[Paper Review] Koszul Property and Frobenius Splitting of Schubert Varieties

Roman Bezrukavnikov|ArXiv.org|Feb 20, 1995

Advanced Combinatorial Mathematics3 references21 citations

TL;DR

This paper establishes that the coordinate ring of any Schubert variety in a projective embedding is Koszul by leveraging Frobenius splitting techniques in positive characteristic. The key contribution is proving that Frobenius splitting compatible with Schubert varieties implies the Koszul property, resolving a conjecture and extending homological properties to singular Schubert varieties via geometric methods.

ABSTRACT

We show how the Frobenius splitting method of Mehta-Ramanathan implies the Koszul property of projective coordinate rings of Schubert varieties.

Motivation & Objective

• To prove that the coordinate ring of any Schubert variety is Koszul, extending known results from smooth flag varieties.
• To demonstrate that Frobenius splitting techniques can be used to establish the Koszul property in commutative algebras.
• To resolve a conjecture by J. Donin on the Koszul property of Schubert varieties.
• To unify homological algebra with algebraic geometry by showing that compatible Frobenius splittings imply Koszulness.

Proposed method

• Utilizes Frobenius splitting in positive characteristic to analyze the cohomology of sheaf ideals on products of flag varieties.
• Constructs a Frobenius splitting on $X^n = (G/P)^n$ compatible with all Schubert, opposite Schubert, and relative Schubert varieties.
• Applies Serre duality and projection formula to identify the space of Frobenius splittings with $H^0(K^{ ensor(1-p)})$, where $K$ is the canonical bundle.
• Uses a section $\sigma \in H^0(K^{-1})$ with divisor $D$ to define $s = \sigma^{p-1}$, which induces a compatible Frobenius splitting.
• Applies semicontinuity and base change to extend results from finite characteristic to characteristic zero.
• Employs local arguments near fixed points of Borel subgroups and automorphisms to reduce the problem to known splitting results on $G/B$.

Experimental results

Research questions

• RQ1 Does the coordinate ring of a Schubert variety satisfy the Koszul property, even when singular?
• RQ2 Can Frobenius splitting techniques be used to prove homological properties of commutative rings?
• RQ3 Is there a geometric method to establish the Koszul property for Schubert varieties beyond the smooth case?
• RQ4 Can compatible Frobenius splittings on $X^n$ induce vanishing of higher cohomology for sheaf ideals?
• RQ5 How does the lattice of subspaces $V_Z$ of global sections relate to the Koszul property?

Key findings

• The coordinate ring of any Schubert variety in a projective embedding is Koszul, generalizing Kostant's quadratic algebra result.
• Frobenius splitting compatible with all Schubert-type subvarieties in $X^n$ exists for any $n$, over fields of positive characteristic.
• The higher cohomology groups $H^i(X^n, J^{Z_2}_{Z_1} \otimes \mathcal{L})$ vanish for $i > 0$, ensuring the Koszul property via homological criteria.
• The vector spaces $V_Z$ of sections vanishing on $Z \in S^n$ form a distributive lattice, a key condition for Koszulity.
• The construction of the Frobenius splitting via $s = \sigma^{p-1}$ with $\sigma$ having divisor $D$ ensures compatibility with all relevant Schubert varieties.
• The result extends from $G/B$ to $G/P$ and from finite to characteristic zero fields via semicontinuity and base change.