Compactly generated t-structures in the derived category of a commutative ring

Abstract

We classify all compactly generated t-structures in the unbounded derived category of an arbitrary commutative ring, generalizing the result of [ATLJS10] for noetherian rings. More specifically, we establish a bijective correspondence between the compactly generated t-structures and infinite filtrations of the Zariski spectrum by Thomason subsets. Moreover, we show that in the case of a commutative noetherian ring, any bounded below homotopically smashing t-structure is compactly generated. As a consequence, all cosilting complexes are classified up to equivalence.

Silting Objects

“…It is shown in that over a commutative Noetherian ring, every homotopically smashing t ‐structure with for some is compactly generated. Combining this with the results in Section 6, one obtains that all cosilting complexes are of cofinite type.…”

Definable coaisles over rings of weak global dimension at most one

“…As in [8], we then switch our focus to the commutative case. The basis for our findings is the structure of compactly generated t-structures which were described in terms of certain filtrations of the Zariski spectrum in [1] and this was further generalized to not necessarily noetherian rings in [22]; see also [45] for a different but related kind of result. The property of being compactly generated localizes well, and this allows us to consider the commutative rings of weak global dimension at most one locally -that is, to confine to valuation domains.…”

“…If R is right semihereditary, then R/I[−n] is a compact object in D(R) for any finitely generated ideal I. If R is commutative, then V can be written as a right orthogonal to a set of suspensions of Koszul complexes (see [22,Lemma 5.4]). In both cases, V is a compactly generated coaisle and therefore is determined on cohomology by Theorem 3.4.…”

“…Finally, suppose that R is commutative and that V is a compactly generated coaisle. Because V n [−n] ⊆ V, we have that V n is closed under injective envelopes by [22,Lemma 3.3].…”

“…This is indeed the case, and should be seen as the correct generalization of the module theoretic case of Proposition 5.10 (cf. [7,Theorem 6.11] and also [22,Proposition 5.10] for a similar type of description in the case of compactly generated t-structures). Proposition 7.10.…”