Exercise 9, § 3

Let $A$ be a ring, $(L, d)$ a bounded complex of free $A$-modules of finite type, zero in degrees < 0. We propose to prove that the following conditions are equivalent: (i) One has $H_i(L) = 0$ for $i > 0$ ; (ii) One has \operatorname{prof}_A(\partial(d_i); A) \geq i and $rg(d_i) + rg(d_{i+1}) = rg_A(L_i)$ for all $i \geq 1$. a) Show that it suffices to prove this statement when the ring $A$ is noetherian (consider the subring of $A$ generated by the coefficients of the matrices $d_i$ for $i \geq 1$ ). We henceforth suppose this condition satisfied. b) If $A$ is a local ring of dimension 0, prove that (ii) implies (i) (use exerc. 8, b)).

c) Suppose that the ring $A$ is local, that condition (ii) is satisfied, and that $\mathrm{Supp}(H_i(L)) = \{ m_A \}$ for $i \geq 1$. Prove that $H_i(L) = 0$ for $i \geq 0$ (let $p$ be the largest integer such that $\mathfrak{d}(d_p) \neq A$; deduce from exerc. 8, b) that $H_i(L_i) = 0$ for $i > p$, then from exerc. 8, a) and cor. 2 of prop. 3 (No. 2) that the complex $0 \to L_p/B_p(L) \to L_{p-1} \to \ldots \to L_0$ is acyclic in degrees $> 0$.

d) Prove the implication (ii) $\Rightarrow$ (i) (reduce to the case where $A$ is local, and reason by induction on $\dim(A)$).

e) Suppose that the complex $L$ is acyclic in degrees $> 0$. Let $p$ be a prime ideal associated with $A$. Show that for every $i \geq 1$ ($\mathrm{Coker}\ d_i$)$_p$ is a free $A_p$-module, of rank $\sum_{p \geq i-1} (-1)^p \mathrm{rg}_A(L_p)$; moreover one has $\mathrm{rg}((d_i)_p) = \sum_{p \geq i} (-1)^{p-i} \mathrm{rg}_A(L_p)$ and $\mathfrak{d}((d_i)_p) = A_p$ (exerc. 8).

f) Deduce that one has $\mathrm{rg}(d_i) = \sum_{p \geq i} (-1)^{p-i} \mathrm{rg}_A(L_p)$ and consequently $\mathrm{rg}(d_i) + \mathrm{rg}(d_{i+1}) = \mathrm{rg}_A(L_i)$ for all $i \geq 1$.

g) Let $i$ be an integer $\geq 1$, and $p$ a prime ideal of $A$. Show that the condition $\mathrm{dp}_{A_p}(H_0(L_p)) \geq i$ is equivalent to $\mathfrak{d}(d_i)A_p \neq A_p$, that is to say $\mathfrak{d}(d_i) \subset p$ (use exerc. 8, a)). With the aid of prop. 8 of § 1, No. 5 and th. 1, deduce that one has $\mathrm{prof}_A(\mathfrak{d}(d_i); A) \geq i$, which proves the implication (i) $\Rightarrow$ (ii).