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From Varieties to Sheaf Cohomology

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From Varieties to Sheaf Cohomology Thomas Goller September 1, 2015 Classical algebraic geometry is the study of geometry using algebraic tools. Modern algebraic geometry is the study of algebra using geometric intuition. 1 Varieties (classical) n Affine varieties are vanishing sets U = (f1,...,fr) A of polynomials fi C[x1,...,xn]. V ⇢ 2 (We write An for affine space instead of Cn because we won’t use the vector space structure; in particular, the origin is not a special point.) The algebraic functions on this affine variety U are the ring C[x1,...,xn]/ (f1,...,fr). Projective varieties are vanishing sets X = n (F1,...,Fr) P of homogenous polynomials Fi C[X0,...,Xn]. The ring of rational V ⇢ p 2 functions on X consists of homogeneous degree 0 rational functions in X0,...,Xn where the denominator cannot be in the ideal (F1,...,Fr)(weneedhomogeneousdegree0to have well-defined functions on n). Projective varieties are compact and have open covers P p by affine varieties. (These open affine ‘charts’ can be obtained by taking the subsets of X where X =0). i 6 Example. P1, the space of lines through the origin in C2,canbedescribedasthesetofpairs (a, b) =(0, 0) modulo the action of scaling by nonzero constants, so we often write points 6 of P1 as (a : b) to emphasize that the ratio of a to b is the critical feature. As a projective 1 variety, P is the vanishing set of the empty set of homogeneous polynomials in C[X0,X1]. b b The subset where X0 =0,namelywherea = 0, can be parametrized by (1 : )for C 6 6 a a 2 and is therefore the affine variety 1 with function ring [ X1 ]. Similarly, the subset where A C X0 1 X0 1 X1 =0isacopyofA with function ring C[ ]. Thus P is covered by two copies of the 6 X1 affine variety A1. (The function rings show how these two copies of A1 are glued together to produce P1.We’llcomebacktothis.) GAGA (Serre, 1956) shows that calculations on a projective variety are equivalent to calculations done on the underlying holomorphic manifold. Classical algebraic geometry is closely tied to complex analysis. 2 Toward schemes Schemes generalize varieties. To see how, let’s think about affine varieties more algebraically. The points of the affine variety U = (f ,...,f )arethemaximumspectrumofthering V 1 r of functions A = C[x1,...,xn]/ (f1,...,fr). These maximal ideals are all of the form m =(x a ,...,x a ) (f ,...,f ), which we were thinking of as a point ~a =(a ,...,a ) 1− 1 n− n ⊃ 1 p r 1 n at which all the f vanish. Think of evaluating a function f A at a point m (which yields a i 2 complex number) as taking the image under the map A A/m C (modding out by m sets ! ' xi = ai,sothisreallyisjustevaluationat~a ). If we replace the words ‘maximal spectrum’ by ‘prime spectrum’ and ‘A = C[x1,...,xn]/ (f1,...,fr)’ by ‘A is any commutative ring with identity’ in the above discussion, then we have arrived at affine schemes. One weird aspect p is that functions f A evaluated at a prime ideal p of A yield values in A/p,whichwillin general not be a field.2 An affine scheme is essentially just a ring A, viewed as functions on its prime spectrum. Morphisms of affine schemes are induced by ring morphisms B A, which by taking preimages of prime ideals (the preimage of a prime ideal is prime!) yield! set maps Spec A Spec B (note that the order of A an B changes, i.e. Spec is a contravariant functor). ! 3 Schemes (modern) General schemes are obtained by gluing affine schemes by isomorphisms along open subsets of those affine schemes. In other words, a scheme is obtained by gluing rings together.Our topology is the Zariski topology of Spec A,whereclosedsetsarealloftheform (I), namely the set of all primes containing a particular ideal I.ThesimplestopensubsetsofSpecV A are the complements Spec A of the closed sets (f)andthegluingoftenhappensalongthese f V (Spec Af consists of all prime ideals of A not containing f). Example. We can glue C[x]andC[y]alongtheopensetsobtainedbyinvertingx and y 1 via the isomorphism C[x]x C[y]y, x 1/y.ThisyieldstheprojectivelineP .(Writing ! 7! x = X1 and y = X0 reveals the connection to the previous example.) X0 X1 From now on, we will view varieties as schemes. We thereby gain a richer topology (more points!) and a general framework that gives us access to more algebraic tools. For instance, consider an affine variety Spec A.ThemaximalidealsofSpecA are in bijection with ring maps A C (the preimage of the zero ideal picks out a maximal ideal of A). More interestingly,! the tangent vectors of an affine scheme are in bijection with ring maps A C[x]/(x2)(thisisn’taclassicalmapofvarietiesduetothepresenceofnilpotents). ! Example. Consider the affine scheme C[x, y], whose maximal ideals (x a, y b)canbe − − viewed as (a, b) A2.RingmapsC[x, y] C[x]/(x2)aredeterminedbyx ↵x + a, 2 ! 7! y βx + b. Now, Spec C[x]/(x2)isasinglepoint(x), which under the given ring map gets 7! sent to the maximal ideal (x a, y b), which can be thought of as the point (a, b) A2. − − @ @ 2 The choice of ↵ and β can be thought of as giving the tangent vector ↵ @x + β @y. Note that the ring map carries more information than the induced topological map on prime ideals. One can also observe generic behavior of a map between varieties by looking at what the map does at the generic point (the zero ideal), which can be thought of as a fuzzy point covering the entire space. Example. Consider the map Spec C[x] Spec C[y]definedbyC[y] C[x], y x2 (this is ! ! 7! analogous to the map C C, z z2). The preimage of a maximal ideal (x a)is(y a2), ! 7! − − 2 so both (x a)and(x+a)inSpecC[x]maptothesamepoint(y a2)ofSpecC[y]. A fancy way of seeing− that this map is generically two-to-one (which we− call degree 2) is to localize at the generic point: since C[x](0) = C(x), we obtain Spec C[x](0) Spec C[y](0) defined by ! C(y) , C(x), y x2,whichisafieldextensionofdegree2. ! 7! 4 Coherent sheaves When we glue together affine varieties Spec A to get a scheme X, we are also gluing together the rings of functions A to get the structure sheaf of rings X , which stores the data of all functions on all open subsets of X.Inparticular,iftheopensubsetisO U =SpecA,then (U)=A,buttoseewhatthefunctionsonU U are we have to look carefully at how OX 1 [ 2 U1 and U2 are glued together. 1 Example. Recall that P is covered by C[x]andC[y]withgluingisomorphismC[y]y 1 ! C[x]x, y .Toseewhattheglobalsectionsof 1 are (i.e. the functions on the open set 7! x OP U = P1), we try to choose sections on an open affine cover that are compatible under the gluing isomorphisms. In our case, we need an element f C[x]andanelementg C[y]that are identified under the map y 1 .Butf and g are polynomials,2 so the only2 compatible 7! x choice is the same constant f = g C and hence the global sections of 1 are just C. 2 OP The structure sheaf X is a sheaf of rings. In commutative algebra, given a ring A,a natural thing to do is toO study A-modules. In algebraic geometry, the natural extension is to study X -modules, which are simply an A-module M for each affine open Spec A together with gluingO isomorphisms that are compatible with the gluing of the rings A. When all the modules M are finitely generated, we call the resulting sheaf of modules a coherent sheaf. Coherent sheaves generalize the notion of vector bundles. When we build the sheaf of modules by gluing free modules, we get a locally free sheaf, which algebraic geometers often call a vector bundle. This is because the same gluing maps can be used to define an algebraic vector bundle over X (instead of gluing free modules of rank r, you could glue copies of Cr), and the sheaf of sections of this vector bundle is exactly the locally free sheaf you started with. Example. The structure sheaf is a locally free sheaf of rank 1 and corresponds to the OX trivial line bundle X C. ⇥ Algebraic geometers usually prefer to work with locally free modules since modules are so natural from the point of view of commutative algebra. Many operations on modules work for coherent sheaves as well (e.g. direct sum, tensor product, push forward, pullback). And we have a powerful tool known as sheaf cohomology. 5 Sheaf cohomology Given a coherent sheaf on a projective variety X (we want X to be compact to ensure that our cohomology vectorF spaces are finite dimensional), we often have a good understanding of what looks like locally (especially if is locally free!), but identifying the global sections F F 3 of is more difficult (even P1 required some work!). The global sections functor, which we’llF call H0,isleftexact,whichmeansthatgivenashortexactsequenceofcoherentsheavesO 0 0 !F1 !F2 !F3 ! 0 0 0 we get an exact sequence of C-vector spaces 0 H ( 1) H ( 2) H ( 3). As we often do when we have a left exact functor, we can! take theF derived! functorF ! of HF0,whichwecall sheaf cohomology and denote by Hi. The point is that our short exact sequence of coherent sheaves then yields a long exact sequence 0 H0( ) H0( ) H0( ) H1( ) H1( ) H1( ) H2( ) ! F1 ! F2 ! F3 ! F1 ! F2 ! F3 ! F1 !··· which often helps us compute H0.
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