An Axiomatic Setup for Algorithmic Homological Algebra and an Alternative Approach to Localization

Barakat, Mohamed; Lange-Hegermann, Markus

doi:10.1142/s0219498811004562

Cited by 23 publications

(38 citation statements)

References 22 publications

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“…A general framework addressing precisely this point has evolved in computational algebraic geometry [40][41][42][43][44][45]. The idea is very simple: the coherent sheaf in question is defined in terms of the (Chow) class of certain points on B 3 .…”

Section: Jhep11(2017)081mentioning

confidence: 99%

Gauge backgrounds and zero-mode counting in F-theory

Bies

Mayrhofer²,

Weigand

2017

J. High Energ. Phys.

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Computing the exact spectrum of charged massless matter is a crucial step towards understanding the effective field theory describing F-theory vacua in four dimensions. In this work we further develop a coherent framework to determine the charged massless matter in F-theory compactified on elliptic fourfolds, and demonstrate its application in a concrete example. The gauge background is represented, via duality with M-theory, by algebraic cycles modulo rational equivalence. Intersection theory within the Chow ring allows us to extract coherent sheaves on the base of the elliptic fibration whose cohomology groups encode the charged zero-mode spectrum. The dimensions of these cohomology groups are computed with the help of modern techniques from algebraic geometry, which we implement in the software gap. We exemplify this approach in models with an Abelian and non-Abelian gauge group and observe jumps in the exact massless spectrum as the complex structure moduli are varied. An extended mathematical appendix gives a self-contained introduction to the algebro-geometric concepts underlying our framework.

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Section: Jhep11(2017)081mentioning

confidence: 99%

Gauge backgrounds and zero-mode counting in F-theory

Bies

Mayrhofer²,

Weigand

2017

J. High Energ. Phys.

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“…Within category theory, this idea was axiomatized in [2] and applied, for instance, to sheaf theory. If D is a very simple domain, in this paper, we show that problems classically studied in module theory (e.g., Serre's splitting-off theorem, Swan's lemma, Bass' theorem), which are based on the solvability of inhomogeneous quadratic equations (see above), can be studied within the same constructive framework.…”

Section: If Rank D (M )mentioning

confidence: 99%

“…For more details, see [11,13,36]. Hence, (2) shows that the properties of the linear system ker F (R.) can be studied by means of hom D (M, F), and thus by means of the left D-modules M and F.…”

Section: Rr N°8225mentioning

confidence: 99%

“…The first entry E[1] of E corresponds to U [1] and the second entry E[2] is U [2]. The residue classes of the rows of the third entry E [3] of E generate ker ϕ 1 , i.e., ker…”

Section: Rr N°8225mentioning

confidence: 99%

“…Then, z = S [2] y is a set of generators {z 1 , z 2 } of L, where y = (y 1 y 2 y 3 ) T and z = (z 1 z 2 ) T . In particular, we have:…”

Section: Stafford's Reductionmentioning

confidence: 99%

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A Constructive Study of the Module Structure of Rings of Partial Differential Operators

Quadrat

Robertz

2014

Acta Appl Math

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To cite this version:Alban Quadrat, Daniel Robertz. A constructive study of the module structure of rings of partial differential operators. [ Abstract:The purpose of this paper is to develop constructive versions of Stafford's theorems on the module structure of Weyl algebras A n (k) (i.e., the rings of partial differential operators with polynomial coefficients) over a base field k of characteristic zero. More generally, based on results of Stafford and Coutinho-Holland, we develop constructive versions of Stafford's theorems for very simple domains D. The algorithmization is based on the fact that certain inhomogeneous quadratic equations admit solutions in a very simple domain. We show how to explicitly compute a unimodular element of a finitely generated left D-module of rank at least two. This result is used to constructively decompose any finitely generated left D-module into a direct sum of a free left D-module and a left D-module of rank at most one. If the latter is torsion-free, then we explicitly show that it is isomorphic to a left ideal of D which can be generated by two elements. Then, we give an algorithm which reduces the number of generators of a finitely presented left D-module with module of relations of rank at least two to two. In particular, any finitely generated torsion left D-module can be generated by two elements and is the homomorphic image of a projective ideal whose construction is explicitly given. Moreover, a non-torsion but non-free left D-module of rank r can be generated by r + 1 elements but no fewer. These results are implemented in the Stafford package for D = A n (k) and their system-theoretical interpretations are given within a D-module approach. Finally, we prove that the above results also hold for the ring of ordinary differential operators with either formal power series or locally convergent power series coefficients and, using a result of Caro-Levcovitz, also for the ring of partial differential operators with coefficients in the field of fractions of the ring of formal power series or of the ring of locally convergent power series. Key-words:Weyl algebras, Stafford's theorems, linear systems of partial differential equations, D-modules, mathematical systems theory, constructive algebra, symbolic computation. A constructive study of the module structure of rings of partial differential operators 4

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A Coq Formalization of Finitely Presented Modules

Cohen

Mörtberg

2014

Interactive Theorem Proving

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International audienceThis paper presents a formalization of constructive module theory in the intuitionistic type theory of Coq. We build an abstraction layer on top of matrix encodings, in order to represent finitely presented modules, and obtain clean definitions with short proofs justifying that it forms an abelian category. The goal is to use it as a first step to get certified programs for computing topological invariants, like homology groups and Betti numbers

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An Axiomatic Setup for Algorithmic Homological Algebra and an Alternative Approach to Localization

Cited by 23 publications

References 22 publications

Gauge backgrounds and zero-mode counting in F-theory

Gauge backgrounds and zero-mode counting in F-theory

A Constructive Study of the Module Structure of Rings of Partial Differential Operators

A Coq Formalization of Finitely Presented Modules

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