[Paper Review] Remarks on Theta-stratifications and derived categories

Daniel Halpern-Leistner|arXiv (Cornell University)|Feb 10, 2015

Homotopy and Cohomology in Algebraic Topology8 references9 citations

TL;DR

This paper establishes a derived $Θ$-stratification framework in derived algebraic geometry to generalize semiorthogonal decompositions of derived categories for global quotient stacks, removing technical hypotheses required in classical GIT. It proves a virtual non-abelian localization theorem in K-theory, showing that Euler characteristics over a stack decompose into contributions from fixed loci via explicit virtual sheaves, valid even for singular or non-smooth stacks.

ABSTRACT

This note extends some recent results on the derived category of a geometric invariant theory quotient to the setting of derived algebraic geometry. Our main result is a structure theorem for the derived category of a derived local quotient stack which admits a stratification of the kind arising in geometric invariant theory. The use of derived algebraic geometry leads to results with pleasingly few hypotheses, even when the stack is not smooth. Using the same methods, we establish a "virtual non-abelian localization theorem" which is a K-theoretic analog of the virtual localization theorem in cohomology.

Motivation & Objective

• To extend semiorthogonal decompositions of derived categories in geometric invariant theory to derived algebraic geometry with minimal hypotheses.
• To resolve the failure of prior results under technical conditions (L+) and (A) by using derived enhancements of quotient stacks.
• To establish a K-theoretic virtual non-abelian localization formula that applies even when the stack is singular or not smooth.
• To show that the Euler characteristic of a perfect complex on a derived quotient stack decomposes into contributions from fixed loci via virtual sheaves.
• To provide a framework for computing equivariant K-theory integrals using stratifications defined by one-parameter subgroups.

Proposed method

• Introduces the notion of a derived $Θ$-stratification in a derived global quotient stack $\mathcal{X} = X/G$ in characteristic 0.
• Applies cohomology with supports to derive a semiorthogonal decomposition of $D^{-}\!\mathop{\rm Coh}\nolimits(\mathcal{X})$ for a single derived $Θ$-stratum.
• Uses Lie algebra cohomology and explicit complexes involving $\bigwedge^\ast(\mathfrak{u}^\vee)$ to compute pushforwards along $U$-gerbes in the stratification.
• Constructs virtual sheaves $E_i$ as infinite direct sums of symmetric powers of negative weight parts of the normal bundle, which are finite in contribution to the Euler characteristic.
• Applies a filtration argument on the complex computing $\pi_*F$, showing that only finitely many terms contribute to the Euler characteristic.
• Combines results from derived algebraic geometry and equivariant K-theory to prove that $\chi(\mathcal{X}, F) = \chi(\mathcal{X}^{ss}, F) + \sum \chi(\mathcal{Z}_i, E_i \otimes \sigma^*F)$.

Experimental results

Research questions

• RQ1 Can semiorthogonal decompositions of derived categories for GIT quotients be generalized to derived algebraic geometry without technical assumptions like (L+) and (A)?
• RQ2 How can a virtual non-abelian localization formula in K-theory be formulated and proven for singular or non-smooth derived quotient stacks?
• RQ3 What is the role of one-parameter subgroups and fixed loci in defining the virtual sheaves $E_i$ that appear in the K-theoretic localization formula?
• RQ4 Can the Euler characteristic of a perfect complex on a derived quotient stack be computed via contributions from fixed loci, even when the semistable locus is empty?
• RQ5 How do derived $Θ$-strata refine the structure of derived categories in the presence of singularities or non-smoothness?

Key findings

• The derived $Θ$-stratification allows a semiorthogonal decomposition of $D^{-}\!\mathop{\rm Coh}\nolimits(\mathcal{X})$ for a derived quotient stack $\mathcal{X} = X/G$ without any technical hypotheses.
• For quasi-smooth derived quotient stacks with vanishing obstruction classes, a semiorthogonal decomposition of $D^b\!\mathop{\rm Coh}\nolimits(\mathcal{X})$ is established via Theorem 3.1.
• When the inclusion of a stratum is a regular embedding, a semiorthogonal decomposition of $\!\mathop{\rm Perf}\nolimits(\mathcal{X})$ is obtained via Theorem 2.1.
• The virtual non-abelian localization theorem (Theorem 5.1) expresses $\chi(\mathcal{X}, F)$ as a sum over fixed loci $\mathcal{Z}_i$, with contributions from virtual sheaves $E_i$ that are infinite sums but only finitely many contribute to the Euler characteristic.
• The virtual sheaves $E_i$ depend on both the one-parameter subgroups $\lambda_i$ and the normal bundles of the fixed loci, making the formula sensitive to the stratification choice.
• The proof relies on a bounded below filtration of the pushforward complex, with associated graded isomorphic to symmetric powers of the negative weight part of the conormal bundle, ensuring finitely many non-zero terms in the Euler characteristic.