An algorithm for computing the rst homology groups of CDGAs with linear di erential

We design here a primary platform for computing the basic homological information of Commutative Di erential Graded Algebras (brie y, CDGAs), endowed with linear di erential. All the algorithms have been implemented in the framework settled by Mathematica, so that we can take advantage of the use of symbolic computation and many other powerful tools this system provides. Working in the context of CDGAs, Homological Perturbation Theory ([8], [9],[17]) supplies immediately a general algorithm computing the 1-homology of these objects at graded module level. But this process, already sketched by Lambe in [12], often bears high computational charges and actually restricts its application to the low dimensional homological calculus. Although we already know the work of Lambe on Axiom (earlier known as \Scratchpad", [11],[13]), our goal is to implement new functions on Mathematica, starting from the theoretical approach settled in [2]. This paper is devoted to design and analyze a particular Mathematica \package" for computing the basic homological information of CDGAs endowed with linear di erential. What we do here, indeed, is to compute the di erential operator and not the homology itself, actually. The full homological information may be reached by only applying Veblen's algorithm (see [19]). In this sense, it may be an Partially supported by PAICYT n. FQM-0143, Spain 2Partially supported by project DGYCI n. PB97-1025-C02-02, Spain 3 Supported by a grant from the Junta de Andaluc a Preprint submitted to IMS'99 27 August 1999 interesting e ort to implement a routine for computing the Smith normal form of a given matrix, which was not our goal here. Nevertheless, homological information is available by knowing the di erential operator. All the functions and formulas below may be implemented in di erent notebooks, so that each of them can be independently executed. The paper is organized as follows. In Section 1, we give some preliminaries of Di erential Homological Algebra. This section is just necessary for getting a basic knowledge about what a homological model of a CDGA is, and how to compute its homology groups. Sections 2 to 8 are devoted to construct the seven elemental functions which lead to de ne the di erential operator on the homological model. From this data, we may de ne the function computing that di erential operator, in the same way it is done in [1]. Last section includes the full code of the programme.