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Coherent functors and asymptotic properties
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis we study properties of the so called coherent functors. Coherent functors were first introduced by Auslander in 1966 in a general setting. Coherent functors have been used since then as powerful tools for different purposes: to describe infinitesimal deformation theory, to describe algebraicity of a stack or to study properties of Rees algebras.

In 1998, Hartshorne proved that half exact coherent functors over a discrete valuation ring 𝐴 are direct sums of the identity functor, Hom-functors of quotient modules of 𝐴 and tensor products of quotient modules of 𝐴. In our first article (Paper A), we obtain a similar characterization for half exact coherent functors over a much wider class of rings: Dedekind domains. This fact allows us to classify half exact coherent functors over Dedekind domains.

In our second article (Paper B), coherent functors over noetherian rings are considered. We study asymptotic properties of sets of prime ideals connected with coherent functors applied to artinian modules or finitely generated modules. Also considering quotient modules M /anM, where an is the nthpower of an ideal 𝑎, one obtains that the Betti and Bass numbers of the images under a coherent functor of the quotient modules above are polynomials in n for large n. Furthermore, the lengths of these image modules are polynomial in 𝑛, for large 𝑛, under the condition that the image modules have finite length.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2019. , p. 43
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1982
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-156730DOI: 10.3384/diss.diva-156730ISBN: 9789176850954 (print)OAI: oai:DiVA.org:liu-156730DiVA, id: diva2:1315353
Public defence
2019-06-04, BL32, B-huset, Campus Valla, Linköping, 13:15 (English)
Opponent
Supervisors
Note

Minor errors has been corrected in the electronic version. See the Errata list for more information what have been corrected.

Available from: 2019-05-13 Created: 2019-05-13 Last updated: 2019-06-20Bibliographically approved
List of papers
1. Half–Exact Coherent Functors over PIDs and Dedekind Domains
Open this publication in new window or tab >>Half–Exact Coherent Functors over PIDs and Dedekind Domains
2016 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

The focus of this thesis is to characterize half–exact coherent functors over principal ideal domains (PIDs) and Dedekind domains. Ever since they where discovered, coherent functors have been useful in the study of some mathematical objects. We aim to explore a little more about them in this thesis.

We first give here a review of the general categorical notions relevant to the characterization. We also review the functors Ext(M,−) and Tor(M,−)  on the category on A–modules, where A is a commutative ring and M is an A–module.

With the assumption that A is a commutative noetherian ring, we introduce coherent functors defined on the category of finitely generated A–modules. It is then shown in the paper that any half–exact coherent functor over a PID, and more generally over a Dedekind domain, arises from a complex of projective modules.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2016. p. 40
Series
Linköping Studies in Science and Technology. Thesis, ISSN 0280-7971 ; 1752
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-128616 (URN)978-91-7685-750-2 (ISBN)
Presentation
2016-06-08, Nobel (BL32), B-huset, Campus Valla, Linköping, 15:15 (English)
Opponent
Supervisors
Note

The thesis serie title Linköping Studies in Science and Technology. Licentiate Thesis is incorrect. The correct title is Linköping Studies in Science and Technology. Thesis.

Available from: 2016-05-25 Created: 2016-05-25 Last updated: 2019-05-13Bibliographically approved
2. Coherent functors and asymptotic stability
Open this publication in new window or tab >>Coherent functors and asymptotic stability
2019 (English)In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 522Article in journal (Refereed) Published
Abstract [en]

Asymptotic properties of high powers of an ideal related to a coherent functor F are investigated. It is shown that when N is an artinian module the sets of attached prime ideals Att(A) F(0 :(N) a(n)) are the same for n large enough. Also it is shown that for an artinian module N if the modules F(0 :(N) a(n)) have finite length and for a finitely generated module M if the modules F(M/a(n) M) have finite length, their lengths are given by polynomials in n, for large n. When A is local it is shown that, the Betti numbers beta(i)(F(M /a(n) M)) and the Bass numbers mu(i)(F(M / a(n) M)) are given by polynomials in n for large n. (C) 2018 Elsevier Inc. All rights reserved.

Place, publisher, year, edition, pages
ACADEMIC PRESS INC ELSEVIER SCIENCE, 2019
Keywords
Asymptotic prime ideal; Coherent functor; Hilbert polynomial; Betti number; Bass number
National Category
Algebra and Logic
Identifiers
urn:nbn:se:liu:diva-154529 (URN)10.1016/j.jalgebra.2018.11.035 (DOI)000457509500001 ()
Note

Funding Agencies|ISP through EAUMP

Available from: 2019-02-20 Created: 2019-02-20 Last updated: 2019-05-13

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