As shown in the examples in the introduction to this chapter, schemes are defined inside some ambient space, either affine or projective space, by a collection of polynomials from the coordinate ring associated with that space. Schemes may also be defined inside other schemes using polynomials from the coordinate ring of the bigger scheme or polynomials from the ambient space.
There is very little difference between creation methods for affine and projective schemes. Of course, in the projective case, the defining polynomials are checked for homogeneity or if an ideal is used, a check is made that its basis contains only homogeneous elements. Otherwise, the only check made at the time of creation is that the polynomials used to define the scheme really do lie in, or are coerced automatically into, the coordinate ring of the chosen ambient space.
Create the scheme (or cluster) inside the scheme X defined by the vanishing of the polynomial f, or the sequence of polynomials F, or the ideal of polynomials I, or the ideal in the denominator of the quotient ring Q = R/I. In each case, the polynomials must be elements of the coordinate ring of A or automatically coercible into it.
> A<x,y,z> := AffineSpace(Rationals(),3); > X := Scheme(A,x-y); > X; Scheme over Rational Field defined by x - y > Y := Scheme(X,[x^2 - z^3,y^3 - z^4]); > Y; Scheme over Rational Field defined by x^2 - z^3 y^3 - z^4 x - y > Ambient(Y) eq A; trueNote that since Y was created as a subscheme of X it inherits the equations of X. The ambient space of Y is still considered to be A.
The scheme ( Spec)(R) associated to the affine algebra R. A new affine space Spec(Generic(R)) will be created as the ambient space of this scheme.
The scheme ( Proj)(R) associated to the affine algebra R which will be interpreted with its grading (which will be the standard grading by degree if no other has been assigned). A new projective space Proj(Generic(R)) will be created as the ambient space of this scheme.
The subscheme of X defined, for an affine scheme X by the trivial polynomial 1, or by maximal ideal (x_1, ..., x_n) for a projective scheme X.
The intersection of schemes X and Y in their common ambient space. This simply concatenates their defining equations without testing for emptiness.
The union of schemes X and Y in their common ambient space. This is formed by creating the intersection of their defining ideals which is done using a Gröbner basis computation.
Returns the scheme that is obtained by taking the closure of the result of removing {( X meet Y)} from X, counting multiplicity. The ideal of the result will be the colon ideal of the ideal of X and the ideal of Y.
> A2<x,y>:=AffineSpace(Rationals(),2); > C:=Scheme(A2,(x*y)); //union of the x- and y-axis > X2:=Scheme(A2,x^2); //y-axis with double multiplicity > Difference(X2,C); //y-axis with mult. 1. Scheme over Rational Field defined by x > O:=Scheme(A2,[x,y]); > Difference(C,O); Scheme over Rational Field defined by x*yRemoving "ambient" spaces is tricky: Everything is removed.
> Difference(C,A2); Scheme over Rational Field defined by 1 > A3<x,y,z>:= AffineSpace(Rationals(),3); > C:=Scheme(A3,Ideal([x,z])*Ideal([y,z])); //again, union of x- and y-axis > Z:=Scheme(A3,[z]); //the x,y plane > Difference(C,Z); Scheme over Rational Field defined by x, y, zAs you see, the Z-plane is removed with multiplicities: all that's left is the origin, which has multiplicity 2 in C and only multiplicity 1 in Z.
Assign the strings in the sequence N to the ambient coordinate functions of the scheme X.
The ith coordinate function of the ambient space of the scheme X. The dot notation X.i may also be used.[Next][Prev] [Right] [Left] [Up] [Index] [Root]