COHOMOLOGY OF COHERENT SHEAVES
K H PARANJAPE AND V SRINIVAS
The basic source for these lectures is Hartshorne s 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 eci ed 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 a ne
schemes The rst is the following vanishing theorem
Theorem Let X Sp ec A be a Noetherian a ne scheme i e A is a
commutative Noetherian ring Let F be a quasi coherent sheaf of O modules
X
i
on X Then H X F for al l i
Proof Recall that any quasi coherent sheaf F on the a ne scheme X Sp ec A
f
is of the form M where M is the A mo 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 quasi coherent sheaves is exact if and only if the corresp onding sequence of
A mo 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
A mo 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 A module I the associated quasi coherent
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 zero divisor Then applying the functor
Hom I to the exact sequence
A
f
A A A f 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
n m 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 zero divisor on A Ann f Hence
n
multiplication by f is surjective on I Hom A Ann 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 non empty 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 ker I I
f
e e
then s U J U I
Now Y supp J satis es Y D f so Y Y is a prop er subscheme
Thus to complete the pro of by induction it su ces to observe that J is also an
injective A mo dule The pro of of this using the Artin Rees 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
quasi coherent O mo dule Let
X
M I I
b e a resolution of M by injective A mo dules Then we have an asso ciated exact
sequence of quasi coherent 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 a ne schemes essentially
due to Serre
Theorem Let X be a Noetherian scheme Then X is a ne H X I
for any coherent sheaf of ideals I O
X
Proof If X is a ne the higher cohomology of any quasi coherent 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 a ne 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 de ned 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
a ne 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 i e 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 ker A B Then for any a ne 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 assume after 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
n m
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 a ne 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 di erential 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 non standard temp orary
notation Then the same formula for the Cech di erential 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 rise to the desired maps H U F H X F
From the de ning prop erties of a sheaf one sees at once that H U F
H X F is always an isomorphism with a little more work one sees also that
H U F H X F is injective for any sheaf of ab elian groups F
Prop osition Leray s theorem Let F be a sheaf of abelian groups on X
j
F for and U fU U g an open covering such that H U U
n i i
p
0
i i
al l fi i g and al l j Then the natural maps H U F H X F are
p
isomorphisms
Proof We give a pro of using sp ectral sequences other pro ofs are p ossible as well
Let F I b e a asque resolution Then for any op en U X the complex
U I computes H U F j
U
q
The Cech complexes of the I determine a double complex of ab elian groups
p q p q
C C U I
p p
with augmentations C U F C From the hypotheses of Leray s theorem
it follows that for each p the sequences
p p p
C U F C C
are exact the cohomology group of this complex in any degree i is
Y
i
F H U U
i i
p
0
i i
p
0
which is given to vanish Hence one of the sp ectral sequences for the dou
p q p
ble complex C degenerates at E and the natural maps H C U F
p
H T ot C are isomorphisms for all p
On the other hand there are augmented complexes
q q q
H X I C C
which may b e viewed as obtained by taking global sections in the exact sequence
lemma of sheaves
q q
I C U I
p q
By construction C U I is asque for each p Hence all sheaves in the
ab ove sequence are asque so that the corresp onding sequence of global sections
is exact This implies that the second sp ectral sequence for the double complex
K H PARANJAPE AND V SRINIVAS
p q p p
C degenerates at E and the natural maps H H X I H T ot C
p p
H X F by de nition are also isomorphisms Note that H H X I
The resulting isomorphism b etween the Cech and derived functor cohomology
of F is given by the natural map b etween these since we may choose I to b e an
injective resolution of F in which case T ot C U I is also an injective resolu
tion of F such that the augmentation map C T ot C U I is a morphism
of complexes lifting the identity map on F
Corollary Let X be a Noetherian separated scheme U fU U g an
n
open covering by a ne open subsets U Sp ec A Then for any quasi coherent
i i
i i
sheaf F of O modules the canonical maps H U F H X F are isomor
X
i
phisms for al l i In particular H X F for al l i n and al l quasi coherent
sheaves F
Proof Since X is separated the intersection of any nite number of a ne op en
subsets of X is again an a ne op en set Hence by Theorem any quasi
coherent sheaf F of O mo dules satis es the hypotheses of Leray s theorem
X
Theorem The vanishing assertion is obvious for the Cech cohomology
i
since C U F for all i n
n
For example if A is a No etherian ring then on P any quasi coherent sheaf has
A
n
vanishing cohomology in degrees n since P has a covering by n a ne
A
op en subsets
Apart from its theoretical signi cance the ab ove corollary is one of the basic
to ols in making concrete calculations with cohomology We illustrate this by de
n
n
termining the cohomology on P of the sheaf O r for any integer r Recall that
P
A
A
n
P Pro j A X X is obtained from a p olynomial algebra over A in n
n
A
variables with the standard grading Let S A X X S where S
n r r r
is the homogeneous comp onent of degree r we may view S as Z graded with van
ishing homogeneous comp onents of negative degree For any r Z let S r de
n
note the ring S considered as a graded S mo dule with S r S then O r
t r t P
A
g
is the corresp onding quasi coherent in fact coherent even invertible sheaf S r
n
n
r Recall By construction there is a canonical homomorphism S P O
r P
A
A
that if M is a Z graded S mo dule and h S is a homogeneous element then
M denotes the th homogeneous comp onent of the also Z graded lo calization
h
n
M Thus D P Sp ec S form a basis consisting of a ne op en sets for the
h
h
A
n
Zariski top ology on P Pro j S
A
Theorem Let A be a Noetherian ring
n
n
r is an isomorphism for a The canonical homomorphism S P O
r P
A
A
n
n
r for r and is a free A module of al l r In particular P O
P
A
A
n r
rank for r
r
i n
n
b H P O r for al l i n and al l r Z
P
A
A
n n
n
n is a free A module of rank and the natural maps c H P O
P
A
A
n n n n n
n n n
n A r H P O n r P O H P O
P A P P
A A A
A A A
are perfect pairings between free A modules
COHOMOLOGY OF COHERENT SHEAVES
Proof The idea is the compute the Cech cohomology groups with resp ect to the
n
standard a ne op en covering U fU g where
i
i
U D X Sp ec A X X X X i n
i i i n i
For i i i n we have
p
Sp ec A X X j i j and j fi i g Sp ec S U U
i j p i i
X X
p
0
i i
p
0
n
with the A submo dule U r on U Next one identi es the sections of O
i i P
p
0
A
of homogeneous elements of degree r in the Z graded A algebra
p
Y
X S S
X X
i i i
p
j
0
j
n
e
Let U Sp ec S X which is an op en subscheme of Sp ec S A Consider
i
i
A
n
e e
the Cech complex for the structure sheaf O for the a ne op en cover U fU g
X i
i
of X Sp ec S n V X X where V X X Sp ec S X X is
n n n
the closed subscheme determined by the ideal S X X The terms in
n
this complex are of the form
Y Y
p
e f f
O S C U O U U
X x x X i i
p
0
i i
p
0
i i n i i n
p p
0 0
Note that this is a complex whose terms are Z graded A algebras and the dif
ferentials are graded A linear maps The sub complex of homogeneous elements
n
r as noted earlier There is also an of degree r is identi ed with C U O
P
A
augmentation
n
Y
e
S C U O S
X X
i
i
induced by the diagonal inclusion
We claim that
n
e e
S C U O C U O
X X
n n
e e
is exact except for the surjectivity of C U O C U O This will
X X
imply a and b of Theorem on considering the homogeneous sub complex
of degree r for each r Z In fact the ab ove complex can b e considered as a
direct limit of the augmented Koszul complexes
n+1
m m n
p
S K K X X S S S
m
n
where K K is induced as follows
m m
Y
n+1
p
S K S
m p
j j n
p
1
on the j j comp onent and K K is multiplication by X X
p m p m p j j
p
1
One veri es that this commutes with the di erentials in the Koszul complex
m m
Since X X form a regular sequence on the p olynomial algebra S each
n
n
of the Koszul complexes is exact except at the right end p oint and H K m
K H PARANJAPE AND V SRINIVAS
m m
S X X By exactness of direct limits we conclude that the complex
n
is exact except at the right end p oint proving a and b of the theorem
and the cokernel of the last arrow is
n
lim H K
m
!
m
Going back to the Cech complex itself the last term is just S
X X
n
0
X X the Laurent p olynomial algebra which is the free A A X X
n
n
Q
n
a
i
X for all a a Z The p enultimate term of mo dule with basis
n
i
i
is
n
S
c
i
X X X
n
0
i
c
where X means that the variable X is omitted The di erential the last map
i i
in is the alternating sum of the natural inclusion maps Clearly the cokernel
a
a
0
n
X is a free A mo dule with basis X with all a which we can rewite
i
n
Q
n
b
b
0
n
as the set of all monomials M b b X X X X
n n
n
i
where b b range over all non negative integers Keeping track of the grad
n
P
ing M b b is homogeneous of degree n b Thus the coho
n i
i
n n
n
mology mo dule H P O n r is a free A mo dule with basis consisting
P
A
A
P
of all monomials M b b satisfying b r in particular the set of such
n i
i
monomials is non empty precisely when r
The pairing
n n n n n
n n n
n n r H P O r H P O H P O
P A P P
A A A
A A A
n
r determines a maps of sheaves is determined as follows a global section of O
P
A
n n
r and hence by tensoring a sheaf map O O
P P
A A
n n
n n r O O
P P
A A
and thus also an A linear map on n th cohomology mo dules
n n n n
n n
n n r H P O H P O
P P
A A
A A
n
Identifying a global section of O r with a homogeneous p olynomial of degree
P
A
r multiplication by this p olynomial on the terms of the Cech complex
n
n
r has a determines the action on the cohomology mo dules Clearly H P O
P
A
A
P
b
b
0
n
basis of monomials M b b X X where b b r Identify
n i i
n
i
Q
n
n n
n
X the image of n with A using the basis element H P O
P
A
i
i
A
the corresp onding monomial M Then fM b b g is visibly the
n
dual basis with resp ect to the pairing to the monomials fM b b g
n
since
Q
n
X if b c for all i
i i
i
i
M b b M c c
n n
otherwise
COHOMOLOGY OF COHERENT SHEAVES
Finite generation of cohomology and related results
In this section we discuss three basic results on cohomology of coherent sheaves
the nite generation theorem the Serre vanishing theorem for ample invertible
sheaves and Serre s criterion for ampleness a sort of converse to Serre s vanishing
theorem
All of these results dep end on the following lemma on quasi coherent sheaves
which has b een established earlier in the lectures on schemes Let X b e a No e
therian scheme L an invertible O mo dule s X L and let X fx X j
X s
s x g where s x denotes the image of s in the dimensional k x vector
space L k x Then X is an op en subset of X which determines an op en
x O s
X x
subscheme The global section s gives an O linear map L O whose im
X X
age is a coherent ideal sheaf corresp onding to a closed subscheme Y of X which
n
we call the scheme of zero es of s Then X X n Y As usual we denote L by
s
n
L
Lemma Let X L s X L be as above Let F be a coherent sheaf on
X
n n
a If t ker X F X L then s t X F L
s
n n
b If t X F then for some n the section s t X F L lies
s s
n n
in the image of X F L i e s t extends to a global section
Theorem Let X be a projective scheme over a Noetherian ring A and
O a very ample invertible sheaf on X relative to Sp ec A Let F be a coherent
X
sheaf on X
i
a For each i H X F is a nitely generated A module
b Serre s Vanishing Theorem There exists an n depending on X O
X
i
and F such that for any n n i we have H X F n
r
r
Proof Let i X P b e a pro jective embedding such that O i O
X P
A
A
Then i is exact and for each j we have
j j r j r
H X F n H P i F n H P i F n
A A
r
Hence we reduce to proving the Theorem for coherent sheaves F on X P
A
r r r
We rst show that for any coherent sheaf F on P we have that H P F is
A A
r r
nitely generated and H P F n for all large n Now F m is generated
A
by its global sections for any su ciently large m Fix such a value of m Since
r
P is No etherian and F m is coherent a nite number of global sections su ce
A
N
to generate F m giving rise to a surjection of sheaves O F m This is
r
P
A
N
r
m F say with kernel G giving rise to an turn induces a surjection O
P
A
exact sequence
r
n m F n G n O
P
A
i r
for each n Z Now H P H for all i r and all quasi coherent sheaves
A
H Corollary Hence from the exact cohomology sequence
N r r r r r r
r
n m H P F n H P G n H P O
P
A A A A
K H PARANJAPE AND V SRINIVAS
r r
we conclude using Theorem that H P F n is a nitely generated A
A
mo dule for all n and that for su ciently large n in fact n m r we have
r r
H P F n
A
Now we pro ceed by descending induction on i to show that for any coherent F
i r i r
we have that H P F is nitely generated and if i then H P F n
A A
for all su ciently large n We have already dealt with the case i r Apply
ing this to G we conclude using the earlier part of the same cohomology exact
sequence
i r N i r i r
r
G n F n H P n m H P O H P
P
A A A
A
i r
combined with Theorem that H P F n is always nitely generated
A
and if i vanishes for all large enough n
Corollary Let X be an integral projective scheme over and algebraically
closed eld k Then H X O k or in other words any global regular func
X
tion on X is a constant
Proof Since X is an integral pro jective k scheme R H X O is an integral
X
domain containing k By Theorem R is a nite dimensional vector space
Hence R must b e a nite extension eld of k and since k k this means R
k
De nition X b e a pro jective scheme over a eld k or more generally an
Artinian ring A Then for any coherent sheaf F on X we can de ne its Euler
X
i i
X F F H X F Z
i
where M denotes the length of any nite A mo dule
Note that if
F F F
is any exact sequence of coherent sheaves then
X F X F X F
from the long exact sequence of cohomology mo dules
n
H X F H X F H X F H X F H X F
where n dim X which is a nite exact sequence of mo dules of nite length for
which the alternating sum of the lengths must vanish
If O is a very ample invertible sheaf on X relative to A then we can also
X
consider the numerical function
H X F n X F n n Z
Theorem Let F be a coherent sheaf on X which is projective over Sp ec A
where A is an Artinian local ring Let O be a very ample invertible sheaf on
X
X We then have the fol lowing
COHOMOLOGY OF COHERENT SHEAVES
a There is a polynomial P t Q t such that
H X F n P n n Z
We call P t the Hilb ert p olynomial of the sheaf F The degree of P equals
the dimension of supp F
N
b If i X P and O is the resulting very ample sheaf then P t is
X
k
the Hilbert polynomial of the homogeneous coordinate ring R X of X more
f
generally if M is any nite graded R X module and F M then P t
coincides with the Hilbert polynomial of M that is H X F n M
n
for al l large n
Proof If
F F F
is an exact sequence of coherent sheaves then H X F n H X F n
H X F n for all n Z Hence if a of the Theorem holds for any two of F
f
F F then it holds for the third If F M for a nite graded R X mo dule
M then M H X F n for all large enough n since we know that
n
H X F e F
n
and any two nite graded R X mo dules determining the same sheaf F must
coincide in large enough degrees Hence H X F n P n holds for all large
enough n where P t is the Hilb ert p olynomial of any such M in particular the
Hilb ert p olynomials of di erent p ossible such mo dules M coincide
We now show that H X F n P n for all n by No etherian induction on
supp F and then by descending induction on n If supp F is a p oint then F is
a skyscrap er sheaf so that H X F n is a constant function which coincides
with a p olynomial function P n for large n then P t must b e a constant and
H X F n P n holds for all n
In general let s R X X O b e a section which is non zero in the
X
stalk of O at every asso ciated p oint of the sheaf F recall that the asso ciated
X
p oints of F are p oints of the subset S X such that for any a ne op en U
f
Sp ec A with F j M the set S U consists of the asso ciated primes of M
U
Then multiplication by s de nes an injective sheaf homomorphism F F
with cokernel G say From the exact sequence F F G it
follows that H X F n H X F n H X G n Since G is not supp orted
at any asso ciated p oint of F which includes the generic p oints of supp F we have
that supp G supp F is a prop er closed subscheme Hence H X G n Q n
for all n Z for a suitable p olynomial Q t Since H X F n P n for all
large n we must have Q t P t P t Hence by descending induction on
n we conclude that H X F n P n for all n Z
Corollary In the above situation
X
i i
H X F n P n H X F n
i
K H PARANJAPE AND V SRINIVAS
Thus the non agreement of the Hilb ert function and Hilb ert p olynomial is ac
counted for by non vanishing higher cohomology groups
Remark If X is a non singular pro jective variety over a eld k the Riemann
Roch theorem of Grothendieck Hirzebruch gives a formula for the p olynomial P t
using intersection theory on X in terms of the Chern classes of F and of the sheaf
of K ahler di erentials
X k
Theorem Let X be proper over a Noetherian a ne scheme Sp ec A and L
an invertible sheaf on X Then L is ample on X if and only if for each coherent
i n
sheaf F on X we have H X F L for al l i and al l su ciently
O
X
large n
m
Proof If L is ample choose m so that L is very ample and apply Serre s
i i n
vanishing theorem to F L for i m We deduce that H X F L
for all i for all su ciently large n
Conversely supp ose the ab ove cohomology vanishing assertion holds By the
n
de nition of ampleness we must show that for any coherent F the sheaf F L is
globally generated for all large enough n Let x X b e a closed p oint M O
x X
its maximal ideal sheaf We have an exact sheaf sequence
M F F F k x
x
where F k x denotes the skyscrap er sheaf at x asso ciated to the nite dimen
n
sional k x vector space F k x Tensoring wih L and taking cohomology
x O
X x
we see that for large n
n n
H X F L F L k x
x O O
x
X x X x
n
is surjective since H X M F L By Nakayama s lemma this means
x
n n
that global sections of F L generate the O mo dule F L at the
X x x O
x
X x
chosen p oint x for all large n how large may dep end on x Thus the quotient
n
of F L by the O submo dule generated by global sections is not supp orted
X
n
at x Since the supp ort of this quotient is closed the global sections of F L
generate the sheaf in an op en neighbourho o d of x for all su ciently large n how
large may dep end on x and the neighbourho o d may dep end on the chosen n
Apply this to F O then there is an n and a neighbourho o d V of x on
X
n
0
which L is generated by its global sections Cho ose n and neighbourho o ds U
i
i n
1
of x i n such that global sections of F L L generate this sheaf on
m
U Then global sections of F L generate this sheaf on U V U U
i x n
0
for all m n since we may write any such m as q n i n for some q
m i n n q
1 0
and i n so that F L F L L is a tensor pro duct of
two sheaves each of whose global sections generate it on U
x
Now since X is No etherian a nite number of such U cover X
x
Serre Duality
We b egin with a discussion of Ext groups and sheaves We let Ab denote the
category of ab elian groups If X is a scheme or more generally any ringed space
COHOMOLOGY OF COHERENT SHEAVES
X O and F is any O mo dule we have two left exact exact functors de ned
X X
on the category Mod X of O mo dules namely
X
Hom F Mod X Ab G Hom F G
O O
X X
and
H om F Mod X Mod X G H om F G
O O
X X
The rst functor is obtained from the second by comp osition with the global
section functor X We de ne
i
Ext F Mod X Ab
to b e the i th right derived functor of Hom F Similarly de ne
X
i
E xt F Mod X Mod X
O
X
to b e the i th right derived functor of H om F
O
X
Lemma For any open set U X
i i
E xt F G j E xt F j G j
U U U
O O
X U
Proof For xed F b oth sides considered as functors in G give functors
i
Mod X Mod U which coincide for i The functor G E xt F G j
U
O
X
i
vanishes if G is injective since E xt F G in that case Hence it is a uni
O
X
versal functor
On the other hand we claim that if G is injective then G j is an injective ob ject
U
of Mod U To see this let j U X b e the inclusion if Q R is an
exact sequence in Mod U and f Q G j is given then j f j Q j G j
U U
e
comp osed with the inclusion j G j G gives rise to a map f j Q G Since
U
e
j Q j R is exact in Mod X and G is injective we can extend f to a
map j R G the restriction of this map to U is the desired extension R G j
U
of f
i
As a consequence we also have E xt F j G j for injective G Hence
U U
O
U
it is also a universal functor
Lemma For any F Mod X we have
a E xt O F F
X
O
X
i
b E xt O F for al l i
X
O
X
i
i
c Ext O F H X F
X
O
X
Proof Since E xt coincides with H om a is clear Since the identity functor is
exact its derived functors vanish which gives b Since Hom O coin
O X
X
cides with the global sections functor X the derived functors of these two
functors must coincide as well
Lemma If F F F is a short exact sequence in Mod X
then for any G Mod X there are long exact sequences functorial in G
F Hom F G Hom F G Hom F G Ext G
O O O
O
X X X
X
Ext F G
O X
K H PARANJAPE AND V SRINIVAS
H om F G H om F G H om F G
O O O
X X X
E xt F G E xt F G
O O
X X
Proof Since Hom I is exact for any injective sheaf I we see that if
O
X
G I is an injective resolution then we have a short exact sequence
of complexes
Hom F I Hom F I Hom F I
O O O
X X X
The corresp onding long exact sequence in cohomology gives the desired exact
sequence of Ext groups To get the analogous sequence of E xt sheaves we remark
that if I is injective then so is I j for any op en U see the pro of of lemma so
U
that H om I is an exact functor and a similar argument applies again
O
X
Lemma Let E E E F be a resolution of an O
X
module F by locally free O modules E of nite rank Then for any O module
X j X
G we have isomorphisms of functors
i i
H H om E G E xt F G
O
O
X
X
i
on the right H denotes the i th cohomology sheaf of the complex of O modules
X
Proof Both sides give functors in the variable G on ModX which clearly
agree for i by the left exactness of H om If G is injective the left side
vanishes and b ecause H om G is exact in this case the right side vanishes
O
X
as well Hence b oth functors are universal and must coincide
For example on a scheme X which is quasi pro jective over a No etherian ring
any coherent sheaf is a quotient of a lo caly free sheaf of nite rank and so we
can always nd such a resolution for any coherent sheaf F
Lemma Let E be a locally free O module and let E H om E O
X O X
X
Then for any O modules F G we have natural isomorphisms for al l i
X
i i
Ext F E G Ext F E G
O
O
X
X
i i i
E xt F G E E xt F E G E xt F E G
O O
O O
X X
X X
Proof First consider the claim for Ext Both sides of the formula de ne
functors which agree for i and for injective G the left side vanishes for
all i On the other hand the functor H om E G is isomorphic to
O
X
H om E G which is exact since G is injective hence E G is
O O O
X X X
also injective and the right side of the formula is as well Thus b oth sides of
the formula de ne universal functors which must hence coincide
For E xt the argument is very similar again using the fact that the three
functors agree for i and all vanish for i when G is injective for the middle
sheaf this is b ecause E G is also injective as noted ab ove
O
X
The ab ove lemmas have all b een in the situation of an arbitrary ringed space
We now prove a lemma which is valid in a more geometric situation It allows
one to make connections b etween the Ext groups and sheaves and notions in
commutative algebra like depth pro jective dimension etc
COHOMOLOGY OF COHERENT SHEAVES
Lemma Let X be a Noetherian scheme and F a coherent sheaf on X Then
for any O module G and any point x X we have isomorphisms for al l i
X
i i
Ext F G E xt F G
x x x
O O
X x X
Proof The question is lo cal so we may assume X Sp ec A is a ne where A
f
is a No etherian ring and F M for some nitely generated A mo dule M Let
f f
A O Let E M b e a free resolution of M so that E M
x X x
f
is a lo cally free O resolution of M Let E E A M M A We
X i x i A x x A x
have a natural isomorphism
e
E G H om E G Hom
i x x O i x A
x
X
since E is a free A mo dule of nite rank Since a free A resolution of M
i x x
i
M the lemma follows using may b e used to compute the Ext groups Ext
x
A
x
lemma
Exercise Let X Sp ec A where A is No etherian and let M N b e A
mo dules with M nitely generated Then
i i
f e
Ext M N Ext M N
O A
X
and
i i
f e
M N E xt M N Ext
O A
X
i
f e
In particular E xt M N is quasi coherent and is coherent if N is also nitely
O
X
generated This implies that on any No etherian scheme if F is coherent and G
i
is quasi coherent then E xt F G is quasi coherent and is coherent whenever
O
X
G is coherent
Prop osition Let X be a projective scheme over a Noetherian ring A and
O a very ample invertible sheaf on X Let F G be coherent O modules
X X
Then there exists n depending on F G and i such that for any n n we
have
i i
Ext F G n X E xt F G n
O O
X X
Proof Fix F Now Hom F X H om F is a comp osition
O O
X X
of functors and for injective G one veri es easily that H om F G is asque
O
X
hence X acyclic Hence there is a Grothendieck sp ectral sequence
p q
p q p q
E R X R H om F G R Hom F G
O O
X X
which we unravel to b e
p q q p q
p
E H X E xt F G Ext F G
O O
X X
Apply this to F and the twist G n of G We compute that by lemma
q
E xt F G n E xt F G n
O
X
O
X
Hence from Serre s vanishing theorem Theorem b we have that for all
su ciently large n how large dep ends on F G and q
q
p
H X E xt F G n p
O X
K H PARANJAPE AND V SRINIVAS
p q
Thus xing i we can choose n such that for all n n we have E for all
p and q i Now the relevant part of the sp ectral sequence degenerates at
E giving isomorphisms
i
i i
E E Ext F G n
O
X
i
i
X E xt F G n But E
O
X
We recall also some results from commutative algebra
Lemma Let A be a ring and M be a nite A module
a M is projective Ext M N for al l modules N
A
i
b M has projective dimension n Ext M N for al l i n for
A
al l modules N
c If A m is a regular local ring of dimension n then every nite A module
M has projective dimension n and A m has projective dimension exactly
n In general the projective dimension of M is n depth M We have
i
p d M n Ext M A for al l i n
A
A
Proof Let
N F M
M N then this sequence is split b e exact with F a free A mo dule If Ext
A
exact and M is pro jective Conversely if M is pro jective Hom M is exact
A
i
and so its derived functors Ext M vanish This proves a The pro of of b
A
is by induction on the pro jective dimension of M if M is not pro jective then in
the ab ove exact sequence the pro jective dimension of N is p d M If
A
i
we already know the vanishing of Ext N for i p d M then from the
i
long exact sequence of Ext s we deduce that Ext M for i p d M
We now discuss the pro of of c Recall that if A m is a regular lo cal ring of
dimension n then m is generated by n elements say a a Then the asso
n
i i
ciated graded ring gr A m m is a quotient of the symmetric algebra
i
m
on m m which is a p olynomial algebra over k A m on the n elements which
are the images of a a in m m It is easy to see that for any quotient of
n
a p olynomial algebra k X X by a non zero homogenous ideal the Hilb ert
n
p olynomial has degree n it su ces to consider the quotient by a homoge
nous principal ideal Since from dimension theory dim A n is the degree of
the Hilb ert p olynomial of gr A we conclude that gr A k X X is a
n
m m
p olynomial algebra over k Using this one can prove left to the reader that
a a form a regular sequence on A so that the residue eld k A m has a
n
nite free resolution by the Koszul complex K a a Using this complex
A n
A
we compute at once that Tor k k for i n and is k if i n In
i
A
M k for all i n particular k has pro jective dimension n Hence Tor
i
for any nite A mo dule M Considering a minimal free resolution F M we
A
have that rank F dim Tor M k and so conclude that p d M n
i k
A
i
Now we show depth M p d M n for any nite A mo dule M We do this
A
by induction on depth M If depth M there is an exact sequence
k M M
COHOMOLOGY OF COHERENT SHEAVES
A
and since p d M n we have Tor M k and from the long exact
n
A
sequence of Tor s we see that Tor M k Hence p d M n and so it must
A
n
b e equal to n If depth M then choose a non zero divisor a nm on M to
get an exact sequence
a
M M M aM
Then depth M aM depth M by standard prop erties of depth If p d M aM
A
A A
r necessarily then Tor M aM k and Tor M aM k as seen
r r
by considering a minimal free resolution of M aM We have an exact sequence
a
A A A
Tor M aM k Tor M k Tor M k Tor M aM k
r
r r r
a
A A
Tor M k Tor M k
r r
A A
Since multiplication by a is the zero map on Tor k we deduce that Tor M k
i r
A
and Tor M k Hence p d M r
A
r
i
Finally we show that if Ext M A for i n then p d M n Let N b e
A
A
i
any nite A mo dule We see easily by induction on p d N that Ext M N
A
A
for all i n Now the criterion in b applies
We now prove the Serre Duality Theorem for coherent sheaves on a pro jective
scheme over a eld We rst consider the case of pro jective space itself then we
reduce the general case to this case If V is a k vector space let V denotes the
dual vector space Hom V k
k
n
n
Theorem Let k be a eld X P Pro j k X X Let
n X
k
X k
n
k we x such an isomorphism a H X
X
b For any coherent sheaf F on X the natural pairing
n n
Hom F H X F H X k
O X k X
X
is a perfect pairing between nite dimensional vector spaces
c For each i there is a natural isomorphism functorial in F
i n i
Ext F H X F
X
O
X
which for i is the pairing in b
Proof One has the Euler exact sequence
O H X O O
X k X X
X k
from which by taking determinants one obtains
n n
O n O H X O
X X k X X
X k
Now a follows from Theorem Since an O linear map F induces a
X X
n n
k linear map H X F H X we have a pairing as describ ed in b To
X
see that it is p erfect rst note that if F O r for some r or a direct sum of
X
such sheaves then Theorem gives the p erfection of the pairing Now we may
write F as a cokernel of a map O n O m for suitable integers n
i X i j X j i
i
m why We also have H X G for any coherent sheaf G and any i n j
K H PARANJAPE AND V SRINIVAS
n
this follows from Corlo oary so that H X is right exact and we have a
presentation
n n n
H X O n H X O m H X F
i X i j X j
This gives rise to a commutative diagram with exact rows
n
n n
H X O m
H X F H X O n
j X j
X i / / / i
O O O
Hom O m
Hom F Hom O n
j O X j X
X i O X i X
/ O / /
X
X X
The middle and right hand vertical arrows are isomorphism and hence so is the
left vertical arrow This proves b
To prove c note that b oth sides de ne contravariant functors on the ab elian
category of coherent sheaves on X which coincide in degree by b Now for
any r we have using lemmas and and the Serre Vanishing theorem
i i
H X r i Ext O r
X X X
O
X
N
We can nd a surjection O r F for any su ciently large r this just
X
n i
means F r is globally generated Also note that H X O r for
X
all i and r This implies that b oth of our contravariant funcotrs are
co e aceable and hence b oth are universal and thus isomorphic
We now consider the general case We b egin with a de nition
De nition Let k b e a eld and let X b e a prop er scheme of dimension n
over Sp ec k A dualizing sheaf on X is a coherent sheaf together with a trace
X
n
map t H X k such that for all coherent sheaves F on X the pairing
X
t
n n
k Hom H X F H X F
O k
X X
X
is a p erfect pairing b etween nite dimensional vector spaces
Since t is de ned by a universal prop erty it is easy to see that it is unique
X
up to unique isomorphism
N
Prop osition Let i X P be a closed subscheme of codimension r
k
Then
r
i E xt i O N
X
P
O X
N
k
P
k
is a dualizing sheaf for X
Proof We rst show that for any coherent O mo dule F we have
X
s
N
s r E E xt i F
s
P
O
N
k
P
k
It su ces to prove that E m has no global sections for all large enough m
s
s
From lemma this is equivalent to showing that Ext i F m N
N
P
P
k
k
N
Theorem this amounts to showing for s r By Serre duality for P
k
j N s N j N
H X F m for all j H P i F m But H P i F m
k k
N s
and N s N r dim X so H X F m as desired
COHOMOLOGY OF COHERENT SHEAVES
Now we note that for any O mo dule F and O N mo dule G we have
X
P
k
Hom F i H om i O G Hom i F G
O O X O
X N N
P P
k k
n
Also if G is injective in Mod P then i H om i O G is injective in Mod X
O X
k N
P
k
Hence there is a Grothendieck sp ectral sequence
q p q p p q
i F G F i E xt i O G Ext Ext E
N X
O O
P N X
P
k
k
p q
Applying this to G N and using that E for all q r and all p since
P
k
q
i E xt i O N we deduce that
X
P
O
N
k
P
k
r
r
r
N
E Ext i F E
P
O
N
k
P
k
N r N N r
H P i F H X F
k
On the other hand we have
r
r
Hom F i E xt E i O N Hom F
O X O
P
O X
X X
N
k
P
k
Hence we have constructed an isomorphism of contravariant vector space valued
N r
functors in F b etween H X F and Hom F
O
X
X
We now give a more concrete description of is some cases
X
N
Prop osition Let i X P be a closed local complete intersection
k
suscheme which is purely of codimension r and let I be the ideal sheaf of X in
r
N
so that I I is an invertible O module Then P
X
k
r
N
I I i
O
P
X
X
k
In particular if X is non singular i e smooth over k of dimension n N r
then
n
n
X X k X k
Proof If A is a No etherian ring I A a complete intersection of height r gen
erated by a regular sequence x x then the Koszul complex K x x
r A r
gives an A free resolution of A I such that the dual complex obtained by ap
plying Hom A is isomorphic to K x x up to reindexing suitably
A A n
i r
In particular we compute that Ext A I A for i r and Ext A I A
A A
r
A I One computes immediately that identifying I I with A I using the
r
generator x x the resulting isomorphism Ext Hom I I A I is
r A I
indep endent of the choice of the regular sequence x x generating I
r
N
is a lo cal complete intersection subscheme purely of co di Thus if X P
k
mension r then using we obtain an isomorphism
r
r
E xt i H om I I O i O O N
O X X
P
O
X
N
k
P
k
r
N
has the description stated in the Prop osition by Now E xt i O
X
P
O
X
N
k
P
k
lemma
K H PARANJAPE AND V SRINIVAS
If X is non singular of dimension n N r then is a lo cally free O
X
X k
mo dule of rank r We have an exact sequence of K ahler di erentials
I I i
N
X k
k P
k
is just N we Taking determinants and noting that the determinant of
N
P
k P
k
k
deduce that
r
n
N
I I i
O
P
X X k
X
k
Now we prove the Serre duality theorem for more general pro jective k schemes
Theorem Let X be a projective scheme over an of dimension n over a
eld k and let t be a dualizing sheaf and trace map for X Let O be a
X
X
very ample invertible sheaf on X
a For al l j there are natural tranformations of contravariant functors in
F
n j j
H X F Ext F
j
X
such that is given by the dualizing property of
X
b the fol lowing are equivalent
i X is Cohen Macaulay and equidimensional
j
ii for any locally free E on X we have H X E q for al l j n
and q
iii the natural transformations are isomorphisms
j
n j
Proof For each j we may identify H X F with
j r
n j N
n
H P i F Ext i F
P
k
n
O
k
P
k
In the sp ectral sequence
p q p q p q
E Ext F i E xt i O N Ext i F N
X
P P
O O O
X N N
k k
P P
k k
p q
we have E if q r N n co dim X as seen in the pro of of Prop osi
tion The natural transformations in a are then just the edge homomor
phisms
j j r j r
j r
n
Ext F E E Ext i F
P
X
n
O O
X
k
P
k
Now we show that any of the three conditions in b is equivalent to
q
N
E xt q r i O
X
P
O
N
k
P
k
N
From the sp ectral sequence and Serre duality for P the ab ove vanishing state
k
p q
ment do es imply iii since we will have E for q r giving isomor
phisms where r N n
j r j
i F N Ext F Ext
N
P
X
O
X
P
k k
COHOMOLOGY OF COHERENT SHEAVES
j
Next if the duality assertion iii holds then for any lo cally free E H X E q
is dual to
n j
n j
H X E q Ext E q
X X
O
X
By the Serre vanishing theorem we deduce ii Now assuming ii we have that
for any lo cally free E on X any integer j n and q we have
N j
j j N
H X E q H P i E q Ext i E q N
P
k
O
N
k
P
k
N j
n j
N
Ext E xt i E N q H P i E N q
P P
k
O
N
k k
P
k
N N
Now any coherent sheaf G on P with H P G q for q must satisfy
k k
G Taking E O we deduce that the vanishing assertion must
X
q
hold Finally we note that the stalk of E xt i O N at any p oint x X
X
P
O
N
k
P
k
q
is isomorphic to Ext O O N since N is an invertible sheaf By
X x
P x P
O
N
k k
P x
k
lemma c we deduce that is equivalent to
p d O r
X x
O
N
P x
k
which in turn is equivalent to
depth r dim O O dim O N
X x X x
x P
O
N
k
P x
k
But we always have depth O dim O Hence this last condition can hold
X x X x
only if
N
r dim O dim O
X x
x P
k
which is the same as saying that X is equidimensional and all the lo cal rings
O have depth equal to their dimension i e are Cohen Macaulay
X x
Corollary Let X be an equidimensional Cohen Macaulay projective k
scheme of dimension n and E a locally free sheaf on X of nite rank Then
there are isomorphisms
n i i n i
H X E H X E H X H om E
O O
X X
X X
Proof This is Theorem combined with Lemma
Remark From b of the ab ove Theorem if X is purely n dimensional and
n i
i
Cohen Macaulay we have a p erfect duality b etween H X F and Ext F
X
O
X
Even without the Cohen Macaulay hypothesis one can get a similar duality as
i n i
sertion b etween H X F and E x t F where is a certain complex of
X X
i
are the hyper ext groups hyperderived functors injective O mo dules and E x t
X
of Hom of pairs of complexes One takes
i H om i O J
O X
X
N
P
k
N
Thus is a complex of O injectives where J is an injective resolution of
X
P
X
k
which is well de ned up to homotopy b ecause J is and whose cohomology
j
sheaves are the sheaves i E xt i O N The sp ectral sequence used in the
X
P
O
N
k
P k
K H PARANJAPE AND V SRINIVAS
pro ofs of Prop osition and Theorem gets re interpreted as a sp ectral
sequence for hyper ext groups Such a complex is called a dualizing complex
X
for X
Remark In fact Serre duality is true for equidimenional Cohen Macaulay
proper k schemes The pro of in this case is more di cult see Ha
Prop osition Lemma of Enriques Severi Zariski Let X be an integral
normal projective scheme over k of dimension and O a very am
X
ple invertible sheaf on X Then for any locally free O module E we have
X
H X E q for al l q
Proof As in the pro of of Theorem b the normality of X implies that
j
E xt i O N for j N From the sp ectral sequence we then
X
P
O
N
k
P
k
obtain for q using Serre vanishing
j
N
Ext i O q N H X i E xt i O N q
X X
N
P
P
O
P
N
k
P
k
k
N
By duality on P we obtain the desired vanishing statement for H
k
Corollary Let X be an integral normal projective scheme of dimension
over an algebraically closed eld k and let D X be the support of an
e ective ample divisor on X Then D is connected
Proof We have H X O k It su ces to show that for a suitable scheme
X
structure on D we have H D O k Cho ose a scheme structure on D so
D
that D is an e ective Cartier divisor which is a su ciently high multiple of a
very ample divisor on X rst choose a strcuture D which is an ample Cartier
divisor then a multiple D mD which is very ample then a su ciently
high multiple D q D Then by Prop osition we have H X O D
X
H X O q D Hence from the long exact sequence in cohomology for
X
the exact sheaf sequence
O D O i O
X X D
where i D X we get that k H X O H D O is surjective
X D
Corollary Let X be an integral normal projective variety of dimension
over an algebraically closed eld k and let D be the support of an ample
divisor on X Then
al g al g
D X
is surjective
Proof From the de nition of the algebraic fundamental group it su ces to prove
that if f Y X is a connected algebraic covering space i e Y is conencted
and f is nite at and unrami ed then for the induced pull back covering space
D Y D D Y is connected But Y is again an irreducible normal k
X X
scheme of nite type and the pull back to Y of any ample Cartier divisor on X
is ample on Y Hence Y is also pro jective and D Y Y is the supp ort of an
X
ample divisor hence by Corollary it is connected
COHOMOLOGY OF COHERENT SHEAVES
We use the Serre duality theorem to discuss the Riemann Ro ch theorem for
non singular pro jective curves Recall that a non singular pro jective curve X
over an algebraically closed eld k is an integral pro jective k scheme of dimension
The rational functions k X on X form a eld which is a nitely generated
extension eld of k of transcendence degree and the p oints of X are in bijection
with the discrete valuation rings of k X which contain k see Ha Chapter
x where the p oint x is asso ciated to the discrete valuation ring O we regard
X x
k X as O where is the generic p oint of x since x lies in the closure of there
X
is a natural homomorphism O O which is an inclusion O k X as
X x X X x
a discrete valuation subring containing k with quotient eld k X
A divisor on X is an element of the free ab elian group Div X on closed
P
p oints of X the degree of a divisor D n x with n Z x X is the
i i i i
i
P
integer deg D n If f k X n f g then we can asso ciate to it the divisor
i
i
P
f ord f x where ord denotes the normalized discrete valuation on
X x x
x
k X asso ciated to the closed p oint x only nitely many of the integers ord f
x
are non zero so the sum do es de ne a divisor
We have an invertible sheaf O D asso ciated to any divisor D on X and
X
this induces an isomorphism b etween the Picard group of isomorphism classes of
invertible sheaves on X and the divisor class group of linear equivalence classes
of divisors a divisor is linearly equivalent to if it is of the form f for some
f k X n f g
If is non zero one can similarly asso ciate to it a divisor as follows
k X k
for each closed p oint x X choose a generator of the free O mo dule
x X x
Since O has quotient eld k X we have
X x
O k
X x
k X
O
k X k O k O k
X x
X x X x
and so f for a unique f k X if we make a di erent choice of the
x x x x
co e cient f is replaced by uf for some unit u O De ne K div
x x X x X
P
ord f Clearly div f f div for any non zero f k X hence
x x X
x
the linear equivalence class of K is clearly well de ned It is called the canonical
X
divisor class on X Any such divisor K is called a canonical divisor on X it
X
determines an isomorphism O K
X X X
X k
Theorem Let X be a non singular projective curve over an algebraically
closed eld k and K a canonical divisor Then for any divisor D on X we
X
have
dim H X O D dim H X O K D deg D g
k X k X X
where g dim H X is the genus of X
k
X k
O D By and so O K D Proof We have O K
X X X X X X X
Serre duality applied to O and O D we then have g dim H X O
X X k X
and dim H X O K D dim H X O D Thus the Theorem is
k X X k X
equivalent to the statement
O D dim H X O D dim H X O D
X k X k X
K H PARANJAPE AND V SRINIVAS
deg D g deg D O
X
P P
Let D n x b e any divisor and de ne jjD jj jn j We prove by
i i i
i i
induction on jjD jj If jjD jj then D and O D O and deg D
X X
so holds trivially For any D let D D x and consider the exact sheaf
sequence
O D O D i k x
X X x
where i k x is the skyscrap er sheaf with stalk k x at x The asso ciated exact
x
sequence in cohomology gives
O D O D i k x O D
X X x X
Hence holds for D it holds for D Now choosing x suitably and
taking D D or D D we deduce by induction on jjD jj
Corollary If X is as above deg K g and dim H X
X k
X k
For any non zero f k X we have deg f
X
dim H X O we have dim H X Proof By Serre duality for
k X k
X k X k
Since dim H X g the Riemann Ro ch theorem for O K implies
k X X
X k
deg K g Finally if D f then O D O so the Riemann Ro ch
X X X X
theorem implies deg D deg
Higher direct images
If f X Y is a morphism and F is a coherent sheaf on X then we
may want to study the cohomology of the restrictions of F to the b ers of f
Ideally one would like to have a quasi coherent sheaf F on Y such that for
i
any p oint y Y if X X Sp ec k y is the scheme theoretic b er then
y Y
i
F k y H X F O
i y y X
y
In general it is imp ossible to nd such a sheaf F one situation where this
i
is p ossible called the base change theorem is discussed later In general the
i
nearest substitute is the i th higher direct image sheaf R f F mentioned in the
sheaf theory lectures
De nition Let f X Y b e a continuous map of top ological spaces and
i
F a sheaf of ab elian groups on X The i th direct image sheaf R f F is the
value on F of the i th derived functor of the left exact functor f Mod Z
X
Mod Z
Y
i
Thus if we have an injective resolution F I then R f F is the i th
cohomology sheaf of the complex
f I f I
For example if Y is a p oint so that sheaves of ab elian groups on Y are identi ed
with ab elian groups then f b ecomes identi ed with the global section functor
i i
and R f b ecome the i th cohomology functor H X
COHOMOLOGY OF COHERENT SHEAVES
i
Lemma If F is any sheaf of abelian groups on X then R f F is the sheaf
on Y associated to the presheaf
i
1
U H f U F j
f U
i
Proof Let H X F denote the sheaf asso ciated to the ab ove presheaf Both
i i
fR f g and fH X g form functors Mod Z Mod Z which
i i X Y
i
agree for i If I is injective then R f I for i by the de nition of
1
derived functors while the ab ove presheaf is b ecause I j is asque for
f U
all U Hence b oth functors are universal and must coincide
Corollary If V Y is open then
i i
1
R f F j R f j F j
V V
f V
i i
Corollary If F is asque then R f F for i Hence R f may be
computed using asque resolutions e g the Godement resolution
Corollary If f X O Y O is a morphism of ringer spaces then
X Y
i
the i th derived functor of f Mod O Mod O coincides with R f
X Y
restricted to O modules
X
Proof Any injective resolution in Mod O is a asque resolution as well
X
The following is slightly weaker than a similar result in Ha but is su cient
for our purp ose
Prop osition Let f X Y be a morphism of between Noetherian sepa
rated schemes
If Y Sp ec A is a ne then for any quasi coherent O module F we have
X
i
i
R f F H X F
i
In general if F is quasi coherent then R f F is quasi coherent
Proof The case with Y a ne implies the general case So assume Y Sp ec A
i
i
There is in any case a map H X F R f F For h A let U Sp ec A b e the
h
i i
1
corrersp onding basic op en set We claim H f U F j H X F
A
f U
i
i
A Assuming the claim lemma implies that H X F R f F is an iso
h
morphism on stalks hence an isomorphism
To prove the claim let U fU U g b e an a ne op en cover of X where
n
U Sp ec B and consider the Cech complex C U F which computes the
i i
i
cohomology groups H X F Then V U U Sp ec B A determines
j j Y i A h
an a ne op en cover V fV V g of V f U and the corresp onding
n
1
Cech complex for F j is just
f V
C U F A C V F j
A h V
Since lo calization is exact the cohomology groups of C U F A are just the
A h
i
lo calizations H X F A
A h
As a corollary to the pro of we also get the following
K H PARANJAPE AND V SRINIVAS
Corollary Let f X Y a morphism of separated Noetherian schemes
and U fU U g be an a ne open cover of X Then for any quasi coherent
n
sheaf F on X
i
R f F i th cohomology sheaf of f C U F
Proof Using the fact that since X Y are separated f V U is a ne whenever
U X V Y are a ne op en subsets the pro of is reduced to the case Y is
a ne which we treat as ab ove
Another corollary is as follows whose conclusion is expressed in words by saying
that cohomology commutes with at base change
Corollary Suppose we have a pul l back square of separated Noetherian schemes
q
W X
g f
p
Z Y
where f is proper and g is at Then for any coherent sheaf F on X there is are
canonical isomorphisms
i i
R g q F i p R f F
Proof The pro of reduces at once to the case when Y Sp ec A Z Sp ec B
are a ne Now if U fU U g is an a ne op en cover of X then V
n
fV V g is one of W where
n
U Sp ec B V Z U q U
i Sp ec A i Y i i
i i
The corollary amounts to the assertion that H W q F H X F B This
A
is true b ecause C V q F C U F B and B is at over A
A
One of the basic results on higher direct images is the following coherence
theorem which is an elab oration of Theorem
Theorem Let f X Y be a proper morphism of Noetherian schemes
Let O be a very ample invertible sheaf on X relative to Y and let F be a
X
coherent O module
X
a For al l n the natural map f f F n F n is surjective
i
b For al l i R f F is a coherent sheaf on Y
i
c For al l i and al l n we have R f F n
Proof The result is lo cal on Y since Y is No etherian hence quasi compact So
we may assume Y is a ne Now the result follows from Prop osition and
Theorem
Exercise Pro jection Formula Let f X Y b e a morphism of ringer
spaces F an O mo dule and E a lo cally free O mo dule of nite rank Prove
X Y
that there are natural isomorphisms for all i
i i
R f f E F E R f F
O O
X Y
We now brie y discuss the Leray sp ectral sequence
COHOMOLOGY OF COHERENT SHEAVES
Theorem Leray Sp ectral Sequence Let f X Y be a continuous map
between topological spaces and F a sheaf of abelian groups on X Then there is
a spectral sequence
p q
p q p q
E H Y R f F H X F
In particular we have a functorial ve term exact sequence of low degree terms
H Y f F H X F Y R f F H Y f F H X F
Proof If I is an injective sheaf on X it is asque and hence so is f I in fact f I
is injective hence Y acyclic Hence there is a Grothendieck sp ectral se
quence of comp osite functors which is just the Leray sp ectral sequence describ ed
E while there is ab ove To get the ve term exact sequence note that E
an exact sequence
E E E E
p q
p q
in other words E E for p q or Since the limit of the
i
sp ectral sequence is the sequence of groups H X F the group H X F ts
into an exact sequence
E H X F E
and there is a step ltration
F H X F F H X F F H X F H X F
where F H X F E F F E and F F E In particular there
is an inclusion E H X F giving a map E H X F Combining
the ab ove we get an exact sequence
E H X F E E H X F
which is the ve term exact sequence
Corollary Let f X Y be an a ne morphism between Noetherian
schemes and let F be quasi coherent on X Then
i
a R f F for al l i and
i i
H X F b there are natural isomorphisms H Y f F
Proof We rst prove a From lemma it su ces to observe that if U Y is
i
a ne then since f U V is also a ne since f is a ne and so H V F j
V
p q
This proves a For b note that the Leray sp ectral sequence has E
for q and hence degenerates at E giving isomorphisms
i
i i i
H Y f F E E H X F
Remark In a similar way if f X Y is any continous map and F
i
a sheaf with R f F for i the Leray sp ectral sequence implies that
i i
H Y f F H X F
K H PARANJAPE AND V SRINIVAS
One application of the Leray Sp ectral Sequence is to the extension of the Co
herence Theorem b to prop er morphisms The pro of uses the Chow Lemma
which states that for any prop er morphism g X S of No etherian schemes
there exists a pro jective morphism h Z S and a morphism of S schemes
p Z X such that for some op en dense subscheme U X the morphism
p U U is an isomorphism The following pro of is prototypical of pro ofs of
results for prop er morphisms by reduction to the case of pro jective morphisms
Theorem Let f X Y be a proper morphism between Noetherian
i
schemes and F a coherent sheaf on X Then R f F is coherent on Y for al l
i
Proof Since the prop erty is lo cal on Y we reduce to the case Y Sp ec A is a ne
i
Now we are reduced to proving H X F is a nite A mo dule for any coherent
F We do this by No etherian induction on supp F the closed subscheme of X
determined by the coherent annihilator ideal sheaf of F If supp F is a closed
p oint x then the residue eld k x is a nitely generated A algebra hence a nite
A mo dule since F is a skyscrap er sheaf whose stalk is a nite dimensional k x
i
vector space we have that H X F is a nite A mo dule and H X F for
i Next if i supp F X is the inclusion then F i i F and we have
i i
H X F H supp F i F So we may assume X supp F
By Chow s Lemma we can nd a morphism p Z X such that the comp osi
tion g p f Z Y is pro jective and for an op en dense U X the morphism
p U U is an isomorphism From lemma the natural map F p p F
has kernel an cokerel supp orted on the complement of U and the higher direct
i
images R p p F are also supp orted on the complement of U By No etherian in
duction we thus know that this kernel and cokernel as well as the higher direct
i
images R p p F have cohomology mo dules on X which are nite A mo dules
In particular from suitable long exact sequences of cohomology mo dules we are
i
reduced to proving that H X p p F is a nite A mo dule for all i
In the Leray sp ectral sequence
s t
s t s t
E H X R p p F H Z p F
n
the limit terms H Z p F are nite A mo dules since Z Y Sp ec A is
s t
pro jective and p F is coherent Hence all the E terms are nite A mo dules
s t
hence E is a nite A mo dule for each s t and all su ciently large r Further
r
s t
s t
we are given the No etherian induction hypothesis that E H X R p p F
is a nite A mo dule whenever t From the exact sequences
i
i r r i
E E E
r r r
i
we conclude by descending induction on r that E is a nite A mo dule for all
r
i
i
H X p p F is the desired i and all r The nite generation of E
conclusion
COHOMOLOGY OF COHERENT SHEAVES
Some relations between coherent sheaf cohomology and local
cohomology
If A is a No etherian ring I an ideal and M an A mo dule let
n
M fm M j I m for some n g H
I
i
Recall that the local cohomology module H M is de ned to b e the value on M
I
of the i th derived functor of M H M By de nition this means that for
I
any injective resolution M J
i
H M i th cohomology of H J
I I
By de nition the lo cal cohomology dep ends only on the radical of I
The category of A mo dules is equivalent to the category of quasi coherent
sheaves on X Sp ec A If Z X is the subscheme de ned by the ideal I
e
or equivalently the coherent ideal sheaf I then we have a natural identi cation
f
H M X M
Z
I
Lemma Let A I be as above For any A module M we have natural iden
ti cations for al l i
i i
f
M H X M H
I Z
f
Proof If M J is an injective resolution then by lemma M
f
J is a asque resolution and asque sheaves are acyclic for cohomology with
supp ort Hence
i i i f i
H M H H J H X J H X F
Z
I I Z
Corollary Let U Sp ec A n Sp ec A I There is an exact sequence
f
H M M H U M j H M
U
I I
and natural isomorphisms
i i
f
H U M j H M i
U
I
Proof There is a long exact sequence of cohomology with supp orts
f f f f f
H X M H X M H U M j H X M H X M
U
Z Z
f f
H U M j H X M
U
Z
from which the corollary follows immediately since on the a ne scheme X we
i
f f
have H X M M and H X M for i Theorem
The ab ove corollary provides the basic link b etween lo cal cohomology mo dules
de ned in commutative algebra and the cohomology theory of quasi coherent
sheaves We will illustrate how this is useful in one imp ortant situation
Let R R b e a graded ring where A R is No etherian each R is
n n n
a nite A mo dule and R is a nitely generated A algebra Let R R b e the
irrelevant graded ideal i e R R ker R A Let X Pro j R b e
n n
K H PARANJAPE AND V SRINIVAS
the asso ciated pro jective A scheme If M M is a graded R mo dule there
n Z n
f f
is an asso ciated quasi coherent sheaf M on X If F M we let F n denote
the sheaf M n where M n is the graded R mo dule with underlying R mo dule
M and shifted grading M n M If R is generated by R as an A algebra
r n r
n
then O is an invertible sheaf on X and F n F O
X O X
X
Theorem In the above situation there is a natural exact sequence
H M M H X F n H M
n Z
R R
+ +
and there are natural isomorphisms
i i
H H X F n M
n Z
R
+
Proof Let Y Sp ec R U Sp ec R n Sp ec A where we regard Sp ec A as the
subscheme de ned by the ideal R There is a canonical morphism of A schemes
f U X given on p oints by asso ciating to a prime ideal A the prime
ideal e generated by all homogenous elements in provided R i e if
U Y this homogeneous prime ideal do es not contain R and is hence a
p oint of Pro j R
At the level of schemes the morphism f may b e de ned as follows if h R is
a homogeneous element we have an a ne op en subset Y Sp ec R Y as well
h h
as an a ne op en subset D h Sp ec R where R is the subring of elements
h h
of degree in the Z graded ring lo calization R Note that Y U and U is
h h
covered by such op en subsets The morphism Y D h is that determined by
h
the incusion R R as the subring of elements of degree
h
h
One sees easily that the diagrams
Y Y
h h h
1 2
i
D h h D h
i
commute for each i where Y Y Y and so these lo cally de ned
h h h h
1 2 1 2
morphisms patch together to de ne f U X One veri es further that
f D h Y this just says h e h so f is an a ne morphism
h
e
Now let F denote the quasi coherent sheaf on Y asso ciated to the R mo dule M
e
ignoring the grading The R mo dule of sections of F on Y is just M which is
h h h
a Z graded mo dule over the Z graded ring R Notice that by de nition of F n
h
the R submo dule of homogeneous elements of degree n in M is naturally
h
h
identi ed with the R mo dule of sections of F n on D h Thus we have a
h
canonical isomorphism of quasi coherent sheaves on X
e
F n f F j
n Z U
i
e
Since f is an a ne morphsim R f F for all i and so by the Leray sp ec
e
tral sequence for f U X and the sheaf F we have a canonical identi cation
i i i
e e
H X F n H X f F j H U F j
n Z U U
Now the Theorem follows from Corollary
COHOMOLOGY OF COHERENT SHEAVES
i
Note that the lo cal cohomology mo dules H M carry natural gradings for
R
+
example as a consequence of the ab ove Theorem
Corollary In the above situation we have an exact sequence of nite A
modules for each n Z
M M M H X F n H H
n n n
R R
+ +
and isomorphisms
i
i
H H X F n M
n
R
+
N
Example Let X P b e a pro jective scheme and R R the ho
n n
k
mogeneous co ordinate ring of X Then from Theorem we have
H R
R
+
coker R H X O n R H
n Z X
R
+
i i
H R H X O n
n Z X
R
+
Thus if X is a normal pro jective variety so that the a ne cone Sp ec R is normal
except p erhaps at the vertex V R then the normality of the cone i e of R
is equivalent to the vanishing of H R or equivalently to the surjectivity for
R
+
each n of the natural maps
N
N
n H X O n H P O
X
P
k
k
whose image is just R H X O n
n X
On the other hand the condition that R is Cohen Macaulay is equivalent to
this surjectivity together with the vanishing result
i
H X O n i dim X n Z
X
For example let E b e a smo oth pro jective plane curve of degree an elliptic
curve and X E E its Segre embedding in P The homogeneous co ordinate
k
ring of X is then easily seen to b e normal but it is not Cohen Macaulay since
H X O
X
Lemma In the situation of Theorem suppose M is a nite graded
R X module and F the corresponding coherent sheaf on X Let P t be the
Hilbert polynomial of M and hence also of F Then
X
i i
H P n M M
n n
R
+
i
P
i i
Proof Since P n F n H X F n the lemma follows
i
from Corollary
The ab ove lemma deals with the case of sheaves on a pro jective scheme over an
Artinian ring There is another situation where one has a siimilar result which
is proved by reduction to the Artinian case Let A m b e a No etherian lo cal
n
ring I an m primary ideal Let R I I b e the Rees algebra of I and
n
let X Pro j R I b e the blow up scheme of I If M is any nite A mo dule we
n
can asso ciate to it a nite graded R I mo dule M I M and hence a
n
K H PARANJAPE AND V SRINIVAS
g
coherent sheaf F M on X Since X Sp ec A is an isomorphism over the
punctured sp ectrum we deduce that for each n we have
i the kernel and cokernel of the natural map M H X F are supp orted
at m i e have nite length and
i
ii the A mo dules H X F n for i are supp orted at m
We also have asso ciated to M a p olynomial P t Q t the Hilbert Samuel
n
polynomial with the prop erty that P n M I M for all n Since
n
X
n j j
M I M I M I M
j
the existence of a Hilb ert Samuel p olynomial as ab ove follows from the exis
n n
tence of one for the nite graded mo dule I M I M over the graded ring
n
n n
R I R I I I to which Theorem applies
n
Theorem Johnston Verma Trivedi For any nite A module M with Hilbert
Samuel polynomial P t Q t we have
X
n i i
P n M I M H M n
n
R I
+
i
P
i i
Proof Let Q n H M For n we have Q n
n
i R I
+
from Corollary b ecause
i the graded R I mo dules M and H X F n coincide in high enough
n
degrees b oth graded mo dules determine the same coherent sheaf F on X
this implies the terms in Q n for i vanish for n
i
ii H X F n for i and n by Serre vanishing so the terms in
Q n for i vanish
n
On the other hand M I M P n for large n since P t is the Hilb ert
Samuel p olynomial of M relative to I Hence the formula in the Theorem is valid
for n
n n
Let gr M I M I M which is a nite graded mo dule over gr A
n
I I
R I R I There is an exact sequence of graded R I mo dules
N M gr M
I
n
where N I M is a graded submo dule of M such that the graded
n
quotient M N is concentrated in degree and vanishes in other degrees
e
Hence N M F and we have an asso ciated exact sequence of coherent
sheaves on X
F F G
where G gr M The sequence of graded mo dules gives rise to a long exact
I
sequence of graded lo cal cohomology mo dules
H N H M H gr M H N
I
R R R R
+ + + +
COHOMOLOGY OF COHERENT SHEAVES
Restricting to the homogeneous comp onents of degree n where n we get a
corresp onding b ounded exact sequence of A mo dules of nite length Hence we
obtain a formula
X
i i
Q n Q n H gr M
n
I
R
+
i
On the other hand by lemma we have
X
n i i
G n gr M H gr M Q n Q n
n
I
I R
+
i
n n n
But gr M R I M R I M P n P n for large n Hence
I
G n P n P n for al l n Z in other words P t P t is the
Hilb ert p olynomial for G This means
n n
P n M I M P n M I M Q n Q n
Hence if the Theorem holds for n then it also holds for n
The formal function theorem
In this section we discuss the Formal Function Theorem and some applications
Theorem Let f X Y be a proper morphism and F a coherent sheaf
n
on X Let y Y and let X X Sp ec O m where m O is the
n Y Y y y Y y
y
maximal ideal let i X X be the given closed immersion Then there is a
n n
canonical isomorphism
i i
b
lim R f F O F H X i
y O Y y n
n
Y y
n
b
where O is the m adic completion of O
Y y y Y y
Proof We will consider only the case of pro jective morphisms though the result
holds more generally for prop er morphisms the prop er case can b e reduced to
the pro jective case as in the pro of of Theorem
We rst reduce to the case Y Sp ec A where A is No etherian lo cal and y is
i
i
the closed p oint Then R f F H X F We may further make the at base
b
b b b b b b
change to Sp ec A if X X Sp ec A X X and f X Sp ec A
Sp ec A
then Corollary we have
i i
b b
H X F A H X F
A
b
and also X X Hence we further reduce the pro of of the Theorem to the case
n n
b
when A A Now we must show that
i i
H X F lim H X i F
n
n
n
is an isomorphism
Since X is pro jective over A we may embed it as a closed subscheme of some
N n N
X is P Let A A m where m is the maximal ideal of A Then X P
n n
A A
n
N
Replacing F and i F by their direct images realized as a closed subscheme of P
n A n
K H PARANJAPE AND V SRINIVAS
N N
resp ectively which do esn t change the corresp onding cohomology on P and P
A A
n
N
mo dules we may assume without loss of generality that X P
A
Now we prove the Theorem by descending induction on i First the Theorem
N
N
r for all r Z by Theorem applied to P clearly holds for the sheaves O
P
A
A
N
and P
A
n
Next we show that the Theorem holds for i N for any coherent F We have
a presentation
q p
N N
b F a O O
P P
A A
which remains exact on applying i This gives corresp onding presentations
n
N N p N N q n N
H P O N a H P O N b H P F
P P
A A A
A A
N N
i F for each n Note and an inverse system of similar presentations for H P
n
A
n
i N
i F are mo dules of nite length for any that the cohomology mo dules H P
n
A
n
coherent F any i and any n We now make use of the following lemma
the pro of is left as an exercise
Lemma Let
A B C
n n n
be an inverse system of exact sequences of A modules of nite length Then
lim A lim B lim C
n n n
n n n
is exact
Thus the functor lim is exact on the category of inverse systems of A mo dules
n
of nite length Applying this lemma to the inverse system of presentations for
N N
i F we see that the Theorem is valid for F for i N H P
n
A
n
Now we pro ceed by descending induction on i Cho ose an exact sequence
r
N
a F G O
P
k
N N
Let F i F with a To simplify notation let X P and X P
n n
n
A A
n
G i G If the functors i were exact the pro of would now b e fairly simple we
n
n n
would have an exact sequence
r N N N N
a H X F H X F H X G H X O
n n n n n n n X
n
and isomorphisms
i i
H X F H X G i N
n n n n
We have unconditionally an analogous exact sequence and isomorphism on X
Taking inverse limits over n we would get an exact sequence
N N r N
lim H X F lim H X G lim a H X O
n n n n n X
n
n n n
N
lim H X F
n n
n
COHOMOLOGY OF COHERENT SHEAVES
and isomorphisms
i i
lim lim H X F H X G i N
n n n n
n n
N N
Now the result for H for G F and O a implies it for H for any coherent
X
i
F Similarly for any i N the result for H for all coherent sheaves implies
i
it for H for all coherent sheaves
So our pro of will b e complete if inspite of the failure of exactness of i we still
n
have and For this we need to make use of a lemma
Lemma Let R be a Noetherian ring I an ideal in R and M a nite R
module For each n there exists m n such that the natural map
R R
m n
Tor M R I Tor M R I
is zero
Proof Cho ose a presentation
N F M
R
M R J JF N J N So it su ces where F is free of nite rank Then Tor
to show that for some m n the natural map
m n
I F N I F N
n
factors through the submo dule I N This is an immediate consequence of the
Artin Rees lemma
Returning to our situation we have exact sequences
O
r
X
a F G O i T or F i O
n n X n X
n n
n
O
X
where for quasi coherent F G the sheaf T or F G is the quasi coherent sheaf
i
such that for any a ne op en U Sp ec A with M U F N U G we
have
O
A
X
M N T or F G j Tor
U
i
i
We break up the ab ove exact sequence into short exact sequences
O
X
G H i T or F i O
n n n X
n
n
r
a F H O
n n X
n
From lemma it follows that
O
i
X
lim i H X i T or F i O
n n X
n
n
n
Hence using lemma and appropriate long exact sequences in cohomology we
deduce that
i i
lim lim H X G H X H i
n n n n
n n
We have an inverse system of exact sequences
r N N N N
a H X F H X F H X H H X O
n n n n n n n X n
K H PARANJAPE AND V SRINIVAS
and isomorphisms
i i
H X F H X H i N
n n n n
Taking inverse limits using lemma and it follows that and
are valid as desired
We now pro ceed to give some applications of the Formal Function theorem
An easy consequence is
Corollary Let f X Y be a proper morphism between Noetherian
i
schemes For each y Y with ber X X Sp ec k y the stalk R f F
y Y y
vanishes for al l i dim X
y
Proof It su ces to prove the completion of this stalk vanishes which is true from
Theorem b ecause all the terms in the inverse limit are
Corollary Let f X Y be a proper morphism between Noetherian
schemes which is quasi nite i e al l set theoretic bers have nite cardinality
Then f is a nite morphism
Proof Since f O is a coherent sheaf on Y it su ces to prove f is an a ne
X
morphism For this we may assume Y is a ne and have to show X is a ne
i
From Corollary we have R f F for all i for all coherent F Since Y
i
is a ne this is the same as saying H X F for all i Now by Serre s
criterion Theorem X is a ne
Theorem Connectedness Theorem Let f X Y be a proper morphism
with f O O Then al l bers of f are connected
X Y
Proof Let y Y and X X Sp ec k y the b er Let X b e the subscheme
y Y n
n
of Y de ned by X X Sp ec O m where m is the maximal ideal of
n Y Y y y
y
O As a top ological space X coincides with X X Hence if X is
Y y n y y
not connected x a connected comp onent Z and let Z b e the corresp onding
n
with a j comp onent of X We can nd a unique function a X O
n Z n n n X
n n
clearly a j a Hence the sequence fa g determines a and a j
n X n n n X nZ
n n
n 1
well de ned element
b b
a lim O f O O H X O
Y y X y Y y n X
n
n
which is clearly a non trivial idemp otent element i e a a with a
b
Since O is a lo cal ring this is a contradiction
Y y
Corollary Zariski s Main Theorem Let f X Y be a birational
proper morphism of Noetherian integral schemes with Y normal Then al l bers
of f are connected
Proof The problem is lo cal on Y so we may assume Y Sp ec A is a ne
By Theorem it su ces to prove f O O which in this case means
X Y
X O A If B X O then since f is birational A and B have
X X
the same quotient eld On the other hand Theorem implies B is a nite
A mo dule Hence B is integral over A and since A is normal B A
COHOMOLOGY OF COHERENT SHEAVES
Corollary Stein factorization Let f X Y be a proper morphism of
Noetherian schemes Then we can uniquely up to isomorphism factorize f as
a composition f g h with h X Z g Z Y such that g is nite and
h O O in particular h has connected bers
X Z
Proof De ne Z Sp ec f O and g h to b e the evident maps Since f O
X X
is O coherent g is nite that h O O follows b ecause g O f O
Y X Z Z X
g h O by construction so that the natural map O h O b eco es an isomor
X Z X
phism on applying g with g nite
Prop osition Let X be a Noetherian scheme Z X a closed local complete
i
intersection subscheme and f Y X the blow up of Z in X Then R f O
Y
for al l i
Proof For simplicity we assume X Z and hence also Y are regular the pro of
in the general case is a little more technical and is left to the reader
The question is lo cal on X so we may assume without loss of generality that
X Sp ec A is a ne and Z V I is a complete intersection where I is
i
generated by a regular sequence a a A Since R f O is a coherent
r Y
i
sheaf it su ces to show that for each closed p oint x Z the stalk R f O
Y x
vanishes
r
P Now the exceptional divisor E f Z as Z schemes and the conormal
Z
r 1
sheaf to E in Y is O This implies that for x Z the b er
P
Z
r
P E Sp ec k x Y Y Sp ec k x
Z x X
k z
mo dule of rank s dim Z Further the conormal sheaf of Y in E is a free O
x Y
x
this uses that Z is regular which implies X is regular at p oints of Z Hence if
I m O is the ideal sheaf of Y in Y where m is the maximal idela of O
x Y x x X x
then the conormal sheaf I I of Y in Y ts into an exact sequence
x
s
O F O
Y
s
Y
x
r
P Since Y this sequence must b e split exact Thus Y is a lo cal complete
x x
k x
s
intersection in Y with conormal sheaf O O
Y
x
Y
x
n
Let Y denote the subscheme de ned by the ideal sheaf I Then Y Y
n x
Y and we have exact sequences
n red
n
S I I O O
Y Y
n
n+1
n
where S I I is the n th symmetric p ower which is a direct sum of invert
r 1
t over a sequence of non negative integers t Hence ible sheaves O
i i i
P
k (x)
i n i
for i we get H Y S I I for all i Since H Y O
x x Y
x
i
lim n i H Y O
n Y
n
n
i
By Theorem we have that R f O for all i
Y x
K H PARANJAPE AND V SRINIVAS
Base Change and Semicontinuity
Let f X Y b e a prop er morphism of No etherian schemes and F a co
herent sheaf on X The Formal Function Theorem gives relations b etween the
i
stalks of R f F and the cohomology along the b ers where we have the take all
p ossible non reduced structures on these b ers into account The Base Change
and Semicontinuity theorems address the situation where F is assumed to b e at
over Y i e for each x X the stalk F is a at O mo dule where we
x
Y f x
can in appropriate situations restrict attention to the scheme theoretic b ers
themselves
i
Since we are interested in comparing the stalks of R f F and the b er coho
mology of the restriction of F we may assume without loss of generality that
Y Sp ec A is a ne
We rst discuss a criterion for atness
Prop osition Let T be an integral Noetherian scheme and F a coherent
N N
sheaf on P Let f P T be the structure morphism For any t T let F
t
T T
N
denote the restriction of F to P and let P x Q x be the Hilbert polynomial
t
k t
of F Then the fol lowing are equivalent
t
a F is at over T
b For al l n f F n is a locally free sheaf on T
c The Hilbert polynomial polynomial P x Q x is independent of the choice
t
of t T
Proof The Theorem is lo cal on T So we may assume T Sp ec A where A is a
No etherian lo cal domain
i i N
For su ciently large n we have that R f F n i e H P F n for
T
N N
all i Let U fU U g b e the standard a ne op en cover of P P
N
T A
Then the Cech complex C U F n determines an exact sequence of A mo dules
N N
H P F n C U F n C U F n C U F n
T
Now if F is at over T so is F n so that C U F n is a complex of at
N
A mo dules The ab ove exact sequence then implies H P F n is also a at
T
A mo dule since it is a nitely generated mo dule it must b e a free A mo dule of
nite rank This means f F n is lo cally free Conversely if f F n is lo cally
N
free for all n n then M H P F n is a free A mo dule of nite rank
n
T
for each n n The graded A X X mo dule M determined by M
N n
N
for n n M H P F n for n n has the prop erties that i M is a
n
T
f
at A mo dule in fact a freee A mo dule ii F M is the corresp onding sheaf
N
on P Pro j A X X In particular the A mo dule of sections of F on
N
T
U is at over A since it is a direct summand of M which is clearly a at
i X
i
X X X
i i i
A mo dule it is the direct limit of M M M which is a direct system
of free A mo dules Thus we have shown that a and b of the Prop osition are
equivalent
N
Let P x Q x denote the Hilb ert p olynomial of F on the generic b er P
k T
where k T is the quotient eld of A Let t T b e the closed p oint We show
that b and c are equivalent to
COHOMOLOGY OF COHERENT SHEAVES
c the Hilb ert p olynomial P x coincides with P x
t
m
We rst show b c Let A A k b e a presentation of the residue
eld k of A This determines a presentation
m
F F i F
t
N N
where i P P is the inclusion of the closed b er For all large n this
A
k
induces by the Serre vanishing theorem a presentation
N m N N
H P F n H P F n H P F
t
A A k
On the other hand there is a similar presentation
N m N N
H P F n A H P F n H P F n k
A A
A A A
Comparing these we deduce that for large n we have that
N N
H P F n H P F n k
t A
k A
Now by Nakayama s lemma
N
M H P F n is a free A mo dule rank M dim M k
n A n k n A
A
since A is a lo cal domain But rank M P n for large n and
A n
N
dim M k dim H P F P n
k n A k t t
A
for all large n Hence c b Clearly c c So it su ces to show
b c So assume b and let s T b e any p oint and T Sp ec A
s s
N N
P b e the where A O is the corresp onding lo cal ring Let g P
s T s
A A
s
i N i N
g F n H P F n A corresp onding morphism We have that H P
A s
A A
s
since the Cech complex computing the cohomology of F n lo calizes to that
computing the cohomology of g F n In particular we reduce to the situation
when b holds and s is the closed p oint of T in which case we have already
proved that c holds
Prop osition Mumford Let f X Sp ec A be a proper morphism where
A is Noetherian and let F be a coherent sheaf on X which is at over A Then
there is a bounded complex
P P
n
where P are projective A modules of nite rank such that for any Noetherian
i
A algebra B if Y X Sp ec B g Y X then there are isomorphisms
Sp ec A
i i
H P B H Y g F
A
functorial in B
Proof Fix an a ne op en cover U fU U g os X Then for any A algebra
n
B there is an induced a ne op en cover U fg U g U g where
B n
g U U Sp ec B Hence the Cech complex C U F is such that there
i i Sp ec A
are canonical isomorphisms
i i
H Y g F H C U F B
A
K H PARANJAPE AND V SRINIVAS
Since F is at over A all terms in the Cech complex C U F are at A
mo dules If P C U F is a sub complex of nitely genrated pro jective A
mo dules such that the the induced maops on cohomology mo dules are all iso
morphisms then the corresp onding mapping cone is an exact sequence of at A
mo dules and so remains exact on tensoring with any A algebra B Forming the
mapping cone commutes with change of rings in fact it commutes with the func
tor on complexes asso ciated to any additive functor b etween ab elian categories
Hence P B C U F B also induces an isomorphism on cohomology
A A
mo dules The desired functoriality in B is manifest from the de nition
Hence using Theorem the Prop osition follows from Lemma b elow
Lemma Let A be a noetherian ring
M M
n
i
a bounded complex of at A modules such that the cohomology modules H M
are nite A modules for al l i Then there exists a complex of nitely generated
projective A modules
P P
n
together with an injective map of complexes P M which induces isomorphisms
on cohomology modules
Proof This is left as an exercise to the reader see Ha I I I Lemma
We call a complex P as ab ove asso ciated to a coherent sheaf F on a prop er
A scheme X a Mumford complex relative to A for F
Theorem Semicontinuity Theorem Let f X Y be a proper morhism
of Noetherian schemes and let F be a coherent sheaf on X which is at over Y
For y Y let X X Sp ec k y be the ber over y F the poull back of F
y Y y
to a coherent sheaf on X Then for each i the function
y
i
y dim H X F
k y y y
is upper semicontinuous on Y i e the subset of Y on which this function is m
is closed for each m
Proof The theorem is lo cal on Y so we may assume Y is a ne Then by Prop o
sition we are reduced to showing the following
if P P is a complex of nitely generated pro jective mo dules
n
over a No etherian ring A then the function on Sp ec A de ned by
i
dim H P k
A
k
is upp er semicontinuous where k is the residue eld of A Further lo calizing
we may assume P are free A mo dules
i
If G is an A linear map b etween free A mo dules then after choosing
bases is determined by an r s matrix M M A where r rank F
r s
s rank G The rank of k F k G k is clearly
A A A
lower semicontinuous on Sp ec A i e the set
f Sp ec A j rank mg
COHOMOLOGY OF COHERENT SHEAVES
is closed for each m which is true b ecause it is the subset of Sp ec A de ned by
the vanishing of all minors of size m of the asso ciated matrix
Applying this to the segment
i 1
i
P P P
i i i
of our complex P of free mo dules if a rank P then
i i
i
f Sp ec A j dim H P k m
k
f j rank a pg f j rank p mg
i i i
k k
a p m
i
This is a closed subset of S pecA
Corollary With the same hypotheses as Theorem assume Y is integral
and suppose that for some i
i
y dim H X F
k y y y
i
is a constant function on Y Then R f F is locally free and the natural maps
i i
R f F k y H X F
y O y y
Y y
are isomorphism for al l y
Proof We may assume Y Sp ec A is a ne and there is a Mumford complex
P for F consisting of free A mo dules of nite rank with rank P a
i i
If m is the constant value of the function in the statement of the Corollary
then we get that
m a rank rank
i i i
k k
Since the functions rank and rank are lower semicon
i i
k k
tinuous but their sum is constant they must b oth actually b e constant This
means the image and kernel of and must b e pro jective A mo dules of nite
i i
i
rank and hence so is the i th cohomology mo dule of P Thus R f F is lo cally
free on Sp ec A Further taking i th cohomology commutes with tensoring with
any A algebra B in this situation in particular letting B k for some prime
ideal we get that
i i i i
H X F k H P B H P k H X F
A A A y y
We give an application of this corollary
Corollary Let Y be an integral scheme of nite type over an algebraically
closed eld f X Y be a proper at morphism with integral bers Let L M
be any invertible sheaves on X such that their restrictions to each ber X are
y
isomorphic Then M L f N for some invertible sheaf N on Y
K H PARANJAPE AND V SRINIVAS
Proof Replacing L by L M we reduce to the statement that if an invertible
O
X
f N for some sheaf L on X has trivial restriction to all b ers of f then L
invertible sheaf N on Y In fact we will show this is true with N f L
for all closed p oints by Since dim H X L dim H X O
k y y y k y y X
y
Corollary the semicontinuity theorem implies that dim H X L
k y y y
for all y and by Corollary f L is a lo cally free sheaf and hence an invertible
sheaf on Y since the generic stalk of f L is a dimensional vector space over
the function eld of Y Further the map y f L H X L is an
y y y
for all y this means that the natural map O isomorphism for all y Since L
X y
y
f f L L is surjective hence an isomorphism
We now give the statement of the technically imp ortant Base Change Theorem
We do not give the pro of here see Ha I I I Theorem
Theorem Base Change Theorem Let f X Y be a proper morphism
of Noetherian schemes and F a coherent sheaf on X such that F is at over Y
Let y Y let X be the ber and F the pul lback of F to X
y y y
a If the natural map
i i
y R f F k y H X F
i y y y
is surjective then it is an isomorphism and the same is true for al l y in
an open neighbourhood of y
b Suppose y is surjective Then
i
i
R f F is locally free in a neighbourhood of y y is also surjective
i
We give an application to illustrate how the Base Change Theorem is used
i
Corollary Let f X Y and F be as above Suppose H X F
y y
i
for some y Then R f F vanishes in an open neighbourhood of y and the map
i i
y R f F k y H X F is an isomorphism
i y y y
Vanishing theorems formal duality and applications
We rst state some imp ortant vanishing theorems for the cohomology of certain
sheaves on non singular pro jective varieties over an algebraically closed eld k
of characteristic Though these theorems now admit algebraic pro ofs using
characteristic p techniques based on the results of Deligne and Illusie the original
pro ofs are by reducing to the case k C and using analytic metho ds For a more
systematic discussion of this imp ortant topic see the b o ok EV
Theorem Ko daira Vanishing Theorem Let L be an ample invertible sheaf
on a non singular irreducible projective variety over a eld k of characteristic
i i
Then H X L for al l i dim X or equivalently H X L for
X
al l i
This can b e generalized in two di erent directions as follows
COHOMOLOGY OF COHERENT SHEAVES
Theorem Ko daira Akizuki Nakano Vanishing With the same hypotheses
j
i
as Theorem we have H X L provided i j dim X Equiva
X k
j
i
lently H X L for al l i j dim X
X k
Thus we have a vanishing result for sheaves more complicated than invertible
sheaves The equivalence of the tweo forms of the theorem follows from Serre
duality since
n i i
H om
O X
X k
X
X k
De nition Let L b e an invertible sheaf on a prop er variety X over a eld
k
i If for each morphism f C X from an irreducible curve C the degree of
the invertible sheaf f L on C is non negative we say L is nef
ii If for some constant A we have
n dim X
dim H X L An
k
for a sequence of integers n tending to in nity we say L is big
Theorem Kawamata Viehweg Let X be a non singular irreducible pro
jective variety over an algebraically closed eld k and L an invertible sheaf on
i
X Assume L is nef and big Then H X L for i dim X
Thus we get the conclusion of Ko daira Vanishing with a weaker hypothesis on
L than ampleness The case dim X of this result is due to C P Ramanujam
Theorem Grauert Riemenschneider Vanishing Let f X Y be a proper
birational morphism of irreducible varieties over a eld k of characteristic
i
where X is non singular Then R f for al l i
X
Proof The question is lo cal on Y so we may assume rst that Y is a ne then
may replace Y by a normal pro jective compati cation Y By Hironaka s theorem
on resolution of singularities we can nd a non singular X with a prop er mor
phism to Y which extends the given map f In other words it su ces to consider
the case when Y is pro jective
First assume X is also pro jective Let L b e a very ample invertible sheaf on Y
Then f L is a nef and big see De nition invertible sheaf on X Hence by
i n i n
Theorem H X f L for all i n or equivalently H X f L
X
for all n
Consider the Leray sp ectral sequence
p q
p q n p q n
H Y R f f L H X L E
X X
i n
By the pro jection formula see we have isomorphisms R f f L
X
i n
R f L for all i Since L is ample on Y Serre vanishing implies that
X
p q
we have E for p for all q for all large n Hence the sp ectral sequence
degenerates at E and we have isomorphisms
i n i n
H X f L for i H Y R f L
X X
i n
But R f L is generated by its global sections for all i for large enough
X
i
n Hence we must have R f for i
X
K H PARANJAPE AND V SRINIVAS
If now X is merely prop er over k we may nd a pro jective birational morphism
g Z X such that f g Z Y is pro jective and birational of course and Z
is non singular In fact if X X is a birational morphism with X pro jective
which exists by Chow s Lemma we may by Hironaka s theorem take Z to b e
a pro jective resolution of singularities of X The morphism g Z X is a
pro jective birational morphism with Z non singular and X normal in fact non
i
singular so by the case already considered we have R g for all i
Z
Also we have a natural map g which is an isomorphism over the op en
X Z
subset where Z X is an isomorphism now applying g to this we obtain an
inclusion g which is an isomorphism outside a co dimension subset of
X Z
X Since is invertible and g is torsion free we must have g
X Z X Z
Now consider the Grothendieck sp ectral sequence for the hihgher direct images
of a comp osition
p q
p q p q
R f R g R f g E
Z Z
p q
We have E for q and all p so that the sp ectral sequence degenerates
and we have isomorphisms
i
i
E R f g i
Z
where we have used the Theorem for f g which is a morphism b etween pro jective
i
i i
varieties But E R f g R f
Z X
We now state a duality theorem for cohomology with supp ort which will allow
us to get an equivalent form of Theorem which is useful for applications
Theorem Formal Duality Let X be an irreducible non singular proper
variety of dimension n over a eld k and let Y be a closed subset Let I be
Y
the ideal sheaf for the corresponding reduced subscheme of X Let F be a locally
free sheaf of nite rank on X Then we have a functorial isomorphism of not
necessarily nite dimensional k vector spaces
n i i m
lim Hom H H X F O I X H om F k
k X O X
Y
X
Y
m
Proof We have natural isomorphisms
j j
m
H X G lim O I G Ext
X
Y
Y O
X
!
m
for any O mo dule G for all j Indeed b oth sides de ne functors in G on
X
Mod X which vanish if G is injective the cohomology with supp ort vanishes
b ecause G is asque and are b oth hence universal functors Since they coincide
when j why they must coincide for all j
Hence if V Hom V k for any k vector space V we have
k
j j
m
H X G lim Ext O I G
X
Y
Y O
X
m
Apply this to G H om F and j n i Since F is lo cally free we have
O X
X
n i n i
m m
Ext O I H om F Ext F O I
X O X X X
Y Y
O X O
X X
i m
H X F O I
X Y
COHOMOLOGY OF COHERENT SHEAVES
by Serre Duality on X see Theorem and Substituting this in the
n i
inverse limit formula for H X G the result follows
Y
Corollary Let f X Y be a surjective morphism with X non singular
projective Let y Y be a closed point with ber X Then for any locally free
y
O module F of nite rank we have an isomorphism
X
i n i
d
H X F O F R f H om
Y y X y O O
X
Y y X
y
Proof This is a consequence of formal duality for the subscheme X of X and
y
the Formal Function Theorem Theorem
Corollary Let f X Y be a birational proper morphism where X is
non singular and proper over a eld k of characteristic Then for any y Y
we have that
i
X O i n H
X
X
y
Proof This follows from Theorem and Corollary
We give two applications of this
Example Let A m b e the lo cal ring of a closed p oint on a normal variety
over a eld of characterstic Supp ose that f X Sp ec A is a resolution of
i
singularities such that H X O for all i we then say Sp ec A has
X
rational singularities Then A is Cohen Macaulay
Indeed let Y Sp ec A X X the b er over the closed p oint U Y n fmg
y
i
V X n X f U Since H X O for i and Y is a ne we
y X
i
have R f O for i From the Leray sp ectral sequence we deduce that
X
i i i
H U O H U f O H V O for all i From the exact sequence
U V V
i i i
H X O H V O H X O
X V X
X
y
and Corollary we deduce that
i i
H V O H U O
V U
i
for i n From Corollary we deduce that H A for
m
i n Since A is normal we deduce that A has depth n and is hence
Cohen Macaulay
Example Let A m b e the lo cal ring of a closed p oint on a normal variety
of dimension n over a eld k of characteristic Assume the punctured sp ectrum
of A is regular Let f X Sp ec A b e a resolution of singularities Then we
have isomorphisms
i i
H X O H A i n
X
m
i
In particular A is Cohen Macaulay H X O for i n
X
i
R f O for i n
X
U so the same Indeed in the notation of the previous example we have V
argument applies giving isomorphisms
i i i i
H A H U O H V O H X O
U V X
m
for i n
K H PARANJAPE AND V SRINIVAS
References
EV H Esnault E Viehweg Lectures on Vanishing Theorems DMV Seminar Band
Birkh auser Basel
Ha R Hartshorne Algebraic Geometry Grad Texts in Math Springer Verlag
Ha R Hartshorne Ample subvarieties of algebraic varieties Lect Notes in Math No
Springer Verlag
Ha R Hartshorne Resdues and Duality Lect Notes in Math Springer Verlag
E mail address kapil imsc ernet in srinivas math tifr res in