加群に関する定義

重要な概念$`\newcommand{\For}{\mbox{For }}%
\newcommand{\hyp}{\mbox{-}}%
\newcommand{\In}{\mbox{ in }}%
\newcommand{\Iff}{\Leftrightarrow}%
\newcommand{\WeDefine}{\mbox{WeDefine } }%
\newcommand{\CAT}{{\bf CAT}}%
\newcommand{\st}{\mbox{ s.t. }}%
\newcommand{\Ch}{\mathscr{C}}%
\newcommand{\cat}[1]{\mathcal{#1}}%
\newcommand{\Imp}{\Rightarrow }%
`$

  1. 階付き加群、G-階付き加群
  2. 階付き微分加群=鎖複体、(G, r)-階付き加群微分加群

加群の平坦性:

$`
\For M \In R\hyp {\bf Mod}\\
\WeDefine IsFlat(M) \\
: \Iff ( (M\otimes \hyp): R\hyp {\bf Mod} \to R\hyp {\bf Mod} \In \CAT) \: isExact
`$

加群の射影性:

$`
\For P \In R\hyp {\bf Mod}\\
\WeDefine IsProjective(P) \\
: \Iff \\
\forall g:M \to N \In R\hyp {\bf Mod} \st isEpi .\\
([id_P, g]: [P, M] \to [P, N] \In R\hyp {\bf Mod}) \: isEpi \\
\Iff \\
([P, \hyp]:R\hyp {\bf Mod} \to R\hyp {\bf Mod}) \: isEpiToEpi
`$

加群の分解:

$`
\For M \In R\hyp {\bf Mod}\\
\For X \In \Ch_\bullet(R\hyp {\bf Mod}) \\
\For \varepsilon : X \to !M \In \Ch_\bullet(R\hyp {\bf Mod}) \\
\WeDefine IsResolutionOf(X, M)\\
: \Iff \\
(X \overset{\varepsilon_0 }{\to} M \to 0) \: isExactSeq
`$

加群の入射性:

$`
\For I \In R\hyp {\bf Mod}\\
\WeDefine IsInjective(I) \\
: \Iff \\
\forall e:L \to M \In R\hyp {\bf Mod} \st isMono .\\
([e, id_I]: [M, I] \to [L, I] \In R\hyp {\bf Mod}) \: isEpi
\Iff \\
([\hyp, I]:R\hyp {\bf Mod} \to R\hyp {\bf Mod}) \: isMonoToEpi
`$

加群の分解(余分解):

$`
\For M \In R\hyp {\bf Mod}\\
\For Y \In \Ch^\bullet(R\hyp {\bf Mod}) \\
\For \eta : !M \to Y \In \Ch^\bullet(R\hyp {\bf Mod}) \\
\WeDefine IsCoResolutionOf(Y, M)\\
: \Iff \\
(0 \to M \overset{\eta^0 }{\to} Y ) \: isExactSeq
`$

F-非輪状対象:

$`
\For \cat{A},\cat{B} \In {\bf AbelianCAT}\\
\For F:\cat{A} \to \cat{B}\In \CAT \: isLeftExactAdditive \\
\For A\In \cat{A} \\
\WeDefine IsAcyclic(A, F)\\
: \Iff \forall p\in {\bf Z}.(p \gt 0 \Imp R^pF(A) = 0)
`$

環が単項イデアル整域:

$`
\For R \In {\bf Rng}\\
\WeDefine IsPID(R)\\
: \Iff
\forall I\in Ideal(R). \exists a\in R.( I = (a)_R )
`$

長いホモロジー完全列の構成:

$`
\mbox{For } z''\in H_q(X'') \\
\mbox{Let } [x''] := z'' \mbox{ s.t. } x'' \in Z_q(X'')\\
\mbox{Let } z'\in Z_{q -1}(X') := \\
\varepsilon\, x'\in Z_{q-1}(X'). (\\
\xymatrix{
{} & x \ar@{|->}[r]^{g_q} \ar@{|->}[d]^{d_q}& x'' \\
x' \ar@{|->}[r]^{f_{q-1}} & y' = y
}\\
)\\
\mbox{Return }[z']
`$

ジャスティフィケーションが別に必要。