Exercise 5, § 7

Commutative Algebra, VIII, p. 104

Let $k$ be a field. Put $R = k[[X, Y, Z]]$ and consider the local ring $A = R/(XY, XZ)$ , of maximal ideal $m$ . Now prove that one has

H_A = (1 - T)(1 - T - T^2) \quad H_R = 1 + \sum_{n \geq 1} (n + 2) T^n.

Deduce that one has $\dim(A) = 2, \dim_k(m/m^2) = 3$ and $e_m(A) = 1$ . (One has $H_A = P_G$ where $G = \operatorname{gr}_m(A) = k[X, Y, Z]/(XY, XZ)$ . We shall prove the existence of an exact sequence of graded $H$ -modules

0 \to H(-3) \to H(-2) \oplus H(-2) \to H \to G \to 0

where $H = k[X, Y, Z].$ )

Solution

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