COHOMOLOGY OF COHERENT SHEAVES K H PARANJAPE AND V SRINIVAS The basic source for these lectures is Hartshornes b o ok Ha particularly Chap ter I I I Following the conventions in this b o ok we will assume that all schemes under consideration are No etherian unless sp ecied otherwise Our treatment more or less follows that in Ha except that we make use of sp ectral sequences to mo dify and simplify certain arguments Cohomology on affine schemes The goal of this section is to discuss two basic results on cohomology for ane schemes The rst is the following vanishing theorem Theorem Let X Sp ec A be a Noetherian ane scheme ie A is a commutative Noetherian ring Let F be a quasicoherent sheaf of O modules X i on X Then H X F for al l i Proof Recall that any quasicoherent sheaf F on the ane scheme X Sp ec A f is of the form M where M is the Amo dule of global sections of F Also recall f that the stalk at x X of M is the lo calization M where is the prime ideal in A corresp onding to x Sp ec A A sequence F G H of quasicoherent sheaves is exact if and only if the corresp onding sequence of Amo dules H X F H X G H X H is exact We recall why the sheaf sequence is exact precisely when its sequence of stalks is exact for each x X which is the same as saying that the lo calizations at all primes of the sequence of global sections is exact but for a sequence of Amo dules exactness at all such lo calizations is equivalent to exactness for the sequence itself The basic lemma needed to prove Theorem is the following e Lemma Let I be an injective Amodule I the associated quasicoherent i e e O module Then I is asque in particular H X I for al l i X Proof We must show that for any op en set U X Sp ec A the restriction e e I X I U I is surjective We rst see that this is true for U D f Sp ec A for any f A f Lectures at Winter School on Cohomological Metho ds in Commutative Algebra HBCSE Mumbai Nov Dec K H PARANJAPE AND V SRINIVAS Supp ose for simplicity that f is a non zerodivisor Then applying the functor Hom I to the exact sequence A f A A Af A the injectivity of I implies that multiplication by f on I is surjective This immediately implies that I I A I is surjective f f n In general even if f is a zero divisor there exist n such that Ann f nm n Ann f for all m since A is No etherian and fAnn f g is an as n n cending chain of ideals Clearly f is a non zerodivisor on AAnn f Hence n multiplication by f is surjective on I Hom AAnn f I I and so A n I I is surjective Since f I I I we have that I I and so I I is f f f f also surjective Now consider the case of an arbitrary op en subset U Sp ec A We use No e therian induction on Y supp I where supp I is the subset of primes of e A with I the overbar denotes Zariski closure If Y is a p oint then I is a skyscrap er sheaf hence it is asque In general we may as well assume e U Y or else U I which is a trivial case Then we can nd f A such that D f Sp ec A U has nonempty intersection with U Y f e If s U I choose t I such that s j t j D f we can do this b ecause D f e I I D f I is surjective Replacing s by s t j we may assume f U s j Thus if D f n J fa I j f a for some n g kerI I f e e then s U J U I Now Y supp J satises Y D f so Y Y is a prop er subscheme Thus to complete the pro of by induction it suces to observe that J is also an injective Amo dule The pro of of this using the ArtinRees lemma is left as an exercise to the reader f We can now easily complete the pro of of Theorem Let F M b e any quasicoherent O mo dule Let X M I I b e a resolution of M by injective Amo dules Then we have an asso ciated exact sequence of quasicoherent O mo dules X f e e M I I e where the sheaves I are all asque Hence there are canonical isomorphisms j j f b etween the cohomology mo dule H X M and the cohomology mo dules of the complex e e X I X I which is just the complex I I COHOMOLOGY OF COHERENT SHEAVES j This complex has H equal to M and vanishing H for j from the exact sequence The second result is a characterization of No etherian ane schemes essentially due to Serre Theorem Let X be a Noetherian scheme Then X is ane H X I for any coherent sheaf of ideals I O X Proof If X is ane the higher cohomology of any quasicoherent sheaf vanishes in particular H of any coherent ideal sheaf vanishes So assume H X I for any coherent ideal sheaf I O The goal is X to prove X Sp ec A where A X O If U Sp ec B is any ane op en X subset of X the ring homomorphism A X O U O B induces X X a morphism U Sp ec B Sp ec A These lo cally dened morphism are easily seen to patch together to give a morphism X Sp ec A We will show that this is an isomorphism We rst show that X can b e covered by op en subsets of the form X for suitable f f A X O where X fx X j f x g here f x denotes the image X f of f in the residue eld k x Let x X b e a closed p oint and U Sp ec B an ane op en neighbourho o d of x in X with complement Y X n U with its reduced structure Then Y fxg determines a reduced closed subscheme of X There is an exact sequence of sheaves I I i k x Y fxg Y x where the rst two terms are the resp ective ideal sheaves and i k x is a x skyscrap er sheaf at x with stalk k x Since H X I we can lift Y fxg k x X i k x to a global section f X I X O A x Y X In other words we can nd f A X O with f x and so x X X f further f vanishes on Y so X U ie X U Sp ec B f f f f We claim that the natural map A B induced by A X O B f f f X f is an isomorphism this will imply that X Sp ec A is an open immersion First supp ose a kerA B Then for any ane op en Sp ec C V X f the image of a in C V X O vanishes hence the image of a in C is f f X annihilated by a p ower of f Covering X by a nite number of such op en sets V we see that a X O A is annihilated by a p ower of f thus A B X f f is injective Also similarly if W X is any op en subset then any element of ker W O W X O is annihilated by a p ower of f Next let X f X n a X O B We claim that for some n af is in the image of f X f X O A B The restriction of a to V X O C is such that X f f X f n f a lifts to C V O for some n A nite number of such op en sets V X i n cover X so we may assumeafter increasing n if needed that f a j lifts to V X f a V O for each i We have that i i X a a j a j V V O ij i V V j V V i j X i j i j m has zero restriction to V V X hence we can nd m with f a i j f ij m for all i j Now the collection of sections f a V O patch up to a unique i i X K H PARANJAPE AND V SRINIVAS nm global section of O which lifts f a We have now shown that A B is X f f surjective as well Thus X has an op en cover by op en subschemes of the form X Sp ec A f f each of which is ane and the natural map X Sp ec A is an op en immersion whose image is the union of these op en subsets Sp ec A Since X is No etherian f Then the functions X this op en cover of X has a nite sub cover say X f f r 1 r f g have no common zero es on X So the sheaf map O O given by r X X g g g f g f for any lo cal sections g g is a surjection of r r r r sheaves giving an exact sequence r F O O X X r i The sheaf O has a ltration by the subsheaves O for i r included X X i through the rst i summands The induced ltration F O of F has graded X pieces which are isomorphic to ideal sheaves Hence by induction on i we i have H X F O and for i r this is just H X F Hence X r r X O X O is surjective in other words A A a a X r X P a f is surjective Hence f f generate the unit ideal in A and so i i r i cover Sp ec A Sp ec A Sp ec A f f r 1 Cech cohomology and the cohomology of projective space We rst review the basics of Cech cohomology Recall that if U fU U g n is a nite op en cover of a top ological space X then for any sheaf F of ab elian groups on X we have an asso ciated Cech complex C U F with terms Y p C U F F U U i i p 0 i i n p 0 p p p and dierential maps C U F C U F given by p X j p j b U U i i 0 p+1 i i i i i 0 p+1 j 0 p+1 j b where i means that the index i is omitted Then C U F is a complex j j called the Cech complex of F with resp ect to U whose cohomology groups are called the Cech cohomology groups of F with resp ect to U and are denoted by i H U F i i U F H X F for each i compare Ha I I I There is a natural map H This is constructed as follows For any op en V X let F denote the V sheaf j j F where j V X is the inclusion this is nonstandard temp orary notation Then the same formula for the Cech dierential gives rise to a complex of sheaves C U F with terms Y p F C U F U U i i p 0 i i n p 0 One also has a natural augmentation F C U F COHOMOLOGY OF COHERENT SHEAVES Lemma The complex of sheaves F C U F C U F is a resolution of F whose complex of global sections is the Cech complex C U F Proof The lemma is proved by showing that the complex of stalks at any p oint is chain contractible the details are left to the reader as an exercise Now if F I is any injective resolution of F the universal prop erty of injectives gives rise to a map unique up to chain homotopy of complexes of sheaves C U F I over the identity map on F The induced maps b etween i i complexes of global sections gives