Realisability and Localisation

TL;DR

This paper investigates the conditions under which modules over the cohomology ring of a differential graded algebra are realizable, establishing local-global principles and connecting localizations with algebra morphisms.

Contribution

It introduces a local-global principle for realizability of modules over graded cohomology rings and constructs localizations of differential graded algebras at prime ideals.

Findings

A finitely presented module is realizable iff all localizations are realizable.

Existence of a morphism of DGAs inducing cohomology localization.

Every smashing localization on derived categories is induced by a DGA morphism.

Abstract

Let $A$ be a differential graded algebra with cohomology ring $H^*A$. A graded module over $H^*A$ is called \emph{realisable} if it is (up to direct summands) of the form $H^*M$ for some differential graded $A$-module $M$. Benson, Krause and Schwede have stated a local and a global obstruction for realisability. The global obstruction is given by the Hochschild class determined by the secondary multiplication of the $A_{\infty}$-algebra structure of $H^*A$. In this thesis we mainly consider differential graded algebras $A$ with graded-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 global obstruction, 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 inducing the canonical map $H^*A \to (H^*A)_{\mathfrak{p}}$ in cohomology. The latter result actually holds in a much more general setting: we prove that every smashing localisation on the derived category of a differential graded algebra is induced by a morphism of differential graded algebras. Finally we discuss the relation between realisability of modules over the group cohomology ring and the Tate cohomology ring.