Braid Groups as Self-Equivalences of Derived Categories

Zimmermann, Alexander

doi:10.1007/978-94-010-0814-3_19

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“…Hence, the action of HD R (A 4 ) on H * (A 4 , R) is the action of Aut R (RA 4 ) on H * (A 4 , R). Again, HD R (A 4 ) contains a braid group on 3 strings as is shown in [12].…”

Section: Recall That the Action Of (ρ T T ) With Equivalence Class mentioning

confidence: 99%

Auto-equivalences of derived categories acting on cohomology

Zimmermann

2002

Archiv der Mathematik

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Let A be a k-algebra which is projective as a k-module, let M be an A-module whose endomorphisms are given by multiplication by central elements of A, and let TrPic k ( A) be the group of standard self-equivalences of the derived category of bounded complexes of A-modules. Then we define an action of the stabilizer of M in TrPic k ( A) on the Ext-algebra of M. In case M is the trivial module for the group algebra kG = A, this defines an action on the cohomology ring of G which extends the well-known action of the automorphism group of G on the cohomology group. Introduction. Let A and B be R-algebras over the commutative ring R so that A is projective as an R-module. If there is an equivalence between the derived categories of bounded complexes of A-modules D b ( A) and the derived category D b (B), Rickard and Kellerequivalences of this type form a group which, in an earlier work with R. Rouquier [8], is called TrPic R ( A). Let M be an A-module. Then, it is reasonable to expect that the set HD M ( A) of elements in TrPic R ( A), which fix M up to isomorphism, acts on the Ext-algebra Ext * A (M, M) of M as ring automorphisms since Ext n A (M, M) = Hom D b (A) (M, M[n]) for any integer n. To get an actual action one has to be a bit more careful. We prove the above statement if any automorphism of M is induced by multiplication by an invertible element of the centre of A. For other modules with more complicated automorphism groups an extension HD M ( A) of HD M ( A) by some quotient of the automorphism group of M acts on Ext * A (M, M). The above defined action is well behaved with respect to change of base rings. In case A is the group ring RG of a group G, the action of HD R (RG) extends the well known action of the outer automorphism group of G on the cohomology ring H * (G, R). This action of HD R (RG) is functorial with respect to the second variable. In further work [10,11] we study the functoriality with respect to the first variable. There, the situation is more complicate and we only have partial answers.

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“…Hence, the action of HD R (A 4 ) on H * (A 4 , R) is the action of Aut R (RA 4 ) on H * (A 4 , R). Again, HD R (A 4 ) contains a braid group on 3 strings as is shown in [12].…”

Section: Recall That the Action Of (ρ T T ) With Equivalence Class mentioning

confidence: 99%

Auto-equivalences of derived categories acting on cohomology

Zimmermann

2002

Archiv der Mathematik

View full text Add to dashboard Cite

show abstract

Braid Groups as Self-Equivalences of Derived Categories

Cited by 1 publication

References 20 publications

Auto-equivalences of derived categories acting on cohomology

Auto-equivalences of derived categories acting on cohomology

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