Morphism of affine algebraic varieties pdf

Explictly, the isomorphism in 22 is induced by the ring homomorphism. Introduction let k he any algebraically closed field and v an algebraic variety defined over k. In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. Combine the two corollaries above, we see that a locally closed subspace intersection of open and. Introduction to algebraic geometry, class 4 contents. Extension of morphism of quasiprojective varieties. A homomorphism of affine algebraic groups is a morphism that is also a group homomorphism, an isomorphism is a homomorphism that has an inverse, and. An open subspace of an algebraic variety is an algebraic variety.

The category of affine algebraic varieties 61 explicit description of morphisms of affine varieties 63. The fibers of a finite morphism of affine varieties are. The fibers of a finite morphism of affine varieties are all finite. Need to check that an open subset of an a ne variety is covered by nitely many a ne varieties. Preimage of smooth curves under morphism of smooth varieties. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Etale endomorphisms of algebraic varieties 349 over the part a y i. But avoid asking for help, clarification, or responding to other answers. A regular map whose inverse is also regular is called biregular, and they are isomorphisms in the category of algebraic varieties. Algebraic varieties, affine or projective, were initially considered over the field of real or complex numbers. Zariski topology, morphisms of affine varieties and affine algebras. A quasia ne variety is an open subset of an a ne variety. A morphism from an algebraic variety to the affine line is also called a regular function. Algebraic sets, affine varieties, and the zariski topology list of topics.

If the address matches an existing account you will receive an email with instructions to reset your password. Injective morphisms of affine varieties mingchang kang communicated by louis j. Then well be ready to define affine varieties, varieties, and morphisms. Algebraic geometry of matrices ii lekheng lim university of chicago july 3, 20. An a ne algebraic variety is an irreducible algebraic set in an, with its induced topology. In this note an elementary proof that every injective morphism from an affine variety into itself is necessarily surjective is given. An a ne algebraic subset of kn is the common zero locus of a collection of polynomials in kx 1x n.