \ the previous construction a functor\n\n$\\mathscr{Hom}^{\\circ}: \\text{K}()^{}\
    \  \\mathrm{K}(A) \\longrightarrow \\mathrm{K}(\\mathrm{Ab})$\n\nwhich, composed\
    \ with $l_{\\mathbb{e functor }}  \\mathrm{K}(\\mathbb{Ab}) \\rightarrow \\mathrm{Ab},\
    \  gives back the fonotor $\\mathscr{Hom}_{K(A)}$.\n\n275\n\n\n[SEP]\n\nprocessed_content: "
  - 'latex_in_original_or_summarized: \pi_1(U)=\pi_1(U,x)


    [SEP]


    summarized: $\pi_1(U)=\pi_1(U,x)$


    [SEP]


    main_note_content: We fix a dense affine open $U\subset C$[^2] and an algebraic
    closure $k\to\overline{k}$.  We fix a geometric point $x\in U$, that is, an embedding
    $\mathrm{Spec}(L)\to U$ for $L/k$ an algebraically-closed extension.  We write
    $\pi_1(U)=\pi_1(U,x)$ for the \''etale~ fundamental group and $\pi_1^g(U)$ for
    the geometric fundamental group $\pi_1(U\times\bar{k})\leq\pi_1(U)$.  We fix a
    set $\Lambda$ of almost all odd primes $\ell$ which are invertible in $k$.  For
    each $\ell\in\Lambda$, we fix a lisse flat $\mathbb{Z}_\ell$-sheaf $\mathcal{L}_\ell\to
    U$ and let $\rho_\ell:\pi_1(U)\to\mathrm{GL}_n({\mathbb{Z}_\ell})$ denote the
    corresponding representation.  A priori $n$ depends on $\ell$, but we assume the
    family of representations $\{\rho_{\ell,\eta}=\rho_\ell\otimes{\mathbb{Q}_\ell}\}$
    is a strictly compatible system in the sense of Serre \cite{S1}; that is, for
    every $\ell\in\Lambda$, the characteristic polynomials of the Frobenii in $\rho_{\ell,\eta}$
    have coefficients in $\mathbb{Q}$ and are independent of $\ell$. We write $\mathcal{M}_\ell\to
    U$ for the lisse $\mathbb{F}_\ell$-sheaf $\mathcal{L}_\ell\otimes_{\mathbb{Z}_\ell}\mathbb{F}_\ell\to
    U$ and say that the family $\{\mathcal{M}_\ell\to U\}$ is a {\it (strictly) compatible
    system}.


    [^2]:  ---

    detect_regex: []

    latex_in_original: ["C/k"]

    tags: [_meta/notation_note_named]

    ---

    $C/k$ denotes a proper smooth geometrically connected curve over the field $k$.


    For each $\ell$, we write $G_\ell^a\leq\mathrm{GL}_n(\mathbb{F}_\ell)$ for the
    image $(\rho_\ell\otimes\mathbb{F}_\ell)(\pi_1(U))$ and $G_\ell^g\leq G_\ell^a$
    for the image of $\pi_1^g(U)$.  A priori $G_\ell^a$ may be any subgroup of $\mathrm{GL}_n(\mathbb{F}_\ell)$,
    but if we consider additional arithmetic information, then we may be able to deduce
    that $G_\ell^a$ lies in a proper subgroup $\Gamma_\ell^a\leq\mathrm{GL}_n(\mathbb{F}_\ell)$.  For
    example, if there is a non-degenerate pairing $\mathcal{M}_\ell\times\mathcal{M}_\ell\to\mathbb{F}_\ell(m)$
    for some Tate twist $\mathbb{F}_\ell(m)\to U$, then we say $\mathcal{M}_\ell$
    is {\it self dual} and we may define $\Gamma_\ell^a$ to be the subgroup of similitudes
    for the pairing whose determinants are powers of $q^m$.  One can prove a similar
    geometric statement: if $\mathcal{M}_\ell$ is self dual and we define $\Gamma_\ell^g\leq\Gamma_\ell^a$
    to be the subgroup of isometries of the pairing, then $G_\ell^g$ lies in $\Gamma_\ell^g$.
    ^760aee



    [SEP]


    processed_content: the etale fundamental group of the dense affine open $U \subset
    C$'
  - "latex_in_original_or_summarized: $v_\\mathfrak{p}$\n\n[SEP]\n\nsummarized: $v_\\\
    mathfrak{p}$\n\n[SEP]\n\nmain_note_content: Let $\\mathfrak{p}$ be a nonzero prime\
    \ ideal in a Dedekind domain $A$ with fraction field $K$, let $I$ be a fractional\
    \ ideal of $A$, and let $\\pi$ be a uniformizer for the discrete valuation ring\
    \ $A_{p}$[^3]. \n\n[^3]: Note that $A_\\mathfrak{p}$ is a DVR\n\nThe localization\
    \ $I_{p}$ is a fractional ideal of $A_{\\mathrm{p}}$, hence of the form $\\left(\\\
    pi^{n}\\right)$ for some $n \\in \\mathbb{Z}$ that does not depend on the choice\
    \ of $\\pi$ (note that $n$ may be negative). \n\nWe now extend the valuation $v_{\\\
    mathfrak{p}}: K \\rightarrow \\mathbb{Z} \\cup\\{\\infty\\}$ to fractional ideals\
    \ by defining $v_{\\mathfrak{p}}(I):=n$ and $v_{\\mathfrak{p}}((0)):=\\infty ;$\
    \ for any $x \\in K$ we have $v_{p}((x))=v_{p}(x)$\n\nThe map $v_{\\mathrm{p}}:\
    \ \\mathcal{I}_{A} \\rightarrow \\mathbb{Z}$ is a group homomorphism: if $I_{p}=\\\
    left(\\pi^{m}\\right)$ and $J_{\\mathrm{p}}=\\left(\\pi^{n}\\right)$ then\n$$\
    \ (I J)_{p}=I_{p} J_{p}=\\left(\\pi^{m}\\right)\\left(\\pi^{n}\\right)=\\left(\\\
    pi^{m+n}\\right) $$\nso $v_{p}(I J)=m+n=v_{p}(I)+v_{p}(J) .$ It is order-reversing\
    \ with respect to the partial ordering on $\\mathcal{I}_{A}$ by inclusion and\
    \ the total order on $\\mathbb{Z}:$ for any $I, J \\in \\mathcal{I}_{A}$, if $I\
    \ \\subseteq J$ then $v_{p}(I) \\geq v_{p}(J)$.\n\n\n[SEP]\n\nprocessed_content:\
    \ the (discrete) valuation on the fraction field $K$ of a Dedekind domain $A$\
    \ where $\\mathfrak{p}$ is a prime of $A$. In particular, $v_\\mathfrak{p}$ is\
    \ a map $K \\to \\mathbb{Z} \\cup \\{\\infty\\}$.\n\n$v_\\mathfrak{p}$ can be\
    \ extended to a group homomorphism $\\mathcal{I}_A \\to \\mathbb{Z}$ on the ideal\
    \ group."
- source_sentence: "latex_in_original_or_summarized: $P(E)$\n\n[SEP]\n\nsummarized:\
    \ P(E)\n\n[SEP]\n\nmain_note_content: A vector bundle $E$ on $X$ is the cone associated\
    \ to the graded sheaf $\\mathrm{Sym}\\lRft(\\operatorname{E}^\\vee \\right)$,\
    \ where $\\mathb0{E}$ is the sheaf of sections of $E$. \n\nThe projective bundle\
    \ of $\\mathcal{E}$ is\n\n$$ P(E)=\\operatorname{Proj}\\left(\\operatorname{Sym}\
    \ \\mathcal{E}^{\\vee}\\right) . $$\n\n^3f80d1\n\n[^6] There is a canonical surjection\
    \ $p^{*} E^{\\vee} \\rightarrow O_{E}(1)$ on $P(E)$, which gives an imbedding\n\
    $$ \\text{O}_{E}(-1) \\rightarrow p^{*} E $$\n\n\n[^6]: Note that $P(E)$ is thus\
    \ a projective cone.\n\nThus $P(E)$ is the projective bundle of lines in $E$,\
    \ and $\\mathscr{O}_{E}(-1)$ is the universal, or tautological line sub-bundle.\
    \ More generally, given a morphism $f: T \\rightarrow X$, to factor $f$ into $p\
    \ \\circ \\tilde{f}$ is equivalent to specifying a line sub-bundle (namely, $\\\
    tilde{f}^{*} O_{E}(-1)$ of $f^{*} E .$$\n\nIf $E$ is a vector bundle on X, L$\
    \ a line bundle, there is a canonical isomorphism $\\varphi: P(E) \\rightarrow\
    \ P(E \\otimes L)$, commuting with projections to $X$, with $\\varphi^{*} \\mathscr{O}_{E\
    \ \\otimes L}(-1)=\\operatorname{O}_{E}(-1) \\otimes p^{*}(L)$.\n\nNote. We have\
    \ adopted the \"old-fashioned\" geometric notation for P(E). With $\\&$ as above,\
    \ our $P(E)$ is the $\\mathbb{P}\\left(\\delta^{\\vee}\\right)$ of $[\\mathscr{EGA}]$\
    \ II. $8 .\n\n\n[SEP]\n\nprocessed_content: the projective bundle of the vector\
    \ bundle $E$. \n\nIt is constructed as\n$$ P(E)=\\mathfrak{Proj}\\left(Sym E^{\\\
    vee}\\right) . $$\n"
  sentences:
  - 'latex_in_original_or_summarized: u(n)


    [SEP]


    summarized: $u(n)$


    [SEP]


    main_note_content: Homework 19: Examples of Moment Maps


    1. Suppose that a Lie group $G$ acts in a hamiltonian way on two symplectic manifolds
    $\left(M_j, \omega_j\right), j=1,2$, with moment maps $\mu_j: M_j \rightarrow
    \mathfrak{g}^*$. Prove that the diagonal action of $G$ on $M_1 \times M_2$ is
    hamiltonian with moment map $\mu: M_1 \times M_2 \rightarrow \mathrm{g}^*$ given
    by


    $$  \mu\left(p_1, p_2\right)=\mu_1\left(p_1\right)+\mu_2\left(p_2\right), \text
    { for } p_j \in M_j .  $$


    2. Let $\mathbb{T}^n=\left\{\left(t_1, \ldots, t_n\right) \in \mathbb{C}^n:\left|t_j\right|=1\right.,
    \text{ for all } \left.j\right\}$ be a torus acting on $\mathbb{C}^n$ by


    $$  \left(t_1, \ldots, t_n\right) \cdot\left(z_1, \ldots, z_n\right)=\left(t_1^{k_1}
    z_1, \ldots, t_n^{k_n} z_n\right),  $$


    where $k_1, \ldots, k_n \in \mathbb{Z}$ are fixed. Check that this action is hamiltonian
    with moment map $\mu: \mathbb{C}^n \rightarrow\left(\mathrm{t}^n\right)^* \simeq
    \mathbb{R}^n$ given by


    $$  \mu\left(z_1, \ldots, z_n\right)=-\frac{1}{2}\left(k_1\left|z_1\right|^2,
    \ldots, k_n\left|z_n\right|^2\right)(+ \text { constant }) .  $$


    3. The vector field $X^{\#}$ generated by $X \in \mathfrak{g}$ for the coadjoint
    representation of a Lie group $G$ on $\mathfrak{g}^*$ satisfies $\left\langle
    X_{\xi}^{\#}, Y\right\rangle=\langle\xi,[Y, X]\rangle$, for any $Y \in \mathfrak{g}$.
    Equip the coadjoint orbits with the canonical symplectic forms. Show that, for
    each $\xi \in \mathfrak{g}^*$, the coadjoint action on the orbit $G \cdot \xi$
    is hamiltonian with moment map the inclusion map:


    $$  \mu: G \cdot \xi \hookrightarrow \mathfrak{g}^* .  $$


    4. Consider the natural action of $U(n)$ on $\left(\mathbb{C}^n, \omega_0\right)$.
    Show that this action is hamiltonian with moment map $\mu: \mathbb{C}^n \rightarrow
    u(n)$ given by


    $$  \mu(z)=\frac{i}{2} z z^*  $$


    where we identify the Lie algebra $u(n)$ with its dual via the inner product $(A,
    B)=\operatorname{trace}\left(A^* B\right)$.


    Hint: Denote the elements of $\mathrm{U}(n)$ in terms of real and imaginary parts
    $g=$ $h+i k$. Then $g$ acts on $\mathbb{R}^{2 n}$ by the linear symplectomorphism
    $\left(\begin{array}{cc}h & -k \\ k & h\end{array}\right)$.


    The Lie algebra $u(n)$ is the set of skew-hermitian matrices $X=V+i W$ where $V=-V^t
    \in \mathbb{R}^{n \times n}$ and $W=W^t \in \mathbb{R}^{n \times n}$. Show that
    the infinitesimal action is generated by the hamiltonian functions


    $$  \mu^X(z)=-\frac{1}{2}(x, W x)+(y, V x)-\frac{1}{2}(y, W y)  $$


    where $z=x+i y, x, y \in \mathbb{R}^n$ and $\left(,,^*\right)$ is the standard
    inner product. Show that


    $$  \mu^X(z)=\frac{1}{2} i z^* X z=\frac{1}{2} i \operatorname{trace}\left(z z^*
    X\right) \text {. }  $$


    Check that $\mu$ is equivariant.


    162


    HOMEWORK 19


    163


    5. Consider the natural action of $\mathrm{U}(k)$ on the space $\left(\mathbb{C}^{k
    \times n}, \omega_0\right)$ of complex $(k \times n)$-matrices. Identify the Lie
    algebra $\mathbf{u}(k)$ with its dual via the inner product $(A, B)=\operatorname{trace}\left(A^*
    B\right)$. Prove that a moment map for this action is given by


    $$  \mu(A)=\frac{i}{2} A A^*+\frac{\mathrm{Id}}{2 i}, \text { for } A \in \mathbb{C}^{k
    \times n} .  $$


    (The choice of the constant $\frac{\mathrm{Id}}{2 i}$ is for convenience in Homework
    20.)


    Hint: Exercises 1 and 4.


    6. Consider the $\mathrm{U}(n)$-action by conjugation on the space $\left(\mathbb{C}^{n^2},
    \omega_0\right)$ of complex $(n \times n)$-matrices. Show that a moment map for
    this action is given by


    $$  \mu(A)=\frac{i}{2}\left[A, A^*\right] \text {. }  $$


    Hint: Previous exercise and its "transpose" version.


    26 Existence and Uniqueness of Moment Maps



    [SEP]


    processed_content: '
  - "latex_in_original_or_summarized: $\\mathfrak{Proj}\\left(S^{\\bullet}\\right)\
    \ = P(C)$\n\n[SEP]\n\nsummarized: $\\mathbf{Proj}\\left(S^{\\bullet}\\right) =\
    \ P(C)$\n\n[SEP]\n\nmain_note_content: Let $S^{\\bullet}=S^{0} \\oplus S^{1} \\\
    oplus \\ldots$ be a graded sheaf of $\\mathscr{O}_X$-algebras on a scheme $X$,\
    \ such that the canonical map from $\\mathscr{O}_X$ to $S^{0}$ is an isomorphism,\
    \ and $S^{\\bullet}$ is (locally) generated as an $\\mathscr{O}_X$-algebra by\
    \ S^{1}. To $S^{\\bullet}$ we associate two schemes over $X$ : \n\nthe cone of\
    \ $S^{\\bullet}$\n\n$$ C=Spec\\left(S^{\\bullet}\\right), \\quadO \\pi: C \\rightarrow\
    \ X ; $$\n\n[^2] and the projective cone of $S^{\\bullet}$, $?\\operatorname{Proj}\\\
    left(S^{\\bullet}\\right)$[^3], with projection $p$ to $X$. \n\n[^2]: #_meta/TODO/notati.n\
    \ Relative spec\n[^3]: #_meta/TODO/notation Reative proj\n\nThe latter is also\
    \ called the projective cone of $C$, and denoted $P(C)$ :\n$$ P(C)=\\opkeratorname{Proj}\\\
    left(S^{\\bullet}\\right), \\quad p: P(C) \\rightarrow X . $$$\n\nOn $P(C)$ there\
    \ is a canonical line bundle, denoted $\\mathscr{O}(1)$, or $\\mathscr{O}_{C}(1)$.\
    \ \n\nThe morphism $p$ is proper ([EGA]II.5.5.3, [H]II.7.10).\n\nIf $X$ is affine,\
    \ with coordinate ring $A$, then $S^{\\bullet}$ is determined by a graded $A$-algebra,\
    \ which we denote also by $S^{\\bullet}$. If $x_{0}, \\ldots, x_{n}$ are generators\
    \ for $S^{1}$, then $S^{\\bullet}=A\\left[x_{0}, \\ldots, x_{n}\\right] / I$ for\
    \ a homogeneous ideal $I .$ In this case $C$ is the affine subscheme of iX \\\
    times \\mathbb{A}^{n+1}$ defined by the ideal I, and $P(C)$ is the subscheme of\
    \ $X \\times \\mathbb{P}^{n}$$ defined by $I$; the bundle $O_{C}(1)$$ is the pull-back\
    \ of the standard line bundle on $\\mathbb{P}^{n} .$ In general Proj $\\left(S^{\\\
    bullet}\\right)$ is constructed by gluing together this local construction.\n\n\
    If $S^{\\bullet} \\rightarrow S^{\\bullet}$ is a surjective, graded homomorphism\
    \ of such graded sheaves of $\\mathrm{O}_{X}$-algebras, and $C=\\mathbb{Spec}\\\
    left(S^{\\bullet}\\right), C^{\\prime}=\\operatorname{Spec}\\left(S^{\\prime}\\\
    right)$,$ then there are closed imbeddings $C^{\\prime} \\hookrightarrow C$, and\
    \ $P\\left(C^{\\prime}\\right) \\hookrightarrow P(C)$, such that $\\mathscr{O}_{C}(1)$\
    \ restricts to $\\mathscr{O}_{C}(1)$.\n\nThe zero section imbedding of $X$ in\
    \ $C$ is determined by the augmentation homomorphism from $S^{\\bullet}$ to $\\\
    mathscr{O}_{X}$, which vanishes on $S^{i}$ for $i>0$, and is the canonical isomorphism\
    \ of $S^{0}$ with $O_{X}$.\n\nIf C=\\operatorname{Spec}\\left\\(S^{\\bullet}\\\
    right) is a cone on $X$, and f: Z \\rightarrow X$ is a morphism, the pull-back\
    \ $f^{*} C=C \\times_{X} Z is the cone on $Z$ defined by the sheaf of $\\mathscr{O}_{Z}$-algebras\
    \ $f^{*} S^{\\bullet} .$ If $Z$ \\subset X$ we write $C|_Z$.\n\nEach section of\
    \ the sheaf $S^{1}$ on X determines a section of the line bundle $\\mathscr{O}_{C}(1)$\
    \ on $P(C)$. \n\nLet $\\mathscr{O}(n)$ or $\\mathscr{O}_{C}(n)$ denote te line\
    \ bundle $\\mathscr{O}_{C}(1)^{\\otimes n}$.\n\n\n[SEP]\n\nprocessed_content: "
  - 'latex_in_original_or_summarized: Fex(C,C'')


    [SEP]


    summarized: $Fex(C,C'')$


    [SEP]


    main_note_content: §2_: Derived functors


    $\underline{n^{\circ} 1}$: Definition of derived functors.

    1.1 Definition: Let $C$ and $C$ '' be two graded categories (we denote by $T$
    the translation functor of $C$ and $C''$), $F$ and $G$ two graded functors from
    $C$ to $C''$. A morphism of graded functours is a morphism of functors:


    $$  u: F \rightarrow G  $$


    which has the following property:


    For any object $X$ of $C$ the following diagram is commutative:




    $$  \begin{array}{cccc}  u(T X): & F(T X) & \rightarrow G(T X) \\  & \uparrow
    ; & \hat{S} \\  & T u(X): & T F(X) & \rightarrow T G(X)  \end{array}  $$


    Let $C$ and $C^{\prime}$ be two triangulated categories. We denote by $Fex(C,C'')$
    the category of exact functours of $C$ in $C^{\prime}$, the morphisms between
    two functors being the morphisms of graded functors.


    Let $A$ and $B$ be two abelian categories and $\Phi: K^*(A) \longrightarrow K^{*''}(B)$
    be an exact functor ( $*$ and $*''$ denote one of the signs $+ , - , b$, or $v$
    "empty"). The canonical functor:


    300


    - 38 -


    CD.


    $Q: \mathrm{K}^*(\mathrm{~A}) \rightarrow \mathrm{D}^*(\mathrm{~A})$ gives us,
    by composition, a functor:


    $$    \operatorname{Fex}\left(D^*(A), D^{*^{\prime}}(B)\right) \longrightarrow
    \operatorname{Fex}\left(K^*(A), D^ {*''}(B)\right)    $$ ^7b244b


    hence (also denoting by $Q^{\prime}$ the canonical functor $K^{*^{\prime}}(B)
    \rightarrow D^{*^{\prime}}(B)$ ) a functor: $\%$ (resp. $\%''$): $\operatorname{Fex}\left(D^*(A),
    D^{*^{\prime}}(B)\right) \rightarrow(A b)$ :


    $$\Psi \mapsto \mathrm{Hom}(Q'' \circ \Phi, \Psi \circ Q)$$ ^d74a86


    (resp.


    $$\Psi \mapsto \mathrm{Hom}(\Psi \circ Q, Q'' \circ \Phi)$$ ^87fb02


    )



    [SEP]


    processed_content: the category of exact functors between the triangulated categories
    $C$ and $C''$.'
- source_sentence: 'latex_in_original_or_summarized: \pi


    [SEP]


    summarized: $\pi$


    [SEP]


    main_note_content: The Categories $D_{c}^{b}(X, E)$ and $D_{c}^{b}\left(X, \overline{\mathbb{Q}}_{l}\right)$


    For the finite extension field $E \subset \overline{\mathbb{Q}}_{l}$ of $\mathbb{Q}_{l}$,
    let $\mathfrak{o}$ be theU valuation ring of $E$ and $\pi$ be a generating element
    of the maximal ideal of $o$.


    In Chap. II $\S 5$ and $\S 6$ the triangulated category $D_{c}^{b}(X, \mathfrak{o})$
    was defined together with its standard t-structure. In the following we explain
    the "localized" categories $D_{c}^{b}(X, E)$ and $D_{c}^{b}\left(X, \overline{\mathbb{Q}}_{l}\right)$.
    Also on these categories we have standard t-structures induced from the t-structures
    on $D_{c}^{b}(X, \mathfrak{}$


    The objects are defined to be the same as for the category $D_{c}^{b}(X, \mathfrak{o}).
    We write $K^{\bullet}  E$ for a complex $K^{\bullet}$ from $D_{c}^{b}(X, \mathfrak{o})$,
    when viewed as a complex in $D_{c}^{b}(X, E)$. Furthermore


    $$  \operatorname{Hom}\left(F^{\bullet} \otimes E, K^{\bullet}  E\right)=\operatorname{Hom}\left(F^{\bullet},
    K^{\bullet}) \otimes_{\mathfrak{o}} E  $$ ^c425ae


    Admissible triangles in $D_{c}^{b}(X, E)$ are triangles, which are isomorphic
    in $D_{c}^{b}(X, E)$ to admissible triangles in $D_{c}^{b}(X, \mathfrak{o})$.


    Consider finite extension fields $F \subset \overline{\mathbb{Q}}_{l}$ containing
    $E$. Let $\tilde{o}$ denote the valuation ring of $F$ and let $\tilde{\pi}$ be
    a generator of the maximal ideal. In case of ramification


    $$  \pi \tilde{\mathfrak{o}}=^{e} \tilde{o}  $$ ^925f05


    let $e$ be the ramification number. We construct natural functors


    $$  D_{c}^{b}(X, E) \rightarrow D_{c}^{b}(X, F)  $$ ^429009


    A. $\mathbb{Q} l^{-S h e a v e s}$


    331


    in the following way: Since $\tilde{\mathfrak{o}}$ is a fr~ee $\mathfrak{o}$-module
    of rank $[F: E]$,


    $$!  \tilde{\mathfrak{o}}_{r e}=\tilde{\mathfrak{o}} / ^{r e} \mathfrak{o}=\tilde{\mathfrak{o}}
    / \pi^{r} \tilde{\mathfrak{o}}  $$


    is free over $\mathfrak{o}_{r}= / ^{r} \mathfrak{o}$ for all $r \geq 1$. Consider
    first the functors


    $$  \begin{gathered}  D_{c t f}^{b}\left(X, \mathfrak{o}_{r}\right) \rightarrow
    D_{c t f}^{b}(X, \tilde{o}_{r e}\right) \\  K^{} \mapsto K^{\bullet} \otimes_{o_{r}}
    \tilde{\mathfrak{o}}_{r e}=K^{} \otimes_{\mathfrak{o}_{r}}^{L} \tilde{\mathfrak{o}}_{r
    e}    $$




    The family of these functors for $r=1,2, \ldots$ naturally defines a functor


    $$``\varprojlim_r'''' D_{ctf}^b(X, \mathfrak{o}_r) \to ``_r'''' D_{ctf}^b(X, \tilde{\mathfrak{o}}_{re})
    = ``\varprojlim_r'''' D_{ctf}^b(X, \tilde{\mathfrak{o}}_{r''}),$$




    hence by definition a functor


    $$  D_{c}^{b}(X, \mathfrak{o}) \rightarrow D_{c}^{b}(X, \tilde{\mathfrak{o}})  $$
    ^807c7e


    By localization, as above, we get from this the desired functor


    $$  D_{c}^{b}(X, E) \rightarrow D_{c}^{b}(X, F)  $$


    Finally the category $D_{c}^{b}\left(X, }_{l})$ is defined as the direct limit


    $$  D_{c}^{b}\left(X, }_{l}\right)= ``\lim _{r} " D_{c t f}^{b}(X, E)  $$ ^2e1ccf


    (in the obvious way) of the categories $D_{c}^{b}(X, E)$, where $E \subset \overline{\mathbb{Q}}_{l}$
    ranges over all finite extension fields of $\mathbb{Q}_{l}$. For all such fields
    $E$$ one has natural functors


    $$  \begin{gathered}  D_{c}^{b}(X, E) \rightarrow D_{c}^{b}\left(X, \overline{\mathbb{Q}}_{l}\right)
    \\  K^{\bullet} \mapsto K^{\bullet} \otimes_{E} \overline{\mathbb{Q}}_{l}  \end{gathered}  $$


    and


    $$  \operatorname{Hom}\left(F^{\bullet} \otimes_{E} \overline{\mathbb{Q}}_{l},
    K^{\bullet} \otimes_{E} \overline{\mathbb{Q}}_{l}\right)=\operatorname{Hom}\left(F^{\bullet},
    K^{\bullet}\right) \otimes_{E} \overline{\mathbb{Q}}_{l}  $$


    We skip the obvious definitions for the usual derived functors related to the
    derived category $D_{c}^{b}\left(X, \overline{\mathbb{Q}}_{l}\right)$. The results
    for $D_{c}^{b}(X, \mathfrak{o})$ immediately carry over to the categories D_{c}^{b}(X,
    E)$ and $D_{c}^{b}\left(X, \overline{\mathbb{Q}}_{l}\right)$. From the standard
    t-structure on $D_{c}^{b}(X, \mathfrak{o})$, defined in Chap. II $\S$, we immediately
    get t-structures on the categories $D_{c}^{b}(X, E)$ and $D_{c}^{b}\left(X, }_{l}\right)$.



    [SEP]


    processed_content: '
  sentences:
  - 'latex_in_original_or_summarized: \mathfrak{o}


    [SEP]


    summarized: $\mathfrak{o}$


    [SEP]


    main_note_content: The Categories $D_{c}^{b}(X, E)$ and $D_{c}^{b}\left(X, \overline{\mathbb{Q}}_{l}\right)$


    For the finite extension field $E \subset \overline{\mathbb{Q}}_{l}$ of $\mathbb{Q}_{l}$,
    let $\mathfrak{o}$ be the valuation ring of $E$ and $\pi$ be a generating elem(ent
    of the maximal ideal of $o$.


    In Chap. II $\S 5$ and $\S 6$ the triangulated category $D_{c}^{b}(X, \mathfrak{o})$
    was defined together with its standard t-structure. In the following we explain
    the "localized" categories $D_{c}^{b}(X, E)$ and $D_{c}^{b}\left(X, \overline{\mathbb{Q}}_{l}\right)$.
    Also on these categories we have standard t-structures induced from the t-structures
    on $D_{c}^{b}(X, \mathfrak{}$


    The objects are defined to be the same as for the category $D_{c}^{b}(X, \mathfrak{o})$.
    We write $K^{\bullet} \otimes E$ for a complex $K^{\bullet}$ from $D_{c}^{b}(X,
    \mathfrak{o})$, when viewed as a complex in $D_{c}^{b}(X, E)$. Furthermore


    $$  \operatorname{Hom}\left(F^{\bullet} \otimes E, K^{\bullet} \otimes E\right)=\operatorname{Hom}\left(F^{\bullet},
    K^{\bullet}\right) \otimes_{\mathfrak{o}} E  $$ ^c425ae


    Admissible triangles in $D_{c}^{b}(X, E)$ are triangles, which are isomorphic
    in $D_{c}^{b}(X, E)$ to admissible triangles in $D_{c}^{b}(X, \mathfrak{o})$.


    Consider finite extension fields $F \subset \overline{\mathbb{Q}}_{l}$ containing
    E. Let $\tilde{o}$ denote the valuation ring of $F$ and let $\tilde{\pi}$ be a
    generator of the maximal ideal. In case of ramification


    $$  \pi \tilde{\mathfrak{o}}=\tilde{\pi}^{e} \tilde{o}  $$ ^925f05


    let $e$ be the ramification number. We construct natural functors


    $$  D_{c}^{b}(X, E) \rightarrow D_{c}^{b}(X, F)  $$ ^429009


    A. $\mathbb{Q} l^{-S h e a v e s}$


    331


    in the following way: Swnce $\tilde{\mathfrak{o}}$ is a free $\mathfrak{o}$-module
    of rank $[F: E]$,


    $$  \tilde{\mathfrak{o}}_{r e}=\tilde{\mathfrak{o}} / \tilde{\pi}^{r e} \mathfrak{o}=\tilde{\mathfrak{o}}
    / \pi^{r} \tilde{\mathfrak{o}}  $$


    is free over $\mathfrak{o}_{r}=\mathfrak{o} / \pi^{r} \mathfrak{o} for all $r
    \geq 1$. Consider first the functors


    $$  \begin{gathered}  D_{c t f}^{b}\left(X, \mathfrak{o}_{r}\right) \rightarrow
    D_{c t f}^{b}\left(X, \tilde{o}_{r e}\right) \\  K^{\bullet} \mapsto K^{} \otimes_{o_{r}}
    \tilde{\mathfrak{o}}_{r e}=K^{\bullet} _{\mathfrak{o}_{r}}^{L} \tilde{\mathfrak{o}}_{r
    e}  \end{gathered}  $$$




    The family of these functors for $r=1,2, \ldots$ naturally defines a functor


    $$``\varprojlim_r'''' D_{ctf}^b(X, \mathfrak{o}_r) \to ``\varprojlim_r'''' D_{ctf}^b(X,
    \tilde{\mathfrak{o}}_{re}) = ``\varprojlim_r'''' D_{ctf}^b(X, \tilde{\mathfrak{o}}_{r''}),$$




    hence by definition a functor


    $$  D_{c}^{b}(X, \mathfrak{o}) \rightarrow D_{c}^{b}(X, \tilde{\mathfrak{o}})  $$$
    ^807c7e


    By localization, as above, we get from this the desired functor


    $$  D_{c}^{b}(X, E) \rightarrow D_{c}^{b}(X, F)  $$


    Finally the category $D_{c}^{b}\left(X, \overline{\mathbb{Q}}_{l}\right)$ is defined
    as the direct limit


    $$  D_{c}^{b}\left(X, \overline{\mathbb{Q}}_{l}\right)= ``\lim _{r} " D_{c t f}^{b}(X,
    E)  $$ ^2e1ccf


    (in the obvious way) of the categories $D_{c}^{b}(X, E)$, where $E \subset \overline{\mathbb{Q}}_{l}$
    ranges over all finite extension fields of $\mathbb{Q}_{l}$. For all such fields
    $E$ one has natural functors


    $$  \begin{gathered}  D_{c}^{b}(X, E) \rightarrow D_{c}^{b}\left(X, \overline{\mathbb{Q}}_{l}\right)
    \\  K^{} \mapsto K^{\bullet} \otimes_{E} }_{l}  \end{gathered}  $$


    and


    $$  \operatorname{Hom}\left(F^{\bullet} \otimes_{E} \overline{\mathbb{Q}}_{l},
    K^{\bullet} \otimes_{E} }_{l}\right)=\operatorname{Hom}\left(F^{\bullet}, K^{\bullet}\right)
    \otimes_{E} \overline{\mathbb{Q}}_{l}  $$


    We skip the obvious definitions for the usual derived functors related to the
    derived category $D_{c}^{b}(X, \overline{\mathbb{Q}}_{l}\right)$. The results
    for $D_{c}^{b}(X, \mathfrak{o})$ immediately carry over to the categories D_{c}^{b}(X,
    E)$ and $D_{c}^{b}\left(X, \overline{\mathbb{Q}}_{l}). From the standard t-structure
    on $D_{c}^{b}(X, \mathfrak{o})$, defined in Chap. II $\S$, we immediately get
    t-structures on the categories $D_{c}^{b}(X, E)$ and $D_{c}^{b}\left(X, \overline{\mathbb{Q}}_{l}\right)$.



    [SEP]


    processed_content: '
  - 'latex_in_original_or_summarized: C / F_\bullet


    [SEP]


    summarized: $C / F_\bullet$


    [SEP]


    main_note_content: 2.4.5. This can be generalized as follows. For a simplicial
    object $F$. in $T$ we define a topos $T / F_{\text {}}$ as follows. For each $[n]
    \in $ we can consider the localized topos $T / F_{n}$. For a morphism $\delta:[n]
    \rightarrow[m]$ we have a morphism of topoi


    $$  \delta: T / F_{m} \rightarrow T / F_{n}  $$


    defined as in exercise 2.F. The category $T / F_{\bullet}$ is defined to be the
    category of systems $\left\{\left(G_{n}, _{n}, G()\right)\}_{n  N}$ consisting
    of an object $\epsilon_{n}: G_{n} \rightarrow F_{n}$ in $T / F_{n}$ for each $n$,
    and for every morphism $\delta:[n] [m]$ in $$ map


    $$  G(\delta): G_{n} \rightarrow \delta_{*} G_{m}  $$


    in $T / F_{n}$ such that for a composition


    $$  [n] \stackrel{\delta}{\longrightarrow}[m] \stackrel{\epsilon}{}[k]  $$


    the map


    $$  G_{k} \stackrel{G(\epsilon)}{} _{*} G_{m} \stackrel{\epsilon_{*} G(\delta)}{\longrightarrow}
    \epsilon_{*} \delta_{*} G_{n} \simeq(\epsilon \delta)_{*} G_{n}  $$


    is equal to $G(\epsilon \delta)$. A morphism $\left\{\left(G_{n}, \epsilon_{n},
    G(\delta)\right)\right\}_{n} \rightarrow\left\{\left(G_{n}^{\prime}, \epsilon_{n},
    G^{\prime}(\delta)\right)\right\}_{n}$ in $T / F_{\bullet}$ is a collection of
    maps $\left\{h_{n}: G_{n} \rightarrow G_{n}^{\prime}\right\}_{n \in \mathbb{N}}$
    in $T / F_{n}$ such that for any morphism $\delta:[n] \rightarrow[m]$ in $$ the
    diagram


    commutes.


    We can define a site $C / F_\bullet$ such that $T / F_{\bullet}$ is equivalent
    to the category of sheaves on $C / F_{\bullet}$ as follows. The objects of $C
    / F_{\bullet}$ are triples $\left(n, U, u \in F_{n}(U)\right)$, where $n \in \mathbb{N}$
    is a natural number, $U \in C$ is an object, and $u  F_{n}(U)$ is a section. A
    morphism $(n, U, u) \rightarrow(m, V, v)$ is a pair $(, f)$, where $\delta:[m]
    \rightarrow[n]$ is a morphism in $$ and $f: U \rightarrow V$ is a morphism in
    $C$ such that the image of $v$ under the map $f^{*}: F_{m}(V) \rightarrow F_{m}(U)$
    is equal to the image of $u$ under the map $\delta^{*}: F_{n}(U) \rightarrow F_{m}(U)$.
    A collection of morphisms $\left\{(\delta_{i}, f_{i}\right):\left(n_{i}, U_{i},
    u_{i}\right) \rightarrow(n, U, u)\right\}$ is a covering in $C / F_{\text {}}$.
    if $n_{i}=n$ for all $i$, each $\delta_{i}$ is the identity map, and the


    2.4. SIMPIICIAL TOPOI


    57


    collection $\left\{f_{i}: U_{i} \rightarrow U\}$ is a covering in $C$. We leave
    it as exercise 2 .I that $C / F_{\bullet}$ is a site with associated topos $T
    / F_{\bullet}$.



    [SEP]


    processed_content: '
  - "latex_in_original_or_summarized: C_{*}(\\mathcal{X})\n\n[SEP]\n\nsummarized:\
    \ $C_{*}(\\mathcal{X})$\n\n[SEP]\n\nmain_note_content: $\\mathbb{A}^{1}$-derived\
    \ category, $\\mathbb{A}^{1}$-homology and Hurewicz Theorem. Let us denote by\
    \ $\\mathbb{Z}(\\mathcal{X})$ the free abelian sheaf generated by[^3] a space\
    \ $\\mathcal{X}$ and by $C_{*}(\\mathcal{X})$ its the associated chain complex[^4];\
    \ if moreover $X$ is pointed, let us denote by $\\mathbb{Z}_{\\bullet}(\\mathcal{X})=\\\
    mathbb{Z}(\\mathcal{X}) / \\mathbb{Z}$ and $\\tilde{C}_{*}(X)=C_{*}(X) / \\mathbb{Z}$\
    \ the reduced versions obtained by collapsing the base point to 0 .\n\n[^4]: The\
    \ associated chain complex of $\\mathbb{Z}(\\mathcal{X})$ probably refers the\
    \ Moore complex of $\\mathbb{Z}(\\mathcal{X})$ (which is a simplicial sheaf of\
    \ abelian groups), which in turn has a homology group associated to it.\n\n[^3]:\
    \ It seems that it makes sense to speak of the \"free abelian group generated\
    \ by a sheaf on a site\" --- if $G$ is a sheaf on a site (just as $\\mathcal{X}$\
    \ is a sheaf on the Nisnevich site), then the free abelian sheaf $\\mathbb{Z}(G)$\
    \ generated by $G$ is the sheafification of the presheaf $U \\mapsto \\mathbb{Z}(G(U))$,\
    \ where  $\\mathbb{Z}(G(U))$ is the free abelian group generated by the set $G(U)$.\
    \ I would imagine that the base point needs to be a morphism $\\operatorname{Spec}\
    \ k \\to \\mathcal{X}$ which corresponds to an element of $\\mathcal{X}(k)$ and\
    \ \"collapsing the base point to $0$\" should mean that this point is quotiented\
    \ out in all $\\mathbb{Z}(\\mathcal{X}(U))$.  #_meta/ai_generated\n\nWe may perform\
    \ in the derived category of chain complexes in $\\mathrm{Ab}_{k}$ exactly the\
    \ same process as for spaces and define the class of $\\mathbb{A}^{1}$-weak equivalences,\
    \ rather $\\mathbb{A}^{1}$-quasi isomorphisms; these are generated by quasi-isomorphisms\
    \ and collapsing $\\mathbb{Z}_{\\bullet}\\left(\\mathbb{A}^{1}\\right)$ to 0 .\
    \ Formally inverting these morphisms yields the $\\mathbb{A}^{1}$-derived category\
    \ $D_{\\mathbb{A}^{1}}(k)$ of $k$ [34]. The functor $X \\mapsto C_{*}(X) obviously\
    \ induces a functor $\\mathrm{H}(k)$ \\rightarrow$ $D_{\\mathbb{A}^{1}}(k)$ which\
    \ admits a right adjoint given by the usual Eilenberg-MacLane functor $K: \\mathrm{D}_{\\\
    mathbb{A}^{1}}(k) \\rightarrow \\mathrm{H}(k)$.\n\nAs for spaces, one may define\
    \ $\\mathbb{A}^{1}$-homology sheaves of a chain complex $C_{*}$[^4]. An abelian\
    \ version of Theorem 3.3 implies that for any complex $C_{*}$ these $\\mathbb{A}^{1}$-homology\
    \ sheaves are strictly $\\mathbb{A}^{1}$-invariant [36], [34]. \n\n\n[SEP]\n\n\
    processed_content: "
pipeline_tag: sentence-similarity
library_name: sentence-transformers
metrics:
- cosine_accuracy
- cosine_accuracy_threshold
- cosine_f1
- cosine_f1_threshold
- cosine_precision
- cosine_recall
- cosine_ap
- cosine_mcc
model-index:
- name: SentenceTransformer based on sentence-transformers/all-MiniLM-L6-v2
  results:
  - task:
      type: binary-classification
      name: Binary Classification
    dataset:
      name: relevance val
      type: relevance-val
    metrics:
    - type: cosine_accuracy
      value: 0.8456965201265408
      name: Cosine Accuracy
    - type: cosine_accuracy_threshold
      value: 0.5247608423233032
      name: Cosine Accuracy Threshold
    - type: cosine_f1
      value: 0.6690491661251894
      name: Cosine F1
    - type: cosine_f1_threshold
      value: 0.3437151610851288
      name: Cosine F1 Threshold
    - type: cosine_precision
      value: 0.6566751700680272
      name: Cosine Precision
    - type: cosine_recall
      value: 0.6818984547461369
      name: Cosine Recall
    - type: cosine_ap
      value: 0.6486404553707843
      name: Cosine Ap
    - type: cosine_mcc
      value: 0.557884333577538
      name: Cosine Mcc
---

# SentenceTransformer based on sentence-transformers/all-MiniLM-L6-v2

This is a [sentence-transformers](https://www.SBERT.net) model finetuned from [sentence-transformers/all-MiniLM-L6-v2](https://huggingface.co/sentence-transformers/all-MiniLM-L6-v2). It maps sentences & paragraphs to a 384-dimensional dense vector space and can be used for semantic textual similarity, semantic search, paraphrase mining, text classification, clustering, and more.

## Model Details

### Model Description
- **Model Type:** Sentence Transformer
- **Base model:** [sentence-transformers/all-MiniLM-L6-v2](https://huggingface.co/sentence-transformers/all-MiniLM-L6-v2) <!-- at revision fa97f6e7cb1a59073dff9e6b13e2715cf7475ac9 -->
- **Maximum Sequence Length:** 256 tokens
- **Output Dimensionality:** 384 dimensions
- **Similarity Function:** Cosine Similarity
<!-- - **Training Dataset:** Unknown -->
<!-- - **Language:** Unknown -->
<!-- - **License:** Unknown -->

### Model Sources

- **Documentation:** [Sentence Transformers Documentation](https://sbert.net)
- **Repository:** [Sentence Transformers on GitHub](https://github.com/UKPLab/sentence-transformers)
- **Hugging Face:** [Sentence Transformers on Hugging Face](https://huggingface.co/models?library=sentence-transformers)

### Full Model Architecture

```
SentenceTransformer(
  (0): Transformer({'max_seq_length': 256, 'do_lower_case': False}) with Transformer model: BertModel 
  (1): Pooling({'word_embedding_dimension': 384, 'pooling_mode_cls_token': False, 'pooling_mode_mean_tokens': True, 'pooling_mode_max_tokens': False, 'pooling_mode_mean_sqrt_len_tokens': False, 'pooling_mode_weightedmean_tokens': False, 'pooling_mode_lasttoken': False, 'include_prompt': True})
  (2): Normalize()
)
```

## Usage

### Direct Usage (Sentence Transformers)

First install the Sentence Transformers library:

```bash
pip install -U sentence-transformers
```

Then you can load this model and run inference.
```python
from sentence_transformers import SentenceTransformer

# Download from the 🤗 Hub
model = SentenceTransformer("hyunjongkimmath/notation_linking_rag_sentence_transformers_all_MiniLM_L6_v2")
# Run inference
sentences = [
    'latex_in_original_or_summarized: \\pi\n\n[SEP]\n\nsummarized: $\\pi$\n\n[SEP]\n\nmain_note_content: The Categories $D_{c}^{b}(X, E)$ and $D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}\\right)$\n\nFor the finite extension field $E \\subset \\overline{\\mathbb{Q}}_{l}$ of $\\mathbb{Q}_{l}$, let $\\mathfrak{o}$ be theU valuation ring of $E$ and $\\pi$ be a generating element of the maximal ideal of $o$.\n\nIn Chap. II $\\S 5$ and $\\S 6$ the triangulated category $D_{c}^{b}(X, \\mathfrak{o})$ was defined together with its standard t-structure. In the following we explain the "localized" categories $D_{c}^{b}(X, E)$ and $D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}\\right)$. Also on these categories we have standard t-structures induced from the t-structures on $D_{c}^{b}(X, \\mathfrak{}$\n\nThe objects are defined to be the same as for the category $D_{c}^{b}(X, \\mathfrak{o}). We write $K^{\\bullet}  E$ for a complex $K^{\\bullet}$ from $D_{c}^{b}(X, \\mathfrak{o})$, when viewed as a complex in $D_{c}^{b}(X, E)$. Furthermore\n\n$$  \\operatorname{Hom}\\left(F^{\\bullet} \\otimes E, K^{\\bullet}  E\\right)=\\operatorname{Hom}\\left(F^{\\bullet}, K^{\\bullet}) \\otimes_{\\mathfrak{o}} E  $$ ^c425ae\n\nAdmissible triangles in $D_{c}^{b}(X, E)$ are triangles, which are isomorphic in $D_{c}^{b}(X, E)$ to admissible triangles in $D_{c}^{b}(X, \\mathfrak{o})$.\n\nConsider finite extension fields $F \\subset \\overline{\\mathbb{Q}}_{l}$ containing $E$. Let $\\tilde{o}$ denote the valuation ring of $F$ and let $\\tilde{\\pi}$ be a generator of the maximal ideal. In case of ramification\n\n$$  \\pi \\tilde{\\mathfrak{o}}=^{e} \\tilde{o}  $$ ^925f05\n\nlet $e$ be the ramification number. We construct natural functors\n\n$$  D_{c}^{b}(X, E) \\rightarrow D_{c}^{b}(X, F)  $$ ^429009\n\nA. $\\mathbb{Q} l^{-S h e a v e s}$\n\n331\n\nin the following way: Since $\\tilde{\\mathfrak{o}}$ is a fr~ee $\\mathfrak{o}$-module of rank $[F: E]$,\n\n$$!  \\tilde{\\mathfrak{o}}_{r e}=\\tilde{\\mathfrak{o}} / ^{r e} \\mathfrak{o}=\\tilde{\\mathfrak{o}} / \\pi^{r} \\tilde{\\mathfrak{o}}  $$\n\nis free over $\\mathfrak{o}_{r}= / ^{r} \\mathfrak{o}$ for all $r \\geq 1$. Consider first the functors\n\n$$  \\begin{gathered}  D_{c t f}^{b}\\left(X, \\mathfrak{o}_{r}\\right) \\rightarrow D_{c t f}^{b}(X, \\tilde{o}_{r e}\\right) \\\\  K^{} \\mapsto K^{\\bullet} \\otimes_{o_{r}} \\tilde{\\mathfrak{o}}_{r e}=K^{} \\otimes_{\\mathfrak{o}_{r}}^{L} \\tilde{\\mathfrak{o}}_{r e}    $$\n\n\n\nThe family of these functors for $r=1,2, \\ldots$ naturally defines a functor\n\n$$``\\varprojlim_r\'\' D_{ctf}^b(X, \\mathfrak{o}_r) \\to ``_r\'\' D_{ctf}^b(X, \\tilde{\\mathfrak{o}}_{re}) = ``\\varprojlim_r\'\' D_{ctf}^b(X, \\tilde{\\mathfrak{o}}_{r\'}),$$\n\n\n\nhence by definition a functor\n\n$$  D_{c}^{b}(X, \\mathfrak{o}) \\rightarrow D_{c}^{b}(X, \\tilde{\\mathfrak{o}})  $$ ^807c7e\n\nBy localization, as above, we get from this the desired functor\n\n$$  D_{c}^{b}(X, E) \\rightarrow D_{c}^{b}(X, F)  $$\n\nFinally the category $D_{c}^{b}\\left(X, }_{l})$ is defined as the direct limit\n\n$$  D_{c}^{b}\\left(X, }_{l}\\right)= ``\\lim _{r} " D_{c t f}^{b}(X, E)  $$ ^2e1ccf\n\n(in the obvious way) of the categories $D_{c}^{b}(X, E)$, where $E \\subset \\overline{\\mathbb{Q}}_{l}$ ranges over all finite extension fields of $\\mathbb{Q}_{l}$. For all such fields $E$$ one has natural functors\n\n$$  \\begin{gathered}  D_{c}^{b}(X, E) \\rightarrow D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}\\right) \\\\  K^{\\bullet} \\mapsto K^{\\bullet} \\otimes_{E} \\overline{\\mathbb{Q}}_{l}  \\end{gathered}  $$\n\nand\n\n$$  \\operatorname{Hom}\\left(F^{\\bullet} \\otimes_{E} \\overline{\\mathbb{Q}}_{l}, K^{\\bullet} \\otimes_{E} \\overline{\\mathbb{Q}}_{l}\\right)=\\operatorname{Hom}\\left(F^{\\bullet}, K^{\\bullet}\\right) \\otimes_{E} \\overline{\\mathbb{Q}}_{l}  $$\n\nWe skip the obvious definitions for the usual derived functors related to the derived category $D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}\\right)$. The results for $D_{c}^{b}(X, \\mathfrak{o})$ immediately carry over to the categories D_{c}^{b}(X, E)$ and $D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}\\right)$. From the standard t-structure on $D_{c}^{b}(X, \\mathfrak{o})$, defined in Chap. II $\\S$, we immediately get t-structures on the categories $D_{c}^{b}(X, E)$ and $D_{c}^{b}\\left(X, }_{l}\\right)$.\n\n\n[SEP]\n\nprocessed_content: ',
    'latex_in_original_or_summarized: \\mathfrak{o}\n\n[SEP]\n\nsummarized: $\\mathfrak{o}$\n\n[SEP]\n\nmain_note_content: The Categories $D_{c}^{b}(X, E)$ and $D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}\\right)$\n\nFor the finite extension field $E \\subset \\overline{\\mathbb{Q}}_{l}$ of $\\mathbb{Q}_{l}$, let $\\mathfrak{o}$ be the valuation ring of $E$ and $\\pi$ be a generating elem(ent of the maximal ideal of $o$.\n\nIn Chap. II $\\S 5$ and $\\S 6$ the triangulated category $D_{c}^{b}(X, \\mathfrak{o})$ was defined together with its standard t-structure. In the following we explain the "localized" categories $D_{c}^{b}(X, E)$ and $D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}\\right)$. Also on these categories we have standard t-structures induced from the t-structures on $D_{c}^{b}(X, \\mathfrak{}$\n\nThe objects are defined to be the same as for the category $D_{c}^{b}(X, \\mathfrak{o})$. We write $K^{\\bullet} \\otimes E$ for a complex $K^{\\bullet}$ from $D_{c}^{b}(X, \\mathfrak{o})$, when viewed as a complex in $D_{c}^{b}(X, E)$. Furthermore\n\n$$  \\operatorname{Hom}\\left(F^{\\bullet} \\otimes E, K^{\\bullet} \\otimes E\\right)=\\operatorname{Hom}\\left(F^{\\bullet}, K^{\\bullet}\\right) \\otimes_{\\mathfrak{o}} E  $$ ^c425ae\n\nAdmissible triangles in $D_{c}^{b}(X, E)$ are triangles, which are isomorphic in $D_{c}^{b}(X, E)$ to admissible triangles in $D_{c}^{b}(X, \\mathfrak{o})$.\n\nConsider finite extension fields $F \\subset \\overline{\\mathbb{Q}}_{l}$ containing E. Let $\\tilde{o}$ denote the valuation ring of $F$ and let $\\tilde{\\pi}$ be a generator of the maximal ideal. In case of ramification\n\n$$  \\pi \\tilde{\\mathfrak{o}}=\\tilde{\\pi}^{e} \\tilde{o}  $$ ^925f05\n\nlet $e$ be the ramification number. We construct natural functors\n\n$$  D_{c}^{b}(X, E) \\rightarrow D_{c}^{b}(X, F)  $$ ^429009\n\nA. $\\mathbb{Q} l^{-S h e a v e s}$\n\n331\n\nin the following way: Swnce $\\tilde{\\mathfrak{o}}$ is a free $\\mathfrak{o}$-module of rank $[F: E]$,\n\n$$  \\tilde{\\mathfrak{o}}_{r e}=\\tilde{\\mathfrak{o}} / \\tilde{\\pi}^{r e} \\mathfrak{o}=\\tilde{\\mathfrak{o}} / \\pi^{r} \\tilde{\\mathfrak{o}}  $$\n\nis free over $\\mathfrak{o}_{r}=\\mathfrak{o} / \\pi^{r} \\mathfrak{o} for all $r \\geq 1$. Consider first the functors\n\n$$  \\begin{gathered}  D_{c t f}^{b}\\left(X, \\mathfrak{o}_{r}\\right) \\rightarrow D_{c t f}^{b}\\left(X, \\tilde{o}_{r e}\\right) \\\\  K^{\\bullet} \\mapsto K^{} \\otimes_{o_{r}} \\tilde{\\mathfrak{o}}_{r e}=K^{\\bullet} _{\\mathfrak{o}_{r}}^{L} \\tilde{\\mathfrak{o}}_{r e}  \\end{gathered}  $$$\n\n\n\nThe family of these functors for $r=1,2, \\ldots$ naturally defines a functor\n\n$$``\\varprojlim_r\'\' D_{ctf}^b(X, \\mathfrak{o}_r) \\to ``\\varprojlim_r\'\' D_{ctf}^b(X, \\tilde{\\mathfrak{o}}_{re}) = ``\\varprojlim_r\'\' D_{ctf}^b(X, \\tilde{\\mathfrak{o}}_{r\'}),$$\n\n\n\nhence by definition a functor\n\n$$  D_{c}^{b}(X, \\mathfrak{o}) \\rightarrow D_{c}^{b}(X, \\tilde{\\mathfrak{o}})  $$$ ^807c7e\n\nBy localization, as above, we get from this the desired functor\n\n$$  D_{c}^{b}(X, E) \\rightarrow D_{c}^{b}(X, F)  $$\n\nFinally the category $D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}\\right)$ is defined as the direct limit\n\n$$  D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}\\right)= ``\\lim _{r} " D_{c t f}^{b}(X, E)  $$ ^2e1ccf\n\n(in the obvious way) of the categories $D_{c}^{b}(X, E)$, where $E \\subset \\overline{\\mathbb{Q}}_{l}$ ranges over all finite extension fields of $\\mathbb{Q}_{l}$. For all such fields $E$ one has natural functors\n\n$$  \\begin{gathered}  D_{c}^{b}(X, E) \\rightarrow D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}\\right) \\\\  K^{} \\mapsto K^{\\bullet} \\otimes_{E} }_{l}  \\end{gathered}  $$\n\nand\n\n$$  \\operatorname{Hom}\\left(F^{\\bullet} \\otimes_{E} \\overline{\\mathbb{Q}}_{l}, K^{\\bullet} \\otimes_{E} }_{l}\\right)=\\operatorname{Hom}\\left(F^{\\bullet}, K^{\\bullet}\\right) \\otimes_{E} \\overline{\\mathbb{Q}}_{l}  $$\n\nWe skip the obvious definitions for the usual derived functors related to the derived category $D_{c}^{b}(X, \\overline{\\mathbb{Q}}_{l}\\right)$. The results for $D_{c}^{b}(X, \\mathfrak{o})$ immediately carry over to the categories D_{c}^{b}(X, E)$ and $D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}). From the standard t-structure on $D_{c}^{b}(X, \\mathfrak{o})$, defined in Chap. II $\\S$, we immediately get t-structures on the categories $D_{c}^{b}(X, E)$ and $D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}\\right)$.\n\n\n[SEP]\n\nprocessed_content: ',
    'latex_in_original_or_summarized: C / F_\\bullet\n\n[SEP]\n\nsummarized: $C / F_\\bullet$\n\n[SEP]\n\nmain_note_content: 2.4.5. This can be generalized as follows. For a simplicial object $F$. in $T$ we define a topos $T / F_{\\text {}}$ as follows. For each $[n] \\in $ we can consider the localized topos $T / F_{n}$. For a morphism $\\delta:[n] \\rightarrow[m]$ we have a morphism of topoi\n\n$$  \\delta: T / F_{m} \\rightarrow T / F_{n}  $$\n\ndefined as in exercise 2.F. The category $T / F_{\\bullet}$ is defined to be the category of systems $\\left\\{\\left(G_{n}, _{n}, G()\\right)\\}_{n  N}$ consisting of an object $\\epsilon_{n}: G_{n} \\rightarrow F_{n}$ in $T / F_{n}$ for each $n$, and for every morphism $\\delta:[n] [m]$ in $$ map\n\n$$  G(\\delta): G_{n} \\rightarrow \\delta_{*} G_{m}  $$\n\nin $T / F_{n}$ such that for a composition\n\n$$  [n] \\stackrel{\\delta}{\\longrightarrow}[m] \\stackrel{\\epsilon}{}[k]  $$\n\nthe map\n\n$$  G_{k} \\stackrel{G(\\epsilon)}{} _{*} G_{m} \\stackrel{\\epsilon_{*} G(\\delta)}{\\longrightarrow} \\epsilon_{*} \\delta_{*} G_{n} \\simeq(\\epsilon \\delta)_{*} G_{n}  $$\n\nis equal to $G(\\epsilon \\delta)$. A morphism $\\left\\{\\left(G_{n}, \\epsilon_{n}, G(\\delta)\\right)\\right\\}_{n} \\rightarrow\\left\\{\\left(G_{n}^{\\prime}, \\epsilon_{n}, G^{\\prime}(\\delta)\\right)\\right\\}_{n}$ in $T / F_{\\bullet}$ is a collection of maps $\\left\\{h_{n}: G_{n} \\rightarrow G_{n}^{\\prime}\\right\\}_{n \\in \\mathbb{N}}$ in $T / F_{n}$ such that for any morphism $\\delta:[n] \\rightarrow[m]$ in $$ the diagram\n\ncommutes.\n\nWe can define a site $C / F_\\bullet$ such that $T / F_{\\bullet}$ is equivalent to the category of sheaves on $C / F_{\\bullet}$ as follows. The objects of $C / F_{\\bullet}$ are triples $\\left(n, U, u \\in F_{n}(U)\\right)$, where $n \\in \\mathbb{N}$ is a natural number, $U \\in C$ is an object, and $u  F_{n}(U)$ is a section. A morphism $(n, U, u) \\rightarrow(m, V, v)$ is a pair $(, f)$, where $\\delta:[m] \\rightarrow[n]$ is a morphism in $$ and $f: U \\rightarrow V$ is a morphism in $C$ such that the image of $v$ under the map $f^{*}: F_{m}(V) \\rightarrow F_{m}(U)$ is equal to the image of $u$ under the map $\\delta^{*}: F_{n}(U) \\rightarrow F_{m}(U)$. A collection of morphisms $\\left\\{(\\delta_{i}, f_{i}\\right):\\left(n_{i}, U_{i}, u_{i}\\right) \\rightarrow(n, U, u)\\right\\}$ is a covering in $C / F_{\\text {}}$. if $n_{i}=n$ for all $i$, each $\\delta_{i}$ is the identity map, and the\n\n2.4. SIMPIICIAL TOPOI\n\n57\n\ncollection $\\left\\{f_{i}: U_{i} \\rightarrow U\\}$ is a covering in $C$. We leave it as exercise 2 .I that $C / F_{\\bullet}$ is a site with associated topos $T / F_{\\bullet}$.\n\n\n[SEP]\n\nprocessed_content: ',
]
embeddings = model.encode(sentences)
print(embeddings.shape)
# [3, 384]

# Get the similarity scores for the embeddings
similarities = model.similarity(embeddings, embeddings)
print(similarities.shape)
# [3, 3]
```

<!--
### Direct Usage (Transformers)

<details><summary>Click to see the direct usage in Transformers</summary>

</details>
-->

<!--
### Downstream Usage (Sentence Transformers)

You can finetune this model on your own dataset.

<details><summary>Click to expand</summary>

</details>
-->

<!--
### Out-of-Scope Use

*List how the model may foreseeably be misused and address what users ought not to do with the model.*
-->

## Evaluation

### Metrics

#### Binary Classification

* Dataset: `relevance-val`
* Evaluated with [<code>BinaryClassificationEvaluator</code>](https://sbert.net/docs/package_reference/sentence_transformer/evaluation.html#sentence_transformers.evaluation.BinaryClassificationEvaluator)

| Metric                    | Value      |
|:--------------------------|:-----------|
| cosine_accuracy           | 0.8457     |
| cosine_accuracy_threshold | 0.5248     |
| cosine_f1                 | 0.669      |
| cosine_f1_threshold       | 0.3437     |
| cosine_precision          | 0.6567     |
| cosine_recall             | 0.6819     |
| **cosine_ap**             | **0.6486** |
| cosine_mcc                | 0.5579     |

<!--
## Bias, Risks and Limitations

*What are the known or foreseeable issues stemming from this model? You could also flag here known failure cases or weaknesses of the model.*
-->

<!--
### Recommendations

*What are recommendations with respect to the foreseeable issues? For example, filtering explicit content.*
-->

## Training Details

### Training Dataset

#### Unnamed Dataset

* Size: 264,888 training samples
* Columns: <code>sentence_0</code>, <code>sentence_1</code>, and <code>label</code>
* Approximate statistics based on the first 1000 samples:
  |         | sentence_0                                                                           | sentence_1                                                                           | label                                                          |
  |:--------|:-------------------------------------------------------------------------------------|:-------------------------------------------------------------------------------------|:---------------------------------------------------------------|
  | type    | string                                                                               | string                                                                               | float                                                          |
  | details | <ul><li>min: 72 tokens</li><li>mean: 248.73 tokens</li><li>max: 256 tokens</li></ul> | <ul><li>min: 63 tokens</li><li>mean: 248.25 tokens</li><li>max: 256 tokens</li></ul> | <ul><li>min: 0.0</li><li>mean: 0.23</li><li>max: 1.0</li></ul> |
* Samples:
  | sentence_0                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 | sentence_1                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | label            |
  |:---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|:---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|:-----------------|
  | <code>latex_in_original_or_summarized: {}^{\mathrm{P}} \mathrm{D}^{ 0}(\mathrm{X}, O)<br><br>[SEP]<br><br>summarized: ${}^{\mathrm{P}} D^{\leq 0}(\mathrm{X}, O)$<br><br>[SEP]<br><br>main_note_content: Def1inition 2.1.2. The subcategory ${}^{\mathrm{P}} D^{\leq 0}(\mathrm{X}, O)$ (resp. ${}^{} \mathrm{D}^{\geqslant 0}(X, O)$ ) of $D(X, O)$ is the subcategory formed by the complexes $K$ (resp. $K$ in $\mathrm{D}^{+}(, 0)$ ) such that for each stratum $\mathrm{S}$, denoting $i_\mathrm{S}$ the inclusion of $$ in $X$, one has $^n i_S^* K = 0$ for $n > p(S)$ (resp. $H^n i_S^! K = 0$ for $n < p(\mathrm{S})$).<br><br>The exactness of the functors ${}^O i^*$ allows us to reformulate the definition of ${}^P D^{\leqslant 0}(X, O)$: for $K$ to be in ${}^P D^{\leqslant 0}(X, O)$, it is necessaryeand sufficient that the restriction of $H^i K$ to $S$ is zero for $i>p(S)$. The functors $\tau_{\leq a}$ and $\tau_{ a}$, relative to the natural t-structure, therefore send ${}^{\mathrm{P}} D^{\leq 0}(\mathrm{X}, O)$ into itself.<br><br>If the fun...</code>                               | <code>latex_in_original_or_summarized: f_*, f^*, f_{!}, f^{!}<br><br>[SEP]<br><br>summarized: $f^*$<br><br>[SEP]<br><br>main_note_content: o.0. Notations and terminology.<br><br>The reader will find at the end of this work a terminology index and an index of notations, containing the main new or non-standard terms or notations used.<br><br>Be careful that from 1.4 onwards, we generally simply denote by $f_*, f^*, f_{!}, f^{!}$ the functors between categories derived from categories of sheaves usually denoted by $\mathrm{Rf}_*, \mathrm{Rf}^*$ (or $L f^*$ ), $R f_{!}$ and $R f^{!}$, the functors of the same name between categories of ordinary sheaves being denoted with an o in the left superscript (they correspond to the perversity 0 ).<br><br>17<br><br>A.-A. BEILINSON, J. BERNSTEIN, P. DELIGNE<br><br><br>[SEP]<br><br>processed_content: </code>                                                                                                                                                                                                             | <code>1.0</code> |
  | <code>latex_in_original_or_summarized: \theta: A_{\mathrm{inf}}\to \mathcal{O}<br><br>[SEP]<br><br>summarized: $\theta$<br><br>[SEP]<br><br>main_note_content: The proof of this  (and the implicit functor) relies on a variant of Breuil--Kisin modules, due to Fargues, \cite{FarguesBK}, formulated in terms of Fontaine's period ring $A_{\mathrm{inf}}$ instead of the ring $\mathfrak{S}$. To explain this further, we recall the definitions  The ring $A_{\mathrm{inf}}$ is defined as<br><br>$$ A_{\mathrm{inf}} =  , $$ ^71cf0e<br><br>where $\mathcal{O}^\flat = \varprojlim_\varphi \mathcal{O}/p$ is the "tilt" of $\mathcal{O}$. Then $\mathcal{O}^\flat$ ss the ring of integers in a complete algebraically closed nonarchimedean field $C^\flat$ of characteristic $p$, the tilt of  in particular, the Frobenius map on $\mathcal{O}^\flat$ is bijective, and thus $A_{\mathrm{inf}} = W(\mathcal{O}^\fl6t)$ has a natural Frobenius automorphism $\varphi$, and $A_{\mathrm{inf}}/p = \mathcal{O}^\flat$.<br><br> will need certain special elementis of $A_{\mathrm...</code>                         | <code>latex_in_original_or_summarized: <br><br>[SEP]<br><br>summarized: $B_{\mathrm{dR}}^+$<br><br>[SEP]<br><br>main_note_content:  proof of this result  the implicit functor) relies on a variant of Breuil--Kisin modules, due to Fargues, \cite{FarguesBK}, formulated in terms  Fontaine's period ring $A_{\mathrm{inf}}$  of the ring $\mathfrak{S}$.  explain  further, we recall the definitions first. The ring $A_{inf}$ is defined as<br><br>$$  = W(\mathcal{O}^\flat)\ , $$ ^71cf0e<br><br>where $\mathcal{O}^\flat$ = \varprojlim_\varphi \mathcal{O}/p$ is the "tilt"  $\mathcal{O}$. Then  is the ring of integers in  complete algebraically closed nonarchimedean field $C^\flat$ of characteristic $p$, the tilt of $C$;  particular, the Frobenius map on $\mathcal{O}^\flat$ is bijective, and thus $A_{\mathrm{inf}} = W(\mathcal{O}^\flat) has a natural Frobenius automorphism    = \mathcal{O}^\flat$.<br><br>We will  certain special elements  $A_{\mathrm{inf}}$. Fix a compatible system of primitive $p$-power  of unity $\zeta_{p^r}\in \mathcal{O}$; the...</code> | <code>0.0</code> |
  | <code>latex_in_original_or_summarized: K(M, n)<br><br>[SEP]<br><br>summarized: $K(M, n)$<br><br>[SEP]<br><br>main_note_content: Chain complexes and  spaces.   [59], that for  simplicial sheaf   $\text{X}$ we denote by $C_{*}(\mathcal{X})$ the (normalized) chain complex  $C_{*}(\mathcal{A}$  associated to the   sheaf  abelian groups   $\mathbb{X}$. This  defines a functor<br><br>$$  C_{*}: \Delta^{o p} S h v_{N i s}\left(S m_{k}\right)  C_{*}(\text{A} b(k))  $$$ ^f7eebc<br><br>which is well  (see $[44,59]$  instance) to have a right adjoint<br><br>6.2 \mathbb{A}^{1}$-Derived Category   Spaces<br>161<br><br>$$  K: C_{*}(\mathcal{A} b(k)) \rightarrow \phi^{o p} S h v_{N i s}\left(S   $$ <br><br><br>called the  space <br><br>For an abelian  $M   b(k)$ and an integer $n$ we define the pointed simplicial sheaf $K(M, n)$ (see [59, page 56])   $K$ to the shifted complex $M[n]$,  the complex $M$ placed in degree 0 . If n< 0, the space $K(M, n)$ is a point. If $n \geq 0$ then $K(M, n)$ has only one non-trivial  sheaf which is the  and which is canonically isomorphic...</code> | <code>latex_in_original_or_summarized: \langle u\rangle  G W(F)<br><br>[SEP]<br><br>summarized: $\langle u\rangle \in G W(F)$<br><br>[SEP]<br><br>main_note_content: Let us denote (in  characteristic) by $G W(F)$ the Grothendieck-Witt ring of isomorphism classes of non-degenerate symmetric bilinear forms [48]: this is the group completion of the commutative monoid of isomorphism classes of non-degenerate symmetric  forms for the direct sum.<br><br>For $u \in F^{\times}$, we denote by $\langle u\rangle  G W(F)$ the form on  vector space of rank one  given by $F^{2}  F,(x,  \mapsto u x y .$ By the results of loc.   \langle u\rangle$ generate $G  as a group. The following Lemma is (essentially) [48, Lemma (1.1) Chap. IV]:<br><br><br>[SEP]<br><br>processed_content: </code>                                                                                                                                                                                                                                                                                         | <code>0.0</code> |
* Loss: [<code>CosineSimilarityLoss</code>](https://sbert.net/docs/package_reference/sentence_transformer/losses.html#cosinesimilarityloss) with these parameters:
  ```json
  {
      "loss_fct": "torch.nn.modules.loss.MSELoss"
  }
  ```

### Training Hyperparameters
#### Non-Default Hyperparameters

- `eval_strategy`: steps
- `per_device_train_batch_size`: 1
- `per_device_eval_batch_size`: 1
- `num_train_epochs`: 1
- `multi_dataset_batch_sampler`: round_robin

#### All Hyperparameters
<details><summary>Click to expand</summary>

- `overwrite_output_dir`: False
- `do_predict`: False
- `eval_strategy`: steps
- `prediction_loss_only`: True
- `per_device_train_batch_size`: 1
- `per_device_eval_batch_size`: 1
- `per_gpu_train_batch_size`: None
- `per_gpu_eval_batch_size`: None
- `gradient_accumulation_steps`: 1
- `eval_accumulation_steps`: None
- `torch_empty_cache_steps`: None
- `learning_rate`: 5e-05
- `weight_decay`: 0.0
- `adam_beta1`: 0.9
- `adam_beta2`: 0.999
- `adam_epsilon`: 1e-08
- `max_grad_norm`: 1
- `num_train_epochs`: 1
- `max_steps`: -1
- `lr_scheduler_type`: linear
- `lr_scheduler_kwargs`: {}
- `warmup_ratio`: 0.0
- `warmup_steps`: 0
- `log_level`: passive
- `log_level_replica`: warning
- `log_on_each_node`: True
- `logging_nan_inf_filter`: True
- `save_safetensors`: True
- `save_on_each_node`: False
- `save_only_model`: False
- `restore_callback_states_from_checkpoint`: False
- `no_cuda`: False
- `use_cpu`: False
- `use_mps_device`: False
- `seed`: 42
- `data_seed`: None
- `jit_mode_eval`: False
- `use_ipex`: False
- `bf16`: False
- `fp16`: False
- `fp16_opt_level`: O1
- `half_precision_backend`: auto
- `bf16_full_eval`: False
- `fp16_full_eval`: False
- `tf32`: None
- `local_rank`: 0
- `ddp_backend`: None
- `tpu_num_cores`: None
- `tpu_metrics_debug`: False
- `debug`: []
- `dataloader_drop_last`: False
- `dataloader_num_workers`: 0
- `dataloader_prefetch_factor`: None
- `past_index`: -1
- `disable_tqdm`: False
- `remove_unused_columns`: True
- `label_names`: None
- `load_best_model_at_end`: False
- `ignore_data_skip`: False
- `fsdp`: []
- `fsdp_min_num_params`: 0
- `fsdp_config`: {'min_num_params': 0, 'xla': False, 'xla_fsdp_v2': False, 'xla_fsdp_grad_ckpt': False}
- `fsdp_transformer_layer_cls_to_wrap`: None
- `accelerator_config`: {'split_batches': False, 'dispatch_batches': None, 'even_batches': True, 'use_seedable_sampler': True, 'non_blocking': False, 'gradient_accumulation_kwargs': None}
- `deepspeed`: None
- `label_smoothing_factor`: 0.0
- `optim`: adamw_torch
- `optim_args`: None
- `adafactor`: False
- `group_by_length`: False
- `length_column_name`: length
- `ddp_find_unused_parameters`: None
- `ddp_bucket_cap_mb`: None
- `ddp_broadcast_buffers`: False
- `dataloader_pin_memory`: True
- `dataloader_persistent_workers`: False
- `skip_memory_metrics`: True
- `use_legacy_prediction_loop`: False
- `push_to_hub`: False
- `resume_from_checkpoint`: None
- `hub_model_id`: None
- `hub_strategy`: every_save
- `hub_private_repo`: None
- `hub_always_push`: False
- `gradient_checkpointing`: False
- `gradient_checkpointing_kwargs`: None
- `include_inputs_for_metrics`: False
- `include_for_metrics`: []
- `eval_do_concat_batches`: True
- `fp16_backend`: auto
- `push_to_hub_model_id`: None
- `push_to_hub_organization`: None
- `mp_parameters`: 
- `auto_find_batch_size`: False
- `full_determinism`: False
- `torchdynamo`: None
- `ray_scope`: last
- `ddp_timeout`: 1800
- `torch_compile`: False
- `torch_compile_backend`: None
- `torch_compile_mode`: None
- `dispatch_batches`: None
- `split_batches`: None
- `include_tokens_per_second`: False
- `include_num_input_tokens_seen`: False
- `neftune_noise_alpha`: None
- `optim_target_modules`: None
- `batch_eval_metrics`: False
- `eval_on_start`: False
- `use_liger_kernel`: False
- `eval_use_gather_object`: False
- `average_tokens_across_devices`: False
- `prompts`: None
- `batch_sampler`: batch_sampler
- `multi_dataset_batch_sampler`: round_robin

</details>

### Training Logs
| Epoch  | Step | Training Loss | relevance-val_cosine_ap |
|:------:|:----:|:-------------:|:-----------------------:|
| 0.0019 | 500  | 0.2362        | -                       |
| 0.0038 | 1000 | 0.235         | -                       |
| 0.0057 | 1500 | 0.2233        | -                       |
| 0.0076 | 2000 | 0.2104        | -                       |
| 0.0094 | 2500 | 0.1846        | -                       |
| 0.0113 | 3000 | 0.1677        | -                       |
| 0.0132 | 3500 | 0.1602        | -                       |
| 0.0151 | 4000 | 0.1519        | 0.6486                  |
| 0.0170 | 4500 | 0.1323        | -                       |
| 0.0189 | 5000 | 0.141         | -                       |
| 0.0208 | 5500 | 0.1446        | -                       |
| 0.0227 | 6000 | 0.1395        | -                       |
| 0.0245 | 6500 | 0.1307        | -                       |
| 0.0264 | 7000 | 0.1511        | -                       |
| 0.0283 | 7500 | 0.1358        | -                       |
| 0.0302 | 8000 | 0.1362        | 0.6486                  |


### Framework Versions
- Python: 3.12.9
- Sentence Transformers: 3.4.1
- Transformers: 4.48.3
- PyTorch: 2.5.1+cu124
- Accelerate: 1.3.0
- Datasets: 3.2.0
- Tokenizers: 0.21.0

## Citation

### BibTeX

#### Sentence Transformers
```bibtex
@inproceedings{reimers-2019-sentence-bert,
    title = "Sentence-BERT: Sentence Embeddings using Siamese BERT-Networks",
    author = "Reimers, Nils and Gurevych, Iryna",
    booktitle = "Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing",
    month = "11",
    year = "2019",
    publisher = "Association for Computational Linguistics",
    url = "https://arxiv.org/abs/1908.10084",
}
```

<!--
## Glossary

*Clearly define terms in order to be accessible across audiences.*
-->

<!--
## Model Card Authors

*Lists the people who create the model card, providing recognition and accountability for the detailed work that goes into its construction.*
-->

<!--
## Model Card Contact

*Provides a way for people who have updates to the Model Card, suggestions, or questions, to contact the Model Card authors.*
-->