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These are lecture notes for the course 18.769 “Tensor categories”, taught by P. Etingof at MIT in the spring of 2009. In these notes we will assume that the reader is familiar with the basic theory of categories and functors; a detailed discussion of this theory can be found in the book [ML]. We will also assume the basics of the theory of abelian categories (for a more detailed treatment see the book [F]). If C is a category, the notation X ∈ C will mean that X is an object of C, and the set of morphisms between X, Y ∈ C will be denoted by Hom(X, Y). Throughout the notes, for simplicity we will assume that the ground field k is algebraically closed unless otherwise specified, even though in many cases this assumption will not be needed. 1. Monoidal categories 1.1. The definition of a monoidal category. A good way of thinking

this paper we develop geometric methods for the study of nite modules over a

This is the first of two papers in which we determine the spectrum of the cohomology algebra of infinitesimal group schemes over a field k of characteristic p> 0. Whereas [SFB] is concerned with detection of cohomology classes, the present paper introduces the graded algebra k[Vr(G)] of functions on the scheme of infinitesimal 1-parameter subgroups of height ≤ r on an affine group scheme G and demonstrates that this algebra is essentially a retract of H ev (G, k) provided that G is an infinitesimal group scheme of height ≤ r. This work is a continuation of [F-S] in which the existence of certain universal extension classes was established, thereby enabling the proof of finite generation of H ∗ (G, k) for any finite group scheme G over k. The role of the scheme of infinitesimal 1-parameter subgroups of G was foreshadowed in [F-P] where H ∗ (G (1), k) was shown to be isomorphic to the coordinate algebra of the scheme of p-nilpotent elements of g = Lie(G) for G a smooth reductive group, G (1) the first Frobenius kernel of G, and p = char(k) sufficiently large. Indeed, p-nilpotent elements of g correspond precisely to infinitesimal 1-parameter subgroups of G (1). Much of our effort in this present paper involves the analysis of the restriction of the universal extension classes to infinitesimal 1-parameter subgroups. In section 1, we construct the affine scheme Vr(G) of homomorphisms from G a(r) to G which we call the scheme of infinitesimal 1-parameter subgroups of height ≤ r in G. In the special case r = 1, this is the scheme of p-nilpotent elements of the p-restricted Lie algebra Lie(G); for various classical groups G, Vr(G) is the scheme of r-tuples of p-nilpotent, pairwise commuting elements of Lie(G). More generally, an embedding G ⊂ GLn determines a closed embedding of Vr(G) into the scheme of r-tuples of p-nilpotent, pairwise commuting elements of gln = Lie(GLn). The relationship between k[Vr(G)], the coordinate algebra of Vr(G), and H ∗ (G, k) is introduced in Theorem 1.14: a natural homomorphism of graded k-algebras ψ: H ev (G, k) → k[Vr(G)] is constructed, a map which we show in [SFB] to induce a bijective map on associated schemes. The universal classes er ∈ H2pr−1 characteristic classes er(G, V) for any affine group scheme G provided with a finite

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Abstract. Support varieties for any finite dimensional algebra over a field were introduced in [20] using graded subalgebras of the Hochschild cohomology. We mainly study these varieties for selfinjective algebras under appropriate finite generation hypotheses. Then many of the standard results from the theory of support varieties for finite groups generalize to this situation. In particular, the complexity of the module equals the dimension of its corresponding variety, all closed homogeneous varieties occur as the variety of some module, the variety of an indecomposable module is connected, periodic modules are lines and for symmetric algebras a generalization of Webb’s theorem is true.

Abstract. We introduce the space P (G) of abelian p-points of a finite group scheme over an algebraically closed field of characteristic p> 0. We construct a homeomorphism ΨG: P (G) → Proj |G | from P (G) to the projectivization of the cohomology variety for any finite group G. For an elementary abelian p-group (respectively, an infinitesimal group scheme), P (G) can be identified with the projectivization of the variety of cyclic shifted subgroups (resp., variety of 1-parameter subgroups). For a finite dimensional G-module M, ΨG restricts to a homeomorphism P (G)M → Proj |G|M, thereby giving a representation-theoretic interpretation of the cohomological support variety. Even though the cohomology groups H i (G, k) of a finite group are typically difficult to compute, D. Quillen in his seminal papers [17] gave a general description of the maximal ideal spectrum |G | of the commutative k-algebra H ev (G, k) in terms

Abstract. We introduce the space Π(G) of equivalence classes of π-points of a finite group scheme G, and associate a subspace Π(G)M to any G-module M. Our results extend to arbitrary finite group schemes G over arbitrary fields k of positive characteristic and to arbitrarily large G-modules the basic results about “cohomological support varieties ” and their interpretation in terms of representation theory. In particular, we prove that the projectivity of any (possibly infinite dimensional) G-module can be detected by its restriction along π-points of G. Unlike the cohomological support variety of a G-module M, the invariant M ↦ → Π(G)M satisfies good properties for all modules, thereby enabling us to determine the thick, tensor-ideal subcategories of the stable

The representation theory of the algebraic supergroup Q(n) has been studied quite intensively over the complex numbers in recent years, especially by Penkov

Let G be a connected reductive algebraic group over an algebraically closed field of characteristic p> 0, G1 be the first Frobenius kernel, and G(Fp) be the corresponding finite Chevalley group. Let M be a rational G-module. In this paper we relate the support variety of M over the first Frobenius kernel with the support variety of M over the group algebra kG(Fp). This provides an answer to a question of Parshall. Applications of our new techniques are presented which allow us to extend results of Alperin-Mason and Janiszczak-Jantzen, and to calculate the dimensions of support varieties for finite Chevalley groups.