Product-complete tilting complexes and Cohen–Macaulay hearts

Abstract

We show that the cotilting heart associated to a tilting complex T is a locally coherent and locally coperfect Grothendieck category (i.e., an Ind-completion of a small artinian abelian category) if and only if T is product-complete. We then apply this to the specific setting of the derived category of a commutative noetherian ring R . If \dim(R)<\infty , we show that there is a derived duality \mathcal{D}^{\textup{b}}_{\textup{fg}}(R) \cong \mathcal{D}^{\textup{b}}(\mathcal{B})^{\textup{op}} between \mod R and a noetherian abelian category \mathcal{B} if and only if R is a homomorphic image of a Cohen–Macaulay ring. Along the way, we obtain new insights about t-structures in \mathcal{D}^{\textup{b}}_{\textup{fg}}(R) . In the final part, we apply our results to obtain a new characterization of the class of those finite-dimensional noetherian rings that admit a Gorenstein complex.