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Space with functions

Invite you to consult the document content, "Space with functions" below. Contents of the document referred to the content you: Affine Varieties, Algebraic Varieties, Nonsingular Varieties, Sheaves, Divisors,. Hopefully document content to meet the needs of learning, work effectively. | 1 Affine Varieties We will begin following Kempf s Algebraic Varieties and eventually will do things more like in Hartshorne. We will also use various sources for commutative algebra. What is algebraic geometry Classically it is the study of the zero sets of polynomials. We will now fix some notation. k will be some fixed algebraically closed field any ring is commutative with identity ring homomorphisms preserve identity and a k-algebra is a ring R which contains k i.e. we have a ring homomorphism L k R . P c R an ideal is prime iff R P is an integral domain. Algebraic Sets We define affine n-space An kn ữi . an ữị 2 kg. Any f f x1 . xn 2 k x1 . xn defines a function f An k a1 . an 1 2 3 f a1 . an . Exercise If f g 2 k x1 . define the same function then f g as polynomials. Definition 1.1 Algebraic Sets . Let S c k x1 . xn be any subset. Then V S a 2 An f a 0 for all f 2 Sg. A subset of An is called algebraic if it is of this form. e.g. a point a1 . an g V x1 a1 . xn an . Exercises 1. I S is the ideal generated by S. Then V S V I . 2. I c J V J c V I . 3. V UaIa V p Ia nV Ia . 4. V I J V I J V I U V J . Definition 1.2 Zariski Topology . We can define a topology on An by defining the closed subsets to be the algebraic subsets. U c An is open iff An U V S for some S c k x1 . . xn . Exercises 3 and 4 imply that this is a topology. The closed subsets of A1 are the finite subsets and A1 itself. Definition 1.3 Ideal of a Subset . If W c An is any subset then I W f 2 k x1 . . xn f a 0 for all a 2 Wg Facts Exercises 1. V c W I W c I V 2. I 0 1 k x1 . xn 3. I An 0 . 1 Definition 1.4 Affine Coordinate Ring . W c An is algebraic. Then A W k W k xi . xn l W We can think of this as the ring of all polynomial functions f W k. Definition 1.5 Radical Ideal . Let R be a ring and I c R be an ideal then the radical of I is the ideal pI f 2 R f 2 I for some i 2 Ng We call I a radical ideal if I pĩ. Exercise If I is an ideal then a I is a radical ideal. Proposition 1.1. W c An any .