Derived categories of Quot schemes of locally free quotients, I

Abstract

This paper studies the derived category of the Quot scheme of rank d locally free quotients of a sheaf G of homological dimension ≤ 1 over a scheme X. In particular, we propose a conjecture about the structure of its derived category and verify the conjecture in various cases. This framework allows us to relax certain regularity conditions on various known formulae -such as the ones for blowups (along Koszul-regular centers), Cayley's trick, standard flips, projectivizations, and Grassmannain-flips -and supplement these formulae with the results on mutations and relative Serre functors. This framework also leads us to many new phenomena such as virtual flips, and structural results for the derived categories of (i) Quot 2 schemes, (ii) flips from partial desingularizations of rank ≤ 2 degeneracy loci, and (iii) blowups along determinantal subschemes of codimension ≤ 4. Generalities on derived categoriesof schemes 3.2. Generators of triangulated categories 3.3. Semiorthogonal decompositions and mutations 3.4. Postnikov systems and convolutions 3.5. Closed monoidal structures 3.6. Linear categories 3.7. Relative Serre functors 3.8. Base change of linear categories 3.9. Relative Fourier-Mukai transforms 3.10. Compositions of relative Fourier-Mukai transforms 3.11. Relative exceptional collections 3.12. Grassmannian bundles Part II. Local geometry 4. Young diagrams and Grassmannians 4.1. Young diagrams and Schur functors 4.2. Borel-Bott-Weil theorem and Kapranov's collections 4.3. Mutations on Grassmannians 5. Local geometry and correspondences 5.1. The key lemma and Lascoux-type resolutions 5.2. First implications 5.3. The case d + = 1: projectivization 5.4. The case ℓ + = 1: standard flips 5.5. The case m = 1: Pirozhkov's theorem 5.6. The cases δ ≤ 3 5.7. The case d + = 2: Quot 2 -formula 5.8. The case ℓ + = 2: flips from resolving rank ≤ 2 degeneracy lociPart III. Global geometry 6. Global situation 6.1. Hom spaces 6.2. Tor-independent conditions and general procedures of base-change 6.3. Blowups along Koszul-regularly immersed centers 6.4. Cayley's trick 6.5. Projectivizations 6.6. Generalized Caylay's trick and Pirozhkov's theorem 6.7. Quot 2 -formula 108 7. Flips, flops and virtual flips 110 7.1. First results: Grassmannian flips and virtual flips 110 7.2. Standard flips revisited 112 7.3. Flips from partial desingularizations of rank ≤ 2 degeneracy loci 114 7.4. The cases rank G ≤ 3, and blowups of determinantal ideals of height ≤ 4 115 Appendix A. Relations in the Grothendieck rings of varieties 121 Appendix B. Characteristic-free results for projective bundles 123 References 127

Moduli Space of Sheaves and Categorified Commutator of Functors

“…The incarnation with derived algebraic geometry is inspired by the work of Porta-Sala [24], Diaconescu-Porta-Sala [6] and Toda [30,31,32,33] on the categorified Hall algebra. Toda [33] proved a recent conjecture of Jiang [15], which obtained a semiorthogonal decomposition the derived category of Grassmanians over a cohomological dimension 1 coherent sheaf. As an application, Koseki [16] obtained a categorical blow-up formula for Hilbert schemes of points on a smooth algebraic surface.…”

Derived categories of Quot schemes of locally free quotients via categorified Hall products

“…The above result is conjectured by Qingyuan Jiang [Jia,Conjecture A.5]. The case of d = 1 is called projectivization formula and proved in [Kuz07, Theorem 5.5], [JL,Theorem 3.4], [Todb, Theorem 4.6.11].…”

“…The case of d = 1 is called projectivization formula and proved in [Kuz07, Theorem 5.5], [JL,Theorem 3.4], [Todb, Theorem 4.6.11]. The d = 2 case is proved in [Jia,Theorem 6.19]. The Quot formula in Theorem 1.1 gives a unified treatment of several known formulas such as Bondal-Orlov standard flip formula [BO], Kapranov exceptional collection for Grassmannian [Kap84], projectivization formula [Kuz07, JL, Todb], semiorthogonal decomposition of Grassmannian flip [BCF + 21, Todc] or so on (see [Jia,Section 1.4.2] for details).…”

“…In [Jia, Conjecture A.5], the conjecture is stated for derived categories of the classical truncations Quot X,d (G ), Quot X,d (H ) under some Tor-independent condition. The Tor-independent condition implies that their dimensions coincide with the virtual dimensions (see [Jia,Lemma 6.7]), so in this case they are equivalent to Quot X,d (G ), Quot X,d (H ). Therefore under the above Torindependent condition, we can replace the derived schemes in Theorem 1.1 with their classical truncations.…”

“…Applications. The Quot formula in Theorem 1.1 has lots of applications on derived categories of classical moduli spaces (see [Jia,Section 1.5]). Here we mention two examples: one is a generalization of [Tod21,Corollary 5.11] and [Jia, Corollary 1.3] on semiorthogonal decompositions of varieties associated with Brill-Noether loci for curves, and the another one is a categorical blow-up formula of Hilbert schemes of points on surfaces obtained by Koseki [Kos].…”