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 = X
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 ])
2 0 0 0 1 1 1
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 = Y
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 information about that.
For more details about varieties, see
Variety.