arXiv subjects

Markus Linckelmann

Publications and source records attributed to Markus Linckelmann.

44 records · Page 3Linked to original sources

arXiv2014

On stable equivalences with endopermutation source

We show that a bimodule between block algebras which has a fusion stable endopermutation module as a source and which induces Morita equivalences between centralisers of nontrivial subgroups of a defect group induces a stable equivalence of Morita type; this is a converse to a theorem of Puig. The special case where the source is trivial has long been known by many authors. The earliest instance for a result deducing a stable equivalence of Morita type from local Morita equivalences with possibly nontrivial endopermutation source is due to Puig, in the context of blocks with abelian defect groups with a Frobenius inertial quotient. The present note is motivated by an application, due to Biland, to blocks of finite groups with structural properties known to hold for hypothetical minimal counterexamples to the $Z_p^*$-Theorem.

math.RT↗

arXiv2013

On the Castelnuovo-Mumford regularity of the cohomology of fusion systems and of the Hochschild cohomology of block algebras

Symonds' proof of Benson's regularity conjecture implies that the regularity of the cohomology of a fusion system and that of the Hochschild cohomology of a p-block of a finite group is at most zero. Using results of Benson, Greenlees, and Symonds, we show that in both cases the regularity is equal to zero.

math.GR↗

arXiv2013

On symmetric quotients of symmetric algebras

We investigate symmetric quotient algebras of symmetric algebras, with an emphasis on finite group algebras over a complete discrete valuation ring ${\mathcal O}$. Using elementary methods, we show that if an ordinary irreducible character $χ$ of a finite group $G$ gives rise to a symmetric quotient over ${\mathcal O}$ which is not a matrix algebra, then the decomposition numbers of the row labelled by $χ$ are all divisible by the characteristic $p$ of the residue field of ${\mathcal O}$.

math.GR↗

arXiv2012

Tate duality and transfer in Hochschild cohomology

We show that dualising transfer maps in Hochschild cohomology of symmetric algebras commutes with Tate duality. This extends a well-known result in group cohomology.

math.RT↗

arXiv2010

Conjectures of Alperin and Broue for 2-blocks with elementary abelian defect groups of order 8

Using the classification of finite simple groups we prove Alperin's weight conjecture and the character theoretic version of Broue's abelian defect group conjecture for 2-blocks of finite groups with an elementary abelian defect group of order 8.

math.RT↗

arXiv2010

Bounds for Hochschild cohomology of block algebras

We show that for any block algebra B of a finite group over an algebraically closed field of prime characteristic the dimension of HH^n(B) is bounded by a function depending only on the nonnegative integer n and the defect of B. The proof uses in particular a theorem of Brauer and Feit which implies the result for n=0.

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arXiv2008

On the graded center of the stable category of a finite $p$-group

We show that for any finite $p$-group $P$ of rank at least 2 and any algebraically closed field $k$ of characteristic $p$ the graded center $Z^*(\modbar(kP))$ of the stable module category of finite-dimensional $kP$-modules has infinite dimension in each odd degree, and if $p=2$ also in each even degree. In particular, this provides examples of symmetric algebras $A$ for which $Z^0(\modbar(A))$ is not finite-dimensional, answering a question raised in [10]

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arXiv2005

Blocks with quaternion defect group over a 2-adic ring: the case \tilde{A}_4

Except for blocks with a cyclic or Klein four defect group, it is not known in general whether the Morita equivalence class of a block algebra over a field of prime characteristic determines that of the corresponding block algebra over a p-adic ring. We prove this to be the case when the defect group is quaternion of order 8 and the block algebra over an algebraically closed field k of characteristic 2 is Morita equivalent to $k\tilde A_4$. The main ingredients are Erdmann's classification of tame blocks and work of Cabanes and Picaronny on perfect isometries between tame blocks.

math.RT↗