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 ${\mathcal{D}}_{\text{fg}}^{\text{b}}(R)\cong {\mathcal{D}}^{\text{b}}(\mathcal{B}{)}^{\text{op}}$ between $\mathrm{mod}R$ 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 ${\mathcal{D}}_{\text{fg}}^{\text{b}}(R)$. 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.