[Paper Review] Categories of modules given by varieties of p-nilpotent operators

Rolf Farnsteiner|arXiv (Cornell University)|Oct 12, 2011

Algebraic structures and combinatorial models37 references3 citations

TL;DR

This paper investigates categories of modules over finite group schemes in positive characteristic p, focusing on equal images modules (EIP), constant Jordan type modules (CJT), and constant j-rank modules (CRj). It establishes that for Frobenius kernels of reductive groups, EIP modules are semisimple and decompose into one-dimensional modules, while for abelian unipotent groups like Z/p × Z/p, EIP modules in Auslander-Reiten components are finite in number and quasi-simple, with structure highly sensitive to group scheme type.

ABSTRACT

For a finite group scheme G over an algebraically closed field k of characteristic p>0 we study G-modules M, which are defined in terms of properties of their pull-backs along p-points of G. We show that the corresponding subcategories strongly depend on the structure of G. The second part of the paper discusses recent work by Carlson-Friedlander-Suslin concerning the subcategory of equal images modules from the vantage point of Auslander-Reiten theory.

Motivation & Objective

• To analyze the structure of equal images modules (EIP), constant Jordan type modules (CJT), and constant j-rank modules (CRj) for finite group schemes over fields of positive characteristic.
• To investigate how these module categories depend on the underlying group scheme structure, especially for Frobenius kernels and abelian unipotent groups.
• To examine the intersection of EIP modules with Auslander-Reiten components using AR theory, particularly in the context of the Kronecker quiver and stable AR quivers.
• To determine finiteness and quasi-simplicity conditions for EIP modules in regular AR components of group schemes with unipotent subgroups of complexity ≥2.
• To clarify the role of Jordan type and Heller shifts in classifying EIP modules within AR components, especially for Z/p × Z/p and reductive group Frobenius kernels.

Proposed method

• Uses π-points and pull-backs to define module categories via nilpotent operators, linking representation theory to support varieties.
• Applies Auslander-Reiten theory to analyze the stable AR quiver Γs(G) of the group algebra kG, focusing on components of tree class A∞.
• Employs the Auslander-Reiten translation and Heller shifts (Ω²) to study module filtrations and irreducible morphisms in AR components.
• Analyzes Jordan type decomposition Jt(M) = ⊕ ai[i] for 1 ≤ i ≤ p, where [i] denotes the i-dimensional indecomposable k[T]/(T^p)-module.
• Uses almost split sequences and module twists via the adjoint action to prove closure of EIP and CJT categories under AR operations.
• Applies results from Carlson-Friedlander-Suslin and Friedlander-Pevtsova on support varieties and p-nilpotent operators to characterize module categories.

Experimental results

Research questions

• RQ1 How do the categories EIP(G), CJT(G), and CRj(G) depend on the structure of the finite group scheme G in positive characteristic?
• RQ2 For Frobenius kernels Gr of reductive groups, are EIP modules semisimple, and do they decompose into one-dimensional modules?
• RQ3 In regular Auslander-Reiten components of Z/p × Z/p, is the intersection with EIP(G) finite, and are all such modules quasi-simple?
• RQ4 Under what conditions does an AR component contain infinitely many EIP modules, and what Jordan type constraints apply?
• RQ5 How do Heller shifts and twists via the adjoint action affect the structure of modules in CJT and EIP categories?

Key findings

• For the r-th Frobenius kernel Gr of a connected reductive group G, every EIP(Gr) module is a direct sum of one-dimensional modules, implying semisimplicity.
• For G1 (the first Frobenius kernel), CRj(G1) = CJT(G1) for all j ∈ {1,…,p−1}, and every CJT(G1) module is isomorphic to its twists under the adjoint action of G.
• In regular AR components of Z/p × Z/p with p ≥ 3, EIP modules are finite in number and all are quasi-simple, provided their Jordan type is ∑_{i=1}^{p−1} ai[i] or ∑_{i=1}^{p−2} ai[i].
• If an AR component Θ contains an EIP module M with Jordan type ∑_{i=1}^{p−1} ai[i], then Θ ∩ EIP(G) is finite and consists only of quasi-simple modules.
• For the group G = Z/p × Z/p, the set EIP(G) ∩ Θ is infinite if and only if the module W_{n,p} is a p-Koszul module, as shown in the remark.
• When n = 2p−1, the AR component Θ_{2p−1,p} contains the infinite family {(r)W_{p−1,p−1} | r ≥ 1} within EIP(G), and no further EIP modules exist beyond this family.