Algebraic Local Cohomology with Parameters and Parametric Standard Bases for Zero-Dimensional Ideals
A computation method of algebraic local cohomology with parameters, associated with zero-dimensional ideal with parameter, is introduced. This computation method gives us in particular a decomposition of the parameter space depending on the structure of algebraic local cohomology classes. This decomposition informs us several properties of input ideals and the output of our algorithm completely describes the multiplicity structure of input ideals. An efficient algorithm for computing a parametric standard basis of a given zero-dimensional ideal, with respect to an arbitrary local term order, is also described as an application of the computation method. The algorithm can always output “reduced” standard basis of a given zero-dimensional ideal, even if the zero-dimensional ideal has parameters.
keywords:standard bases, algebraic local cohomology, multiplicity structure, systems of parametric polynomials
Msc:13D45, 32C37, 13J05, 32A27
Local cohomology was introduced by A. Grothendieck in (Grothendieck, 1967). Subsequent development to a great extent has been motivated by Grothendieck’s ideas Brodmann, M. P. and Sharp, R. Y. (1998); Lyubeznik, G. (2002). Nowadays, local cohomology is a key ingredient in algebraic geometry, commutative algebra, topology and D-modules, and is a fundamental tool for applications in several fields.
In (Tajima et al., 2009), we proposed, with Y. Nakamura, an algorithmic method to compute algebraic local cohomology classes, supported at a point, associated with a given zero-dimensional ideal. We described therein an efficient method for computing standard bases of zero-dimensional ideals, that utilize algebraic local cohomology classes. The underlying idea of the proposed method comes from the fact that algebraic local cohomology classes can completely describe the multiplicity structure of a zero-dimensional ideal via the Grothendieck local duality theorem. More recently in our result of ISSAC2014 (Nabeshima and Tajima, 2014), we considered the Jacobi ideal, with deformation parameter, of a semi-quasihomogeneous hypersurface isolated singularity. By adopting the same approach presented in Tajima et al. (2009), we constructed an algorithm for computing algebraic local cohomology classes, with parameters, that are annihilated by the Jacobi ideal. As an application, we obtained a new method to compute parametric standard bases of Jacobi ideals associated with a deformation of semi-quasihomogeneous hypersurface isolated singularities.
In this paper, we address the problem of finding an effective method to treat algebraic local cohomology classes with parameters associated with a given zero-dimensional ideal with parameters, that works in general cases.
In order to state precisely the problem, let be an open neighborhood of the origin of the -dimensional complex space with coordinates . We assume that a set of polynomials in satisfying generically are given where are parameters. Let be a set of algebraic local cohomology classes supported at the origin that are annihilated by the ideal generated by . Then is a finite-dimensional vector space if and only if the ideal generated by is zero-dimensional in the rings of formal power series. In such cases, there is a possibility that (the same meaning is that is not zero-dimensional) for some values of parameters, because of parameters. As our aim is to construct algorithms for studying the structure of and the multiplicity structure of on , it is necessary, beforehand if possible, to detect these values of parameters, that constitute constructible sets, from the parameter space for computing algebraic local cohomology classes.
In the first part of this paper, we introduce a new notion of parametric local cohomology system as an analogue of comprehensive system to deal with parametric problems. We describe a new effective method to compute parametric local cohomology systems. The resulting algorithms compute in particular a suitable decomposition of parameter space to a finite set of constructible sets according to the structure of algebraic local cohomology classes in question. The key of the algorithm for decomposing is the use of a comprehensive Gröbner bases computation in a polynomial ring with parameters. The algorithms for computing bases of , is designed as dynamic algorithm in consideration of computational efficiency. The output of our algorithm, has the abundant information of the input ideal and provides a complete description of the multiplicity structures of parametric zero-dimensional ideals.
In the second part of this paper, we describe algorithms for computing parametric standard bases as an application of parametric local cohomology systems. We show that the use of algebraic local cohomology provides an efficient algorithm for computing standard bases. Furthermore, the use of algebraic local cohomology transforms a standard basis of a dimensional ideal with respect to any given local term order into a standard basis with respect to any other ordering, without computing the standard basis, again. In general, the computation complexity of standard bases, is strongly influenced by the term order, like Gröbner bases computation. Thus, this property is useful to compute a standard basis.
Especially, our algorithm can output always “reduced” standard basis of a given zero-dimensional ideal, even if has parameters. Note that, an algorithm implemented in the computer algebra system
Singular (Decker, W. et al., 2012) that compute standard bases does not enjoy this property. Moreover, in general, comprehensive Gröbner basis (Nabeshima, 2012; Weispfenning, V., 1992) in a polynomial ring does not have this property, too.
As we mentioned above, there are several applications of algebraic local cohomology. For examples, our algorithm can be used to analyze properties of singularities and deformations of Artin algebra (Iarrobino and Emsalem, 1978; Iarrobino, 1984). It is a powerful tool to study several problems relevant to zero-dimensional ideals.
All algorithms in this paper, have been implemented in the computer algebra system
Risa/Asir (Noro and Takeshima, 1992).
This paper is organized as follows. Section 2 briefly reviews algebraic local cohomology, and gives notations and definitions used in this paper. Section 3 is the discussion of the new algorithm for algebraic local cohomology classes with parameters. This section is the main part of this paper. Section 4 gives algorithms for computing parametric standard bases for a given zero-dimensional ideals.
In this section, first we briefly review algebraic local cohomology. Second, we introduce a term order for computing algebraic local cohomology classes and algebraically constructible sets, which will be exploited several times in this paper. Throughout this paper, we use the notation as the abbreviation of variables . The set of natural number includes zero. is the field of rational numbers or the field of complex numbers .
2.1 Algebraic local cohomology
Let denote the set of algebraic local cohomology classes supported at the origin with coefficients in , defined by
where is the maximal ideal generated by .
Let be a neighborhood of the origin of . Consider the pair and its relative Čech covering. Then, any section of can be represented as an element of relative Čech cohomology. We use the notation for representing an algebraic local cohomology class in where , with . Note that the multiplication is defined as
where and .
We represent an algebraic local cohomology class as a polynomial in variables to manipulate algebraic local cohomology classes efficiently (on computer), where is the abbreviation of variables . We call this representation “polynomial representation”. For example, let be an algebraic local cohomology class where are variables. Then, the polynomial representation of , is where variables are corresponding to variables . That is, we have the following table for variables:
where . The multiplication for polynomial representation is defined as follows:
where , and . We use “ ” for polynomial representation.
After here, we adapt polynomial representation to represent an algebraic local cohomology class. We use mainly the following term order to compute algebraic local cohomology classes.
For two multi-indices and in , we denote if , or if and there exists so that for and where . In general, this term order is called a total degree lexicographic term order.
For a given algebraic local cohomology class of the form, , we call the head term and , the lower terms. We denote the head term of a cohomology class by .
2.2 Strata and specialization
We use the notation as the abbreviation of variables . (One can also regard as parameters.) Let be an algebraic closure field of . For , denotes the affine variety of , i.e., and
We use an algebraically constructible set that has a form where . We call the form a stratum. (Notation are frequently used to represent strata.)
When we treat with systems of parametric equations, then it is necessary to check consistency of their parametric consistents. In several papers (Kapur et al., 2010; Montes, 2002; Suzuki, A. and Sato, Y., 2003), algorithms for checking consistency have been already introduced. Thus, it is possible to decide whether is an empty set or not, by these algorithms where . The details are in the papers.
We define the localization of w.r.t. a stratum as follows: . Then, for every , we can define the canonical specialization homomorphism (or ). When we say that makes sense for , it has to be understood that for some with . We can regard as substituting into variables .
3 Algebraic local cohomology with Parameters
Let us assume that a set of polynomials in satisfying generically are given where is a neighborhood of the origin of . Here, we regard as parameters, and , are the main variables.
We define a set to be the set of algebraic local cohomology classes in that are annihilated by the ideal generated by , where
The ideal at is a zero-dimensional ideal if and only if is a finite-dimensional vector space. In this section we describe an algorithm for computing bases of the vector space . More precisely, we describe algorithms for computing parametric local cohomology systems (see Definition 5 in this section).
The new algorithm consists of the following three parts.
Decompose the parameter space into safe strata and danger strata.
Compute bases of the vector space on safe strata.
Compute bases of the vector space on danger strata.
3.1 An algorithm for testing dimensions of a parametric ideal
Since polynomials have parameters, there is a possibility that . As our aim is to construct algorithms for studying the system on , it is necessary, beforehand, to take away these values of parameters that constitute constructible sets from the parameter space for computing local cohomology.
Here, we describe an algorithm for decomposing into and where is zero-dimensional on and nonzero-dimensional on in a polynomial ring, for . This decomposition is possible by mainly computing a comprehensive Gröbner system of . We adopt the following definition of comprehensive Gröbner systems, because this definition is suitable to compute dimensions of ideals in the algorithm ZeroDimension. (The following definition is different from the original one).
For any and , (resp. , ) is the head term (resp. the head monomial, the head coefficient, the multidegree) of a polynomial so that and hold and where is or .
Definition 2 (Comprehensive Gröbner system (CGS)).
Let fix a term order. Let be a subset of , strata in and subsets of . A finite set of pairs is called a comprehensive Gröbner system (CGS) on for if is a Gröbner basis of the ideal in and for each and . Each is called a segment of . We simply say is a comprehensive Gröbner system for if .
After obtaining a CGS of w.r.t a total degree term order, as each segment of the CGS has the property , the dimension of is easily decided in . Since an algorithm for computing a CGS terminates, the following algorithm clearly terminates.
Algorithm 1. (ZeroDimension)
Testing dimensions of a parametric ideal on .
Input: : a set of parametric polynomials in
Output: : is a CGS on for s.t. for all , is zero-dimensional in , for each . is a CGS on for such that for all , is not zero-dimensional in , for each . .
; ; compute a CGS on of w.r.t. a total degree term order
select from ; ; compute the dimension of in
if then else end-if
In our implementation, we adopt Nabeshima’s algorithm (Nabeshima, 2012) for computing comprehensive Gröbner systems, because the algorithm is much more useful than others for computing dimensions of parametric ideals.
Using the same notation as in the above algorithm, let be an output of ZeroDimension. Then, for each , is called a safe stratum, and for each , is called a danger stratum.
Let be a polynomial with a parameter in . A CGS of w.r.t. the total degree reverse lexicographic term order s.t. , is .
If the parameter belongs to or , then is zero-dimensional. If the parameter belongs to or , then is one-dimensional. Therefore, and . That is, are safe strata, and are danger strata.
Let denote an output of ZeroDimension where and (notation is from the algorithm ZeroDimension). Since for all , is zero-dimensional in , is also zero-dimensional in . However, in general, for all , it is NOT possible for us to say that is not zero-dimensional in . For some , may be zero-dimensional in .
After decomposing the parameter space into safe strata and danger strata by the algorithm ZeroDimension, we compute bases of the vector space on safe strata and danger strata, separately. Actually, this decomposition lets us construct an efficient algorithm for computing the bases. (See section 3.3).
As the set has parameters, the structure of the vector spaces depends on the values of parameters . Here, we introduce a definition of parametric local cohomology system of .
Using the same notation as in the above, let strata in and a subset of where and . Set and . Then, a pair is called a parametric local cohomology system of on , if for all and , is a basis of the vector space , and for all and , where
After here, we represent “a parametric local cohomology system of on ” as simply “” which is the abbreviation. Similarly, we call “a parametric local cohomology system of on a stratum ” “ bases of (the vector space) on ”.
As this section 3 presents thirteen algorithms for computing bases of the vector space , Fig. 1 illustrates the relations of the all algorithms. The main algorithm is ALCohomolog.
First, we introduce in section 3.2 an algorithm for computing bases of the vector space on safe strata. Second, we describe in section 3.3 an algorithm for computing bases of the vector space on danger strata.
3.2 Computation of algebraic local cohomology with parameters on safe strata
Here, we present an algorithm for computing bases of algebraic local cohomology classes , on safe strata. This section consists of three parts. In section 3.2.1, an algorithm for computing monomial elements of bases of is introduced. In section 3.2.2 and 3.2.3, an algorithm for treating with elements, which form linear combination , of bases of , is given.
3.2.1 Monomial elements
Here, we give an algorithm for computing monomial elements of bases of . Before describing the algorithm, we define some notation.
Let be a set of polynomials in and .
The set of monomials of is denoted by , i.e., Moreover, the set of monomials of the set is denoted by , i.e., .
For all , a map is defined as changing variables into . The inverse map is defined as changing variables into . That is, for any , is in . The set is also defined as .
For instance, for , then and in where variables are corresponding to variables .
Let be an output of and . Assume that is a CGS of the monomial ideal in on , and . Then, a monic monomial which does not belong to , has the property for each . Namely, all terms which do not belong to , are members of bases of on .
Proof. Let be a monomial s.t. . By Definition 2, for all , . As is a zero-dimensional ideal and , for all , there always exists such that where . Therefore, by the multiplication, .
This proposition gives rise to the following algorithm to compute monomial elements of bases of on . Since the termination of Nabeshima’s algorithm (Nabeshima, 2012) is guaranteed, the following algorithm terminates.
Algorithm 2. (MonoSafe)
Computing monomial elements of bases of on a safe stratum .
Input: : a segment of a CGS of such that for all , is zero-dimensional in . (This is from ZeroDimension().)
Output: : a finite set of triples such that the set includes all monomial elements of bases of on , and the elements of do not belong to .
; compute a CGS of on
select from ; ;
compute monomial elements which do not belong to in
Let us remark that as is a zero-dimensional ideal on , the set consists of finitely many monomial elements. Note that monomial elements, on danger strata, will be considered in section 3.3.
We illustrate the algorithm MonoSafe with the following example.
Let be a polynomial with a parameter in . Set . Then, satisfies generically where is a neighborhood of the origin of . From Example 4, and can be inputs of the algorithm MonoSafe.
Take as an input of the algorithm MonoSafe. Then, a CGS of on , is . Set . Then, all elements of do not belong to . See in Fig. 2. can be a subset of bases of on .
Take where . As , a CGS of on is . Set and compute monomial elements which do not belong to . Then, we obtain which can be a subset of bases of on . See in Fig. 3.
3.2.2 Head terms of linear combination elements and the main algorithm
Here, we illustrate an algorithm for computing bases of . Before describing the algorithm, first we treat with elements, which form linear combination , of bases of . Especially, we discuss how to decide head terms of the linear combination elements . Second, an algorithm for computing bases of on safe strata, is given. Note that an algorithm for deciding lower terms, will be described in section 3.2.3.
Let us recall the following lemma which follows from the fact that if , so is for each . This lemma informs us candidates of head terms in .
Lemma 9 (Tajima and Nakamura (2009)).
Let denote the set of exponents of head terms in and . Let denote a subset of : If , then, for each is in , provided .
Let be a term where . We call a neighbor of for each . Then, .
Let be a set of terms in . Then, we define the neighbor of as Neighbor, i.e., Neighbor