# Derived Equivalences for Group Rings by Steffen König

By Steffen König

A self-contained creation is given to J. Rickard's Morita conception for derived module different types and its fresh purposes in illustration thought of finite teams. specifically, Broué's conjecture is mentioned, giving a structural reason behind relatives among the p-modular personality desk of a finite workforce and that of its "p-local structure". The e-book is addressed to researchers or graduate scholars and will function fabric for a seminar. It surveys the present country of the sphere, and it additionally offers a "user's consultant" to derived equivalences and tilting complexes. effects and proofs are offered within the generality wanted for workforce theoretic applications.

Best symmetry and group books

Rotations, quaternions, and double groups

This exact monograph treats finite aspect teams as subgroups of the whole rotation team, delivering geometrical and topological tools which enable a distinct definition of the quaternion parameters for all operations. an incredible characteristic is an straightforward yet accomplished dialogue of projective representations and their program to the spinor representations, which yield nice benefits in precision and accuracy over the extra classical double team technique.

The theory of groups

Might be the 1st actually well-known e-book committed essentially to finite teams was once Burnside's e-book. From the time of its moment version in 1911 till the looks of Hall's ebook, there have been few books of comparable stature. Hall's publication continues to be thought of to be a vintage resource for basic effects at the illustration conception for finite teams, the Burnside challenge, extensions and cohomology of teams, \$p\$-groups and lots more and plenty extra.

Extra info for Derived Equivalences for Group Rings

Sample text

Id) : Rk ;! R g Rk where i is the embedding of the closed submanifold Graph into R ( id) C 1 (R Since 0 = r(0) = Rk f0g) = g i C 1 (R Rk Rk . Therefore f0g): (0) = (0), we see that g(0) = 0. So we also have C 1 (Rk f0g) = g C 1 (Graph f0g): Therefore it remains to prove that i C 1 (R f0g) = C 1 (Graph f0g): Now take an arbitrary f 2 C 1 (Graph f0g). There is a smooth extension f~ of f Rk on R Rk but it need not be at at zero. So consider a submanifold chart ( U ) of Graph around 0 and de ne fU : U ;!

We de ne the normal slice at x by ; Sx := expx Norr=2(G:x) x : July 31, 1997 P. 2 6. 3. Lemma. Under these conditions we have (1) (2) Sg:x = g:Sx . Sx is a slice at x. Proof. 1(1) : ; ; ; Sg:x = expg:x Tx`g Norr=2 (G:x) x = `g expx Norr=2 (G:x) x = g:Sx (2) r : G:Sx ;! G:x : expg:x X 7! g:x de nes a smooth equivariant retraction (note that Sx and Sy are disjoint if x 6= y). 4. De nition. Let M be a G-manifold and x 2 M , then there is a representation of the isotropy group Gx Gx ;! GL(TxM ) : g 7!

Again, it will do to nd a slice S at x with only a nite number of Gx -orbit types. If dim S < dim M , this follows from the induction hypothesis. Now suppose dim S = n. S is equivariantly di eomorphic to an open ball in Tx M under the slice representation (by exp). Since the slice representation is orthogonal, it restricts to a Gx -action on the sphere S n;1 . By the induction hypothesis, locally, S n;1 has only nitely many Gx -orbit types. Since S n;1 is compact, it has only nitely many orbit types globally.

Rated 4.38 of 5 – based on 18 votes