caa a22 4500 606160779 CHVBK 20210128100631.0 cr unu---uuuuu 210128e20150601xx s 000 0 eng 10.1007/s10468-014-9514-7 doi (NATIONALLICENCE)springer-10.1007/s10468-014-9514-7 Bounding Cohomology for Finite Groups and Frobenius Kernels [Elektronische Daten] [Christopher Bendel, Daniel Nakano, Brian Parshall, Cornelius Pillen, Leonard Scott, David Stewart] Let G be a simple, simply connected algebraic group defined over an algebraically closed field k of positive characteristic p. Let σ :G → G be a strict endomorphism (i.e., the subgroup G(σ) of σ-fixed points is finite). Also, let G σ be the scheme-theoretic kernel of σ, an infinitesimal subgroup of G. This paper shows that the dimension of the degree m cohomology group Hm(G(σ),L) for any irreducible k G(σ)-module L is bounded by a constant depending on the root system Φ of G and the integer m. These bounds are actually established for the degree m extension groups Ext G ( σ ) m ( L , L ′ ) $ Ext^{m}_{G(\sigma )}(L,L^{\prime })$ between irreducible k G(σ)-modules L , L ′ $L,L^{\prime }$ , with a similar result holding for G σ . In these Extm results, the bounds also depend on the highest weight associated to L, but are, nevertheless, independent of the characteristic p. We also show that one can find bounds independent of the prime for the Cartan invariants of G(σ) and G σ , and even for the lengths of the underlying PIMs. Springer Science+Business Media Dordrecht, 2015 Algebraic groups nationallicence Cohomology nationallicence Finite groups of Lie type nationallicence Frobenius kernels nationallicence Bendel Christopher Department of Mathematics, Statistics and Computer Science, University of Wisconsin-Stout, 54751, Menomonie, WI, USA aut Nakano Daniel Department of Mathematics, University of Georgia, 30602, Athens, GA, USA aut Parshall Brian Department of Mathematics, University of Virginia, 22903, Charlottesville, VA, USA aut Pillen Cornelius Department of Mathematics and Statistics, University of South Alabama, 36688, Mobile, AL, USA aut Scott Leonard Department of Mathematics, University of Virginia, 22903, Charlottesville, VA, USA aut Stewart David Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, UK aut Algebras and Representation Theory Springer Netherlands 18/3(2015-06-01), 739-760 1386-923X 18:3<739 2015 18 10468 https://doi.org/10.1007/s10468-014-9514-7 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10468-014-9514-7 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Bendel Christopher Department of Mathematics, Statistics and Computer Science, University of Wisconsin-Stout, 54751, Menomonie, WI, USA aut NATIONALLICENCE 700 1- Nakano Daniel Department of Mathematics, University of Georgia, 30602, Athens, GA, USA aut NATIONALLICENCE 700 1- Parshall Brian Department of Mathematics, University of Virginia, 22903, Charlottesville, VA, USA aut NATIONALLICENCE 700 1- Pillen Cornelius Department of Mathematics and Statistics, University of South Alabama, 36688, Mobile, AL, USA aut NATIONALLICENCE 700 1- Scott Leonard Department of Mathematics, University of Virginia, 22903, Charlottesville, VA, USA aut NATIONALLICENCE 700 1- Stewart David Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, UK aut NATIONALLICENCE 773 0- Algebras and Representation Theory Springer Netherlands 18/3(2015-06-01), 739-760 1386-923X 18:3<739 2015 18 10468