Bourbaki

Exercise 7, § 3

Commutative Algebra, X

Let AA be a local noetherian ring, MM a finitely generated a-module, TT a finitely generated a-module and of finite injective dimension. Exercise prove the equality

prof⁡(A)=prof⁡A(M)+sup⁡{i∣Ext⁡Ai(M,T)≠0} \operatorname{prof}(A) = \operatorname{prof}_A(M) + \sup \{ i \mid \operatorname{Ext}_A^i(M, T) \neq 0 \}

(reason by induction on \operatorname{prof}_A(M), using Prop. 9).

Solution

No solution has been written yet.

Machine translation of the English, by gpt-5-mini. Not checked by a person. That model is a cut down version of the one the rest of this language was written by.