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 $\mathfrak{m}$ in a commutative ring $R$ we show how solving (in)homogeneous linear systems over $R_{\mathfrak{m}}$ can be reduced to solving associated systems over $R$. Hence, the computability of $R$ implies that of $R_{\mathfrak{m}}$. As a corollary we obtain the computability of the category of finitely presented $R_{\mathfrak{m}}$-modules as an Abelian category, without the need of a Mora-like algorithm. The reduction also yields, as a by-product, 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 Mora's algorithm.Comment: Fixed a typo in the proof of Lemma 4.3 spotted by Sebastian Posu

Constructive Algebraic Topology

The classical ``computation'' methods in Algebraic Topology most often work by means of highly infinite objects and in fact +are_not+ constructive. Typical examples are shown to describe the nature of the problem. The Rubio-Sergeraert solution for Constructive Algebraic Topology is recalled. This is not only a theoretical solution: the concrete computer program +Kenzo+ has been written down which precisely follows this method. This program has been used in various cases, opening new research subjects and producing in several cases significant results unreachable by hand. In particular the Kenzo program can compute the first homotopy groups of a simply connected +arbitrary+ simplicial set.Comment: 24 pages, background paper for a plenary talk at the EACA Congress of Tenerife, September 199

The phantom menace in representation theory

Our principal goal in this overview is to explain and motivate the concept of a phantom in the representation theory of a finite dimensional algebra $\Lambda$. In particular, we exhibit the key role of phantoms towards understanding how a full subcategory $\cal A$ of the category $\Lambda\text{-mod}$ of all finitely generated left $\Lambda$-modules is embedded into $\Lambda\text{-mod}$, in terms of maps leaving or entering $\cal A$. Contents: 1. Introduction and prerequisites; 2. Contravariant finiteness and first examples; 3. Homological importance of contravariant finiteness and a model application of the theory; 4. Phantoms. Definitions, existence, and basic properties; 5. An application: Phantoms over string algebras

Resolution of Stringy Singularities by Non-commutative Algebras

In this paper we propose a unified approach to (topological) string theory on certain singular spaces in their large volume limit. The approach exploits the non-commutative structure of D-branes, so the space is described by an algebraic geometry of non-commutative rings. The paper is devoted to the study of examples of these algebras. In our study there is an auxiliary commutative algebraic geometry of the center of the (local) algebras which plays an important role as the target space geometry where closed strings propagate. The singularities that are resolved will be the singularities of this auxiliary geometry. The singularities are resolved by the non-commutative algebra if the local non-commutative rings are regular. This definition guarantees that D-branes have a well defined K-theory class. Homological functors also play an important role. They describe the intersection theory of D-branes and lead to a formal definition of local quivers at singularities, which can be computed explicitly for many types of singularities. These results can be interpreted in terms of the derived category of coherent sheaves over the non-commutative rings, giving a non-commutative version of recent work by M. Douglas. We also describe global features like the Betti numbers of compact singular Calabi-Yau threefolds via global holomorphic sections of cyclic homology classes.Comment: 36 pages, Latex, 5 figures. v2:Reference adde

Finitary Topos for Locally Finite, Causal and Quantal Vacuum Einstein Gravity

Previous work on applications of Abstract Differential Geometry (ADG) to discrete Lorentzian quantum gravity is brought to its categorical climax by organizing the curved finitary spacetime sheaves of quantum causal sets involved therein, on which a finitary (:locally finite), singularity-free, background manifold independent and geometrically prequantized version of the gravitational vacuum Einstein field equations were seen to hold, into a topos structure. This topos is seen to be a finitary instance of both an elementary and a Grothendieck topos, generalizing in a differential geometric setting, as befits ADG, Sorkin's finitary substitutes of continuous spacetime topologies. The paper closes with a thorough discussion of four future routes we could take in order to further develop our topos-theoretic perspective on ADG-gravity along certain categorical trends in current quantum gravity research.Comment: 49 pages, latest updated version (errata corrected, references polished) Submitted to the International Journal of Theoretical Physic

Macaulay inverse systems revisited

Since its original publication in 1916 under the title "The Algebraic Theory of Modular Systems", the book by F. S. Macaulay has attracted a lot of scientists with a view towards pure mathematics (D. Eisenbud,...) or applications to control theory (U. Oberst,...).However, a carefull examination of the quotations clearly shows that people have hardly been looking at the last chapter dealing with the so-called "inverse systems", unless in very particular situations. The purpose of this paper is to provide for the first time the full explanation of this chapter within the framework of the formal theory of systems of partial differential equations (Spencer operator on sections, involution,...) and its algebraic counterpart now called "algebraic analysis" (commutative and homological algebra, differential modules,...). Many explicit examples are fully treated and hints are given towards the way to work out computer algebra packages.Comment: From a lecture at the International Conference : Application of Computer Algebra (ACA 2008) july 2008, RISC, LINZ, AUSTRI