Representations of finite groups. Local cohomology and support. (Q642884)

Let \(G\) be a finite group, \(k\) be a field of prime characteristic \(p\), and \(kG\) be the associated group algebra. Consider the category \(\text{Mod}(kG)\) of (arbitrary) \(kG\)-modules and the associated stable module category \(\text{StMod}(kG)\) (where homomorphisms are considered modulo projective modules). While \(\text{Mod}(kG)\) is an Abelian category, the stable module category is not. It is however a (tensor) triangulated category. In [Ann. Math. (2) 174, No. 3, 1643-1684 (2011; Zbl 1261.20057)], the authors classified the non-zero tensor ideal localising subcategories of \(\text{Mod}(kG)\) in terms of subsets of prime ideals of the cohomology ring \(H^*(G,k)\). This follows from the fact that \(\text{StMod}(kG)\) is shown to be ``stratified'' by the action of \(H^*(G,k)\) which gives a classification of tensor ideal localising subcategories of \(\text{StMod}(kG)\). That work also led to the recovery (and improvement) of the classification of thick subcategories of the stable module category of finitely generated \(kG\)-modules due to \textit{D. J. Benson, J. F. Carlson,} and \textit{J. Rickard} [Fundam. Math. 153, No. 1, 59-80 (1997; Zbl 0886.20007)]. These classifications for \(kG\)-modules parallel in a sense similar classification theorems for derived categories of modules over a commutative ring \(R\). \textit{M. J. Hopkins} classified the thick subcategories of the derived category of bounded complexes of finitely generated projective \(R\)-modules [in Lond. Math. Soc. Lect. Note Ser. 117, 73-96 (1987; Zbl 0657.55008)]. That result was clarified by \textit{A. Neeman} [in Topology 31, No. 3, 519-532 (1992; Zbl 0793.18008)] wherein Neeman also gave a classification of the localising subcategories of the derived category of unbounded complexes of \(R\)-modules. Building upon the methods of these prior results, the key to the recent classification theorem for \(kG\)-modules is the notion of ``support'' in a general triangulated category. The ideas developed by the authors have already seen application in other triangulated categories and are likely to be useful in various representation theory contexts. The manuscript under review provides a quite nice introduction to the tools used in these classification theorems and offers an excellent starting point for someone new to the area. The manuscript is based on a week-long series of lectures given by the authors to introduce people to the ideas involved in the proof of the classification of localising subcategories of \(\text{Mod}(kG)\). The book contains five chapters based on the five days of lectures along with an appendix. Four of the five chapters include exercises as well (based on discussions from the workshop). As mentioned above, the problem for \(\text{Mod}(kG)\) is first reduced to a stratification problem for \(\text{StMod}(kG)\). This is then translated to a stratification problem for the homotopy category of injective \(kG\)-modules. That problem is then reduced to the case of an elementary Abelian \(p\)-group. From here, using a Bernstein-Gelfand-Gelfand correspondence, the problem is further reduced to a problem over a graded polynomial algebra where the problem connects with Neeman's classification results. In this manuscript, the reduction to a polynomial algebra is done only for the case \(p=2\). The first chapter provides an historical overview and a review of related ideas from modular representation theory. This includes information on the structure of the group ring \(kG\), the classification of modules, the cohomology ring \(H^*(G,k)\), rank varieties, categories of modules, the stable module category, a discussion of localising and thick subcategories in this context, and statements of some versions of the main results. In addition, this chapter contains a discussion of triangulated categories and related ideas: Frobenius categories, localising subcategories, thick subcategories, compact objects, and cohomology functors. In the second chapter, the authors review in more detail some of the key historical techniques. For a commutative Noetherian ring \(A\), the authors introduce the derived category of \(A\)-modules and the bounded derived category of finitely generated \(A\)-modules, thick subcategories, perfect complexes, and the concept of support of an \(A\)-module or complex thereof. The notion of support for a commutative ring is supplemented by an appendix devoted to this topic. Hopkins' theorem relating thick subcategories of perfect complexes to support is given which leads to the classification of such thick subcategories. Next some ideas from homotopy theory are introduced for triangulated categories: Brown representability, localisation and colocalisation functors, and related properties. Lastly, the authors return to the finite group setting discussing in more detail the cohomology ring \(H^*(G,k)\), Quillen stratification thereof with respect to elementary Abelian \(p\)-groups, and cohomological varieties and their properties. Cohomological varieties are first considered for finitely generated modules. Then the arbitrary case is considered including a discussion of Rickard idempotents and finally arriving at the aforementioned classification of thick subcategories of the stable module category. In the third chapter, the theory of support for triangulated categories is developed using local cohomology functors. This is followed by techniques for computing support along with the notion of a Koszul object and its computational usefulness. Lastly, the authors introduce the aforementioned homotopy category of injective modules \(K(\text{Inj\,}kG)\) and discuss its relationship to the derived category of \(\text{Mod}(kG)\) and to \(\text{StMod}(kG)\) which is given by a recollement. Varieties for objects in \(K(\text{Inj\,}kG)\) are also introduced. It is further observed that \(K(\text{Inj\,}kG)\) is related to the derived category of co-chains on the classifying space of \(G\). The fourth chapter is aimed at giving an outline of the proof of the classification theorem. In the context of a triangulated category, there is a map from the localising subcategories to subsets of the support of the category (relative to a ring). To say that one has a ``classification'' of the localising subcategories means that this map is a bijection. When the map is a bijection one has a ``local-global principle'' (relating the global problem to subcategories defined by local cohomology functors associated to prime ideals) and a ``minimality condition'' (that these subcategories associated to prime ideals are \textit{minimal} localising subcategories). A triangulated category is said to be ``stratified'' by the action of a ring if these two conditions hold. The key result towards classification is that, conversely, stratification implies that the above map is a bijection. These ideas are developed and discussed in the context of the appropriate categories of interest. Following this, a number of consequences of the classification theorem are presented, and lastly the specific case of the Klein four group (over a field of characteristic 2) is discussed with concepts worked out in detail. Following the outline previously presented, the remaining details of the proof of the classification theorem are given in the final chapter (with complete details of one step given only for characteristic 2).