Unbounding Ext

TL;DR

This paper constructs examples in algebraic group cohomology demonstrating unbounded extension dimensions and exponential growth of cohomology dimensions, answering two open questions for the case G=SL_2.

Contribution

It provides explicit examples showing unbounded Ext^2 dimensions and exponential growth of cohomology in algebraic groups, addressing open questions by Parshall and Scott.

Findings

xt_G^2(L,L) can be arbitrarily large

The sequence x_{L- ext{irred}} ext{dim} H^k(G,L) grows exponentially

Answers two open questions for G=SL_2

Abstract

We produce examples in the cohomology of algebraic groups which answer two questions of Parshall and Scott. Specifically, if $G=SL_2$, then we show: (a) $\dim \Ext_G^2(L,L)$ can be arbitrarily large for a simple module $L$; and (b) the sequence $\max_{L-\text{irred}}\dim H^k(G,L)$ grows exponentially fast with $k$.