Derived equivalences and Gorenstein algebras

In this note, we introduce the notion of Gorenstein algebras. Let R be a commutative Gorenstein ring and A a noetherian R-algebra. We call A a Gorenstein R-algebra if A has Gorenstein dimension zero as an R-module (see [2]), add(D(AA)) = PA, where D = HomR(−, R), and Ap is projective as an Rpmodule for all p ∈ Spec R with dim Rp < dim R. Note that if dim R = ∞ then a Gorenstein R-algebra A is projective as an R-module and that A is a Gorenstein R-algebra if A is projective as an R-module and add(D(AA)) = PA. Also, in case R is equidimensional and Ap 6= 0 for all p ∈ Spec R, a Gorenstein R-algebra A with A ' DA in Mod-A is a Gorenstein R-order in the sense of [1]. In Section 3, we see that a Gorenstein R-algebra A enjoys properties similar to those of R. Especially, A satisfies the Auslander condition (see [5]) and for any nonzero P • ∈ K−(PA) we have HomK(Mod-A)(P •, A[i]) 6= 0 for some i ∈ Z. Unfortunately, the class of Gorenstein R-algebras is not closed under derived equivalence in general (see Example 4.9). In Section 4, for a tilting complex P • over a Gorenstein R-algebra A we show that B = EndK(Mod-A)(P •) is also a Gorenstein R-algebra if and only if add(P •) = add(νP •), where ν = D ◦ HomA(−, A). In particular, the class of Gorenstein R-algebras A with A ' DA in Mod-A is closed under derived equivalence. More precisely, for any partial tilting complex P • over a Gorenstein R-algebra A with A ' DA in Mod-A, B = EndK(Mod-A)(P •) is also a Gorenstein R-algebra with B ' DB in Mod-B. Then, in Section 5, we provide a construction of such tilting complexes. Namely, we show that tilting complexes P • associated with a certain sequence of idempotents in a Gorenstein R-algebra A satisfy the condition add(P •) = add(νP •). In Sections 6 and 7, we deal with the case where R is a complete local ring and A is free as an R-module. For a tilting complex P • constructed in Section 5, we show that B = EndK(Mod-A)(P •) is also free as an R-module and then provide a way to construct a two-sided tilting complex corresponding to P •.