algebraic varieties

We may use Spec to create an affine scheme (or algebraic variety) with a specified coordinate ring and ring to recover the ring.

i1 : R = ZZ/2[x,y,z]



o1 = R



o1 : PolynomialRing

i2 : X = Spec R



o2 = Spec R



o2 : AffineVariety

i3 : ring X



o3 = R



o3 : PolynomialRing

i4 : dim X



o4 = 3

The variety X is a 3-dimensional affine space.

We may form products.

i5 : X * X



          ZZ

o5 = Spec(--[x,y,z,x,y,z,MonomialOrder => GRevLex])

           2



o5 : AffineVariety

i6 : dim oo



o6 = 6

We may use Proj to create a projective scheme (or algebraic variety) with a specified homogeneous coordinate ring.

i7 : Y = Proj R



o7 = Proj R



o7 : ProjectiveVariety

i8 : ring Y



o8 = R



o8 : PolynomialRing

i9 : dim Y



o9 = 2

The most important reason for introducing the notion of algebraic variety into a computer algebra system is to support the notion of coherent sheaf. See coherent sheaves for further information.