翻訳と辞書 Words near each other ・ Grotesque (1988 film) ・ Grotesque (2009 film) ・ Grotesque (After the Gramme) ・ Grotesque (band) ・ Grotesque (chess) ・ Grotesque (disambiguation) ・ Grotesque (novel) ・ Grotesque (The X-Files) ・ Grotesque body ・ Grotesque dance ・ Grotesque Impalement ・ Grotesque Wedlock ・ Grotesquerie ・ Groth ・ Grothe ・ Grothendieck category ・ Grothendieck connection ・ Grothendieck construction ・ Grothendieck duality ・ Grothendieck existence theorem ・ Grothendieck group ・ Grothendieck inequality ・ Grothendieck local duality ・ Grothendieck space ・ Grothendieck spectral sequence ・ Grothendieck topology ・ Grothendieck trace formula ・ Grothendieck universe ・ Grothendieck's connectedness theorem ・ Grothendieck's Galois theory Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Grothendieck category ： ウィキペディア英語版 Grothendieck category In mathematics, a Grothendieck category is a certain kind of abelian category, introduced in Alexander Grothendieck's Tôhoku paper of 1957〔. (English translation ).〕 in order to develop the machinery of homological algebra for modules and for sheaves in a unified manner. To every algebraic variety $V$ one can associate a Grothendieck category $\backslash operatorname(V)$, consisting of the quasi-coherent sheaves on $V$. This category encodes all the relevant geometric information about $V$, and $V$ can be recovered from $\backslash operatorname(V)$. This example gives rise to one approach to noncommutative algebraic geometry: the study of "non-commutative varieties" is then nothing but the study of Grothendieck categories.〔(【引用サイトリンク】author=Izuru Mori )〕 ==Definition== By definition, a Grothendieck category $\backslash mathcal$ is an AB5 category with a generator. Spelled out, this means that * $\backslash mathcal$ is an abelian category; * every (possibly infinite) family of objects in $\backslash mathcal$ has a coproduct (a.k.a. direct sum) in $\backslash mathcal$; * direct limits (a.k.a. filtered colimits) of exact sequences are exact; this means that if a direct system of short exact sequences in $\backslash mathcal$ is given, then the induced sequence of direct limits is a short exact sequence as well. (Direct limits are always right-exact; the important point here is that we require them to be left-exact as well.) * $\backslash mathcal$ possesses a generator, i.e. there is an object $G$ in $\backslash mathcal$ such that $\backslash operatorname(G,-)$ is a faithful functor from $\backslash mathcal$ to the category of sets. (In our situation, this is equivalent to saying that every object $X$ of $\backslash mathcal$ admits an epimorphism $G^\backslash rightarrow\; X$, where $G^$ denotes a direct sum of copies of $G$, one for each element of the (possibly infinite) set $I$.) 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「Grothendieck category」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.