# 数学代写|交换代数代写commutative algebra代考|Complexes and Exact Sequences

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## 数学代写|交换代数代写commutative algebra代考|Complexes and Exact Sequences

When we have successive linear maps
$$M \stackrel{\alpha}{\longrightarrow} N \stackrel{\beta}{\longrightarrow} P \stackrel{\gamma}{\longrightarrow} Q$$
we say that they form a complex if the composition of any two successive linear maps is null. We say that the sequence is exact in $N$ if $\operatorname{Im} \alpha=\operatorname{Ker} \beta$. The entire sequence is said to be exact if it is exact in $N$ and $P$. This extends to sequences of arbitrary length.
This “abstract” language has an immediate counterpart in terms of systems of linear equations when we are dealing with free modules of finite rank. For example if $N=\mathbf{A}^n, P=\mathbf{A}^m$ and if we have an exact sequence
$$0 \rightarrow M \stackrel{\alpha}{\longrightarrow} N \stackrel{\beta}{\longrightarrow} P \stackrel{\gamma}{\longrightarrow} Q \rightarrow 0$$
The linear map $\beta$ is represented by a matrix associated with a system of $m$ linear equations with $n$ unknowns, the module $M$, isomorphic to $\operatorname{Ker} \beta$, represents the defect of injectivity of $\beta$ and the module $Q$, isomorphic to Coker $\beta$, represents its defect of surjectivity of $\beta$.
An exact complex of the type
$$0 \rightarrow M_m \stackrel{u_m}{\longrightarrow} M_{m-1} \longrightarrow \cdots \cdots \cdots \stackrel{u_1}{\longrightarrow} M_0 \rightarrow 0$$
with $m \geqslant 3$ is called a long exact sequence (of length $m$ ).

## 数学代写|交换代数代写commutative algebra代考|Localization and Exact Sequences

Fact Let $S$ be a monoid of a ring $\mathbf{A}$.
If $M$ is a submodule of $N$, we have the canonical identification of $M_S$ with a submodule of $N_S$ and of $(N / M)_S$ with $N_S / M_S$.
In particular, for every ideal $\mathfrak{a}$ of $\mathbf{A}$, the $\mathbf{A}$-module $\mathfrak{a}_S$ is canonically identified with the ideal $\mathfrak{a} \mathbf{A}_S$ of $\mathbf{A}_S$.
If $\varphi: M \rightarrow N$ is an $\mathbf{A}$-linear map, then:
a. $\operatorname{Im}\left(\varphi_S\right)$ is canonically identified with $(\operatorname{Im}(\varphi))_S$,
b. $\operatorname{Ker}\left(\varphi_S\right)$ is canonically identified with $(\operatorname{Ker}(\varphi))_S$,
c. $\operatorname{Coker}\left(\varphi_S\right)$ is canonically identified with $(\operatorname{Coker}(\varphi))_S$.
If we have an exact sequence of $\mathbf{A}$-modules
$$M \stackrel{\varphi}{\longrightarrow} N \stackrel{\psi}{\longrightarrow} P$$
then the sequence of $\mathbf{A}S$-modules $$M_S \stackrel{\varphi_S}{\longrightarrow} N_S \stackrel{\psi_S}{\longrightarrow} P_S$$ is also exact. 6.5 Fact If $M_1, \ldots, M_r$ are submodules of $N$ and $M=\bigcap{i=1}^r M_i$, then by identifying the modules $\left(M_i\right)S$ and $M_S$ with submodules of $N_S$ we obtain $M_S=\bigcap{i=1}^r\left(M_i\right)_S$.
6.6 Fact Let $M$ and $N$ be two submodules of an $\mathbf{A}$-module $P$, with $N$ finitely generated. Then, the conductor ideal $\left(M_S: N_S\right)$ is identified with $(M: N)_S$, via the natural maps of $(M: N)$ in $\left(M_S: N_S\right)$ and $(M: N)_S$.
This is particularly applied to the annihilator of a finitely generated ideal.

# 交换代数代考

## 数学代写|交换代数代写commutative algebra代考|Complexes and Exact Sequences

$$M \stackrel{\alpha}{\longrightarrow} N \stackrel{\beta}{\longrightarrow} P \stackrel{\gamma}{\longrightarrow} Q$$

$$0 \rightarrow M \stackrel{\alpha}{\longrightarrow} N \stackrel{\beta}{\longrightarrow} P \stackrel{\gamma}{\longrightarrow} Q \rightarrow 0$$

$$0 \rightarrow M_m \stackrel{u_m}{\longrightarrow} M_{m-1} \longrightarrow \cdots \cdots \cdots \stackrel{u_1}{\longrightarrow} M_0 \rightarrow 0$$

## 数学代写|交换代数代写commutative algebra代考|Localization and Exact Sequences

A. $\operatorname{Im}\left(\varphi_S\right)$通常等同于$(\operatorname{Im}(\varphi))_S$，
B. $\operatorname{Ker}\left(\varphi_S\right)$通常等同于$(\operatorname{Ker}(\varphi))_S$，
C. $\operatorname{Coker}\left(\varphi_S\right)$通常被标识为$(\operatorname{Coker}(\varphi))_S$。

$$M \stackrel{\varphi}{\longrightarrow} N \stackrel{\psi}{\longrightarrow} P$$

6.6事实设$M$和$N$是$\mathbf{A}$ -模块$P$的两个子模块，其中$N$有限生成。然后，通过$\left(M_S: N_S\right)$和$(M: N)_S$中的$(M: N)$的自然映射，将理想导体$\left(M_S: N_S\right)$与$(M: N)_S$确定。

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