Exercise 4, § 4

Place ourselves in the situation of Theorem 1 of § 4, No. 2, and denote by $C$ the set of classes of finite type graded $H$-modules with components of finite length over $H_0$, and by $K(C)$ the corresponding grothendieck group. a) For every $H$-module $M$ as above, now prove that there exists a finite length graded $H_0$-module $M_0$ and a surjective graded homomorphism $p : H \otimes_{H_0} M_0 \to M$. Deduce that every $H$-module $M$ as above admits a graded resolution (A, X, p. 56) by $H$-modules:

where, for every $i$, $M_i$ is a finite length graded $H_0$-module. b) Using a Koszul resolution of the $H$-module $H_0$ (A, X, p. 152), now prove that for every graded $H$-module $M$, $\operatorname{Tor}_j^H(H_0, M)$ is endowed canonically with a graduation, that $\operatorname{Tor}_j^H(H_0, M) = 0$ for $j > \operatorname{Card}(I)$ and that $\operatorname{Tor}_j^H(H_0, P \otimes_{H_0} H)$ is a finite length $H_0$-module for every $j$ when $P$ is a finite length $H_0$-module.

c) Show that for every $j$, $\mathrm{Tor}_j^H(H_0, M)$ is a graded $H_0$-module of finite length when $M$ is of type $\mathscr{C}$. (It may be shown by increasing induction on $j$. If $j = 0$, use $a$) and b). In the general case, use an exact sequence $0 \to N \to H \otimes_{H_0} M_0 \to M \to 0$, the induction hypothesis and b).) d) For every $H$-module $M$ of type $\mathscr{C}$, put

thus obtaining an element of $K(\mathscr{C}_0)[T, T^{-1}]$ where $K(\mathscr{C}_0)$ is the grothendieck group of $H_0$-modules of finite length. Show that $M \mapsto e(M)$ is additive on exact sequences of graded $H$-modules, and that consequently $M \mapsto e(M)$ determines a group homomorphism For every graded $H_0$-module $P$ of finite length, put Show that $P \mapsto \tau(P)$ is additive on exact sequences and that consequently $P \mapsto \tau(P)$ determines a homomorphism between the corresponding grothendieck groups Show that $e$ and $\tau$ are inverse group isomorphisms. (First show that $e \circ \tau$ is the identity. To prove the assertion, it is therefore enough to show that $\tau$ is surjective, that is to show that for every $H$-module $M$ of type $\mathscr{C}$, $[M]$ belongs to the image of $\tau$. Notice that $M$ admits a finite filtration whose successive quotients are annihilated by a maximal ideal of $H_0$ and that an $H$-module $M$ of type $\mathscr{C}$, annihilated by a maximal ideal $m$ of $H_0$, admits a finite graded resolution by free $H/mH$-modules of finite type (A, X, p. 56).) e) Let $M$ be an $H$-module of type $\mathscr{C}$. Show that one has in $K(\mathscr{C})$ f) Let $\Gamma$ be a commutative group and $\Phi : K(\mathscr{C}) \to \Gamma$ a group homomorphism. For every $H_0$-module of finite length $P$, put Show that for every $H$-module $M$ of type $\mathscr{C}$, one has In particular, with the notation of Remark 4 of § 4, No. 2 show that one has and consequently that g) Give, using the preceding results, a new proof of Theorem 1 of § 4, No. 1. h) Let $\Gamma$ be a commutative group. Describe all additive mappings $M \mapsto \Phi(M)$ on the $H$-modules of type $\mathscr{C}$ with values in $\Gamma$, in terms of mappings from $\mathrm{Specmax}(H_0) \times \mathbf{Z}$ into $\Gamma$. i) Suppose that $H_0$ is a field. Let $A$ be a ring. Describe all additive mappings $M \mapsto C(M)$ with values in $A$ such that $C(M).C(N) = \sum_j (-1)^j C(\mathrm{Tor}_j^H(M, N))$ for every pair $M$ and $N$ of graded $H$-modules of finite type.