190 research outputs found

    An algebraic model for chains on ΩBG^p

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    Peer reviewedPublisher PD

    Symmetric tensor categories in characteristic 2

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    We construct and study a nested sequence of finite symmetric tensor categories Vec=C0C1Cn{\rm Vec}=\mathcal{C}_0\subset \mathcal{C}_1\subset\cdots\subset \mathcal{C}_n\subset\cdots over a field of characteristic 22 such that C2n\mathcal{C}_{2n} are incompressible, i.e., do not admit tensor functors into tensor categories of smaller Frobenius--Perron dimension. This generalizes the category C1\mathcal{C}_1 described by Venkatesh and the category C2\mathcal{C}_2 defined by Ostrik. The Grothendieck rings of the categories C2n\mathcal{C}_{2n} and C2n+1\mathcal{C}_{2n+1} are both isomorphic to the ring of real cyclotomic integers defined by a primitive 2n+22^{n+2}-th root of unity, On=Z[2cos(π/2n+1)]\mathcal{O}_n=\mathbb Z[2\cos(\pi/2^{n+1})].Comment: 27 pages, latex; in v2 corrections made suggested by the referee and a number of results added, in particular Corollary 2.5 and Proposition 2.6; introduction expanded; in v.4 small errors in Propositions 3.3, 3.4, Corollary 3.5, and Propositions 3.9, 3.16 (proofs) fixed, and the reference to [CEH] adde

    Local cohomology and support for triangulated categories

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    We propose a new method for defining a notion of support for objects in any compactly generated triangulated category admitting small coproducts. This approach is based on a construction of local cohomology functors on triangulated categories, with respect to a central ring of operators. Suitably specialized one recovers, for example, the theory for commutative noetherian rings due to Foxby and Neeman, the theory of Avramov and Buchweitz for complete intersection local rings, and varieties for representations of finite groups according to Benson, Carlson, and Rickard. We give explicit examples of objects whose triangulated support and cohomological support differ. In the case of group representations, this allows us to correct and establish a conjecture of Benson.Comment: 40 pages. Relatively minor changes (clarifications, corrections, new references, etc). This article will appear in the Ann. Sci. Ecole Norm. Su

    Localising subcategories for cochains on the classifying space of a finite group

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    The localising subcategories of the derived category of the cochains on the classifying space of a finite group are classified. They are in one to one correspondence with the subsets of the set of homogeneous prime ideals of the cohomology ring H(G,k)H^*(G,k).Comment: 5 pages, minor changes, accepted for publication in C. R. Math. Acad. Sci. Pari

    Stratifying modular representations of finite groups

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    We classify localising subcategories of the stable module category of a finite group that are closed under tensor product with simple (or, equivalently all) modules. One application is a proof of the telescope conjecture in this context. Others include new proofs of the tensor product theorem and of the classification of thick subcategories of the finitely generated modules which avoid the use of cyclic shifted subgroups. Along the way we establish similar classifications for differential graded modules over graded polynomial rings, and over graded exterior algebras.Comment: 37 pages; minor changes from the first version. This is slated to appear in the January 2012 issue of Annals of Math., volume 175, no.

    Colocalizing subcategories and cosupport

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    The Hom closed colocalizing subcategories of the stable module category of a finite group are classified. Along the way, the colocalizing subcategories of the homotopy category of injectives over an exterior algebra, and the derived category of a formal commutative differential graded algebra, are classified. To this end, and with an eye towards future applications, a notion of local homology and cosupport for triangulated categories is developed, building on earlier work of the authors on local cohomology and support.Comment: 38 pages; minor changes; to appear in J. Reine. Angew. Mat

    Cohomology of symplectic groups and Meyer's signature theorem

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    Meyer showed that the signature of a closed oriented surface bundle over a surface is a multiple of 44, and can be computed using an element of H2(Sp(2g,Z),Z)H^2(\mathsf{Sp}(2g, \mathbb{Z}),\mathbb{Z}). Denoting by 1ZSp(2g,Z)~Sp(2g,Z)11 \to \mathbb{Z} \to \widetilde{\mathsf{Sp}(2g,\mathbb{Z})} \to \mathsf{Sp}(2g,\mathbb{Z}) \to 1 the pullback of the universal cover of Sp(2g,R)\mathsf{ Sp}(2g,\mathbb{R}), Deligne proved that every finite index subgroup of Sp(2g,Z)~\widetilde{\mathsf {Sp}(2g, \mathbb{Z})} contains 2Z2\mathbb{Z}. As a consequence, a class in the second cohomology of any finite quotient of Sp(2g,Z)\mathsf{Sp}(2g, \mathbb{Z}) can at most enable us to compute the signature of a surface bundle modulo 88. We show that this is in fact possible and investigate the smallest quotient of Sp(2g,Z)\mathsf{Sp}(2g, \mathbb{Z}) that contains this information. This quotient H\mathfrak{H} is a non-split extension of Sp(2g,2)\mathsf {Sp}(2g,2) by an elementary abelian group of order 22g+12^{2g+1}. There is a central extension 1Z/2H~H11\to \mathbb{Z}/2\to\tilde{{\mathfrak{H}}}\to\mathfrak{H}\to 1, and H~\tilde{\mathfrak{H}} appears as a quotient of the metaplectic double cover Mp(2g,Z)=Sp(2g,Z)~/2Z\mathsf{Mp}(2g,\mathbb{Z})=\widetilde{\mathsf{Sp}(2g,\mathbb{Z})}/2\mathbb{Z}. It is an extension of Sp(2g,2)\mathsf{Sp}(2g,2) by an almost extraspecial group of order 22g+22^{2g+2}, and has a faithful irreducible complex representation of dimension 2g2^g. Provided g4g\ge 4, H~\widetilde{\mathfrak{H}} is the universal central extension of H\mathfrak{H}. Putting all this together, we provide a recipe for computing the signature modulo 88, and indicate some consequences.Comment: 18 pages. Minor corrections. The most important one is in the table for g=1g=1 on page 16: two columns had been swapped in the previous version. This is the version accepted for publication in Algebraic and Geometric Topolog
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