## affine varieties

22 examples (0.01 sec)
• Let G be a unipotent group acting on an affine variety.
• If it is a smooth affine variety, then all extensions of locally free sheaves split, so the group has an alternative definition.
• Using regular functions from an affine variety to A 1, we can define regular maps from one affine variety to another.
• An affine variety over an algebraically closed field is conceptually the easiest type of variety to define, which will be done in this section.
• For that, it is enough to show: are affine varieties.
• Affine varieties of non-zero dimension are never complete.
• Two affine varieties are birationally equivalent if there two rational functions between them which are inverse one to the other in the regions where both are defined.
• An affine variety is a rational variety if it is birationally equivalent to an affine space.
• In particular, a regular map between affine varieties corresponds contravariantly in one-to-one to a ring homomorphism between the coordinate rings.
• A Zariski open subspace of an affine variety is called a quasi-affine variety.
• Affine varieties can be given a natural topology by declaring the closed sets to be precisely the affine algebraic sets.
• This definition works over any field, It allows you to glue affine varieties (along common open sets) without worrying whether the resulting object can be put into some projective space.
• In a similar way one can define an affine algebraic supergroup as a group object in the category of super algebraic affine varieties.
• An important example is for R a field and S a unital algebra over R (such as the coordinate ring of an affine variety).
• More generally, there is an antitone Galois connection between ideals in the ring and subschemes of the corresponding affine variety.
• Notice that every affine variety is quasi-projective.
• In algebraic geometry, a regular map between affine varieties is a mapping which is given by polynomials.
• On the other hand the definitions extend naturally to projective varieties (next section), as an affine variety and its projective completion have the same field of functions.
• An affine variety is, up to an equivalence of categories a special case of an affine scheme, which is precisely the spectrum of a ring.
• As affine spaces can be embedded in projective spaces, all affine varieties can be embedded in projective spaces too.