Hearts of t-structures in the derived category of a commutative Noetherian ring

Abstract

Let R be a commutative Noetherian ring and let D(R) be its (unbounded) derived category. We show that all compactly generated t-structures in D(R) associated to a left bounded filtration by supports of Spec(R) have a heart which is a Grothendieck category. Moreover, we identify all compactly generated t-structures in D(R) whose heart is a module category. As geometric consequences for a compactly generated t-structure (U, U ⊥ [1]) in the derived category D(X) of an affine Noetherian scheme X, we get the following: 1) If the sequence (U[−n]∩D ≤0 (X)) n∈N is stationary, then the heart H is a Grothendieck category; 2) If H is a module category, then H is always equivalent to Qcoh(Y), for some affine subscheme Y ⊆ X; 3) If X is connected, then: a) when ⋂ k∈Z U[k] = 0, the heart H is a module category if, and only if, the given t-structure is a translation of the canonical t-structure in D(X); b) when X is irreducible, the heart H is a module category if, and only if, there are an affine subscheme Y ⊆ X and an integer m such that U consists of the complexes U ∈ D(X) such that the support of H j (U ) is in X ∖ Y, for all j > m. ) and from the Fundación 'Séneca' of Murcia (04555/GERM/06), with a part of FEDER funds. The authors thank these institutions for their help. We also thank Leovigildo Alonso and Ana Jeremías for their remarks and for calling our attention on Ana's thesis. Finally, we thank the referee for the careful reading of the manuscript and his/her suggestions and comments.1. If H is a module category, then there exists an affine subscheme Y of X such that H ≅ Qcoh(Y).2. When X is connected, the following statements hold true:. Then H is a module category if, and only if, there exists an integer m such that (U, U ⊥ [1]) = (D ≤m (X), D ≥m (X)).(b) Suppose that X is irreducible. Then H is a module category if, and only if, there exist an integer m and a nonempty affine subscheme Y ⊆ X such that U consists of the complexes U ∈ D(X) such that Supp(H i (U )) ⊆ X ∖ Y, for all i > m. In this case H is equivalent to Qcoh(Y).The organization of the paper goes as follows. The reader is referred to Section 2 for the pertinent definitions. In Section 2 we introduce and recall the concepts and results which are most relevant for the latter sections of the paper. In particular, we recall the bijection (see [AJS, Theorem 3.1]) between compactly generated t-structures of D(R) and filtrations by supports of Spec(R). In Section 3 we give some properties of the heart H of a compactly generated t-structure in the derived category D(R) of a commutative Noetherian ring, showing in particular that H always has a generator (Proposition 3.10) and that the AB5 condition on

On locally coherent hearts

“…Let L =H |U : U −→ H be the left adjoint to the inclusion functor H ֒→ U. A slight modification of the proof of [24,Proposition 3.10] shows that X := L(U ′ ) is a skeletally small class of generators of H. By lemma 6.2, we get that X ⊆ H ∩ D b f g (R), which ends this first step.…”

“…when φ(i) = ∅, for some i ∈ Z): We shall check all conditions 1-4 of Proposition 4.5. Without loss of generality, we assume that the filtration is [24,Theorem 4.9] we also get that condition 4 holds.…”

Local Coherence of Hearts Associated with Thomason Filtrations

“…by applying the hom functor of X[−r] ∈ U[2] on the rotation of the above triangle. Slightly diverting from [PS17], we fix the following notation: given a Thomason filtration Φ, for any k ∈ Z we set…”

“…Indeed, it is known e.g. from [PS17,Sao17] that this occurs in the case of a noetherian commutative ring. On the other hand, not all non-coherent commutative rings satisfy the equivalent conditions of Theorem 4.7, as we will show in the next example.…”

$t$-Structures with Grothendieck hearts via functor categories

“…The problem of identifying the t-structures whose heart is a Grothendieck category has deserved a lot of attention since it first arose for the Happel-Reiten-Smalø t-structure associated to a torsion pair in a Grothendieck or module category [CGM07,CMT11]. For the general question, several strategies have been used to tackle the problem, including ad hoc arguments [PS17,Baz19], functor categories [Bon16,Bon19] and suitable enhancements of the ambient triangulated category, such as stable 8-categories [Lur17,Lur18] or derivators [S ŠV17,Lak18].…”