---
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)
- **Maximum Sequence Length:** 256 tokens
- **Output Dimensionality:** 384 dimensions
- **Similarity Function:** Cosine Similarity
### 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]
```
## Evaluation
### Metrics
#### Binary Classification
* Dataset: `relevance-val`
* Evaluated with [BinaryClassificationEvaluator](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 |
## Training Details
### Training Dataset
#### Unnamed Dataset
* Size: 264,888 training samples
* Columns: sentence_0, sentence_1, and label
* Approximate statistics based on the first 1000 samples:
| | sentence_0 | sentence_1 | label |
|:--------|:-------------------------------------------------------------------------------------|:-------------------------------------------------------------------------------------|:---------------------------------------------------------------|
| type | string | string | float |
| details | - min: 72 tokens
- mean: 248.73 tokens
- max: 256 tokens
| - min: 63 tokens
- mean: 248.25 tokens
- max: 256 tokens
| - min: 0.0
- mean: 0.23
- max: 1.0
|
* Samples:
| sentence_0 | sentence_1 | label |
|:---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|:---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|:-----------------|
| latex_in_original_or_summarized: {}^{\mathrm{P}} \mathrm{D}^{ 0}(\mathrm{X}, O)
[SEP]
summarized: ${}^{\mathrm{P}} D^{\leq 0}(\mathrm{X}, O)$
[SEP]
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 $n < p(\mathrm{S})$).
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.
If the fun... | latex_in_original_or_summarized: f_*, f^*, f_{!}, f^{!}
[SEP]
summarized: $f^*$
[SEP]
main_note_content: o.0. Notations and terminology.
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.
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 ).
17
A.-A. BEILINSON, J. BERNSTEIN, P. DELIGNE
[SEP]
processed_content: | 1.0 |
| latex_in_original_or_summarized: \theta: A_{\mathrm{inf}}\to \mathcal{O}
[SEP]
summarized: $\theta$
[SEP]
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
$$ A_{\mathrm{inf}} = , $$ ^71cf0e
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 $A_{\mathrm{inf}}/p = \mathcal{O}^\flat$.
will need certain special elementis of $A_{\mathrm... | latex_in_original_or_summarized:
[SEP]
summarized: $B_{\mathrm{dR}}^+$
[SEP]
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
$$ = W(\mathcal{O}^\flat)\ , $$ ^71cf0e
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 $A_{\mathrm{inf}} = W(\mathcal{O}^\flat) has a natural Frobenius automorphism = \mathcal{O}^\flat$.
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... | 0.0 |
| latex_in_original_or_summarized: K(M, n)
[SEP]
summarized: $K(M, n)$
[SEP]
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
$$ C_{*}: \Delta^{o p} S h v_{N i s}\left(S m_{k}\right) C_{*}(\text{A} b(k)) $$$ ^f7eebc
which is well (see $[44,59]$ instance) to have a right adjoint
6.2 \mathbb{A}^{1}$-Derived Category Spaces
161
$$ K: C_{*}(\mathcal{A} b(k)) \rightarrow \phi^{o p} S h v_{N i s}\left(S $$
called the space
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... | 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: | 0.0 |
* Loss: [CosineSimilarityLoss](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
Click to expand
- `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