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1同调代数习题

11.1	Let $\mathcal{C}$ be a category and $X,Y$ be objects in $\mathcal{C}$. Suppose we have a natural isomorphism between funtors$${\eta }_X:\mathrm{Hom}(-,X)\to \mathrm{Hom}(-,Y).$$Then we have$$\mathrm{Hom}(X,X)\cong \mathrm{Hom}(X,Y)\text{and}\mathrm{Hom}(Y,X)\cong \mathrm{Hom}(Y,Y).$$Let $f\in \mathrm{Hom}(X,Y)$ be the image of $1_X$ and $g\in \mathrm{Hom}(Y,X)$ be the preimage of $1_Y$ . Use the naturality of $\eta$ to show that $fg=1_Y$.
证明:
11.2	Let $F:\mathcal{C}\to \mathcal{D}$ be an additive functor between abelian categories and suppose $G$ is its right adjoint. Show that $G$ is also an additive functor. [Hint: By definition, the zero object $0$ is both the initial and the terminal object. Use this to show that $G$ takes zero morphisms to zero morphisms. Then consider direct sums.]
证明:
11.3	Recall that a chain map $f:\mathcal{C}\to \mathcal{D}$ is a chain-homotopy equivalence if there is a chain map $g:\mathcal{D}\to \mathcal{C}$ such that $fg$ and $gf$ are both chain-homotopic to the identity map. We say $\mathcal{C}$ and $\mathcal{D}$ are chain-homotopy equivalent if there exists a chain-homotopy equivalence from $\mathcal{C}$ to $\mathcal{D}$. This is obviously an equivalence relation. Show that the complex$$\cdots \to 0\to A\to B\to C\to 0\to \cdots$$is chain-homotopy equivalent to the $0$ complex if and only if$$0\to A\to B\to C\to 0$$is a split short exact sequence.
证明:
11.4	Let $G$ be a finite group and let $\mathcal{A}$ be the category of $G$-representations over $\mathbb{C}$. Prove that every object of $\mathcal{A}$ is projective.
证明:	$\mathcal{A}$ 与 $\mathbb{C}[G]-\text{Mod}$ 等价. $\mathbb{C}[G]$ 是半单环, 半单环上的模都是投射模.
11.5	Recall that a subgroup of a free abelian group is free. What are the projective objects in the category of abelian groups?
证明:	自由群.
11.6	Let $\mathcal{A}$ be the category of abelian groups and let $n$ be a positive integer. Let $F:\mathcal{A}\to \mathcal{A}$ be the functor $F(A)=A/nA$. Directly from definition of derived functors, show that $L_{i}F(A)=0$ for $i⩾2$ and $L_{1}F(A)=\{a\in A\mid na=0\}$ is the subgroup of $n$-torsion. Do not forget to describe how $L_1(F)$ acts on morphisms. (We will have a better way of computing this soon.)
证明:
13.1	Suppose $(\underset{i,j⩾0}{⨁}{A}^{i,j},d)$ is a double complex, i.e. $$d(A^{i,j})\subseteq A^{i+1,j}\oplus A^{i,j+1},d^2=0.$$Suppose all rows are exact. (a) Prove that the total complex $\mathrm{Tot}(A)$ is exact. (b) Is it necessary that all columns are exact?
证明:
13.2	Let $R$ be a ring such that every submodule of a projective module is projective. Show that every quotient of a injective module is injective.
证明:
13.3	In the following two exercises, let $X$ be a topological space and $\text{Sh}(X)$ be the category of sheaves of abelian groups on $X$. Let $\varphi :\mathcal{F}\to \mathcal{G}$ be a morphism in $\text{Sh}(X)$. Suppose that for each $x\in X$, the induced morphism on stalks$${\mathcal{F}}_x\to {\mathcal{G}}_x$$is surjective. Show that $\varphi$ is an epimorphism in the category $\text{Sh}(X)$.
证明:
13.4	Let $x\in X$ and $I$ be an injective Abelian group. Denote by $I_{x,X}$ the skyscraper sheaf at $x$ with stalk $I$. Show that $I_{x,X}$ is an injective object in $\text{Sh}(X)$. [Hint: Having fixed $x$, find the functor adjoint to the "skyscraper sheaf functor" $\mathcal{A}b\to \text{Sh}(X)$. Then mimic the proof that ${\mathrm{Hom}}_{\mathcal{A}b}(R,\mathbb{Q}/\mathbb{Z})$ is injective.]
证明: