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Milnor K-theory via commuting automorphisms
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Grech, Daniel (2019) Milnor K-theory via commuting automorphisms. PhD thesis, University of Warwick.
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WRAP_Theses_Grech_2018.pdf - Submitted Version - Requires a PDF viewer. Download (1099Kb) | Preview |
Official URL: http://webcat.warwick.ac.uk/record=b3439156~S15
Abstract
In this thesis, we give a presentation for Milnor K-theory of a field F whose generators are tuples of commuting automorphisms. This is similar to a presentation for Milnor K-theory given by the cohomology groups of Grayson. The main difference is that, in our presentation, we do not use a homotopy invariance relation, which we should not expect to hold for non-regular rings R.
We go on to study this presentation for R a local ring. We conjecture that it agrees with the usual definition of Milnor K-theory for any local ring. We give some evidence towards this, including showing that the natural map Kn(R)→K ̃n(R) is injective when n = 0, 1, 2 or when R is a regular, local ring containing an infinite field. We also show a reciprocity result for K ̃n^M (R) any ring R, which, when R is a field, allows us to deduce surjectivity of the map.
We prove a version of the additivity, resolution, devissage and cofinality theorems for the groups K ̃n^M (R). We also construct a comparison homomorphsim from K ̃n^M (R) to the presentation of Quillen K-theory given by Grayson.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | K-theory, Automorphisms, Cohomology operations, Algebraic topology | ||||
Official Date: | March 2019 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Schlichting, Marco | ||||
Sponsors: | Economic and Social Research Council (ESRC) | ||||
Format of File: | |||||
Extent: | 116 leaves: illustrations | ||||
Language: | eng |
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