# 交换式代数 Commutative Algebra MATH32012

if $\zeta \in H^{n}(G, k)$ defines a hypersurface $V$ in $V_{G}$ and $\mathscr{\gamma}$ is the set of subvarieties of $V$ then we write $E_{\zeta}$ and $F_{\zeta}$ instead of $E_{\mathscr{\gamma}}$ and $F_{\gamma}$. If $\zeta$ is represented by a cocycle $\hat{\zeta}: \Omega^{n}(k) \rightarrow k$, then the module $F_{\zeta}$ can be constructed as follows. We can dimension shift $\hat{\zeta}$ to give maps
$$k \stackrel{\Omega^{-n} \hat{\xi}}{\longrightarrow} \Omega^{-n}(k) \stackrel{\Omega^{-2 n} \hat{\zeta}}{\longrightarrow} \Omega^{-2 n}(k) \stackrel{\Omega^{-3 n} \hat{\xi}}{\longrightarrow} \cdots$$

and the colimit is $F_{\zeta}$. the cohomology of $F_{\zeta}$,
$$\hat{H}^{}\left(G, F_{\zeta}\right) \cong \hat{H}^{}(G, k){\zeta} \cong H^{*}(G, k){\zeta}$$
is the localization of either Tate or ordinary cohomology with respect to $\zeta$.
If we talee the map from the first term in the sequence to the colimit and complete to a triangle, we get the module $E_{\zeta}$ and the triangle
$$E_{\zeta} \rightarrow k \rightarrow F_{\zeta} \rightarrow \Omega^{-1} E_{\zeta} .$$
If $L_{\zeta}$ i is the kernel of $\hat{\zeta}^{i}$ then $E_{\zeta}$ can be written as the colimit of
$$\Omega^{-n} L_{\zeta} \rightarrow \Omega^{-2 n} L_{\zeta}^{2} \rightarrow \Omega^{-3 n} L_{\zeta}^{3} \rightarrow \cdots$$

## MATH32012 COURSE NOTES ：

$$x^{q}-1=\prod_{i=1}^{d} f_{i}(x), \quad \text { with } \sum_{i=1}^{d} \operatorname{deg}\left(f_{i}(x)\right)=q .$$
Since the ideals $\left(f_{i}(x)^{\mathrm{p}^{n}}\right)$, where $1 \leq i \leq d$, in $k[x]$ are pair wise comaximal, the Chinese Remainder Theorem yields
$$R \cong \prod_{i=1}^{d} R_{i}, \quad \text { where } R_{i}=\frac{k[x]}{\left(f_{i}(x)^{p^{n}}\right)}$$