Jeremy Rickard

Equivalences of derived categories for symmetric algebras

We give a characterization of the sets of objects of the derived category of a block of a finite group algebra (or other symmetric algebra) that occur as the set of images of simple modules under an equivalence of derived categories. We give some applications to proving that certain blocks have equivalent derived categories.

Cocovers and tilting modules

We recall [ 5 ] that a module T for a finite-dimensional algebra Λ is called a tilting module if (i) T has protective dimension one; (ii) (iii) there is a short exact sequence 0 → Λ → T 0 → T 1 → 0 with T 0 and T 1 in add ( T ), the category of direct summands of direct sums of copies of T .

Translation Functors and Equivalences of Derived Categories for Blocks of Algebraic Groups

We prove the existence of many self-equivalences of the derived categories of blocks of reductive groups in prime characteristic that are not induced by self-equivalences of the module categories of the blocks. We conjecture that our examples are just the simplest special case of a much more general phenomenon. This is analogous to conjectures about derived categories of blocks of symmetric groups.

Bousfield Localization for Representation Theorists

Bousfield localization is a technique that has been used extensively in algebraic topology, specifically in stable homotopy theory, over the last quarter century. Its name derives from the fundamental work of Bousfield [3], although this is based on earlier work of Brown [4] and Adams [1]. In abstract terms, Bousfield localization deals with the inclusion of a thick subcategory (i.e., a triangulated subcategory closed under direct summands) into a triangulated category and the existence of adjoint functors to such an inclusion. In the original topological setting, the triangulated category was typically the stable homotopy category, and the thick subcategory was defined by the vanishing of some homology theory, but the techniques involved work much more generally. In particular, the triangulated category can be one of interest to representation theorists, such as a stable module category or the derived category of a module category: since the techniques rely heavily on limiting procedures, however, one is forced to work with infinite dimensional modules.

Representations of Finite Groups

We provide a formal framework for the theory of representations of finite groups, as modules over the group ring. Along the way, we develop the general theory of groups (relying on the group add class for the basics), modules, and vector spaces, to the extent required for theory of group representations. We then provide formal proofs of several important introductory theorems in the subject, including Maschke’s theorem, Schur’s lemma, and Frobenius reciprocity. We also prove that every irreducible representation is isomorphic to a submodule of the group ring, leading to the fact that for a finite group there are only finitely many isomorphism classes of irreducible representations. In all of this, no restriction is made on the characteristic of the ring or field of scalars until the definition of a group representation, and then the only restriction made is that the characteristic must not divide the order of the group.