Realisability and Localisation - Mathematics > Representation TheoryReport as inadecuate

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Abstract: Let $A$ be a differential graded algebra with cohomology ring $H^*A$. Agraded module over $H^*A$ is called \emph{realisable} if it is up to directsummands of the form $H^*M$ for some differential graded $A$-module $M$.Benson, Krause and Schwede have stated a local and a global obstruction forrealisability. The global obstruction is given by the Hochschild classdetermined by the secondary multiplication of the $A {\infty}$-algebrastructure of $H^*A$.In this thesis we mainly consider differential graded algebras $A$ withgraded-commutative cohomology ring. We show that a finitely presented graded$H^*A$-module $X$ is realisable if and only if its $\mathfrak{p}$-localisation$X {\mathfrak{p}}$ is realisable for all graded prime ideals $\mathfrak{p}$ of$H^*A$.In order to obtain such a local-global principle also for the globalobstruction, we define the \emph{localisation of a differential graded algebra$A$ at a graded prime $\mathfrak{p}$ of $H^*A$}, denoted by $A {\mathfrak{p}}$,and show the existence of a morphism of differential graded algebras inducingthe canonical map $H^*A \to H^*A {\mathfrak{p}}$ in cohomology. The latterresult actually holds in a much more general setting: we prove that everysmashing localisation on the derived category of a differential graded algebrais induced by a morphism of differential graded algebras.Finally we discuss the relation between realisability of modules over thegroup cohomology ring and the Tate cohomology ring.

Author: Birgit Huber


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