Algebraic K-Theory and its Applications

Bass, Hyman; Kuku, Aderemi O.; Pedrini, Claudio

doi:10.1142/9789814528474

Cited by 39 publications

(96 citation statements)

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“…The first (resp., last) point above can be traced back to Serre's splitting-off theorem ( [39]) (resp., Bass' cancellation theorem ( [5])), not requiring, however, any projectivity condition on M . These results are well-known in commutative algebra and they have been studied in [17,27] (see also the references therein) within a constructive commutative algebra approach.…”

Section: If Rank D (M )mentioning

confidence: 99%

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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Section: If Rank D (M )mentioning

confidence: 99%

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

Quadrat

Robertz

2014

Acta Appl Math

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“…Then A is a commutative algebra and we have a right H-coaction ρ on A such that [1] b [1] , for all a, b ∈ A. Here we use the Sweedler-Heyneman notation for the coaction ρ: ρ(a) = a [0] ⊗ a [1] , with summation implicitly understood. For the comultiplication on H, we use the notation ∆(h) = h (1) ⊗ h (2) .…”

Section: Relative Hopf Modulesmentioning

confidence: 99%

“…A relative Hopf module M is a k-module, together with a right A-action and a right C-coaction ρ M such that [1] a [1] , for all a ∈ A and m ∈ M . The category of relative Hopf modules and A-linear H-colinear maps will be denoted by…”

Section: Relative Hopf Modulesmentioning

confidence: 99%

“…In this paper we will discuss these two problems in a more general situation: we will assume that A is a commutative H-comodule algebra, with H an arbitrary commutative bialgebra over a commutative ring k. We then ask for an algebraic interpretation of the first Harrison cohomology group H 1 Harr (H, A, G m ) (with notation as in [6]). If H is finitely generated and projective, then this Harrison cohomology group is isomorphic to a Sweedler cohomology group Z 1 Harr (H, A, G m ), and if H = ZG with G a finite group, then it reduces to the cohomology group H 1 (G, G m (A)).…”

Section: Introductionmentioning

confidence: 99%

“…If H is finitely generated and projective, then this Harrison cohomology group is isomorphic to a Sweedler cohomology group Z 1 Harr (H, A, G m ), and if H = ZG with G a finite group, then it reduces to the cohomology group H 1 (G, G m (A)).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Relative Picard Group of a Comodule Algebra and Harrison Cohomology

Caenepeel

Guédénon

2005

Proceedings of the Edinburgh Mathematical Society

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Let A be a commutative comodule algebra over a commutative bialgebra H. The group of invertible relative Hopf modules maps to the Picard group of A, and the kernel is described as a quotient group of the group of invertible group-like elements of the coring A ⊗ H, or as a Harrison cohomology group. Our methods are based on elementary K-theory. The Hilbert 90 theorem follows as a corollary. The part of the Picard group of the coinvariants that becomes trivial after base extension embeds in the Harrison cohomology group, and the image is contained in a well-defined subgroup E. It equals E if H is a cosemisimple Hopf algebra over a field.

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Multiplicative K-theory and K-theory of Functors

Felisatti

2008

MedJM

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Algebraic K-Theory and its Applications

Cited by 39 publications

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

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

The Relative Picard Group of a Comodule Algebra and Harrison Cohomology

Multiplicative K-theory and K-theory of Functors

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