Semi-dualizing complexes and their Auslander categories

Abstract

Abstract. Let R be a commutative Noetherian ring. We study R-modules, and complexes of such, with excellent duality properties. While their common properties are strong enough to admit a rich theory, we count among them such, potentially, diverse objects as dualizing complexes for R on one side, and on the other, the ring itself. In several ways, these two examples constitute the extremes, and their well-understood properties serve as guidelines for our study; however, also the employment, in recent studies of ring homomorphisms, of complexes "lying between" these extremes is incentive.

Sequences for complexes II

“…The final Theorem 3.9 of this paper generalizes the result above in two directions: the dualizing complex is replaced by a Cohen-Macaulay semidualizing complex (see [3,Sec. 2] or 3.8 below for definitions), and the finite module is replaced by a complex with finite homology.…”

“…Semi-dualizing Complexes 3.8. We recall two basic definitions from [3]: Relations between dimension and depth for C-reflexive complexes are studied in [3, sec. 3], and the next result is an immediate consequence of [3, (3.1) and (2.10)].…”

“…The first biconditional follows by 3.5 (b) and the last by 3.5 (a). Since K (x) is a bounded complex of free modules (hence of finite projective dimension), it follows from [3,Thm. (3.17)] that also K(x; Y ) is C-reflexive, and the second biconditional is then immediate by the CMD-formula [3,Cor.…”

“…The definition of parameters for complexes is based on a notion of anchor prime ideals. These do for complexes what minimal prime ideals do for modules, and the quantitative relations between dimension and depth under dagger duality-studied in [3]-have a qualitative description in terms of anchor and associated prime ideals.…”

Restricted homological dimensions over local homomorphisms and Cohen-Macaulayness

“…where the second isomorphism follows from [2], Lemma 4.4. It follows from [6], Theorem 2.13, that pd S RHom S (P, S) = − inf P is finite and inf RHom S (P, S) = −pd S P . Thus we have the following equalities.…”