Exercise 2, § 3

Commutative Algebra, X, p. 157

Let $A$ be a noetherian ring, $M$ an $A$ -module admitting a finite length resolution $(L, p)$ by finite type free modules.

a) Now prove that one has, for every prime ideal $p$ of $A$ ,

\sum_{i \geq 0} (-1)^i \operatorname{rg}_A(L_i) = \sum_{i \geq 0} (-1)^i [\operatorname{Tor}_i^A(M, \kappa(p))] = \sum_{i \geq 0} (-1)^i [\operatorname{Ext}_A^i(M, \kappa(p))] ;

we denote this integer by $\chi(M)$ .

b) Prove that one has $\chi(M) \geq 0$ (consider a prime ideal $p$ associated with $A$ ).

c) Prove that the following conditions are equivalent:

(i) $\chi(M) = 0$ ; (ii) the annihilator of $M$ is not reduced to zero ; (iii) the annihilator of $M$ contains a cancellable element.

(If $\chi(M) = 0$ , prove that $M_p$ is zero for every $p \in \operatorname{Ass}(A)$ ; if $\chi(M) > 0$ , prove analogously that the annihilator of $\operatorname{Ann}(M)$ contains a cancellable element, which implies $\operatorname{Ann}(M) = 0$ .)

Solution

No solution has been written yet.