From Varieties to Sheaf Cohomology
Thomas Goller
September 1, 2015
Classical algebraic geometry is the study of geometry using algebraic tools. Modern algebraic
geometry is the study of algebra using geometric intuition.
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Varieties (classical)
Affine varieties are vanishing sets U = V(f1 , . . . , fr ) ⇢ An of polynomials fi 2 C[x1 , . . . , xn ].
(We write An for affine space instead of Cn because we won’t use the vector space structure;
in particular, the origin is not apspecial point.) The algebraic functions on this affine variety
U are the ring C[x1 , . . . , xn ]/ (f1 , . . . , fr ). Projective varieties are vanishing sets X =
V(F1 , . . . , Fr ) ⇢ Pn of homogenous polynomials Fi 2 C[X0 , . . . , Xn ]. The ring of rational
functions on X consists of homogeneouspdegree 0 rational functions in X0 , . . . , Xn where
the denominator cannot be in the ideal (F1 , . . . , Fr ) (we need homogeneous degree 0 to
have well-defined functions on Pn ). Projective varieties are compact and have open covers
by affine varieties. (These open affine ‘charts’ can be obtained by taking the subsets of X
where Xi 6= 0).
Example. P1 , the space of lines through the origin in C2 , can be described as the set of pairs
(a, b) 6= (0, 0) modulo the action of scaling by nonzero constants, so we often write points
of P1 as (a : b) to emphasize that the ratio of a to b is the critical feature. As a projective
variety, P1 is the vanishing set of the empty set of homogeneous polynomials in C[X0 , X1 ].
The subset where X0 6= 0, namely where a 6= 0, can be parametrized by (1 : ab ) for ab 2 C
1
and is therefore the affine variety A1 with function ring C[ X
]. Similarly, the subset where
X0
X0
1
1
X1 6= 0 is a copy of A with function ring C[ X1 ]. Thus P is covered by two copies of the
affine variety A1 . (The function rings show how these two copies of A1 are glued together to
produce P1 . We’ll come back to this.)
GAGA (Serre, 1956) shows that calculations on a projective variety are equivalent to
calculations done on the underlying holomorphic manifold. Classical algebraic geometry is
closely tied to complex analysis.
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Toward schemes
Schemes generalize varieties. To see how, let’s think about affine varieties more algebraically.
The points of the affine variety U = V(f1 , . . . , fr ) are the maximum spectrum of the ring
p
of functions A = C[x1 , . . . , xn ]/ (f1 , . . . , fr ). These maximal ideals are all of the form
m = (x1 a1 , . . . , xn an ) (f1 , . . . , fr ), which we were thinking of as a point ~a = (a1 , . . . , an )
at which all the fi vanish. Think of evaluating a function f 2 A at a point m (which yields a
complex number) as taking the image under the map A ! A/m ' C (modding out by m sets
xi = ai , so this really is just evaluation at p
~a). If we replace the words ‘maximal spectrum’ by
‘prime spectrum’ and ‘A = C[x1 , . . . , xn ]/ (f1 , . . . , fr )’ by ‘A is any commutative ring with
identity’ in the above discussion, then we have arrived at affine schemes. One weird aspect
is that functions f 2 A evaluated at a prime ideal p of A yield values in A/p, which will in
general not be a field. An affine scheme is essentially just a ring A, viewed as functions on
its prime spectrum. Morphisms of affine schemes are induced by ring morphisms B ! A,
which by taking preimages of prime ideals (the preimage of a prime ideal is prime!) yield set
maps Spec A ! Spec B (note that the order of A an B changes, i.e. Spec is a contravariant
functor).
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Schemes (modern)
General schemes are obtained by gluing affine schemes by isomorphisms along open subsets
of those affine schemes. In other words, a scheme is obtained by gluing rings together. Our
topology is the Zariski topology of Spec A, where closed sets are all of the form V(I), namely
the set of all primes containing a particular ideal I. The simplest open subsets of Spec A are
the complements Spec Af of the closed sets V(f ) and the gluing often happens along these
(Spec Af consists of all prime ideals of A not containing f ).
Example. We can glue C[x] and C[y] along the open sets obtained by inverting x and y
via the isomorphism C[x]x ! C[y]y , x 7! 1/y. This yields the projective line P1 . (Writing
X0
1
x= X
and y = X
reveals the connection to the previous example.)
X0
1
From now on, we will view varieties as schemes. We thereby gain a richer topology
(more points!) and a general framework that gives us access to more algebraic tools. For
instance, consider an affine variety Spec A. The maximal ideals of Spec A are in bijection
with ring maps A ! C (the preimage of the zero ideal picks out a maximal ideal of A).
More interestingly, the tangent vectors of an affine scheme are in bijection with ring maps
A ! C[x]/(x2 ) (this isn’t a classical map of varieties due to the presence of nilpotents).
Example. Consider the affine scheme C[x, y], whose maximal ideals (x a, y b) can be
viewed as (a, b) 2 A2 . Ring maps C[x, y] ! C[x]/(x2 ) are determined by x 7! ↵x + a,
y 7! x + b. Now, Spec C[x]/(x2 ) is a single point (x), which under the given ring map gets
sent to the maximal ideal (x a, y b), which can be thought of as the point (a, b) 2 A2 .
@
@
The choice of ↵ and can be thought of as giving the tangent vector ↵ @x
+ @y
. Note that
the ring map carries more information than the induced topological map on prime ideals.
One can also observe generic behavior of a map between varieties by looking at what the
map does at the generic point (the zero ideal), which can be thought of as a fuzzy point
covering the entire space.
Example. Consider the map Spec C[x] ! Spec C[y] defined by C[y] ! C[x], y 7! x2 (this is
analogous to the map C ! C, z 7! z 2 ). The preimage of a maximal ideal (x a) is (y a2 ),
2
so both (x a) and (x + a) in Spec C[x] map to the same point (y a2 ) of Spec C[y]. A fancy
way of seeing that this map is generically two-to-one (which we call degree 2) is to localize
at the generic point: since C[x](0) = C(x), we obtain Spec C[x](0) ! Spec C[y](0) defined by
C(y) ,! C(x), y 7! x2 , which is a field extension of degree 2.
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Coherent sheaves
When we glue together affine varieties Spec A to get a scheme X, we are also gluing together
the rings of functions A to get the structure sheaf of rings OX , which stores the data of all
functions on all open subsets of X. In particular, if the open subset is U = Spec A, then
OX (U ) = A, but to see what the functions on U1 [ U2 are we have to look carefully at how
U1 and U2 are glued together.
Example. Recall that P1 is covered by C[x] and C[y] with gluing isomorphism C[y]y !
C[x]x , y 7! x1 . To see what the global sections of OP1 are (i.e. the functions on the open set
U = P1 ), we try to choose sections on an open affine cover that are compatible under the
gluing isomorphisms. In our case, we need an element f 2 C[x] and an element g 2 C[y] that
are identified under the map y 7! x1 . But f and g are polynomials, so the only compatible
choice is the same constant f = g 2 C and hence the global sections of OP1 are just C.
The structure sheaf OX is a sheaf of rings. In commutative algebra, given a ring A, a
natural thing to do is to study A-modules. In algebraic geometry, the natural extension is to
study OX -modules, which are simply an A-module M for each affine open Spec A together
with gluing isomorphisms that are compatible with the gluing of the rings A. When all the
modules M are finitely generated, we call the resulting sheaf of modules a coherent sheaf.
Coherent sheaves generalize the notion of vector bundles. When we build the sheaf of
modules by gluing free modules, we get a locally free sheaf, which algebraic geometers often
call a vector bundle. This is because the same gluing maps can be used to define an algebraic
vector bundle over X (instead of gluing free modules of rank r, you could glue copies of Cr ),
and the sheaf of sections of this vector bundle is exactly the locally free sheaf you started
with.
Example. The structure sheaf OX is a locally free sheaf of rank 1 and corresponds to the
trivial line bundle X ⇥ C.
Algebraic geometers usually prefer to work with locally free modules since modules are so
natural from the point of view of commutative algebra. Many operations on modules work
for coherent sheaves as well (e.g. direct sum, tensor product, push forward, pullback). And
we have a powerful tool known as sheaf cohomology.
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Sheaf cohomology
Given a coherent sheaf F on a projective variety X (we want X to be compact to ensure that
our cohomology vector spaces are finite dimensional), we often have a good understanding of
what F looks like locally (especially if F is locally free!), but identifying the global sections
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of F is more difficult (even OP1 required some work!). The global sections functor, which
we’ll call H 0 , is left exact, which means that given a short exact sequence of coherent sheaves
0 ! F1 ! F2 ! F3 ! 0
we get an exact sequence of C-vector spaces 0 ! H 0 (F1 ) ! H 0 (F2 ) ! H 0 (F3 ). As we often
do when we have a left exact functor, we can take the derived functor of H 0 , which we call
sheaf cohomology and denote by H i . The point is that our short exact sequence of coherent
sheaves then yields a long exact sequence
0 ! H 0 (F1 ) ! H 0 (F2 ) ! H 0 (F3 ) ! H 1 (F1 ) ! H 1 (F2 ) ! H 1 (F3 ) ! H 2 (F1 ) ! · · ·
which often helps us compute H 0 .
Like other cohomology theories, sheaf cohomology is a way of assigning algebraic invariants (in this case C-vector spaces) to objects we want to study (in this case coherent
sheaves). These invariants tell us something about the objects (H 0 is always global sections;
higher cohomology groups often have a nice interpretation too). Sheaf cohomology is hard to
compute from the definition (usually via C̆ech cohomology), but once some simple cases are
computed (e.g. cohomology of all line bundles on Pn , see Hartshorne III.5), many others can
be deduced from long exact sequences as above and from other nice facts about cohomology,
such as
• X an affine scheme: all cohomology vanishes in degree
1 for all coherent sheaves;
• X a general scheme: all cohomology vanishes in degree > dim X for all coherent
sheaves;
• Serre duality: if X is a smooth projective variety of dimension n and F is a locally
free coherent sheaf on X, then
H i (F) ⇠
= H n i (F _ ⌦ !X ),
where !X is the canonical line bundle.
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Summary
• Schemes generalize both classical algebraic varieties and commutative ring theory.
• We glue rings together to get schemes; we glue finitely-generated modules to get coherent sheaves.
• Sheaf cohomology, the derived functor of global sections, is an algebraic tool for studying coherent sheaves.
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