Cohomologies of derived intersections

Finegan, Oscar 2024. Cohomologies of derived intersections. PhD Thesis, Cardiff University. Item availability restricted.

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Abstract

The intersection of derived schemes carries as structure complex the derived tensor product of structure sheaves of the schemes we are intersecting ([17][10][1] ). For intersections of underived schemes, the cohomologies of the intersection structure complex carries important geometric information about the intersection. Computations using these derived tensor products also arise naturally in the context of Fourier-Mukai transforms. In §2 we provide the background material for this thesis. This includes a construction of the derived tensor product, as well as an overview of the necessary results and definitions for Koszul complexes and local complete intersections. In §3 we give proofs of results in the literature on the cohomologies of derived intersections. This includes a novel proof of the excess intersection formula in the local complete intersection case. In §4 we provide new results on the cohomologies of derived intersections in the non-local complete intersection case. In the case that we study, we provide a precise description of the cohomologies as a glued object over the components of the intersection.

Item Type:	Thesis (PhD)
Date Type:	Completion
Status:	Unpublished
Schools:	Mathematics
Subjects:	Q Science > QA Mathematics
Funders:	EPSRC
Date of First Compliant Deposit:	9 September 2024
Last Modified:	09 Sep 2024 08:45
URI:	https://orca.cardiff.ac.uk/id/eprint/171924

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