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Projective Modules and Cohomology for Integral Basic Algebras

David J. Benson, Kay Jin Lim

TL;DR

The paper develops a framework to compare characteristic-zero and good-prime behavior for finite-dimensional ${\mathcal{O}}$-algebras by introducing strong hypotheses that control Ext groups, radical filtrations, and base-change phenomena. It proves a chain of implications among these hypotheses, showing that ${\text{Hyp.freeJ}}$ implies the basic structure and Ext-freeness, which in turn imply Ext-base-change compatibility and equality of radical-layer multiplicities across ${\mathcal{O}}$, ${\mathbb{F}}$, and ${\mathbb{K}}$. The authors apply the theory to Solomon's descent algebra, nilCoxeter algebras, and certain semigroup algebras, establishing conditions under which Ext-quivers are preserved and primitive idempotents lift integrally. Their results explain why, in many cases, cohomological dimensions and Cartan-related invariants behave similarly in characteristic zero and at suitable primes, and they develop constructive idempotent theories and quiver descriptions to support these conclusions. The work provides a principled explanation for modular-phenomena in a broad class of basic algebras, with concrete consequences for representation theory and the structure of descent algebras and related monoid algebras.

Abstract

Algebras defined over fields of characteristic zero and positive characteristic usually do not behave the same way. However, for certain algebras, for example the group algebras, they behave the same way as the characteristic zero case at "good enough" prime. In this paper, we initiate the study of this topic by imposing increasingly strong hypotheses on basic algebras. When the algebras satisfy the right hypotheses, we have equalities of the dimensions of their cohomology groups between simple modules and equalities of graded Cartan numbers. The examples include the Solomon descent algebras of finite Coxeter groups at large enough primes, nilCoxeter algebra, and certain finite semigroup algebras at an arbitrary prime.

Projective Modules and Cohomology for Integral Basic Algebras

TL;DR

The paper develops a framework to compare characteristic-zero and good-prime behavior for finite-dimensional -algebras by introducing strong hypotheses that control Ext groups, radical filtrations, and base-change phenomena. It proves a chain of implications among these hypotheses, showing that implies the basic structure and Ext-freeness, which in turn imply Ext-base-change compatibility and equality of radical-layer multiplicities across , , and . The authors apply the theory to Solomon's descent algebra, nilCoxeter algebras, and certain semigroup algebras, establishing conditions under which Ext-quivers are preserved and primitive idempotents lift integrally. Their results explain why, in many cases, cohomological dimensions and Cartan-related invariants behave similarly in characteristic zero and at suitable primes, and they develop constructive idempotent theories and quiver descriptions to support these conclusions. The work provides a principled explanation for modular-phenomena in a broad class of basic algebras, with concrete consequences for representation theory and the structure of descent algebras and related monoid algebras.

Abstract

Algebras defined over fields of characteristic zero and positive characteristic usually do not behave the same way. However, for certain algebras, for example the group algebras, they behave the same way as the characteristic zero case at "good enough" prime. In this paper, we initiate the study of this topic by imposing increasingly strong hypotheses on basic algebras. When the algebras satisfy the right hypotheses, we have equalities of the dimensions of their cohomology groups between simple modules and equalities of graded Cartan numbers. The examples include the Solomon descent algebras of finite Coxeter groups at large enough primes, nilCoxeter algebra, and certain finite semigroup algebras at an arbitrary prime.
Paper Structure (13 sections, 41 theorems, 69 equations)

This paper contains 13 sections, 41 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. The hypotheses and main results
  3. Quivers and relations over ${\mathcal{O}}$
  4. ${\mathcal{O}}$-freeness of $\operatorname{\sf Ext}$
  5. Comparing $\operatorname{\sf Ext}$ over ${\mathcal{O}}$, ${\mathbb{F}}$ and $\mathbb{K}$
  6. ${\mathcal{O}}$-freeness of $J^n/J^{n+1}$
  7. The descent algebras
  8. Idempotents for the descent algebras
  9. Sufficient conditions for the descent algebras
  10. Verifying \ref{['hyp:rad']}
  11. Verifying \ref{['hyp:freeJ']}
  12. Independence of choice of basis for $n_W$
  13. nilCoxeter Algebras satisfy \ref{['hyp:freeJ']}

Key Result

Theorem 2.3

Theorems & Definitions (79)

  • Theorem 2.3
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • Example 3.3
  • Theorem 3.4
  • proof
  • Lemma 4.1
  • proof
  • ...and 69 more