Mohamed Barakat

homalg – A META-PACKAGE FOR HOMOLOGICAL ALGEBRA

The central notion of this work is that of a functor between categories of finitely presented modules over so-called computable rings, i.e. rings R where one can algorithmically solve inhomogeneous linear equations with coefficients in R. The paper describes a way allowing one to realize such functors, e.g. HomR, ⊗R, , , as a mathematical object in a computer algebra system. Once this is achieved, one can compose and derive functors and even iterate this process without the need of any specific knowledge of these functors. These ideas are realized in the ring independent package homalg. It is designed to extend any computer algebra software implementing the arithmetics of a computable ring R, as soon as the latter contains algorithms to solve inhomogeneous linear equations with coefficients in R. Beside explaining how this suffices, the paper describes the nature of the extensions provided by homalg.

AN AXIOMATIC SETUP FOR ALGORITHMIC HOMOLOGICAL ALGEBRA AND AN ALTERNATIVE APPROACH TO LOCALIZATION

In this paper we develop an axiomatic setup for algorithmic homological algebra of Abelian categories. This is done by exhibiting all existential quantifiers entering the definition of an Abelian category, which for the sake of computability need to be turned into constructive ones. We do this explicitly for the often-studied example Abelian category of finitely presented modules over a so-called computable ring R, i.e. a ring with an explicit algorithm to solve one-sided (in)homogeneous linear systems over R. For a finitely generated maximal ideal 𝔪 in a commutative ring R, we show how solving (in)homogeneous linear systems over R𝔪 can be reduced to solving associated systems over R. Hence, the computability of R implies that of R𝔪. As a corollary, we obtain the computability of the category of finitely presented R𝔪-modules as an Abelian category, without the need of a Mora-like algorithm. The reduction also yields, as a byproduct, a complexity estimation for the ideal membership problem over local polynomial rings. Finally, in the case of localized polynomial rings, we demonstrate the computational advantage of our homologically motivated alternative approach in comparison to an existing implementation of Moras algorithm.

Characterizing Serre Quotients with no Section Functor and Applications to Coherent Sheaves

We prove an analogon of the the fundamental homomorphism theorem for certain classes of exact and essentially surjective functors of Abelian categories

On the Ext-computability of Serre quotient categories

\mathcal{Q}:\mathcal{A} \to \mathcal{B}

ON MONADS OF EXACT REFLECTIVE LOCALIZATIONS OF ABELIAN CATEGORIES

. It states that

Jets. A Maple-Package for Formal Differential Geometry

\mathcal{Q}

A Constructive Approach to the Module of Twisted Global Sections on Relative Projective Spaces

is up to equivalence the Serre quotient

conley: Computing connection matrices in Maple

\mathcal{A} \to \mathcal{A} / \ker \mathcal{Q}

LocalizeRingForHomalg: localize commutative rings at maximal ideals

, even in cases when the latter does not admit a section functor. For several classes of schemes X, including projective and toric varieties, this characterization applies to the sheafification functor from a certain category

The existence of Cartan connections and geometrizable principle bundles

\mathcal{A}