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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Behrend's formula ： ウィキペディア英語版 Behrend's trace formula In algebraic geometry, Behrend's formula is a generalization of the Grothendieck–Lefschetz trace formula to a smooth algebraic stack over a finite field, conjectured in 1993 〔Behrend, K. (The Lefschetz Trace Formula for the Moduli Stack of Principal Bundles. ) PhD dissertation.〕 and proven in 2003 〔Behrend, K. (Derived l-adic categories for algebraic stacks. ) Memoirs of the American Mathematical Society Vol. 163, 2003〕 by Kai Behrend. Unlike the classical one, the formula counts points in the "stacky way"; it takes into account the presence of nontrivial automorphisms. The desire for the formula comes from the fact that it applies to the moduli stack of principal bundles on a curve over a finite field (in some instances indirectly, via the Harder–Narasimhan stratification, as the moduli stack is not of finite type.〔K. Behrend, A. Dhillon, (Connected components of moduli stacks of torsors via Tamagawa numbers )〕〔http://www.math.harvard.edu/~lurie/282ynotes/LectureIII-Cohomology.pdf〕) See the moduli stack of principal bundles and references therein for the precise formulation in this case. Deligne found an example that shows the formula may be interpreted as a sort of the Selberg trace formula. A proof of the formula in the context of the six operations formalism developed by Laszlo and Olsson〔 *〕 is given by Shenghao Sun. == Formulation == By definition, if ''C'' is a category in which each object has finitely many automorphisms, the number of points in $C$ is denoted by :$\backslash \#\; C\; =\; \backslash sum\_p\; ,$ with the sum running over representatives ''p'' of all isomorphism classes in ''C''. (The series may diverge in general.) The formula states: for a smooth algebraic stack ''X'' of finite type over a finite field $\backslash mathbf\_q$ and the "arithmetic" Frobenius $\backslash phi^:\; X\; \backslash to\; X$, i.e., the inverse of the usual geometric Frobenius $\backslash phi$ in Grothendieck's formula,〔To define Frobenius $\backslash phi$ on a stack ''X'', let $\backslash phi:\; \backslash overline\; \backslash to\; \backslash overline,\; x\; \backslash mapsto\; x^q$. Then we have $id\; \backslash times\; \backslash phi:\; X\; \backslash times\_\; \backslash overline\; \backslash to\; X\; \backslash times\_\; \backslash overline$, which is the Frobenius on ''X'', also denoted by $\backslash phi$.〕 :$\backslash \#\; X\; (\backslash mathbf\_q)\; =\; q^\; \backslash sum\_^\; (-1)^i\; \backslash operatorname(\backslash phi^;\; H^i(X,\; \backslash mathbb\_l)).$ Here, it is crucial that the cohomology of a stack is with respect to the smooth topology (not etale). When ''X'' is a variety, the smooth cohomology is the same as etale one and, via the Poincaré duality, this is equivalent to Grothendieck's trace formula. (But the proof relies on Grothendieck's formula, so this does not subsume Grothendieck's.) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Behrend's trace formula」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.