190 research outputs found

### Symmetric tensor categories in characteristic 2

We construct and study a nested sequence of finite symmetric tensor
categories ${\rm Vec}=\mathcal{C}_0\subset \mathcal{C}_1\subset\cdots\subset
\mathcal{C}_n\subset\cdots$ over a field of characteristic $2$ such that
$\mathcal{C}_{2n}$ are incompressible, i.e., do not admit tensor functors into
tensor categories of smaller Frobenius--Perron dimension. This generalizes the
category $\mathcal{C}_1$ described by Venkatesh and the category
$\mathcal{C}_2$ defined by Ostrik. The Grothendieck rings of the categories
$\mathcal{C}_{2n}$ and $\mathcal{C}_{2n+1}$ are both isomorphic to the ring of
real cyclotomic integers defined by a primitive $2^{n+2}$-th root of unity,
$\mathcal{O}_n=\mathbb Z[2\cos(\pi/2^{n+1})]$.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

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

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 $H^*(G,k)$.Comment: 5 pages, minor changes, accepted for publication in C. R. Math. Acad.
Sci. Pari

### Stratifying modular representations of finite groups

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

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

Meyer showed that the signature of a closed oriented surface bundle over a
surface is a multiple of $4$, and can be computed using an element of
$H^2(\mathsf{Sp}(2g, \mathbb{Z}),\mathbb{Z})$. Denoting by $1 \to \mathbb{Z}
\to \widetilde{\mathsf{Sp}(2g,\mathbb{Z})} \to \mathsf{Sp}(2g,\mathbb{Z}) \to
1$ the pullback of the universal cover of $\mathsf{ Sp}(2g,\mathbb{R})$,
Deligne proved that every finite index subgroup of $\widetilde{\mathsf {Sp}(2g,
\mathbb{Z})}$ contains $2\mathbb{Z}$. As a consequence, a class in the second
cohomology of any finite quotient of $\mathsf{Sp}(2g, \mathbb{Z})$ can at most
enable us to compute the signature of a surface bundle modulo $8$. We show that
this is in fact possible and investigate the smallest quotient of
$\mathsf{Sp}(2g, \mathbb{Z})$ that contains this information. This quotient
$\mathfrak{H}$ is a non-split extension of $\mathsf {Sp}(2g,2)$ by an
elementary abelian group of order $2^{2g+1}$. There is a central extension
$1\to \mathbb{Z}/2\to\tilde{{\mathfrak{H}}}\to\mathfrak{H}\to 1$, and
$\tilde{\mathfrak{H}}$ appears as a quotient of the metaplectic double cover
$\mathsf{Mp}(2g,\mathbb{Z})=\widetilde{\mathsf{Sp}(2g,\mathbb{Z})}/2\mathbb{Z}$.
It is an extension of $\mathsf{Sp}(2g,2)$ by an almost extraspecial group of
order $2^{2g+2}$, and has a faithful irreducible complex representation of
dimension $2^g$. Provided $g\ge 4$, $\widetilde{\mathfrak{H}}$ is the universal
central extension of $\mathfrak{H}$. Putting all this together, we provide a
recipe for computing the signature modulo $8$, and indicate some consequences.Comment: 18 pages. Minor corrections. The most important one is in the table
for $g=1$ 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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