Product-complete tilting complexes and Cohen–Macaulay hearts

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 (R) <, we show that there is a derived duality D^b₅₆ (R) D^b (B) ^op between R and a noetherian abelian category 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 D^b₅₆ (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.