A (complex) singularity is, by definition, nothing but a complex analytic germ together with its analytic local ring , where is the convergent power series ring in . For an arbitrary field let for some ideal in the formal power series ring . We call Spec or just a singularity ( denotes the maximal ideal of the local ring ) and write for the convergent and for the formal power series ring if the statements hold for both.
If is an ideal with then the singularity of at is, using the above notation, . However, we may also consider the local ring with the localisation of at , as the singularity of at 0. Geometrically, for , the difference is the following: describes the variety in an arbitrary small neighbourhood of 0 in the Euclidean topology while describes in an arbitrary small neighbourhood of 0 in the (much coarser) Zariski topology.
At the moment, we can compute efficiently only in as we shall explain below. In many cases of interest, we are happy since invariants of at 0 can be computed in as well as in . There are, however, others (such as factorisation), which are completely different in both rings.
is called non-singular or regular or smooth if is isomorphic (as local ring) to a power series ring , or if is a regular local ring.
By the implicit function theorem, or by the Jacobian criterion, this is equivalent to the fact that has a system of generators such that the Jacobian matrix of has rank in some neighbourhood of 0. is called an isolated singularity if there is a neighbourhood of 0 such that is regular everywhere.
In order to compute with singularities, we need the notion of standard basis which is a generalisation of the notion of Gröbner basis, cf. GP1,GP2.
A monomial ordering is a total order on the set of monomials satisfying
Any can be written uniquely as , with and for any non-zero term of . We set lm , the leading monomial of and lc, the leading coefficient of .
For a subset we define the leading ideal of as
So far, the general case is not different to the case of a well-ordering. However, the following definition provides something new for non-global orderings:
For a monomial ordering define the multiplicatively closed set
Note that Loc and Loc if and only if is global and Loc if and only if is local (which justifies the names).
Let be a fixed monomial ordering. In order to have a short notation, I write
Let be an ideal. A finite set is called a standard basis of if and only if , that is, for any there exists a satisfying lmlm.
If the ordering is a well-ordering, then a standard basis is called a Gröbner basis. In this case and, hence, .
Standard bases can be computed in the same way as Gröbner bases except that we need a different normal form. This was first noticed by Mo for local orderings (called tangent cone orderings by Mora) and, in general, by GP1,Getal.
Let denote the set of all finite and ordered subsets . A map
NF is called a weak normal form if, instead of (ii), only the following condition (ii') holds:
Moreover, we need (in particular for computing syzygies) (weak) normal forms with standard representation: if , we can write
Indeed, if and consist of polynomials, we can compute, in finitely many steps, weak normal forms with standard representation such that and NF are polynomials and, hence, compute polynomial standard bases which enjoy most of the properties of Gröbner bases.
Once we have a weak normal form with standard representation, the general standard basis algorithm may be formalised as follows:
STANDARDBASIS(G,NF) [arbitrary monomial ordering]
Input: a finite and ordered set of polynomials, NF a weak normal form with standard representation.
Output: a finite set of polynomials which is a standard basis of .
- return ;
Here spoly denotes the -polynomial of and where lm lm.
The algorithm terminates by Dickson's lemma or by the noetherian property of the polynomial ring (and since NF terminates). It is correct by Buchberger's criterion, which generalises to non-well-orderings.
If we use Buchberger's normal form below, in the case of a well-ordering, STANDARDBASIS ist just Buchberger's algorithm:
Input: a finite ordered set of polynomials, a polynomial.
Output: a normal form of with respect to with standard representation.
- while and exist so that lmlm
choose any such ;
- return ;
For an algorithm to compute a weak normal form in the case of an arbitrary ordering, we refer to GP1.
To illustrate the difference between local and global orderings, we compute the dimension of a variety at a point and the (global) dimension of the variety.
The dimension of the singularity , or the dimension of at 0, is, by definition, the Krull dimension of the analytic local ring , which is the same as the Krull dimension of the algebraic local ring in case is generated by polynomials, which follows easily from the theory of dimensions by Hilbert-Samuel series.
Using this fact, we can compute by computing a standard basis of the ideal generated in Loc with respect to any local monomial ordering on . The dimension is equal to the dimension of the corresponding monomial ideal (which is a combinatorial problem).
For example, the dimension of the affine variety is 2 but the dimension of the singularity (that is, the dimension of at the point 0) is 1:
Using SINGULAR we compute first the global dimension with the degree reverse lexicographical ordering denoted by dp and then the local dimension at 0 using the negative degree reverse lexicographical ordering denoted by ds. Note that in the local ring (represented by the ordering ds) is a unit.
ring R = 0,(x,y,z),dp; //global ring ideal i = yx-y,zx-z; ideal si = groebner(i); si; ==> si=xz-z, //leading ideal of i is <xz,xy> ==> si=xy-y dim(si); ==> 2 //global dimension = dim R/<xz,xy> ring r = 0,(x,y,z),ds; //local ring ideal i = yx-y,zx-z; ideal si = groebner(i); si; ==> si=y //leading ideal of i is <y,z> ==> si=z dim(si); ==> 1 //local dimension = dim r/<y,z>