Exercise 15, § 3

Let $A$ be a noetherian local ring, $m$ its maximal ideal, $k = A/m$. Let $P$ be a minimal projective resolution of the a-module $k$; for $n \geqslant 0$, denote by $b_n$ the rank of the free a-module $P_n$. (a) Show that one has $b_0 = 1$ and that $b_1 = \dim_k(m/m^2)$ is the minimal number of generators of $m$. (b) If $A$ is commutative, one has $b_2 \geqslant \frac{1}{2} b_1(b_1 - 1)$ (consider the exact sequence $P_2 \to mP_1 \to m^2 \to 0$). (c) Suppose that the ring $A$ is artinian (not necessarily commutative). Prove the inequality $b_2 \geqslant \frac{1}{4} b_1^2$ (let $M_t$ be the submodule of $P_2$ formed by the elements x such that $d_2(x) \in m^t P_1$; show that one has

and that there exists an exact sequence $0 \to M_t/M_{t+1} \to m^t P_1/m^{t+1} P_1 \to m^{t+1}/m^{t+2} \to 0$. Deduce that the polynomial $p(t) = \sum_l (\dim_k(m^l/m^{l+1})) t^l$ satisfies $(b_2 t^2 - b_1 t + 1) p(t) \geqslant 0$ for $0 \leqslant t \leqslant 1$. Conclude by examining separately the cases $b_1 = 1, b_1 \geqslant 2$.