We classify complactly generated t-structures on the derived category of modules over a commutative Noetherian ring R in terms of decreasing filtrations by supports on Spec(R). A decreasing filtration by supports \phi : Z -> Spec(R) satisfies the weak Cousin condition if for any integer i \in Z, the set \phi(i) contains all the inmediate generalizations of each point in \phi(i+1). Every t-structure on D^b_fg(R) (equivalently, on D^-_fg(R)) is induced by complactly generated t-structures on D(R) whose associated filtrations by supports satisfy the weak Cousin condition. If the ring R has dualizing complex we prove that these are exactly the t-structures on D^b_fg(R). More generally, if R has a pointwise dualizing complex we classify all compactly generated t-structures on D_fg(R).
Deep Dive into Classifying Compactly generated t-structures on the derived category of a Noetherian ring.
We classify complactly generated t-structures on the derived category of modules over a commutative Noetherian ring R in terms of decreasing filtrations by supports on Spec(R). A decreasing filtration by supports \phi : Z -> Spec(R) satisfies the weak Cousin condition if for any integer i \in Z, the set \phi(i) contains all the inmediate generalizations of each point in \phi(i+1). Every t-structure on D^b_fg(R) (equivalently, on D^-_fg(R)) is induced by complactly generated t-structures on D(R) whose associated filtrations by supports satisfy the weak Cousin condition. If the ring R has dualizing complex we prove that these are exactly the t-structures on D^b_fg(R). More generally, if R has a pointwise dualizing complex we classify all compactly generated t-structures on D_fg(R).
The concept of t-structure on a triangulated category arises as a categorical framework for Goresky-MacPherson's intersection homology. Through this construction Beȋlinson, Bernstein, Deligne and Gabber extended intersection cohomology to the étale context. A t-structure provides a homological functor with values in a certain abelian category contained in the original triangulated category denominated the heart of the t-structure. In Grothendiecks's terms, this study accounts for the study of extraordinary cohomology theories for discrete coefficients. Intersection cohomology and its variants have been studied successfully with these methods over the last twenty years. Let us point out [BBD] and [GN] and references therein.
On the side of continuous coefficients the development has proceeded at a slower pace. Deligne was first to make contributions to the problem of constructing t-structures on the bounded derived category of coherent sheaves on a Noetherian scheme under certain hypothesis, namely the existence of a dualizing complex and of global locally free resolutions. His work was not available until the expository e-print by Bezrukavnikov [Be]. Later Kashiwara [Ka] constructs from a decreasing family of supports satisfying certain condition a t-structure on the bounded derived category of coherent sheaves on a complex manifold, which corresponds in the algebraic case to a smooth separated scheme of finite type over C. Also, Yekutieli and Zhang [YZ] considered the Grothendieck dual t-structure of the canonical one on the derived category of finitely generated modules over a Noetherian ring with dualizing complex. This t-structure is called the Cohen-Macaulay t-structure in the present paper and it is shown to exist in the whole unbounded derived category.
Deligne, Bezrukavnikov and Kashiwara built, on the derived category of bounded complexes with finitely generated homologies, a t-structure starting with a finite filtration by supports
Z n = ∅ in the corresponding topological space X. In Bezrukavnikov’s paper Noetherian induction is used. Kashiwara constructs the triangle of the corresponding t-structure using in an essential way that the complexes are bounded. The filtrations by supports used by both authors are finite and satisfy a condition that we call in this paper the weak Cousin condition (for any integer i, the set Z i contains all the immediate generalizations of each point in Z i+1 ). This name refers to a weakening and reformulation of the notion of codimension function introduced by Grothendieck, see [H, V.7] and §4 below. An equivalent notion was used in [Be] under the name comonotone perversity.
In the aforementioned papers, the authors rely on finite step-by-step constructions and do not take into account the possibility allowed by infinite constructions if one considers the unbounded derived category. This approach makes sense after [AJS2] where it is proved that for a collection of objects in the unbounded derived category of a Grothendieck category there is a t-structure whose aisle is generated by this set of objects.
A logical next step is to try to classify t-structures on D(R) by filtrations of subsets of Spec (R) extending the fact, proved in the key paper [N1], that Bousfield localizations (the class of triangulated t-structures) are classified by subsets of Spec (R). However, a counterexample by Neeman and further developments by the third author made clear that residue fields of prime ideals were not the right objects to use in order to achieve the classification of t-structures (see remark on page 19).
Stanley in his preprint [Sta] treated the problem of studying t-structures on D b fg (R), the subcategory of D(R) of bounded complexes with finitely generated homologies, where R is a commutative Noetherian ring. He showed that it is not possible to classify all t-structures on D(R) because the class of t-structures on D(Z) is not a set [Sta,Corollary 8.4]. Then it is not possible to put t-structures on D(R) in correspondence with collections of subsets of Spec (R).
On the positive side, Stanley showed that there is an order preserving bijection between filtrations of Spec(R) by stable under specialization subsets and nullity classes in D b fg (R). Theorem B in [Sta] states that the weak Cousin condition is a necessary condition over a filtration for the corresponding nullity class in D b fg (R) to be an aisle, and he conjectured that the converse is true. However, the proof of Theorem B in [Sta] does not seem to be complete. We give here an alternate approach to Stanley’s result encompassing the unbounded category and, in addition, we answer Stanley’s conjecture in the affirmative.
Specifically, the category D fg (R) is skeletally small so any t-structure on D fg (R) is the restriction of a t-structure on D(R) (see Lemma 1.3 and Proposition 1.4). We look at t-structures on D(R) generated by complexes with finitely generated homologies and characterize those that restr
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