D-modules and Cohomology of Varieties

Abstract

In this chapter we introduce the reader to some ideas from the world of differential operators. We show how to use these concepts in conjunction with Macaulay 2 to obtain new information about polynomials and their algebraic varieties.Gröbner bases over polynomial rings have been used for many years in computational algebra, and the other chapters in this book bear witness to this fact. In the mid-eighties some important steps were made in the theory of Gröbner bases in non-commutative rings, notably in rings of differential operators. This chapter is about some of the applications of this theory to problems in commutative algebra and algebraic geometry.Our interest in rings of differential operators and D-modules stems from the fact that some very interesting objects in algebraic geometry and commutative algebra have a finite module structure over an appropriate ring of differential operators. The prime example is the ring of regular functions on the complement of an affine hypersurface. A more general object is theČech complex associated to a set of polynomials, and its cohomology, the local cohomology modules of the variety defined by the vanishing of the polynomials. More advanced topics are restriction functors and de Rham cohomology.With these goals in mind, we shall study applications of Gröbner bases theory in the simplest ring of differential operators, the Weyl algebra, and develop algorithms that compute various invariants associated to a polynomial f . These include the Bernstein-Sato polynomial b f (s), the set of differential operators J(f s ) which annihilate the germ of the function f s (where s is a new variable), and the ring of regular functions on the complement of the variety of f .For a family f 1 , . . . , f r of polynomials we study the associatedČech complex as a complex in the category of modules over the Weyl algebra. The algorithms are illustrated with examples. We also give an indication what other invariants associated to polynomials or varieties are known to be computable at this point and list some open problems in the area.

Algorithmic semi-algebraic geometry and topology—recent progress and open problems

“…• We should also mention at least one other approach towards computation of Betti numbers (of complex varieties) that we do not describe in detail in this survey. Using the theory of local cohomology and D-modules, Oaku and Takayama [55] and Walther [66,67], have given explicit algorithms for computing a sub-complex of the algebraic de Rham complex of the complements of complex affine varieties (quasi-isomorphic to the full complex but of much smaller size) from which the Betti numbers of such varieties as well as their complements can be computed easily using linear algebra. For readers familiar with de Rham cohomology theory for differentiable manifolds, the algebraic de Rham complex is an algebraic analogue of the usual de Rham complex consisting of vector spaces of differential forms.…”

“…The computational complexities of these procedures are not analyzed very precisely in the papers cited above. However, these algorithms use Gröbner basis computations over non-commutative rings (of differential operators), and as such are unlikely to have complexity better than double exponential (see [67,Section 2.4]). Also, these techniques are applicable only over algebraically closed fields, and not immediately useful in the semi-algebraic context which is our main interest in this paper, and as such we do not discuss these algorithms any further.…”

Richardson varieties, projected Richardson varieties and positroid varieties

“…Proposition 3.21 is useful because there are standard algorithms for computing the cohomology and mixed Hodge structure on open sets of affine space, see [OakuTakayama99,Walther02], and the Macaulay 2 command deRham in the Dmodules package. It also provides a concise proof of the following results:…”