Product-complete tilting complexes and Cohen-Macaulay hearts

TL;DR

This paper explores the conditions under which tilting complexes induce certain Grothendieck categories and applies these results to derived categories of rings, providing new dualities and characterizations related to Cohen-Macaulay rings and Gorenstein complexes.

Contribution

It establishes a characterization of product-complete tilting complexes and connects derived dualities with Cohen-Macaulay ring properties, advancing understanding of t-structures and Gorenstein complexes.

Findings

Cotilting heart is locally coherent if and only if the tilting complex is product-complete.

Derived duality exists between certain derived categories if and only if the ring is a homomorphic image of a Cohen-Macaulay ring.

New characterization of finite-dimensional Noetherian rings admitting Gorenstein complexes.

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 $\mathcal{D}^b_{fg}(R) \cong \mathcal{D}^b(\mathcal{B})^{op}$ 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 $\mathcal{D}^b_{fg}(R)$. 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.