Hearts for commutative noetherian rings: torsion pairs and derived equivalences

Abstract

Over a commutative noetherian ring R, the prime spectrum controls, via the assignment of support, the structure of both Mod(R) and D(R). We show that, just like in Mod(R), the assignment of support classifies hereditary torsion pairs in the heart of any nondegenerate compactly generated t-structure of D(R). Moreover, we investigate whether these t-structures induce derived equivalences, obtaining a new source of Grothendieck categories which are derived equivalent to Mod(R).

Singular equivalences to locally coherent hearts of commutative noetherian rings

“…The two claims of the implication (ii) ⇒ (i) are proven in [31,Corollary 6.17] and [38, Theorem 6.3], respectively. It remains to show (i) ⇒ (ii).…”

“…Let V = {m}, consider the associated hereditary torsion pair t = (T, F) in Mod-R, and let H be the HRS-tilt of Mod-R with respect to t; namely, H = F[1] * T, we refer to [31] for terminology and details. Notice that since D(H) ∼ = D(Mod-R) (by [31,Corollary 5.11]) the former is compactly generated.…”

“…Let V = {m}, consider the associated hereditary torsion pair t = (T, F) in Mod-R, and let H be the HRS-tilt of Mod-R with respect to t; namely, H = F[1] * T, we refer to [31] for terminology and details. Notice that since D(H) ∼ = D(Mod-R) (by [31,Corollary 5.11]) the former is compactly generated. Also, H is the heart of the Happel-Reiten-Smalø t-structure corresponding to the torsion pair (T, F), and this is an intermediate t-structure which is compactly generated and restrictable ([31, Remark 4.8 and Theorem 2.16(3)]).…”