General Theory of Natural Equivalences

General Theory of Natural Equivalences

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K-theory and Derived Equivalences

We show that if two rings have equivalent derived categories then they have the same algebraic K-theory. Similar results are given for G-theory, and for a large class of abelian categories.

Optimal Natural Dualities. Ii: General Theory

A general theory of optimal natural dualities is presented, built on the test algebra technique introduced in an earlier paper. Given that a set R of finitary algebraic relations yields a duality on a class of algebras A = ISP(M), those subsets R′ of R which yield optimal dualities are characterised. Further, the manner in which the relations in R are constructed from those in R′ is revealed in...

Rickard Equivalences and Block Theory

De nition We say that a subgroup H of G controls the fusion of p subgroups of G if the following two conditions are ful lled C H contains a Sylow p subgroup Sp of G C whenever P is a subgroup of Sp and g is an element of G such that gPg Sp there exist z in the centralizer CG P of P in G and h in H such that g hz Basic example We denote by Op G the largest normal subgroup of G with order prime t...

Block Theory via Stable and Rickard Equivalences