Cohomological Mackey Functors Research Articles

COHOMOLOGICAL MACKEY 2-FUNCTORS

AbstractWe show that the bicategory of finite groupoids and right-free permutation bimodules is a quotient of the bicategory of Mackey 2-motives introduced in [2], obtained by modding out the so-called cohomological relations. This categorifies Yoshida’s theorem for ordinary cohomological Mackey functors and provides a direct connection between Mackey 2-motives and the usual blocks of representation theory.

The dominant dimension of cohomological Mackey functors

We show that a separable equivalence between symmetric algebras preserves the dominant dimensions of certain endomorphism algebras of modules. We apply this to show that the dominant dimension of the category [Formula: see text] of cohomological Mackey functors of a [Formula: see text]-block [Formula: see text] of a finite group with a nontrivial defect group is [Formula: see text].

On Morita and derived equivalences for cohomological Mackey algebras

By results of the second author, a source algebra equivalence between two p-blocks of finite groups induces an equivalence between the categories of cohomological Mackey functors associated with these blocks, and a splendid derived equivalence between two blocks induces a derived equivalence between the corresponding categories ofcohomological Mackey functors. The main result of this paper proves a partial converse: an equivalence (resp. Rickard equivalence) between the categories of cohomological Mackey functors of two blocks of finite groups induces a permeable Morita (resp. derived) equivalence between the two block algebras.

On the Projective Dimensions of Mackey Functors

We examine the projective dimensions of Mackey functors and cohomological Mackey functors. We show over a field of characteristic p that cohomological Mackey functors are Gorenstein if and only if Sylow p-subgroups are cyclic or dihedral, and they have finite global dimension if and only if the group order is invertible or Sylow subgroups are cyclic of order 2. By contrast, we show that the only Mackey functors of finite projective dimension over a field are projective. This allows us to give a new proof of a theorem of Greenlees on the projective dimension of Mackey functors over a Dedekind domain. We conclude by completing work of Arnold on the global dimension of cohomological Mackey functors over ℤ.

On equivalences for cohomological Mackey functors

By results of Rognerud, a source algebra equivalence between two p p -blocks of finite groups induces an equivalence between the categories of cohomological Mackey functors associated with these blocks, and a splendid derived equivalence between two blocks induces a derived equivalence between the corresponding categories of cohomological Mackey functors. We prove this by giving an intrinsic description of cohomological Mackey functors of a block in terms of the source algebras of the block, and then using this description to construct explicit two-sided tilting complexes realising the above mentioned derived equivalence. We show further that a splendid stable equivalence of Morita type between two blocks induces an equivalence between the categories of cohomological Mackey functors which vanish at the trivial group. We observe that the module categories of a block, the category of cohomological Mackey functors, and the category of cohomological Mackey functors which vanish at the trivial group arise in an idempotent recollement. Finally, we extend a result of Tambara on the finitistic dimension of cohomological Mackey functors to blocks.

The extension algebra of some cohomological Mackey functors

Let k be a field of characteristic p. We construct a new inflation functor for cohomological Mackey functors for finite groups over k. Using this inflation functor, we give an explicit presentation of the graded algebra of self-extensions of the simple functor S1G, when p is odd and G is an elementary abelian p-group.

Correction to “Voevodsky’s motives and Weil reciprocity,” Duke Math. J. 162 (2013), 2751–2796

We correct a mistake in Section 2.4 of the said paper. In [1, Section 2.4], we wrote: “The category Z Span is isomorphic to the full subcategory of Cor consisting of smooth k-schemes of dimension 0.” Tom Bachmann kindly pointed out to us that this statement is incorrect. Here we clarify the relationship between the two categories and show that it does not affect any argument about cohomological Mackey functors (the only Mackey functors appearing in [1]). We retain the notation of [1]. 1 Let Cor0 be the full subcategory of Cor given by 0-dimensional smooth schemes (D etale k-schemes). If f W X ! Y is a surjective morphism of degree d of etale k-schemes, then we have the formula in Cor0, f i f D d: (1) 2 There is a canonical functor W Z Span!Cor0 (2) which is the identity on objects and sends a span (2.1) from [1], X g Z f ! Y; (3) to f i g. DUKE MATHEMATICAL JOURNAL Vol. 164, No. 10, © 2015 DOI 10.1215/00127094-3146068 Received 23 October 2014. Revision received 14 March 2015. 2010 Mathematics Subject Classification. Primary 19E15; Secondary 19A22, 18D10.

Cohomological finiteness conditions for Mackey and cohomological Mackey functors

We study cohomological finiteness conditions for groups associated to Mackey and cohomological Mackey functors, proving that the cohomological dimension associated to cohomological Mackey functors is always equal to the F-cohomological dimension, and characterising the conditions Mackey-FPn and cohomological Mackey-FPn.

Lattices and cohomological Mackey functors for finite cyclic [formula omitted]-groups

For a finite cyclic p-group G and a discrete valuation domain R of characteristic 0 with maximal ideal pR the R[G]-permutation modules are characterized in terms of the vanishing of first degree cohomology on all subgroups (cf. Theorem A). As a consequence any R[G]-lattice can be presented by R[G]-permutation modules (cf. Theorem C). The proof of these results is based on a detailed analysis of the category of cohomological G-Mackey functors with values in the category of R-modules. It is shown that this category has global dimension 3 (cf. Theorem E). A crucial step in the proof of Theorem E is the fact that a gentle R-order category (with parameter p) has global dimension less than or equal to 2 (cf. Theorem D).

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.

Cohomological Mackey functors in number theory

We prove relations between the evaluations of cohomological Mackey functors over complete discrete valuation rings or fields and apply this to Mackey functors that arise naturally in number theory. This provides relations between λ- and μ-invariants in Iwasawa theory, between Mordell–Weil groups, Shafarevich–Tate groups, Selmer groups and zeta functions of elliptic curves, and between ideal class groups and regulators of number fields.