[Paper Review] A realization theorem for modules of constant Jordan type and vector bundles

David J. Benson, Julia Pevtsova|arXiv (Cornell University)|Jul 22, 2010

Advanced Algebra and Geometry4 references4 citations

TL;DR

This paper establishes a realization theorem linking finitely generated modules of constant Jordan type over the group algebra of an elementary abelian $ p $-group to vector bundles on $ \mathbb{P}^{r-1} $. It proves that for any vector bundle $ \mathcal{F} $ of rank $ s $, there exists a $ kE $-module $ M $ of stable constant Jordan type $[1]^s$ such that $ \mathcal{F}_1(M) \cong \mathcal{F} $ when $ p = 2 $, and $ \mathcal{F}_1(M) \cong F^*(\mathcal{F}) $ when $ p $ is odd, where $ F $ is the Frobenius morphism. The result is sharp: for odd $ p $, Chern numbers $ c_1, \dots, c_{p-2} $ of $ \mathcal{F}_1(M) $ are always divisible by $ p $, so the theorem cannot be improved.

ABSTRACT

Let E be an elementary abelian p-group of rank r and let k be a field of characteristic p. We introduce functors F_i from finitely generated kE-modules of constant Jordan type to vector bundles over projective space of dimension r-1. The fibers of these functors encode complete information about the Jordan type of the module. We prove that given any vector bundle of rank s on P^{r-1}, there is a kE-module M of stable constant Jordan type [1]^s such that the functor F_1 applied to M yields the original vector bundle for p=2 and the Frobenius twist of the original vector bundle for p>2. We also prove that the theorem cannot be improved if p is odd, because if M is any module of stable constant Jordan type [1]^s then the Chern numbers c_1, ... ,c_{p-2} of F_1(M) are divisible by p.

Motivation & Objective

• To establish a correspondence between finitely generated $ kE $-modules of constant Jordan type and algebraic vector bundles on $ \mathbb{P}^{r-1} $, where $ E $ is an elementary abelian $ p $-group and $ k $ is a field of characteristic $ p $.
• To define and study functors $ \mathcal{F}_i $ that assign vector bundles to such modules, encoding information about the Jordan block structure.
• To prove that every vector bundle on $ \mathbb{P}^{r-1} $ arises as $ \mathcal{F}_1(M) $ for some module $ M $ of stable constant Jordan type $[1]^s$, up to Frobenius pullback when $ p $ is odd.
• To show that the theorem cannot be strengthened for odd $ p $, due to arithmetic constraints on Chern classes of the resulting bundles.

Proposed method

• Functors $ \mathcal{F}_i $ are defined from $ kE $-modules to coherent sheaves on $ \mathbb{P}^{r-1} $, capturing the sum of the socles of Jordan blocks of length $ i $.
• The construction uses the action of the augmentation ideal $ J(kE) $ on modules, with elements $ X_\alpha = \sum \lambda_i X_i $ acting nilpotently.
• The functors $ \mathcal{F}_i $ are defined as subquotients of the trivial bundle $ \widetilde{M} $ via the map $ \theta_M $, which encodes the action of $ X_i $ on $ M $.
• For modules of constant Jordan type, the $ \mathcal{F}_i $ produce algebraic vector bundles, and their behavior under Heller shifts and duality is analyzed.
• The proof of the main theorem uses the Chern class formula for twists of vector bundles and the identity $ x(x+1)\cdots(x+p-1) \equiv x^p - x \pmod{p} $.
• A key step is showing that the product of Chern classes of $ \widetilde{M} $'s filtration quotients satisfies $ c(\widetilde{M}, h) \equiv c(\mathcal{F}_1(M), h) \pmod{(p, h^{p-1})} $, leading to divisibility of Chern numbers by $ p $.

Experimental results

Research questions

• RQ1 Can every vector bundle on $ \mathbb{P}^{r-1} $ be realized as $ \mathcal{F}_1(M) $ for some $ kE $-module $ M $ of stable constant Jordan type $[1]^s$?
• RQ2 How does the Frobenius morphism affect the realizability of vector bundles in the context of modules of constant Jordan type?
• RQ3 What arithmetic constraints exist on the Chern classes of $ \mathcal{F}_1(M) $ when $ M $ has stable constant Jordan type $[1]^s$?
• RQ4 Why is the realization theorem for odd $ p $ necessarily weaker than for $ p = 2 $, and what limits its strength?
• RQ5 Are there obstructions to realizing certain vector bundles as $ \mathcal{F}_1(M) $, and if so, what are they?

Key findings

• For $ p = 2 $, every vector bundle $ \mathcal{F} $ of rank $ s $ on $ \mathbb{P}^{r-1} $ is realized as $ \mathcal{F}_1(M) $ for some $ kE $-module $ M $ of stable constant Jordan type $[1]^s$.
• For odd $ p $, every vector bundle $ \mathcal{F} $ is realized as $ \mathcal{F}_1(M) \cong F^*(\mathcal{F}) $, where $ F $ is the Frobenius morphism.
• The theorem cannot be improved for odd $ p $, as $ c_m(\mathcal{F}_1(M)) $ is divisible by $ p $ for all $ 1 \leq m \leq p-2 $.
• This divisibility condition is sharp: for $ p \geq 7 $, the Horrocks–Mumford bundle and its twists cannot be realized as $ \mathcal{F}_1(M) $ for any $ M $ of type $[1]^2$.
• The Chern class formula for twists of vector bundles, combined with Fermat’s little theorem modulo $ p $, underlies the proof of the Chern class divisibility.
• The result shows a fundamental obstruction in the geometry of vector bundles arising from modules of constant Jordan type, particularly in odd characteristic.