Perverse coherent t-structures through torsion theories
📝 Abstract
Bezrukavnikov (later together with Arinkin) recovered the work of Deligne defining perverse $t $-structures for the derived category of coherent sheaves on a projective variety. In this text we prove that these $t $-structures can be obtained through tilting torsion theories as in the work of Happel, Reiten and Smal\o. This approach proves to be slightly more general as it allows us to define, in the quasi-coherent setting, similar perverse $t $-structures for certain noncommutative projective planes.
💡 Analysis
Bezrukavnikov (later together with Arinkin) recovered the work of Deligne defining perverse $t $-structures for the derived category of coherent sheaves on a projective variety. In this text we prove that these $t $-structures can be obtained through tilting torsion theories as in the work of Happel, Reiten and Smal\o. This approach proves to be slightly more general as it allows us to define, in the quasi-coherent setting, similar perverse $t $-structures for certain noncommutative projective planes.
📄 Content
arXiv:0911.0343v4 [math.RT] 7 Aug 2013 PERVERSE COHERENT T-STRUCTURES THROUGH TORSION THEORIES JORGE VITÓRIA Abstract. Bezrukavnikov, later together with Arinkin, recovered Deligne’s work defining perverse t-structures in the derived category of coherent sheaves on a projective scheme. We prove that these t-structures can be obtained through tilting with respect to torsion theories, as in the work of Happel, Reiten and Smalø. This approach allows us to define, in the quasi-coherent setting, similar perverse t-structures for certain noncommutative projective planes.
- Introduction A t-structure in a triangulated category D ([9]) is a pair of strict full subcate- gories, (D≤0, D≥0), such that, for D≤n := D≤0[−n] and D≥n := D≥0[−n] (n ∈Z), (1) Hom(X, Y ) = 0, ∀X ∈D≤0, ∀Y ∈D≥1; (2) D≤0 ⊆D≤1; (3) For all X ∈D, there are A ∈D≤0, B ∈D≥1 and a triangle A −→X −→B −→A[1]. The intersection D≤0 ∩D≥0 is an abelian category ([9]), called the heart. Also, it is well known ([21]) that D≤0, called the aisle, determines the t-structure by setting D≥0 = (D≤0)⊥[1]. A t-structure (D≤0, D≥0) has associated truncation functors τ ≤i : D →D≤i, τ ≥i : D →D≤i and cohomological functors Hi : D →D≤0 ∩D≥0, for all i ∈Z (see [9] for details). If A is an abelian category, its derived category D(A) has a standard t-structure, denoted throughout by (D≤0 0 , D≥0 0 ), defined by D≤0 0 := {X ∈D : Hi 0(X) = 0, ∀i > 0}, D≥0 0 := {X ∈D : Hi 0(X) = 0, ∀i < 0}, where Hi 0 is the usual complex cohomology functor. We denote the associated truncation functors by τ ≤i 0 and τ ≥i 0 and the associated cohomological functor is precisely the complex cohomology functor Hi 0, for all i ∈Z. The standard t- structure restricts to the bounded derived category Db(A) and we shall use the same notations for the restriction, when appropriate. Let K be an algebraically closed field. For a scheme X over K, Arinkin and Bezrukavnikov ([4],[10]) constructed perverse coherent t-structures in Db(coh(X)) Most of this work was developed at the University of Warwick and supported by FCT (Portu- gal), research grant SFRH/BD/28268/2006. Later, this project was also supported by DFG (SPP
- in Stuttgart and by SFB 701 in Bielefeld. The author would like to thank Steffen Koenig, Qunhua Liu, Dmitriy Rumynin, Jan Šťovíček and the anonymous referee for valuable comments on the previous versions of this paper. 1 2 JORGE VITÓRIA as follows. Let Xtop denote the set of generic points of all closed irreducible sub- schemes of X. A perversity is a map p : Xtop −→Z satisfying (1.1) y ∈¯x ⇒p(y) ≥p(x) ≥p(y) −(dim(x) −dim(y)). Note that the image of p has at most dim(X) + 1 elements. The perverse coherent t-structure associated with p ([4], [10]) is defined by the aisle Dp,≤0 = n F • ∈Db(coh(X)) : ∀x ∈Xtop, Li∗ x(F •) ∈D≤p(x) 0 (O{x}-mod) o , where Li∗ x is the left derived functor of the pullback by the inclusion of schemes ix : {x} −→X. In our notation, we identify modules over the residue field k(x) with quasi-coherent sheaves over {x}. Still, we choose to use the notation O{x}-mod for coherent sheaves over {x} to be consistent with the notation in [10]. Our main theorem gives an alternative description of this aisle. Theorem (Theorem 4.6) Let X be a smooth projective scheme over K, R = Γ∗(X) its homogeneous coordinate ring and p a perversity on X. Suppose that R is a commutative connected, noetherian, positively graded K-algebra generated in degree
- Let Ti denote the torsion class cogenerated in T ails(R) by πEi, where Ei = Y {x∈Xtop:p(x)≤i} Eg(R/Ix), with Ix standing for the defining ideal of x ∈Xtop in R. Then we have: Dp,≤0 = F • ∈Db(T ails(R)) : Hi 0(F •) ∈Tj, ∀i > j
∩Db(tails(R)). We clarify some notation. Denoting by OX the structure sheaf of X and by Γ the functor of global sections, the homogeneous coordinate ring R is defined by R = Γ∗(X) := M n∈Z Γ(X, OX(n)). Throughout, R will be assumed to be noetherian. We denote the injective enve- lope of a graded module M in the category of graded (right) R-modules, Gr(R), by Eg(M). The category T ails(R) is the quotient Gr(R)/T ors(R) (we denote the projection functor to this quotient by π) where T ors(R) is the full subcategory of modules M in Gr(R) such that for all x in M there is N ≥0 with xRj = 0, for all j > N. This category is equivalent to Qcoh(X), the category of quasi- coherent sheaves over X, as shown by Serre in [29]. When written in the lower case, tails(R) = gr(R)/tors(R) denotes the subcategory of finitely generated ob- jects in T ails(R), thus being equivalent to coh(X). Throughout we will use these equivalences without mention. We will show in section 3 that, for some rings R, X• ∈Db(T ails(R)) : Hi 0(X•) ∈Tj, ∀i > j
is an aisle of Db(T ails(R)), obtained through a suitable iteration of tilting with re- spect to torsion theories (as in the work of Happel, Reiten and Smalø, [18]), for some torsion classes {Ta, …, Ta+s} (see also [3], [22] and [30] for similar constructions). The categories T ails(R) are also define
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