# 11/06/2020: Algebraic Geometry (Robin Hartshorne)
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Last Annotation: 11/06/2020
## Highlights
- We define affine n-space over k, denoted Aj or simply A”, to be the set of all n-tuples of elements of k\. (Robin Hartshorne 18)
- 4 = &{x,\.\.\.\.\.x,| b (Robin Hartshorne 19)
- nterpret the semen of A as functions from the affine n-space to k, by defining f\(P\) = f\(a, \.\.\.\.qa,\), where f € A and Pe A"\. (Robin Hartshorne 19)
- Z\(f\)= [PeA"f\(P\) = 0]\. (Robin Hartshorne 19)
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- A 1s a noetherian ring, (Robin Hartshorne 19)
- Z\(T\) can be expressed as the common zeros of the finite set of polynomials f,,\.\.\.\./,\. (Robin Hartshorne 19)
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- Definition\. We define the Zariski topology on A" by taking the open subsets to be the complements of the algebraic sets\. (Robin Hartshorne 19)
- Every ideal in A = k[ x] is principal, (Robin Hartshorne 19)
- he open sets are the empty set and the complements of finite subsets\. (Robin Hartshorne 19)
- Definition\. A nonempty subset Y of a topological space X is irreducible if it cannot be expressed as the union Y = Y, u Y, of two proper subsets, each one of which is closed in Y\. (Robin Hartshorne 20)
- k is algebraically closed, hence infinite\)\. (Robin Hartshorne 20)
- ny nonempty open subset of an irreducible space 1s irreducible and dense\. (Robin Hartshorne 20)
- Definition\. An affine algebraic variety \(or simply affine variety\) is an irreducible closed subset of A” \(with the induced topology\)\. An open subset of an affine variety 1s a quasi-affine variety\. (Robin Hartshorne 20)
- So for any subset Y = A”, let us define the ideal of Y in A by I\(Y\)={feA|f\(P\) =0forall Pe Y]\. (Robin Hartshorne 20)
- Now we have a function Z which maps subsets of A to algebraic sets, and a function I which maps subsets of A” to ideals\. (Robin Hartshorne 20)
- va=\(feA fe aforsomer > 0]\. (Robin Hartshorne 20)
- Theoren:t 1\.3A \(Hilbert's Nullstellensatz\)\. Let k be an algebraically closed field, let a be an ideal in A = k[ xy\.\.\. \.\. \, |\. and let fe A be a polynomial which vanishes at all points of Z{a\)\. Then "ea for some integer r > 0\. (Robin Hartshorne 21)
- Corollary 1\.4\. There is «a one-to-one inclusion-reversing correspondence hetween algebraic sets in A" and radical ideals \(1\.e\., ideals which are equal to their own radical\) in A, given by Y + I\(Y\) and a +— Z\(a\)\. Furthermore, an algebraic set is irreducible if and only if its ideal is a prime ideal\. (Robin Hartshorne 21)
- If Y is irreducible, we show that I\(Y\) is prime\. (Robin Hartshorne 21)
- Let / be an irreducible polynomial in 4 = k[ x\.y]\. Then f generates a prime ideal in 4, since 4 1s a unique factorization domain\. so the zero set Y = Z\(f\) is irreducible\. We call it the affine curve defined by the equation f\(x\.yv\) = 0\. (Robin Hartshorne 21)
- More generally, if f 1s an irreducible polynomial im 4 = NET v, |\. we obtain an affine variety Y = Z\(f\), which is called a surface ifn = 3\. or a hypersurface ifn > 3\. (Robin Hartshorne 21)
- This shows that every maximal ideal of 4 is of the form N= \(Ny — dye\.\.\. \ «,\)\. for some «ay, \.\.\.\.u, ek\. (Robin Hartshorne 21)
- the curve x? + 2 + 1 = 0in Ag has no points\. (Robin Hartshorne 21)
- Definition\. If Y © A” is an affine algebraic set, we define the uffine coordinate ring ACY\) of Y\.to be A I\(Y\)\. (Robin Hartshorne 21)
- Definition\. In a ring 4\. the /icight of a prime ideal p 1s the supremum of all integers 1 such that there exists a chain p, cp, =\.\.\. cp, =p of distinct prime ideals\. We define the dimension \(or Krull dimension\) of 4 to be the supremum of the heights of all prime ideals\. (Robin Hartshorne 23)
- Now if X is a scheme of finite type over C, we define the associated complex analytic space X, as follows\. Cover X with open affine subsets Y; = Spec A;\. Each A; is an algebra of finite type over C, so we can write it as A; = Clxy,\.\.\.\.x,\)/\(f1,\.\.\.\.f,\)\. Here fi, \.\.\. f arepolynomialsin x, \.\.\. x,\. We can regard them as holomorphic functions on C”, so that their set of common zeros is a complex analytic subspace \(Y;\), & C"\. The scheme X 1s obtained by glueing the open sets Y;, so we can use the same glueing data to glue the analytic spaces \(Y;\), into an analytic space X,\. This is the associated complex analytic space of X\. (Robin Hartshorne 456)
- If X 1s a compact complex manifold, then one can show that a scheme X such that X, = X, if it exists, 1s unique\. So if such an X exists, we will simply say X is algebraic\. (Robin Hartshorne 458)