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] |
i2 : X = Spec R |
i3 : ring X |
i4 : dim X |
The variety X is a 3-dimensional affine space.
We may form products.
i5 : X * X |
i6 : dim oo |
We may use Proj to create a projective scheme (or algebraic variety) with a specified homogeneous coordinate ring.
i7 : Y = Proj R |
i8 : ring Y |
i9 : dim Y |
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 information about that.
For more details about varieties, see Variety.