Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stable sheaf cohomology on flag varieties

Claudiu Raicu, Keller VandeBogert

TL;DR

The paper develops an effective stabilization framework for the cohomology of line bundles on flag varieties Fl$_n$ and for polynomial functors of the cotangent bundle on projective space, in positive characteristic with fixed $p$ as $n\to\infty$. It proves that stable cohomology groups can be described by polynomial functors in the base vector space, and it establishes a robust set of structural tools, including a Künneth formula, Frobenius invariance, and explicit resolutions (Akin–Buchsbaum) for key Schur functors. The authors provide explicit stable computations, reveal intricate phenomena (e.g., disconnected nonvanishing degrees and super-polynomial growth in multiplicities), and connect these to applications in commutative algebra via Koszul modules. The work yields both broad theoretical insights into modular representation theory and concrete computational methods for stable invariants, with implications for homological invariants and representation theory in all characteristics.

Abstract

We prove an effective stabilization result for the sheaf cohomology groups of line bundles on flag varieties parametrizing complete flags in k^n, as well as for the sheaf cohomology groups of polynomial functors applied to the cotangent sheaf Omega on projective space. In characteristic zero, these are natural consequences of the Borel-Weil-Bott theorem, but in characteristic p>0 they are non-trivial. Unlike many important contexts in modular representation theory, where the prime characteristic p is assumed to be large relative to n, in our study we fix p and let n go to infinity. We illustrate the general theory by providing explicit stable cohomology calculations in a number of cases of interest. Our examples yield cohomology groups where the number of indecomposable summands has super-polynomial growth, and also show that the cohomological degrees where non-vanishing occurs do not form a connected interval. In the case of polynomial functors of Omega, we prove a Kunneth formula for stable cohomology, and show the invariance of stable cohomology under Frobenius, which combined with the Steinberg tensor product theorem yields calculations of stable cohomology for an interesting class of simple polynomial functors arising in the work of Doty. The results in the special case of symmetric powers of Omega provide a nice application to commutative algebra, yielding a sharp vanishing result for Koszul modules of finite length in all characteristics.

Stable sheaf cohomology on flag varieties

TL;DR

The paper develops an effective stabilization framework for the cohomology of line bundles on flag varieties Fl and for polynomial functors of the cotangent bundle on projective space, in positive characteristic with fixed as . It proves that stable cohomology groups can be described by polynomial functors in the base vector space, and it establishes a robust set of structural tools, including a Künneth formula, Frobenius invariance, and explicit resolutions (Akin–Buchsbaum) for key Schur functors. The authors provide explicit stable computations, reveal intricate phenomena (e.g., disconnected nonvanishing degrees and super-polynomial growth in multiplicities), and connect these to applications in commutative algebra via Koszul modules. The work yields both broad theoretical insights into modular representation theory and concrete computational methods for stable invariants, with implications for homological invariants and representation theory in all characteristics.

Abstract

We prove an effective stabilization result for the sheaf cohomology groups of line bundles on flag varieties parametrizing complete flags in k^n, as well as for the sheaf cohomology groups of polynomial functors applied to the cotangent sheaf Omega on projective space. In characteristic zero, these are natural consequences of the Borel-Weil-Bott theorem, but in characteristic p>0 they are non-trivial. Unlike many important contexts in modular representation theory, where the prime characteristic p is assumed to be large relative to n, in our study we fix p and let n go to infinity. We illustrate the general theory by providing explicit stable cohomology calculations in a number of cases of interest. Our examples yield cohomology groups where the number of indecomposable summands has super-polynomial growth, and also show that the cohomological degrees where non-vanishing occurs do not form a connected interval. In the case of polynomial functors of Omega, we prove a Kunneth formula for stable cohomology, and show the invariance of stable cohomology under Frobenius, which combined with the Steinberg tensor product theorem yields calculations of stable cohomology for an interesting class of simple polynomial functors arising in the work of Doty. The results in the special case of symmetric powers of Omega provide a nice application to commutative algebra, yielding a sharp vanishing result for Koszul modules of finite length in all characteristics.
Paper Structure (23 sections, 27 theorems, 216 equations, 2 figures)

This paper contains 23 sections, 27 theorems, 216 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quick review of polynomial functors
  4. Young diagrams, cores, and strong linkage
  5. Polynomial stabilization for cohomology of line bundles
  6. Polynomial functors of $\Omega$
  7. Trivial $\operatorname{GL}_n$-action and vanishing for negative twists
  8. Stabilization of cohomology
  9. Künneth Formula
  10. Truncated powers
  11. Cohomology of Weyl functors
  12. Invariance under Frobenius
  13. Vanishing for $p$-cores with multiple columns
  14. Akin--Buchsbaum resolutions by tensor products of exterior powers
  15. Ribbon shapes
  16. ...and 8 more sections

Key Result

Theorem 3.1

Suppose that $\lambda\in\mathbb{Z}^r$ and We have that $H^j\left(Fl_n,\mathcal{O}_{Fl_n}(\lambda^{[n]})\right)$ is a polynomial $\operatorname{GL}_n$-representation for all $j\geq 0$.

Figures (2)

  • Figure 1: The complex $C_{\bullet}(w_0,w_1,w_2,w_3)$, $w_0,w_1,w_2\geq 0$, $w_3\in\mathbb{Z}$.
  • Figure 2: The complex $G_{\bullet}(m,3)$, also the dual of $C_{\bullet}(1,1,1,-m-4)$

Theorems & Definitions (68)

  • Example 1.1
  • Example 1.2
  • Remark 1.3
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • Theorem 3.3
  • proof
  • Corollary 3.4
  • ...and 58 more