# 11/06/2020: Algebraic Geometry (Robin Hartshorne) /home/zack/Dropbox/Library/Robin Hartshorne/Algebraic Geometry (607)/Algebraic Geometry - Robin Hartshorne.pdf 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) - IV oY IC ¥ hie aeal ar 4 gane 3 Eg \. 3PNY § ¥ He Sy mESE ody ee 41S CIA Of Aa EONCrai DY §, IRCn Li 4\) = (Robin Hartshorne 19) - 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) - A SEY ME 1 MEERANC EY INE F, ¥ J NE FA COAT; AT £3 se ¥ ey I Yok TAside SA seadyiva SR I : fier cerry Nod rifoad FE there TET\) ES NER e YX oo ‘ W hg Nt TE§ 8 { ee — A nd2X SEY MEERANC INE F ¥ J NE FA AT £3 se ¥ ey I TAside SA seadyiva SR I fier cerry Nod FE TET\) ES NER YX ‘ W Nt 8 { — Definition\. AUSUUSTD FO A ES GRP ONG Lr her BL LIL C\$ ist § a Su SE L008 MM Lr 1 PY — FTO in3 yf : x7 ¥ && 84&8 3 CY > ¥ { fR—— S48 y ¥ && 84&8 3 CY > ¥ SU Cf oii { 4 : mn fR—— S48 Loy 8Fd F y (Robin Hartshorne 19) - 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)