Let $A$ be a Noetherian (not necessarily local) ring and $M$ a finitely generated $A$-moduel. Is the length of the minimal injective resolution of $M$ always equal to the injective dimension of $M$? (Just like the projective dimension and minimal free resolution.) I suspect the formula for the Bass number $$\mu_i(\mathfrak{p},M)=\mbox{dim}_{\kappa(\mathfrak{p})}\mbox{Ext}^i_{A_{\mathfrak{p}}}(\kappa(\mathfrak{p}),M_{\mathfrak{p}})$$ might hold the key, but I can't seem to go anywhere.
Does the minimal injective resolution have the smallest length?
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commutative-algebra
homological-algebra