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Stratifying modular representations of finite groups. (English) Zbl 1261.20057

Let \(G\) be a finite group and let \(k\) be a field of characteristic \(p\). Denote by \(\mathsf{Mod}(kG)\) the category of all \(kG\)-modules. The cohomology ring \(H^*(G,k)\) is graded commutative, and therefore it is possible to form the projective space \(\text{Proj}(H^*(G,k))\) of all non trivial homogeneous prime ideals on this ring. The paper under review shows that there is a bijection between the subsets of \(\text{Proj}(H^*(G,k))\) and those non zero subcategories of \(\mathsf{Mod}(kG)\) which are invariant under direct summands, direct sums, tensoring with simple modules, and satisfy that whenever two objects in an exact sequence are in the category, then is the third.
The proof is highly involved and passes through various intermediate steps which are always interesting in their own right. The bijection is defined in the following way. Let \(R\) be a graded commutative ring. Then, under certain assumptions, in any triangulated \(R\)-category \(T\) we get a localisation functor \(L_V\) so that \(L_V\) vanishes precisely on those \(X\) with \(R_q\otimes_R\operatorname{Hom}_T(-,X[n])=0\) for all \(n\) and all primes \(q\) not in \(V\). It can be shown that there is a functor \(\Gamma_V\) and an exact triangle \(\Gamma_VX\rightarrow X\rightarrow L_V\rightarrow\). For a graded commutative ring \(R\) and a prime ideal \(p\), let \(V\) and \(W\) be two sets of primes, both closed under taking primes bigger than any given in the set, so that \(V\setminus W\) has only the element \(p\). Then define \(\Gamma_p:=\Gamma_VL_W\). Given a localising subcategory \(C\) of \(T\), then define \(\sigma(T)\) the primes \(p\) of \(R\) so that \(\Gamma_pC\neq 0\). The map \(\sigma\) gives the bijection, for various \(R\) and \(T\) adapted to the situation in the proof. The inverse is given explicitly in a similar fashion.
In the first step \(T\) is the derived category of a graded polynomial algebra \(R\) in \(r\) variables. The second step uses as \(T\) the homotopy category of injective modules over the exterior algebra with \(r\) variables, and \(R\) the Ext-algebra of the trivial module. Then in further steps one passes to the homotopy category of injective modules over a maximal elementary Abelian \(p\)-subgroup of \(G\), and further to homotopy category of injective modules over \(kG\), and finally to the module category of \(kG\). The bijection is always given by some \(\sigma\), adapted to each case.

MSC:

20J06 Cohomology of groups
20C20 Modular representations and characters
16E35 Derived categories and associative algebras
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
18E30 Derived categories, triangulated categories (MSC2010)
13D09 Derived categories and commutative rings
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References:

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