Finite Dimensional DGAs

Isambard Goodbody (University of Glasgow)

Wednesday 26th April, 2023 16:00-17:00 Maths 311B

Abstract

A DGA is a chain complex with a compatible multiplication. These appear in algebra and geometry as any derived category which is generated by a single object is equivalent to the derived category of a DGA. A finite dimensional DGA is one whose underlying chain complex is finite dimensional and so they generalise fd algebras. Orlov has constructed a radical filtration for fd DGAs and we use this to prove some properties reminiscent of fd algebras. One property is that the simples can detect modules with finite projective dimension; this gives a new characterisation of perfect DG-modules. Another is the projective-simple bijection; this fails to generalise fully due to the existence of phantoms. However we introduce a class of fd DGAs with an idempotent lifting property for which this does hold. Using this we can answer some open questions about Grothendieck groups and phantoms.

 

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