Some homological invariants of local PI algebras (Q1972430)

The following main results for a Noetherian PI-algebra \(A\) are proved: (1) if \(A\) is a local algebra then the Auslander-Buchsbaum formula holds; i.e., if \(X\in D^b_{fg}(\text{Mod }A)\) with \(\text{pd }X<\infty\), then \(\text{pd }X+\text{depth }X=\text{depth }A\); (2) if \(A\) is a semilocal algebra and \(\widehat A\) its completion then (i) for all finite \(A\)-modules \(M\): \(\text{K-dim }M=\text{K-dim}_{\widehat A}\widehat M\); (ii) \(\text{id }X=\text{id}_{\widehat A}\widehat X\) for all \(X\in D^b_{fg}(\text{Mod }A)\); (3) if an \((A,T)\)-bimodule \(_AE_T\) defines a Morita duality between \(A\) and \(T\) then \(T\) is a Noetherian PI-algebra; (4) if \(A\) is a semilocal algebra and \(M\) is a finite \(A\)-module then \(\text{lcd}(M)=\text{K-dim }M\) and \(H^d_{\mathcal M}(M)\) has dual Krull dimension \(d\) where \(d=\text{K-dim }M\).