Complexity and cohomology of cohomological Mackey functors
Copyright policy )Complexity and cohomology of cohomological Mackey functors
Let $k$ be a field of characteristic $p>0$. Call a finite group $G$ a poco group over $k$ if any finitely generated cohomological Mackey functor for $G$ over $k$ has polynomial growth. The main result of this paper is that $G$ is a poco group over $k$ if and only if the Sylow $p$-subgroups of $G$ are cyclic, when $p>2$, or have sectional rank at most 2, when $p=2$. A major step in the proof is the case where $G$ is an elementary abelian $p$-group. In particular, when $p=2$, all the extension groups between simple functors can be determined completely, using a presentation of the graded algebra of self extensions of the simple functor $S_1^G$, by explicit generators and relations.
Mathematics(all), Mackey functor, Ext and Tor, generalizations, Künneth formula (category-theoretic aspects), Modular representations and characters, restriction, Growth, Group Theory (math.GR), Resolutions; derived functors (category-theoretic aspects), Module categories in associative algebras, Growth rate, Gelfand-Kirillov dimension, FOS: Mathematics, Category Theory (math.CT), Cohomology of groups, complexities of modules, induction, functor categories, Frobenius induction, Burnside and representation rings, cohomological Mackey functors, module categories, Mathematics - Category Theory, K-Theory and Homology (math.KT), Complexity, finite groups, modules of polynomial growth, Finite nilpotent groups, \(p\)-groups, sectional ranks, Cohomological, Mathematics - K-Theory and Homology, cyclic Sylow subgroups, projective resolutions, transfer, Mathematics - Group Theory, 16P90, 18G10, 18G15, 20J05
Mathematics(all), Mackey functor, Ext and Tor, generalizations, Künneth formula (category-theoretic aspects), Modular representations and characters, restriction, Growth, Group Theory (math.GR), Resolutions; derived functors (category-theoretic aspects), Module categories in associative algebras, Growth rate, Gelfand-Kirillov dimension, FOS: Mathematics, Category Theory (math.CT), Cohomology of groups, complexities of modules, induction, functor categories, Frobenius induction, Burnside and representation rings, cohomological Mackey functors, module categories, Mathematics - Category Theory, K-Theory and Homology (math.KT), Complexity, finite groups, modules of polynomial growth, Finite nilpotent groups, \(p\)-groups, sectional ranks, Cohomological, Mathematics - K-Theory and Homology, cyclic Sylow subgroups, projective resolutions, transfer, Mathematics - Group Theory, 16P90, 18G10, 18G15, 20J05
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