Many of the names of intrinsics in this section come from the usual terminology of algebraic geometry. A reference for them is Hartshorne's book [Har77], especially Chapter II, Section 3.
The dimension of the scheme X. If X is irreducible then the meaning of this is clear, but in general it returns only the dimension of the highest dimensional component of X. The dimension of an empty scheme will be returned as -1. If the dimension is not already known, a Gröbner basis calculation is employed. If X is projective in a multi-graded ambient then it is saturated before this calculation takes place. The computation method involves computing the Dimension of the Ideal of the scheme, and then (for projective schemes) subtracting the number of gradings.
The codimension of the scheme X in its ambient space. In fact, this number is calculated as the difference of Dimension(A) and Dimension(X) where A is the ambient space, so if X is not irreducible this number is the codimension of a highest dimensional component of X.
The degree of the scheme X.
The arithmetic genus of a scheme X. The ambient space of X must be ordinary projective space.
Returns true if and only if the scheme X has no points over any algebraic closure of its base field. This intrinsic tests if the ideal of X is trivial (in a sense to be interpreted separately according to whether X is affine or projective) and then applies the Nullstellensatz.
Returns true if and only if the scheme X is nonsingular and equidimensional over an algebraic closure of its base field. The test IsEmpty for the emptiness of the scheme is applied to the scheme defined by the vanishing of appropriately sized minors of the jacobian matrix of X.
Returns true if and only if the scheme X either has a singular point or fails to be equidimensional over an algebraic closure of its base field.
The subscheme of the scheme X defined by the vanishing of the appropriately sized minors of the jacobian matrix of X. If X is not equidimensional, its lower dimensional components will be contained in this scheme whether they are singular or not.
A sequence containing the irredundant prime components of the scheme X.
A sequence containing the irredundant primary components of the scheme X.
The subscheme of X with reduced scheme structure, followed by the map of schemes to X. This function uses a Gröbner basis to compute the radical of the defining ideal of X.
Returns true if and only if the scheme X has a unique prime component. If X is not a hypersurface, a Gröbner basis calculation is necessary and X is saturated before this occurs if it is projective.
Returns true if and only if the defining ideal of the scheme X equals its radical. If X is a hypersurface the evaluation of this intrinsic uses only derivatives so works more generally than the situations where a Gröbner basis calculation is necessary. In the latter case, X is saturated before the calculation if it is projective.
CheckEqui: BoolElt Default: false
These intrinsics currently only apply to schemes in ordinary projective space. The first two intrinsics return whether X is (locally) Cohen-Macaulay/Gorenstein, meaning that the local ring of every the scheme-theoretic point on X satisfies the property. The second two intrinsics return whether the coordinate ring of X (the polynomial coordinate ring of the projective ambient quotiented by the maximal defining ideal of X) satisfies the corresponding property. The arithmetic version implies the local version. The results are stored internally with X for future reference. Also, if X is known to be non-singular, we can immediately deduce the local version of the properties is true and this check is also performed internally.There is a further slight restriction in that X has to be equidimensional (each irreducible component having the same dimension and there being no other scheme-theoretic "associated points" beside the generic points of the irreducible components : true if X is also reduced). This is not checked by default in order to save some computation time. If the user is unsure whether X is equidimensional, the CheckEqui parameter should be set to true which forces a check.
The implementations use the minimal free polynomial resolution of the maximal defining ideal of X. The arithmetic versions are actually faster than the plain versions. IsGorenstein may be particularly heavy computationally as it has to check whether the canonical sheaf is locally free of rank 1 after the Cohen-Macaulay property has been verified.
> A<x,y,z> := AffineSpace(Rationals(),3); > X := Scheme(A,[x*y^3,x^3*z]); > Dimension(X); 2 > IsReduced(X); false > PrimaryComponents(X); [ Scheme over Rational Field defined by x, Scheme over Rational Field defined by x^3 y^3, Scheme over Rational Field defined by y^3 z ] > ReducedSubscheme(X); Scheme over Rational Field defined by x*y x*zThe reduced scheme of X is clearly the union of a line and a plane. The scheme X itself is more complicated, having another line embedded in the plane component.