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Title: C-infinity algebraic geometry with corners
Author: Francis-Staite, Kelli
ISNI:       0000 0004 7971 6481
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2019
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C-schemes are a generalisation of manifolds that have nice properties such as the existence of fibre products. C-schemes have been used as a model for synthetic differential geometry, as in Dubuc, Kock, and Moerdijk and Reyes, and for defining derived differential geometry as in Lurie, and Spivak. Manifolds with corners are a generalisation of manifolds locally modelled on [0,∞)k × Rn-k, and their smooth maps behave well with respect to the corners as in Melrose. In particular, Joyce describes a corner functor from the category of manifolds with corners to the category of 'interior' manifolds with corners with mixed dimension. C-algebraic geometry with corners is the study of C-rings and C-schemes with corners, which we define in this thesis. We define (local/interior/firm) C-rings with corners, and study categorical properties such as the existence of limits and colimits using various adjoint functors. We describe a spectrum functor from C-rings with corners to local C-ringed spaces with corners, and show this a right adjoint to a global sections functor. We define C-schemes with corners using this spectrum functor. We show there is a full and faithful embedding of the category of manifolds with corners into the category of firm C-schemes with corners, and that fibre products of firm $C^\infty$-schemes with corners exist. We show that manifolds with corners are affine under geometric conditions. We define (b-)cotangent sheaves of C-schemes with corners and show they correspond to the (b-)cotangent bundles of manifolds with corners of Joyce. We describe the categories of interior local C-ringed spaces with corners and interior firm C-schemes with corners. We construct corner functors for both of these categories, which are right adjoint to the inclusion of these interior spaces/schemes into the non-interior ones. We show that these corner functors correspond to the corner functor for manifolds with corners. We expect applications of this work in defining derived spaces with corners in derived differential geometry, and we explore the connections of this work to log geometry and the positive log differentiable spaces of Gillam and Molcho.
Supervisor: Joyce, Dominic Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Geometry ; Mathematics ; Algebraic Geometry ; Category Theory ; Differential Geometry