[Paper Review] Derived categories of Quot schemes of locally free quotients via categorified Hall products

Yukinobu Toda|arXiv (Cornell University)|Oct 6, 2021

Algebraic structures and combinatorial models21 references4 citations

TL;DR

This paper proves Qingyuan Jiang's conjecture on semiorthogonal decompositions of derived categories of Quot schemes of locally free quotients by applying categorified Hall products to Grassmannian flips. The key result establishes a semiorthogonal decomposition of $ D^b(\mathbf{Quot}_{X,d}(\mathscr{G})) $ into $ \binom{\delta}{i} $ copies of $ D^b(\mathbf{Quot}_{X,d-i}(\mathscr{H})) $, generalizing known formulas like Kapranov's exceptional collection and the projectivization formula.

ABSTRACT

We prove Qingyuan Jiang's conjecture on semiorthogonal decompositions of derived categories of Quot schemes of locally free quotients. The author's result on categorified Hall products for Grassmannian flips is applied to prove the conjecture.

Motivation & Objective

• To prove Qingyuan Jiang's conjecture on semiorthogonal decompositions of derived categories of Quot schemes of locally free quotients.
• To generalize known results such as the projectivization formula and Kapranov's exceptional collection for Grassmannians.
• To establish a derived categorical structure for Quot schemes using categorified Hall products and Grassmannian flips.
• To extend the framework to families of moduli stacks of quiver representations with super-potentials.

Proposed method

• Utilizes categorified Hall products for families of Grassmannian flips, constructed via Koszul duality on $(-1)$-shifted cotangent derived stacks.
• Applies the theory of d-critical Grassmannian flips to interpret derived Quot schemes as moduli of representations with super-potentials.
• Employs lexicographical ordering on weight sequences to define semiorthogonal components in the derived category.
• Uses fully-faithful functors induced by categorified Hall products to embed $ D^b(\mathbf{Quot}_{X,d-i}(\mathscr{H})) $ into $ D^b(\mathbf{Quot}_{X,d}(\mathscr{G})) $.
• Relies on global constructions of categorified Hall products with right adjoints, ensuring semiorthogonal decompositions via formal local arguments.
• Applies the determinant character $ \chi_0 $ to define line bundles on moduli stacks, enabling weight-based classification of summands.

Experimental results

Research questions

• RQ1 Does the derived category of the Quot scheme $ \mathbf{Quot}_{X,d}(\mathscr{G}) $ admit a semiorthogonal decomposition into copies of $ D^b(\mathbf{Quot}_{X,d-i}(\mathscr{H})) $, as conjectured by Jiang?
• RQ2 Can the categorified Hall product formalism be extended to families of Grassmannian flips over a base scheme $ X $, preserving semiorthogonal structures?
• RQ3 Is the derived Quot scheme $ \mathbf{Quot}_{X,d}(\mathscr{G}) $ equivalent to a d-critical stack that models a Grassmannian flip under suitable conditions?
• RQ4 How does the virtual dimension of $ \mathbf{Quot}_{X,d}(\mathscr{G}) $ relate to the rank $ \delta $ of $ \mathscr{E}^0 $ and the integer $ d $?
• RQ5 Can the derived category of $ \mathbf{Quot}_{X,d}(\mathscr{G}) $ be decomposed into $ \binom{\delta}{i} $ copies of $ D^b(\mathbf{Quot}_{X,d-i}(\mathscr{H})) $ when $ \delta \geq 0 $?

Key findings

• The derived category $ D^b(\mathbf{Quot}_{X,d}(\mathscr{G})) $ admits a semiorthogonal decomposition into $ \binom{\delta}{i} $ copies of $ D^b(\mathbf{Quot}_{X,d-i}(\mathscr{H})) $ for $ 0 \leq i \leq \min\{d, \delta\} $, proving Jiang's conjecture.
• The result generalizes Kapranov's exceptional collection for Grassmannians by setting $ X $ to be a point and $ \mathscr{G} $ locally free of rank $ \delta $.
• The projectivization formula for $ d=1 $ is recovered as a special case of the main theorem.
• When $ \mathscr{G} $ has homological dimension $ \leq 1 $, the derived Quot schemes $ \mathbf{Quot}_{X,d}(\mathscr{G}) $ and $ \mathbf{Quot}_{X,d}(\mathscr{H}) $ are equivalent to their classical counterparts under Tor-independence.
• The categorified Hall product construction yields fully-faithful functors $ D^b(\mathbf{Quot}_{X,d-i}(\mathscr{H})) \hookrightarrow D^b(\mathbf{Quot}_{X,d}(\mathscr{G})) $, with kernels not explicitly described.
• The decomposition is compatible with the $ \mathbb{C}^* $-action and the determinant character $ \chi_0 $, ensuring weight-graded structure on the moduli stack.