Algebraic Geometry 2, Summer term 2023.
Lecture course notes.

Content

6 Cohomology of \({\mathscr O}_X\)-modules

General references: [ We ] , [ HS ] , [ Gr ] , [ KS ] , [ GW2 ] .

June 7,
2023

The formalism of derived functors

 

(6.1) Complexes in abelian categories

Reference: [ We ] Ch. 1, [ GW2 ] App. F.

Let \({\mathcal A}\) be an abelian category (e.g., the category of abelian groups, the category of \(R\)-modules for a ring \(R\), the category of abelian sheaves on a topological space \(X\), the category of \({\mathscr O}_X\)-modules on a ringed space \(X\), or the category of quasi-coherent \({\mathscr O}_X\)-modules on a scheme \(X\); see  [ GW2 ] Section (F.7)).

A complex in \({\mathcal A}\) is a sequence of morphisms

\begin{tikzcd} \cdots \rar & A^i \arrow[r, "d^i"] & A^{i+1} \arrow[r, "d^{i+1}"] & A^{i+2} \rar & \cdots \end{tikzcd}

in \({\mathcal A}\) (\(i\in \mathbb {Z}\)), such that \(d^{i+1}\circ d^i = 0\) for every \(i\in \mathbb {Z}\). The maps \(d^i\) are called the differentials of the complex.

Given complexes \(A^\bullet \), \(B^\bullet \), a morphism \(A^\bullet \to B^\bullet \) of complexes is a family of morphisms \(f^i\colon A^i\to B^i\) such that the \(f^i\) commute with the differentials of \(A^\bullet \) and \(B^\bullet \) in the obvious way. With this notion of morphisms, we obtain the category \(C({\mathcal A})\) of complexes in \({\mathcal A}\). This is an abelian category (kernels, images, …are formed degree-wise); see  [ We ] Thm. 1.2.3.

Definition 6.1
Let \(A^\bullet \) be a complex in \({\mathcal A}\). For \(i\in \mathbb {Z}\), we call
\[ H^i(A^\bullet ) := \operatorname{Ker}(d^{i}) / \mathop{\rm im}(d^{i-1}) \]
the \(i\)-th cohomology object of \(A^\bullet \). We obtain functors \(H^i\colon C({\mathcal A})\to {\mathcal A}\). We say that \(A^\bullet \) is exact at \(i\), if \(H^i(A^\bullet ) = 0\). We say that \(A^\bullet \) is exact, if \(H^i(A^\bullet )=0\) for all \(i\).

Remark 6.2
Let \(0\to A^\bullet \to B^\bullet \to C^\bullet \to 0\) be a sequence of morphisms of complexes. The sequence is exact (in the sense that at each point the kernel and image in the category \(C({\mathcal A})\) coincide) if and only if for every \(i\), the sequence \(0\to A^i\to B^i\to C^i\to 0\) is exact.

Proposition 6.3
[ We ] Thm. 1.3.1 Let \(0\to A^\bullet \to B^\bullet \to C^\bullet \to 0\) be an exact sequence of complexes in \({\mathcal A}\). Then there are maps \(\delta ^i\colon H^i(C^\bullet ) \to H^{i+1}(A^\bullet )\) (called boundary maps) such that, together with the maps induced by functoriality of the \(H^i\), we obtain the long exact cohomology sequence
\[ \cdots H^i(A^\bullet ) \to H^i(B^\bullet ) \to H^i(C^\bullet ) \to H^{i+1}(A^\bullet ) \to \cdots . \]

We need a criterion which ensures that two morphisms between complexes induce the same maps on all cohomology objects. See  [ We ] 1.4.

Definition 6.4
Let \(f, g\colon A^\bullet \to B^\bullet \) be morphisms of complexes. We say that \(f\) and \(g\) are homotopic, if there exists a family of maps \(k^i\colon A^i\to B^{i-1}\), \(i\in \mathbb {Z}\), such that
\[ f - g = dk + kd, \]
which we use as short-hand notation for saying that for every \(i\),
\[ f^i - g^i = d_B^{i-1}\circ k^i + k^{i+1}\circ d_A^i. \]
In this case we write \(f\sim g\). The family \((k^i)_i\) is called a homotopy.

Proposition 6.5
Let \(f, g\colon A^\bullet \to B^\bullet \) be morphisms of complexes which are homotopic. Then for every \(i\), the maps \(H^i(A^\bullet )\to H^i(B^\bullet )\) induced by \(f\) and \(g\) are equal.

In particular, if \(A^\bullet \) is a complex such that \(\operatorname{id}_{A^\bullet }\sim 0\), then \(H^i(A^\bullet )=0\) for all \(i\), i.e., \(A^\bullet \) is exact.

Definition 6.6
Let \(A^\bullet \) and \(B^\bullet \) be complexes. We say that \(A^\bullet \) and \(B^\bullet \) are homotopy equivalent, if there exist morphisms \(f\colon A^\bullet \to B^\bullet \) and \(g\colon B^\bullet \to A^\bullet \) of complexes such that \(g\circ f \sim \operatorname{id}_A\) and \(f\circ g\sim \operatorname{id}_B\). In this case, \(f\) and \(g\) induce isomorphisms \(H^i(A^\bullet )\cong H^i(B^\bullet )\) for all \(i\).

(6.2) Left exact functors

Let \({\mathcal A}\), \({\mathcal B}\) be abelian categories. All functors \(F\colon {\mathcal A}\to {\mathcal B}\) that we consider are assumed to be additive, i.e., they induce group homomorphisms \(\operatorname{Hom}_{{\mathcal A}}(A, A')\to \operatorname{Hom}_{{\mathcal B}}(F(A), F(A'))\) for all \(A\), \(A'\) in \({\mathcal A}\).

Definition 6.7
A (covariant) functor \(F\colon {\mathcal A}\to {\mathcal B}\) is called left exact, if for every short exact sequence \(0\to A'\to A\to A''\to 0\) in \({\mathcal A}\), the sequence
\[ 0 \to F(A') \to F(A) \to F(A'') \]
is exact.

Definition 6.8
A contravariant functor \(F\colon {\mathcal A}\to {\mathcal B}\) is called left exact if for every short exact sequence \(0\to A'\to A\to A''\to 0\) in \({\mathcal A}\), the sequence
\[ 0 \to F(A'') \to F(A) \to F(A') \]
is exact.

Similarly, we have the notion of right exact functor. A functor which is left exact and right exact (and hence preserves exactness of arbitrary sequences) is called exact.

Let \(A_0\in {\mathcal A}\). Then the functors \(A\mapsto \operatorname{Hom}_{{\mathcal A}}(A, A_0)\) and \(A\mapsto \operatorname{Hom}_{{\mathcal A}}(A_0, A)\) are left exact.

(6.3) \(\delta \)-functors

Reference: [ We ] 2.1, [ GW2 ] Section (F.48).

Let \({\mathcal A}\), \({\mathcal B}\) be abelian categories.

Definition 6.9
A \(\delta \)-functor from \({\mathcal A}\) to \({\mathcal B}\) is a family \((T^i)_{i\ge 0}\) of functors \({\mathcal A}\to {\mathcal B}\) together with morphisms \(\delta ^i\colon T^i(A'')\to T^{i+1}(A')\) (called boundary morphisms) for every short exact sequence \(0\to A'\to A\to A''\to 0\) in \({\mathcal A}\), such that the sequence
\[ 0\to T^0(A') \to T^0(A) \to T^0(A'') \to T(A') \to \cdots \]
is exact, and such that the \(\delta ^i\) are compatible with morphisms of short exact sequences in the obvious way.

Definition 6.10
A \(\delta \)-functor \((T^i)_i\) from \({\mathcal A}\) to \({\mathcal B}\) is called universal, if for every \(\delta \)-functor \((S^i)_i\) and every morphism \(f^0\colon T^0\to S^0\) of functors, there exist unique morphisms \(f^i\colon T^i\to S^i\) of functors for al \(i {\gt} 0\), such that the \(f^i\), \(i\ge 0\) are compatible with the boundary maps \(\delta ^i\) of the two \(\delta \)-functors for each short exact sequence in \({\mathcal A}\).

The definition implies that given a (left exact) functor \(F\), any two universal \(\delta \)-functors \((T^i)_i\), \(({T’}^i)_i\) with \(T^0 = {T’}^{0} = F\) are isomorphic (in the obvious sense) via a unique isomorphism.

Definition 6.11
A functor \(F\colon {\mathcal A}\to {\mathcal B}\) is called effaceable, if for every \(X\) in \({\mathcal A}\) there exists a monomorphism \(\iota \colon X\hookrightarrow A\) with \(F(\iota )=0\).

A particular case is the situation where each \(X\) admits a monomorphism to an object \(I\) with \(F(I)=0\).

Proposition 6.12
( [ We ] Thm. 2.4.7, Ex. 2.4.5.) Let \((T^i)_i\) be a \(\delta \)-functor from \({\mathcal A}\) to \({\mathcal B}\) such that for every \(i{\gt}0\), the functor \(T^i\) is effaceable. Then \((T^i)_i\) is a universal \(\delta \)-functor.

(6.4) Injective objects

June 12,
2023

Let \({\mathcal A}\) be an abelian category.

Definition 6.13
An object \(I\) in \({\mathcal A}\) is called injective, if the functor \(X\mapsto \operatorname{Hom}_{{\mathcal A}}(X, I)\) is exact.

If \(I\) is injective, then every short exact sequence \(0\to I\to A\to A''\to 0\) in \({\mathcal A}\) splits. Conversely, if \(I\) is an object such that every short exact sequence \(0\to I\to A\to A''\to 0\) splits, then \(I\) is injective.

Definition 6.14
Let \(X\in {\mathcal A}\). An injective resolution of \(X\) is an exact sequence
\[ 0 \to X \to I^0 \to I^1 \to I^2 \to \cdots \]
in \({\mathcal A}\), where every \(I^i\) is injective.

Definition 6.15
We say that the category \({\mathcal A}\) has enough injectives if for every object \(X\) there exists a monomorphism \(X\hookrightarrow I\) from \(X\) into an injective object \(I\). Equivalently: Every object has an injective resolution.

Remark 6.16

  1. The categories of abelian groups, of \(R\)-modules for a ring \(R\), of abelian sheaves on a topological space, and more generally of \({\mathscr O}_X\)-modules on a ringed space \(X\) all have enough injective objects. For these categories, it is not too hard to show this statement directly (for \({\mathscr O}_X\)-modules on a ringed space \(X\), e.g., see  [ H ] Proposition III.2.2). Alternatively, one can show that they all are “Grothendieck abelian categories”, and that those have enough injective objects (see  [ GW2 ] Sections (F.12), (21.2)).

  2. Dually, we have the notion of projective object (i.e., \(P\) such that \(\operatorname{Hom}_{{\mathcal A}}(P, -)\) is exact), of projective resolution \(\cdots \to P_1\to P_0 \to A \to 0\), and of abelian categories with enough projective objects. For a ring \(R\), the category of \(R\)-modules clearly has enough projectives, since every free module is projective, and every module admits an epimorphism from a free module. Categories of sheaves of abelian groups or \({\mathscr O}_X\)-modules typically do not have enough projectives.

Lemma 6.17

( [ We ] Theorem 2.3.7, see also Porism 2.2.7) Let \({\mathcal A}\) be an abelian category and let \(f\colon A\to B\) be a morphism in \({\mathcal A}\). Let \(I^i\in {\mathcal A}\) be injective, \(i\ge 0\), and let \(0\to A\to A^0\to A^1\to \cdots \) be exact with \(A^i\in {\mathcal A}\). Then there exists a morphism \(g\colon A^\bullet \to I^\bullet \) of complexes such that the diagram

\begin{tikzcd} 0 \ar[r] & A \ar[r]\ar[d] & A^0 \ar[r]\ar[d] & A^1 \ar[r]\ar[d] & \cdots\\ 0 \ar[r] & B \ar[r] & I^0 \ar[r] & I^1 \ar[r] & \cdots \end{tikzcd}

is commutative, and \(g\) is uniquely determined up to homotopy.

Lemma 6.18
(cf.  [ We ] Lemma 2.2.8, “Horseshoe lemma”) Let \(0\to A_1\to A_2\to A_3\to 0\) be an exact sequence in an abelian category \({\mathcal A}\). Let \(0\to A_1\to I_1^\bullet \) and \(0\to A_3\to I_3^\bullet \) be injective resolutions. Let \(I_2^i := I_1^i\times I_3^i\) and define \(A_2\to I_2^0\) by using the composition \(A_2\to A_3\to I_3^0\) and lifting the map \(A_1\to I_1^0\) to a map \(A_3\to I^1_0\), using the injectivity of \(I^1_0\). Then we obtain an injective resolution \(0\to A_2\to I_2^\bullet \), and a term-wise split short exact sequence
\[ 0\longrightarrow I_1^\bullet \longrightarrow I_2^\bullet \longrightarrow I_3^\bullet \longrightarrow 0 \]
of complexes.

(6.5) Right derived functors

Theorem 6.19

Let \(F\colon {\mathcal A}\to {\mathcal B}\) be a left exact functor, and assume that \({\mathcal A}\) has enough injectives.

For each \(A\in {\mathcal A}\), fix an injective resolution \(0\to A\to I^\bullet \), and define

\[ R^iF(A) = H^i(F(I^\bullet )),\qquad i\ge 0, \]

where \(F(I^\bullet )\) denotes the complex obtained by applying the functor to all \(I^i\) and to the differentials of the complex \(I^\bullet \). Then:

  1. The \(R^i F\) are additive functors \({\mathcal A}\to {\mathcal B}\), and \(R^iFX\) is independent of the choice of injective resolution of \(X\) up to natural isomorphism.

  2. We have an isomorphism \(F\cong R^0F\) of functors.

  3. For \(I\) injective, we have \(R^i FI=0\) for all \(i{\gt}0\).

  4. The family \((R^i F)_i\) admits natural boundary maps making it a universal \(\delta \)-functor.

We call the \(R^iF\) the right derived functors of \(F\).

Remarks on the proof

For part (1), the main ingredient is Lemma 6.17. Parts (2) and (3) are easy. To construct the boundary maps giving rise to long exact cohomology sequences in Part (4), use Lemma 6.18; since the exact sequence \(0\to I_1^\bullet \to I_2^\bullet \to I_3^\bullet \to 0\) is term-wise split, the exactness is preserved when applying the functor \(F\). Finally, this \(\delta \)-functor is effaceable by (c) and hence universal.

June 14,
2023

Definition 6.20
Let \(F\) be a left exact functor as above. We say that an object \(A\in {\mathcal A}\) is \(F\)-acyclic, if \(R^iF(A) = 0\) for all \(i{\gt}0\).

Definition 6.21
Let \(F\) be a left exact functor as above, and let \(A\in {\mathcal A}\). An \(F\)-acyclic resolution of \(A\) is an exact sequence \(0\to A\to J^0\to J^1\to \cdots \) where all \(J^i\) are \(F\)-acyclic.

Proposition 6.22
Let \(F\) be a left exact functor as above, and let \(A\in {\mathcal A}\). Let \(0\to A\to J^0\to J^1\to \cdots \) be an \(F\)-acyclic resolution. Then we have natural isomorphisms \(R^iF(A) = H^i(F(J^\bullet ))\), i.e., we can compute \(R^iF(A)\) by an \(F\)-acyclic resolution.

Sketch of the proof

The assertion is easy to see for \(i=0\). For \(i=1\), note that \(\operatorname{Ker}(F(J^1)\to F(J^2)) = F(J^0/A)\) by the left exactness of \(F\). From this, one gets \(R^1F(A) = H^1(F(J^\bullet ))\).

From the long exact cohomology sequence attached to the short exact sequence \(0\to A\to J^0 \to J^0/A \to 0\), we get isomorphisms \(R^iF(J^0/A)\cong R^{i+1}F(A)\) for \(i \ge 1\). Since \(0\to J^0/A\to J^1\to \cdots \) is an acyclic resolution of \(J^0/A\), we get

\[ R^{i+1}F(A) \cong R^iF(J^0/A) \cong H^{i+1}(F(J^\bullet )), \]

where the second isomorphism holds by induction.

(6.6) Derived categories

Sometimes it is useful to employ the language of derived categories which gives rise to a notion of derived functor \(RF\) that contains slightly more information than the family \((R^iF)_i\) discussed above (for a left exact functor \(F\colon {\mathcal A}\to {\mathcal B}\)). In fact, \(RF\) attaches to each object of \({\mathcal A}\), and more generally to each complex \(A\) in \(C({\mathcal A})\), a complex \(RF(A)\) (up to some equivalence relation; more precisely, an object \(RF(A)\in D({\mathcal B})\) in the derived category of \({\mathcal B}\)) whose cohomology objects are the \(R^iF(A)\), i.e., \(H^i(RF(A)) = R^iF(A)\).

See  [ GW2 ] (and the references given there) for a systematic treatment: Appendix F for the general theory, in particular Sections (F.37) for the definition of the derived category of an abelian category, (F.42), (F.43), (F.44) for the notion of derived functor, (F.48) for a comparison with the notion of \(\delta \)-functor defined above, Chapter 21 for general results on cohomology of \({\mathscr O}_X\)-modules on a ringed space \(X\) (specifically, the derived functors of the global section and direct image functors), Chapter 22 for cohomology of quasi-coherent \({\mathscr O}_X\)-modules on a scheme \(X\), Chapter 23 for cohomology of proper schemes and the derived direct image functor of a proper morphism. While in these chapters one sometimes obtains stronger or clearer statements using derived categories, for a large part the key arguments are exactly the same as with the “naive approach” using derived functors that we will take. In Chapter 24 on the theorem on formal functions and in particular in Chapter 25 on Grothendieck duality, the machinery of derived categories really shows its full power (but it is unlikely that we will get there in this class).

Cohomology of sheaves

 

General reference: [ H ] Ch. III, [ Stacks ] , [ GW2 ] Chapter 21.

(6.7) Cohomology groups

Let \(X\) be a topological space. Denote by \(({\rm Ab}_{X})\) the category of abelian sheaves (i.e., sheaves of abelian groups) on \(X\). We have the global section functor

\[ \Gamma \colon ({\rm Ab}_{X}) \to ({\rm Ab}),\quad {\mathscr F}\mapsto \Gamma (X, {\mathscr F}), \]

to the category of abelian groups. This is a left exact functor, and we denote its right derived functors by \(H^i(X, -)\). We call \(H^i(X, {\mathscr F})\) the \(i\)-th cohomology group of \(X\) with coefficients in \({\mathscr F}\).

Example 6.23
For a field \(k\), \(H^1(\mathbb {P}^1_k, {\mathscr O}(-2)) \ne 0\).

(6.8) Flasque sheaves

Reference: [ GW2 ] Section (21.7).

June 19,
2023

Definition 6.24
Let \(X\) be a topological space. A sheaf \({\mathscr F}\) on \(X\) is called flasque (or flabby), if all restriction maps \({\mathscr F}(U)\to {\mathscr F}(V)\) for \(V\subseteq U\subseteq X\) open are surjective.

For the next lemma, we use the extension by zero functor: Let \(j\colon U\to X\) be the inclusion of an open subspace into a topological space \(X\) (similarly one can work with an open immersion of ringed spaces). For an abelian sheaf \({\mathscr F}\) on \(U\) we define \(j_!{\mathscr F}\) as the sheafification of the presheaf

\[ V\mapsto \begin{cases} {\mathscr F}(V) & \text{if}\ V\subseteq U,\\ 0 & \text{otherwise.} \end{cases} \]

We obtain a left adjoint to the restriction functor \(j^*\).

Lemma 6.25
Let \(X\) be a ringed space. Let \({\mathscr F}\) be an injective object in the category of \({\mathscr O}_X\)-modules. Then \({\mathscr F}\) is flasque.

Sketch of proof

Identify \({\mathscr F}(U) = \operatorname{Hom}(j_{U, !}{\mathscr O}_U, {\mathscr F})\), where \(j_U\colon U\to X\) is the inclusion of an open subset \(U\) into \(X\), and \(j_{U, !}\) is the extension by zero functor, using the adjunction between \(j_{U, !}\) and \(j_U^*\).

Proposition 6.26
Let \(X\) be a topological space, and let \({\mathscr F}\) be a flasque abelian sheaf on \(X\). Then \({\mathscr F}\) is \(\Gamma \)-acyclic, i.e., \(H^i(X, {\mathscr F}) = 0\) for all \(i{\gt}0\).

Proof

Given a flasque sheaf \({\mathscr F}\); embed it into an injective abelian sheaf and use the results of Problem 40 and dimension shift.

Corollary 6.27

Let \(X\) be a ringed space. The right derived functors of the global section functor from the category of \({\mathscr O}_X\)-modules to the category of abelian groups can naturally be identified with \(H^i(X, -)\).

It follows that for an \({\mathscr O}_X\)-module \({\mathscr F}\) the cohomology groups \(H^i(X, {\mathscr F})\) carry a natural \(\Gamma (X, {\mathscr O}_X)\)-module structure.

(6.9) Grothendieck vanishing

Reference: [ H ] III.2, [ GW2 ] Section (21.12).

Lemma 6.28
Let \(X\) be a topological space, and let \(\iota \colon Y\to X\) be the inclusion of a closed subset \(Y\) of \(X\). Let \({\mathscr F}\) be an abelian sheaf on \(X\). Then there are natural isomorphisms
\[ H^i(Y, {\mathscr F}) = H^i(X, \iota _*{\mathscr F}), \quad i\ge 0. \]

Sketch of proof

Use that \(\iota _*\) preserves the property of being flasque, and that the cohomology groups can be computed using a flasque resolution.

Proposition 6.29
Let \(X\) be a noetherian topological space, and let \(({\mathscr F}_i)_i\) be a filtered inductive system of abelian sheaves on \(X\). Then the natural homomorphism
\[ \mathop{\rm colim}\limits _i H^n(X, {\mathscr F}_i)\to H^n(X, \mathop{\rm colim}\limits _i{\mathscr F}) \]
is an isomorphism for all \(n\ge 0\).

Sketch of proof

First show the statement for global sections, i.e., for \(n=0\). Since filtered colimits are exact, both sides give \(\delta \)-functors from the abelian category of inductive systems of abelian sheaves to the category of abelian groups. To conclude the proof, it is enough to show that they are both effaceable (because this implies they are universal, and universal \(\delta \)-functors are determined by their \(0\)-th term).

To show effaceability, take a system \(({\mathscr F}_i)_i\) and consider functorial flasque resolutions of the \({\mathscr F}_i\) (such as the Godement resolution). Since filtered colimits are exact and, on noetherian spaces, preserve flasqueness, this gives rise to a flasque resolution of \(\mathop{\rm colim}\limits {\mathscr F}_i\). With this at hand, it is not hard to finish the proof.

June 21,
2023

Let \(X\) be a topological space, \(U\subseteq X\) open and \(Z = X\setminus U\) its closed complement. Denote by \(j\colon U\to X\) and \(i\colon Z\to X\) the inclusions. Recall the extension by zero functor \(j_!\colon ({\rm Ab}_{U})\to ({\rm Ab}_{X})\) (which is left adjoint to the pull-back \(j^{-1}\)). From this adjunction and the one between \(i^*\) and \(i_*\) we obtain the maps in the short exact sequence

\[ 0\to j_!({\mathscr F}_{|U}) \to {\mathscr F}\to i_*({\mathscr F}_{|Z})\to 0 \]

of abelian sheaves on \(X\). The exactness can be checked on stalks where it is clear.

Theorem 6.30 (Grothendieck)
Let \(X\) be a noetherian topological space (i.e., the descending chain condition holds for closed subsets of \(X\)), let \(n = \dim X\), and let \({\mathscr F}\) be a sheaf of abelian groups on \(X\). Then
\[ H^i(X, {\mathscr F}) = 0\quad \text{for all}\ i {\gt} n. \]

Proof

The proof is omitted in these notes. It consists of a series of reduction steps using the above ingredients, ultimately reducing to the fact that the cohomology of the constant sheaf \(\mathbb {Z}_X\) on an irreducible space vanishes in positive degrees (since there a constant sheaf is flasque). See the references given above for more.

Čech cohomology

 

Reference: [ GW2 ] Sections (21.14) to (21.17), [ H ] III.4, [ Stacks ]  01ED (and following sections); a classical reference is [ Go ] .

(6.10) Čech cohomology groups

Let \(X\) be a topological space, and let \({\mathscr F}\) be an abelian sheaf on \(X\). (The definitions below can be made more generally for presheaves.)

Let \({\mathscr U}= (U_i)_{i\in I}\) be an open cover of \(X\). We fix a total order of the index set \(I\) (but see below for a sketch that the results are independent of this). For \(i_0, \dots , i_p\in I\), we write \(U_{i_0\dots i_p} := \bigcap _{\nu =0}^p U_{i_\nu }\).

We define

\[ C^p({\mathscr U}, {\mathscr F}) = \prod _{i_0{\lt} \cdots {\lt} i_p} \Gamma (U_{i_0\dots i_p}, {\mathscr F}) \]

and

\[ d\colon C^p({\mathscr U}, {\mathscr F})\to C^{p+1}({\mathscr U}, {\mathscr F}),\quad (s_{\underline{i}})_{\underline{i}} \mapsto \left(\sum _{\nu = 0}^{p+1} (-1)^\nu {s_{i_0\dots \widehat{i_\nu }\dots i_p}}_{|U_{\underline{i}}} \right)_{\underline{i}}, \]

where \(\widehat{\cdot }\) indicates that the corresponding index is omitted. One checks that \(d\circ d =0\), so we obtain a complex, the so-called Čech complex for the cover \({\mathscr U}\) with coefficients in \({\mathscr F}\).

June 26,
2023

Definition 6.31
The Čech cohomology groups for \({\mathscr U}\) with coefficients in \({\mathscr F}\) are defined as
\[ \check{H}^p({\mathscr U}, {\mathscr F}) = H^p(C^\bullet ({\mathscr U}, {\mathscr F})),\quad p\ge 0. \]

Since \({\mathscr F}\) is a sheaf, we have \(\check{H}^0({\mathscr U}, {\mathscr F}) = \Gamma (X, {\mathscr F}) = H^0(X, {\mathscr F})\). In fact, a presheaf \({\mathscr F}\) is a sheaf if and only if for all open subsets \(U\subseteq X\) and all covers \({\mathscr U}\) of \(U\) the natural map \(\Gamma (U, {\mathscr F})\to \check{H}^0({\mathscr U}, {\mathscr F})\) is an isomorphism.

(6.11) The “full” Čech complex

Instead of the “alternating” (or “ordered”) Čech complex as above, we can also consider the “full” Čech complex

\[ C^p_f({\mathscr U}, {\mathscr F}) = \prod _{i_0, \dots , i_p} \Gamma (U_{i_0\dots i_p}, {\mathscr F}), \]

with differentials defined by the same formula as above. Then the projection \(C^\bullet _f({\mathscr U}, {\mathscr F}) \to C^\bullet ({\mathscr U}, {\mathscr F})\) is a homotopy equivalence, with “homotopy inverse” given by

\[ (s_{\underline{i}})_{\underline{i}}\mapsto (t_{\underline{i}})_{\underline{i}}, \]

where \(t_{\underline{i}} = 0\) whenever two entries in the multi-index \(\underline{i}\) coincide, and otherwise \(t_{\underline{i}} = \operatorname{sgn}(\sigma ) s_{\sigma (\underline{i})}\), where \(\sigma \) is the permutation such that \(\sigma (\underline{i})\) is in increasing order.

In particular, we have natural isomorphisms between the cohomology groups of the two complexes. So we also see that the Čech cohomology groups as defined above are independent of the choice of order on \(I\).

(6.12) Passing to refinements

Definition 6.32
A refinement of a cover \({\mathscr U}= (U_i)_i\) of \(X\) is a cover \({\mathscr V}= (V_j)_{j\in J}\) (with \(J\) totally ordered) together with a map \(\lambda \colon J\to I\) respecting the orders on \(I\) and \(J\) such that \(V_j\subseteq U_{\lambda (j)}\) for every \(j\in J\).

Given a refinement \({\mathscr V}\) of \({\mathscr U}\), one obtains a natural map (using restriction of sections to smaller open subsets)

\[ \check{H}^p({\mathscr U}, {\mathscr F}) \to \check{H}^p({\mathscr V}, {\mathscr F}). \]

We can pass to the colimit over all these maps given by refinements, and define

\[ \check{H}^p(X, {\mathscr F}) := \mathop{\rm colim}\limits _{{\mathscr U}} \check{H}^p({\mathscr U}, {\mathscr F}), \]

the \(p\)-th Čech cohomology group of \(X\) with coefficients in \({\mathscr F}\). More precisely, we here take the colimit over the category of refinements (see  [ GW2 ] Section (F.3)). Working with the full Čech complex, one can view this as a colimit over a partially ordered set “as usual”, for the cohomology groups and even for the Čech complexes, if one considers them in the homotopy category (i.e., the map between complexes attached to a refinement is independent of \(\lambda \) up to homotopy), cf.  [ GW2 ] Section (21.16), also for the discussion of set-theoretic issues.

Proposition 6.33
Let \(0\to {\mathscr F}' \to {\mathscr F}\to {\mathscr F}''\to 0\) be a short exact sequence of abelian sheaves on \(X\). Then there exists a homomorphism \(\delta \colon \Gamma (X, {\mathscr F}'')\to \check{H}^1(X, {\mathscr F}')\) such that the sequence
\begin{align*} 0\to & \Gamma (X, {\mathscr F}’) \to \Gamma (X, {\mathscr F})\to \Gamma (X, {\mathscr F}”) \\ \to & \check{H}^1(X, {\mathscr F}’) \to \check{H}^1(X, {\mathscr F}) \to \check{H}^1(X, {\mathscr F}”) \end{align*}
is exact. (But note that the sequence does not continue after \(\check{H}^1(X, {\mathscr F}'')\).)

Proof

This can be checked “directly”. For instance, to construct the connecting homomorphism \(\delta \), take an element \(s\in \Gamma (X, {\mathscr F}'')\). Locally on \(X\), we can lift it to sections of \({\mathscr F}\), so we obtain an element \((s_i)_i\in C^1({\mathscr U}, {\mathscr F})\). Its image \((s_{ij})_{i,j}\) in \(C^2({\mathscr U}, {\mathscr F})\) will usually be different from \(0\) (in fact it is \(=0\) if and only if \(s\) is in the image of \(\Gamma (X, {\mathscr F})\)), but has image \(0\) in \(C^2({\mathscr U}, {\mathscr F}'')\), and hence comes from an element \((t_{ij})_{i,j}\in C^2({\mathscr U}, {\mathscr F}')\). Then \((t_{ij})_{i,j}\) has image \(0\) in \(C^3({\mathscr U}, {\mathscr F}')\) (because that is obviously true in \(C^3({\mathscr U}, {\mathscr F})\) and the morphism \({\mathscr F}'\to {\mathscr F}\) is injective), and so gives rise to a class in \(\check{H}^1({\mathscr U}, {\mathscr F}')\). Its image in \(\check{H}^1(X, {\mathscr F}')\) is the image of \(s\) under \(\delta \). One checks that this procedure is independent of choices and gives rise to the exact sequence in the proposition.

(6.13) Comparison of cohomology and Čech cohomology

In degrees \(0\) and \(1\), cohomology and Čech cohomology coincide. For degree \(0\), we have already shown this, so we proceed to the case of degree \(1\). We start with some preparations.

We define a sheaf version of the Čech complex as follows:

\[ {\mathscr C}^p({\mathscr U}, {\mathscr F}) = \prod _{\underline{i}=(i_0{\lt} \cdots {\lt} i_p)} j_{\underline{i}, *}({\mathscr F}_{|U_{\underline{i}}}), \]

with differentials defined by (basically) the same formula as above. Here \(j_{\underline{i}}\) denotes the inclusion \(U_{\underline{i}}\hookrightarrow X\).

We have a natural map \({\mathscr F}\to {\mathscr C}^0({\mathscr U}, {\mathscr F})\), which on an open \(V\) is given by \(s\mapsto (s_{|U_i\cap V})_i\).

Proposition 6.34
The sequence \(0\to {\mathscr F}\to {\mathscr C}^0({\mathscr U}, {\mathscr F})\to {\mathscr C}^1({\mathscr U}, {\mathscr F})\to \cdots \) is exact.

Proof

The exactness can be checked on stalks, and one can show that for each point of \(X\), the stalks of the above complex form a complex that is homotopy equivalent to \(0\). We omit the details.

Proposition 6.35
If \({\mathscr F}\) is flasque, then all \({\mathscr C}^p({\mathscr U}, {\mathscr F})\) are flasque, and \(\check{H}^p({\mathscr U}, {\mathscr F})=0\) for all \(p{\gt}0\).

Proof

It is not hard to check that the sheaves \({\mathscr C}^p({\mathscr U}, {\mathscr F})\) are flasque, since all the constructions involved preserve flasqueness.

We then obtain

\[ \check{H}^p({\mathscr U}, {\mathscr F})= H^p(\Gamma ({\mathscr C}^\bullet ({\mathscr U}, {\mathscr F}))) = H^p(X, {\mathscr F}) = 0, \]

where the second equality holds since \({\mathscr C}^\bullet ({\mathscr U}, {\mathscr F})\) is a flasque resolution of \({\mathscr F}\) by the above, and the third one follows since \({\mathscr F}\) itself is flasque.

Proposition 6.36
Let \(X\) be a topological space and let \({\mathscr F}\) be an abelian sheaf on \(X\).

  1. Let \({\mathscr U}\) be an open cover of \(X\). For every \(i\), there is a natural map \(\check{H}^i({\mathscr U}, {\mathscr F})\to H^i(X, {\mathscr F})\).

  2. These maps are compatible with refinements, so we obtain a natural map \(\check{H}^i(X, {\mathscr F})\to H^i(X, {\mathscr F})\). These maps are functorial in \({\mathscr F}\).

  3. For \(i=0, 1\), the natural map \(\check{H}^i(X, {\mathscr F})\to H^i(X, {\mathscr F})\) is an isomorphism.

Proof

Part (1) follows from Proposition 6.17 applied to the resolution \({\mathscr C}^\bullet ({\mathscr U}, {\mathscr F})\) of \({\mathscr F}\) and any injective resolution. We omit the proof of Part (2).

For Part (3), it remains to consider the case \(i=1\). Embed \({\mathscr F}\) into a flasque sheaf \({\mathscr G}\). We obtain short exact sequences

\[ \Gamma (X, {\mathscr G})\to \Gamma (X, {\mathscr G}/{\mathscr F}) \to \check{H}^1(X, {\mathscr F}) \to 0 \]

by Proposition 6.33 Proposition 6.35 and

\[ \Gamma (X, {\mathscr G})\to \Gamma (X, {\mathscr G}/{\mathscr F}) \to H^1(X, {\mathscr F}) \to 0 \]

since flasque sheaves are \(\Gamma (X, -)\)-acyclic. The statement follows from this.

One can also show that the natural map \(\check{H}^2(X, {\mathscr F})\to H^2(X, {\mathscr F})\) is always injective.

The following result will allow us to compute cohomology groups of separated schemes with coefficients in quasi-coherent modules as Čech cohomology.

Theorem 6.37
(Cartan’s Theorem) Let \(X\) be a ringed space, and let \({\mathscr B}\) be a basis of the topology of \(X\) which is stable under finite intersections. Let \({\mathscr F}\) be an \({\mathscr O}_X\)-module. Assume that \(\check{H}^i(U, {\mathscr F})=0\) for all \(U\in {\mathscr B}\) and \(i{\gt} 0\). Then

  1. we have \(H^i(U, {\mathscr F}) = 0\) for all \(U\in {\mathscr B}\) and \(i{\gt}0\),

  2. The natural homomorphisms \(\check{H}^i({\mathscr U},{\mathscr F})\to H^i(X, {\mathscr F})\) are isomorphisms for all \(i\ge 0\) and all covers \({\mathscr U}\) of \(X\) consisting of elements of \({\mathscr B}\).

  3. The natural homomorphisms \(\check{H}^i(X,{\mathscr F})\to H^i(X, {\mathscr F})\) are isomorphisms for all \(i\ge 0\).

See e.g., [ Go ] II Thm. 5.9.2, [ Stacks ]  01EO or  [ GW2 ] Section (21.17).

Cohomology of affine schemes

 

General references: [ H ] Ch. III, [ Stacks ] , [ GW2 ] Chapter 22.

June 28,
2023

(6.14) Vanishing of cohomology of quasi-coherent sheaves on affine schemes

Theorem 6.38
Let \(X\) be an affine scheme, and let \({\mathscr F}\) be a quasi-coherent \({\mathscr O}_X\)-module. Then \(\check{H}^i(X, {\mathscr F})= 0\) for all \(i {\gt} 0\).

Proof

It is enough to show that \(\check{H}^i({\mathscr U}, {\mathscr F}) = 0\) for all covers \({\mathscr U}\) of \(X\) by principal open subsets. This can be proved using “direct computation” (see  [ GW1 ] Lemma 12.33), or can be viewed as a consequence of the theory of “faithfully flat descent” (specifically Problem 37, see also  [ GW1 ] Lemma 14.64).

From this theorem, it follows immediately (using the above results) that \(H^1(X, {\mathscr F})=0\) for \(X\) affine and \({\mathscr F}\) quasi-coherent. In particular, the global section functor on \(X\) preserves exactness of every short exact sequence where the left hand term is a quasi-coherent \({\mathscr O}_X\)-module. But using Cartan’s Theorem, Theorem 6.37, we get more:

Theorem 6.39
Let \(X\) be an affine scheme, and let \({\mathscr F}\) be a quasi-coherent \({\mathscr O}_X\)-module. Then \(H^i(X, {\mathscr F})= 0\) for all \(i {\gt} 0\).

Theorem 6.40
Let \(X\) be a separated scheme, and let \({\mathscr U}\) be a cover of \(X\) by affine open subschemes. Let \({\mathscr F}\) be a quasi-coherent \({\mathscr O}_X\)-module. Then the natural homomorphisms \(\check{H}^i({\mathscr U}, {\mathscr F})\to H^i(X, {\mathscr F})\) are isomorphisms for all \(i\ge 0\).

Remark 6.41

For \(X\) noetherian, the use of Cartan’s Theorem can be avoided by using the result (see  [ H ]  III.3) that for a noetherian ring \(A\), \(X=\operatorname{Spec}A\), and \(I\) an injective \(A\)-module, the \({\mathscr O}_X\)-module \(\widetilde{I}\) is a flasque \({\mathscr O}_X\)-module.

Together with the fact that for an affine scheme \(X\) the global section functor is exact on the category of quasi-coherent \({\mathscr O}_X\)-modules, this gives the vanishing of \(H^i(X, {\mathscr F})\) for \(X\) affine, \({\mathscr F}\) quasi-coherent and \(i {\gt} 0\).

It also implies that any quasi-coherent \({\mathscr O}_X\)-module on a noetherian scheme can be embedded into a flasque quasi-coherent sheaf. From this one can prove Theorem 6.40.

See  [ H ] III.3, Theorem III.4.5; cf. also [ GW2 ] Section (22.18).

Corollary 6.42
Let \(X\) be a separated scheme which can be covered by \(n+1\) affine open subschemes. Then \(H^i(X, {\mathscr F})=0\) for every quasi-coherent \({\mathscr O}_X\)-module \({\mathscr F}\) and every \(i {\gt} n\).

Lemma 6.43

Let \(X\) be a scheme. For \(f\in \Gamma (X, {\mathscr O}_X)\) write

\[ X_f = \{ x\in X;\ f(x)\ne 0\in \kappa (x)\} , \]

an open subset of \(X\) which we consider as an open subscheme.

If there exist \(f_1, \dots , f_n\in \Gamma (X, {\mathscr O}_X)\) such that \(X_{f_i}\) is affine for \(i=1,\dots , n\) and such that \(f_1, \dots , f_n\) generate the unit ideal in the ring \(\Gamma (X, {\mathscr O}_X)\), then \(X\) is affine.

Theorem 6.44
(Serre’s criterion for affineness) Let \(X\) be a quasi-compact scheme. The following are equivalent:

  1. The scheme \(X\) is affine.

  2. For every quasi-coherent \({\mathscr O}_X\)-module \({\mathscr F}\) and every \(i {\gt} 0\), \(H^i(X, {\mathscr F})=0\).

  3. For every quasi-coherent ideal sheaf \({\mathscr I}\subseteq {\mathscr O}_X\), \(H^1(X, {\mathscr I})=0\).

Proof

See  [ H ] Theorem III.3.7 or  [ GW1 ] Theorem 12.35.

Cohomology of projective schemes

 

(6.15) The cohomology of line bundles on projective space

References: [ H ] III.5, [ GW2 ] Section (22.6), [ Stacks ]  01XS.

July 3,
2023

Using Čech cohomology, we can compute the cohomology of line bundles on projective space. It is best to aggregate the results for all \({\mathscr O}(d)\), as we have already seen for their global sections, a result which we repeat as the first statement below.

Theorem 6.45
Let \(A\) be a ring, \(n\ge 1\), \(S=A[T_0, \dots , T_n]\), \(X = \mathbb {P}^n_A\). Then

  1. There is a natural isomorphism \(S \cong \bigoplus _{d\in \mathbb {Z}} H^0(X, {\mathscr O}(d))\).

  2. For \(i\ne 0, n\) and all \(d\in \mathbb {Z}\) we have \(H^i(X, {\mathscr O}(d)) = 0\).

  3. There is a natural isomorphism \(H^n(X, {\mathscr O}(-n-1)) \cong A\).

  4. For every \(d\), there is a perfect pairing

    \[ H^0(X, {\mathscr O}(d)) \times H^n(X, {\mathscr O}(-d-n-1)) \to H^n(X, {\mathscr O}(-n-1))\cong A, \]

    i.e., a bilinear map which induces isomorphisms

    \[ H^0(X, {\mathscr O}(d)) \cong H^n(X, {\mathscr O}(-d-n-1))^\vee \]

    and

    \[ H^0(X, {\mathscr O}(d))^\vee \cong H^n(X, {\mathscr O}(-d-n-1)) \]

    (where \(-^\vee = \operatorname{Hom}_A(-, A)\) denotes the \(A\)-module dual).

Sketch of proof

We compute the cohomology groups as Čech cohomology groups for the standard cover \({\mathscr U}= (D_+(T_i))_i\) of \(\mathbb {P}^n_A\). It simplifies the reasoning to do the computation for all \({\mathscr O}(d)\) at once, i.e., to compute the cohomology groups of \({\mathscr F}:=\bigoplus _{d\in \mathbb {Z}} {\mathscr O}(d)\) (and to – implicitly – keep track of the grading by \(d\)). Since cohomology is compatible with direct sums (cf. Proposition 6.29; we proved that for noetherian schemes, but it holds more generally for quasi-compact separated schemes, cf.  [ GW2 ] Corollary 21.56), this also gives the result for the individual \({\mathscr O}(d)\).

The Čech complex \(C^\bullet ({\mathscr U}, {\mathscr F})\) is

\[ 0\to \prod _i S_{T_i} \to \prod _{i,j} S_{T_iT_j}\to \cdots \to S_{T_0\cdots T_n}\to 0 \]

(with non-zero entries in degrees \(0, \dots , n\)), cf. the computation of the global sections of the sheaves \({\mathscr O}(d)\), Proposition 3.19. Also note that that proposition proves Part (1) of the theorem here. From this, we see that we can identify

\[ H^n(\mathbb {P}^n_A, {\mathscr F}) = \bigoplus _{i_0,\dots , i_n {\lt} 0} A\cdot T_0^{i_0}\cdots T_n^{i_n}\qquad \subseteq S[T_0^{-1}, \dots , T_n^{-1}]. \]

This easily implies Parts (3) and (4).

It remains to prove the vanishing statement of Part (2) for \(0 {\lt} i {\lt} n\). We do this by induction on \(n\). Note that in view of Part (1), all the cohomology groups \(H^i(\mathbb {P}^n_A, {\mathscr F})\) carry a natural \(S\)-module structure.

First note that for the localization we have \(H^i(\mathbb {P}^n_A, {\mathscr F})_{T_n} = 0\). In fact, this localization is the \(i\)-th cohomology group of the localized Čech complex \(C^\bullet ({\mathscr U}, {\mathscr F})_{T_n}\) which computes the cohomology \(H^\bullet (D_+(T_n), {\mathscr F}_{|D_+(T_n)})\) which vanishes in positive degrees. Therefore it suffices to show that multiplication by \(T_n\) is a bijection \(H^i(\mathbb {P}^n_A, {\mathscr F})\to H^i(\mathbb {P}^n_A, {\mathscr F})\) for all \(0{\lt}i{\lt}n\).

The global section \(T_n\in H^0(\mathbb {P}^n_A, {\mathscr O}(1))\) gives rise to a short exact sequence

\[ 0\to {\mathscr O}(-1)\to {\mathscr O}_{\mathbb {P}^n_A}\to {\mathscr O}_{V_+(T_n)}\to 0, \]

and tensoring with the locally free module \({\mathscr F}\) we obtain a short exact sequence

\[ 0\to {\mathscr F}\otimes {\mathscr O}(-1)\to {\mathscr F}\to {\mathscr F}' \to 0 \]

where \({\mathscr F}' = {\mathscr F}\otimes _{{\mathscr O}_{\mathbb {P}^n_A}}{\mathscr O}_{V_+(T_n)}\) is (the push-forward from \(V_+(T_n) \cong \mathbb {P}^{n-1}_A\) to \(\mathbb {P}^n_A\)) of the sheaf analogous to \({\mathscr F}\) on \(\mathbb {P}^{n-1}_A\).

Multiplication by \(T_n\in \Gamma (\mathbb {P}^n_A, {\mathscr O}(1))\) gives an isomorphism \({\mathscr F}\otimes {\mathscr O}(-1)\to {\mathscr F}\). Therefore the long exact cohomology sequence attached to the above short exact sequence can be written as

\[ \cdots \to H^i(\mathbb {P}^n_A, {\mathscr F}) \to H^i(\mathbb {P}^n_A, {\mathscr F}) \to H^i(\mathbb {P}^{n-1}_A, {\mathscr F}')\to \cdots \]

where the maps \(H^i(\mathbb {P}^n_A, {\mathscr F}) \to H^i(\mathbb {P}^n_A, {\mathscr F})\) are given by multiplication by \(T_n\) (i.e., what we want to show is that these maps are isomorphisms). The induction hypothesis together with Lemma 6.28 and the observations that the map \(H^0(\mathbb {P}^n_A, {\mathscr F})\to H^0(\mathbb {P}^{n-1}_A, {\mathscr F}')\) is surjective (cf. Part (1)) and that the map \(H^{n-1}(\mathbb {P}^{n-1}_A, {\mathscr F}')\to H^n(\mathbb {P}^n_A,{\mathscr F})\) is injective, then allow us to conclude. For the injectivity cf. the above proof for Part (3). Looking at individual graded pieces of \(H^{n-1}(\mathbb {P}^{n-1}_A, {\mathscr F}')\) and the kernel of \(H^n(\mathbb {P}^n_A,{\mathscr F})\to H^n(\mathbb {P}^n_A,{\mathscr F})\), we have a surjective homomorphism of free \(A\)-modules of the same rank which is necessarily an isomorphism. (The kernel of \(H^n(\mathbb {P}^n_A,{\mathscr F})\to H^n(\mathbb {P}^n_A,{\mathscr F})\) is the free \(A\)-module spanned by all monomials of the form \(T_0^{i_0}\cdots T_{n-1}^{i_{n-1}}T_n^{-1}\) with all \(i_\nu {\lt} 0\).)

Remark 6.46
The homomorphism \(H^{n-1}(\mathbb {P}^{n-1}_A, {\mathscr F}')\to H^n(\mathbb {P}^n_A,{\mathscr F})\) at the end of the proof of the previous theorem is given by mapping \(T_0^{i_0}\cdots T_{n-1}^{i_{n-1}}\) to \(T_0^{i_0}\cdots T_{n-1}^{i_{n-1}}T_n^{-1}\). To verify this, one can use that the long exact cohomology sequence for the short exact sequence \(0\to {\mathscr F}\otimes {\mathscr O}(-1)\to {\mathscr F}\to {\mathscr F}' \to 0\) can be computed in terms of Čech cohomology, namely as the long exact sequence attached to the (exact!) sequence \(0\to C^\bullet ({\mathscr U}, {\mathscr F}\otimes {\mathscr O}(-1))\to C^\bullet ({\mathscr U}, {\mathscr F})\to C^\bullet ({\mathscr U}, {\mathscr F}')\to 0\).

(6.16) Finiteness of cohomology of coherent \({\mathscr O}_X\)-modules on projective schemes

References: [ H ] III.5; [ GW2 ] Sections (23.1), (23.2).

July 5,
2023

Definition 6.47
Let \(X\) be a noetherian scheme. An \({\mathscr O}_X\)-module \({\mathscr F}\) is called coherent, if it is quasi-coherent and of finite type.

Let \(A\) be a noetherian ring. For an \({\mathscr O}_{\mathbb {P}^n_A}\)-module \({\mathscr F}\), we write \({\mathscr F}(d) := {\mathscr F}\otimes _{{\mathscr O}_{\mathbb {P}^n_A}}{\mathscr O}(d)\). We need the following general lemma.

Lemma 6.48
Let \(X\) be a quasi-compact and separated scheme, let \({\mathscr L}\) be an invertible \({\mathscr O}_X\)-module, and let \(s \in \Gamma (X,{\mathscr L})\) be a global section. Let \({\mathscr F}\) be a quasi-coherent \({\mathscr O}_X\)-module.

  1. Let \(t \in \Gamma (X,{\mathscr F})\) be a global section such that \(t{}_{\vert }{}_{X_s} = 0\). Then there exists an integer \(n {\gt} 0\) such that \(t \otimes s^{\otimes n} = 0 \in \Gamma (X,{\mathscr F}\otimes {\mathscr L}^{\otimes n})\).

  2. For every section \(t' \in \Gamma (X_s,{\mathscr F})\) there exist \(n {\gt} 0\) and a section \(t \in \Gamma (X,{\mathscr F}\otimes {\mathscr L}^{\otimes n})\) such that \(t{}_{\vert }{}_{X_s} = t' \otimes s^{\otimes n}\).

Proof

If \(X\) is affine and \({\mathscr L}= {\mathscr O}_X\), then this follows immediately from our results on quasi-coherent \({\mathscr O}_X\)-modules; namely we know that then \(\Gamma (X_s,{\mathscr F}) = \Gamma (X, {\mathscr F})_s\). For the general case, let \(X=\bigcup _i U_i\) be a finite affine open cover such that \({\mathscr L}_{|U_i}\cong {\mathscr O}_{U_i}\) for all \(i\) (and fix such isomorphisms). Then (1) can be checked on each \(U_i\) individually and thus follows from what was said in the beginning. To prove Part (2), we construct \(t\) by considering sections \(t_i\) on the \(U_i\) obtained from the restrictions \(t_{|U_i\cap X_s}\), using the result in the affine case. The \(t_i\) may not glue, but applying Part (1) to the intersections \(U_i\cap U_j\) and the elements \(t_{i|U_i\cap U_j}-t_{j|U_i\cap U_j}\) we find that for \(n\) sufficiently large, the \(t_i\otimes s^{\otimes n}\) will glue to a section of \({\mathscr F}\otimes {\mathscr L}^{\otimes n}\). See  [ GW1 ] Theorem 7.22 or  [ H ] Lemma II.5.14 for more details.

Proposition 6.49
Let \(A\) be a noetherian ring, \(n\ge 1\), let \(X = \mathbb {P}^n_A\), and let \({\mathscr F}\) be a coherent \({\mathscr O}_X\)-module.

  1. There exist integers \(d_1, \dots , d_s\) and a surjective \({\mathscr O}_X\)-module homomorphism

    \[ \bigoplus _{i=1}^n {\mathscr O}(d_i)\twoheadrightarrow {\mathscr F}. \]

  2. For \(d\) sufficiently large, the \({\mathscr O}_{\mathbb {P}^n_A}\)-module \({\mathscr F}(d)\) is globally generated, i.e., there exist \(N\ge 0\) and a surjective homomorphism \({\mathscr O}^N_{\mathbb {P}^n_A}\to {\mathscr F}(d)\).

Sketch of proof

It is easy to see that (1) and (2) are equivalent. We prove (2). For \(i\in \{ 0,\dots , n\} \), \({\mathscr F}_{|D_+(T_i)}\) has the form \(\widetilde{M_i}\) for some finitely generated \(A[\frac{T_0}{T_i},\cdots , \frac{T_n}{T_i}]\)-module \(M_i\). For any \(s\in M_i = \Gamma (D_+(T_i), {\mathscr F})\), for \(d\) sufficiently large, \(X_i^ds\) extends to a global section of \({\mathscr F}(d)\), by Lemma 6.48. This implies the claim.

Theorem 6.50
Let \(A\) be a noetherian ring, \(X\) be a projective \(A\)-scheme, and let \({\mathscr F}\) be a coherent \({\mathscr O}_X\)-module. Then for all \(i\ge 0\), the \(A\)-module \(H^i(X, {\mathscr F})\) is finitely generated.

Sketch of proof

Using Lemma 6.28, we reduce to the case that \(X=\mathbb {P}^n_A\). By Proposition 6.49, the vanishing result Corollary 6.42 and descending induction, we may further reduce to the case that \({\mathscr F}\) is a finite direct sum of sheaves of the form \({\mathscr O}(d)\), but for these we already know the result.

At this point it is not hard to prove that higher derived images \(R^if_*{\mathscr F}\) of a coherent \({\mathscr O}_X\)-module under a projective morphism \(f\colon X\to Y\) are coherent (see  [ H ]  III.8).

Another useful result is the following vanishing statement.

Proposition 6.51
Let \(A\) be a noetherian ring, \(\iota \colon X\to \mathbb {P}^n_A\) a closed immersion, \({\mathscr L}= \iota ^*{\mathscr O}_{\mathbb {P}^n_A}(1)\), and let \({\mathscr F}\) be a coherent \({\mathscr O}_X\)-module. For \(n\in \mathbb {Z}\) we write \({\mathscr F}(n):= {\mathscr F}\otimes _{{\mathscr O}_X}{\mathscr L}^{\otimes n}\). Then there exists \(n_0\in \mathbb {Z}\) such that for all \(n\ge n_0\) and all \(i{\gt} 0\), we have \(H^i(X, {\mathscr F}(n)) = 0\).

Sketch of proof

Using the projection formula (see below), one reduces to the case \(X=\mathbb {P}^n_A\). It is then clear from the above, that the statement holds whenever \({\mathscr F}\) is a direct sum of line bundles \({\mathscr O}(d)\). The general statement follows from this by descending induction, similarly as above.

Proposition 6.52
(Projection formula) Let \(f\colon X\to Y\) be a morphism of schemes, let \({\mathscr F}\) be an \({\mathscr O}_X\)-module, and let \({\mathscr G}\) be an \({\mathscr O}_Y\)-module. There is a natural homomorphism
\[ (f_*{\mathscr F})\otimes _{{\mathscr O}_Y}{\mathscr G}\to f_*({\mathscr F}\otimes _{{\mathscr O}_X} f^*{\mathscr G}) \]
which is an isomorphism if one of the following conditions is satisfied:

  1. \({\mathscr G}\) is a locally free \({\mathscr O}_Y\)-module,

  2. \(f\) is a closed immersion,

  3. \({\mathscr F}\) and \({\mathscr G}\) are quasi-coherent, and for every affine open \(V\subseteq Y\), \(f^{-1}(V)\) is affine,

  4. \({\mathscr F}\) and \({\mathscr G}\) are quasi-coherent, \(f\) is quasi-compact and separated, and \({\mathscr G}\) is a flat \({\mathscr O}_Y\)-module.

Sketch of proof

The homomorphism is obtained formally using the adjunction between \(f^*\) and \(f_*\). The statement that it is an isomorphism is local on \(Y\). Thus for (1), we may assume that \({\mathscr G}\) is free, and then the claim is easy to check. In situation (2) one can check that the homomorphism induces an isomorphism on each stalk, and hence is an isomorphism. Under the assumptions in (3), one may assume that \(Y\) and \(X\) are affine and the claim then follows easily from the description of pushforward and pullback in this case (Proposition 2.15). For further details, and for an argument to prove the statement in case (4), see  [ GW2 ] Proposition 22.80.

Serre duality

 

July 10,
2023

(6.17) The Theorem of Riemann–Roch revisited

References: [ H ] III.7, IV.1; [ GW2 ] Chapters 25, 26.

Recall the Theorem of Riemann–Roch that we stated above (Theorem 3.13). In this section, we prove a preliminary version, which also gives a more conceptual view on the “error term” \(\dim \Gamma (X, {\mathscr O}(K-D))\) (with notation as above).

Let \(k\) be an algebraically closed field.

Definition 6.53
Let \(X\) be a projective \(k\)-scheme, and let \({\mathscr F}\) be a coherent \({\mathscr O}_X\)-module. We call
\[ \chi ({\mathscr F}) = \sum _{i\ge 0} (-1)^i \dim _k H^i(X, {\mathscr F}) \]
the Euler characteristic of \({\mathscr F}\).

Note that the sum is finite (by the Grothendieck vanishing theorem, Theorem 6.30) and that each term is finite by the results of the previous section.

Now let \(X/k\) be a smooth, projective, connected curve. Then \(\chi ({\mathscr F}) = \dim _k H^0(X, {\mathscr F}) - \dim _k H^1(X, {\mathscr F})\).

The following theorem is the preliminary version of the Theorem of Riemann–Roch mentioned above.

Theorem 6.54
Let \({\mathscr L}\) be a line bundle on \(X\). Then
\[ \chi ({\mathscr L}) = \deg ({\mathscr L}) + \chi ({\mathscr O}_X). \]

Sketch of proof

The statement is clear for \({\mathscr L}= {\mathscr O}_X\). Since every line bundle is isomorphic to the line bundle attached to a Weil divisor, it is therefore enough to prove that for every closed point \(x\in X\), and every line bundle \({\mathscr L}\) on \(X\), we have

\[ \chi ({\mathscr L}) = \chi ({\mathscr L}\otimes _{{\mathscr O}_X}{\mathscr O}_X(-[x])) + 1 \]

We write \({\mathscr L}(D):={\mathscr L}\otimes _{{\mathscr O}_X}{\mathscr O}_X(D)\) for any divisor \(D\). The short exact sequence

\[ 0\to {\mathscr O}_X(-[x])\to {\mathscr O}_X\to \kappa (x)\to 0 \]

remains exact after tensoring with \({\mathscr L}\), so we obtain a short exact sequence

\[ 0\to {\mathscr L}(-[x])\to {\mathscr L}\to \kappa (x)\to 0. \]

Since the Euler characteristic is additive in short exact sequences (use the long exact cohomology sequence) and since \(\chi (\kappa (x)) = 1\), the claim follows.

Now we can define the genus of \(X\) as \(g:= 1-\chi ({\mathscr O}_X) = \dim _k H^1(X, {\mathscr O}_X)\). From the above, we immediately get

Corollary 6.55 (Theorem of Riemann)
Let \({\mathscr L}\) be a line bundle on \(X\). Then
\[ \dim _k H^0(X, {\mathscr L}) \ge \deg ({\mathscr L}) + 1-g. \]

Furthermore, the Theorem of Riemann–Roch, Theorem 3.13, follows from the above result on Euler characteristics and the Serre duality theorem (which however we cannot prove in this class).

Theorem 6.56 (Serre duality)
Let \(k\) be an algebraically closed field and let \(X\) be a connected smooth proper \(k\)-scheme. (At this point we can take smooth to mean that all local rings \({\mathscr O}_{X,x}\) are regular; in fact it is enough to assume that \(X\) is Cohen-Macaulay, i.e., that all local rings of \(X\) are Cohen-Macaulay rings.) There is a unique (up to isomorphism) coherent \({\mathscr O}_X\)-module \(\omega \), the so-called dualizing sheaf, such that for every locally free \({\mathscr O}_X\)-module \({\mathscr E}\) of finite rank on \(X\), there is a natural isomorphism

July 12,
2023

\[ H^{n-i}(X, {\mathscr E}) \cong H^i(X, {\mathscr E}^{-1} \otimes \omega )^\vee \]
of \(k\)-vector spaces (where \(-^\vee \) denotes the dual \(k\)-vector space).

If \(X\) is smooth over \(k\), then the dualizing sheaf is a line bundle and coincides with the so-called canonical bundle, the top exterior power of the sheaf \(\Omega ^1_{X/k}\) of differentials of \(X\) over \(k\), see Chapter 4.

A fairly elementary approach in the case where \(X\) is projective is to prove a similar duality theorem for the cohomology of line bundles on projective space \(\mathbb {P}^n_k\) (this we have basically done above, Theorem 6.45 (4), with \(\omega = {\mathscr O}(-n-1)\)), and then to derive the statement for (certain) closed subschemes of projective space; see [ H ] III.7. A more general (but technically more sophisticated) approach is to derive this duality result from the existence of a right adjoint functor \(f^\times \) of the derived push-forward functor \(Rf_*\colon D_{\rm qcoh}(X)\to D_{\rm qcoh}(\operatorname{Spec}k)\), where \(f\colon X\to \operatorname{Spec}k\) is the structure morphism and \(D_{\rm qcoh}(X)\) denotes the full triangulated subcategory of the derived category \(D(X)\) of the category of \({\mathscr O}_X\)-modules (and this generalizes to the case of arbitrary proper morphisms \(f\) between noetherian schemes). In fact, one shows that for \(X\) Cohen-Macaulay and equidimensional of dimension \(n\) the complex \(f^\times k\) is concentrated in degree \(-n\), and we denote by \(\omega \) the unique non-vanishing cohomology object. For every quasi-coherent \({\mathscr O}_X\)-module \({\mathscr F}\) and integer \(i\) one then has

\begin{align*} H^i(X, {\mathscr F})^\vee & = \operatorname{Hom}_k(Rf_*{\mathscr F}[i], k) = \operatorname{Hom}_{D(X)}({\mathscr F}[i], \omega [n])\\ & = \mathop{\rm Ext}\nolimits ^{n-i}_{{\mathscr O}_X}({\mathscr F}, \omega ), \end{align*}

and if \({\mathscr F}\) is locally free of finite rank, then we can identify the right hand side with

\[ \mathop{\rm Ext}\nolimits ^{n-i}_{{\mathscr O}_X}({\mathscr O}_X, {\mathscr F}^\vee \otimes _{{\mathscr O}_X} \omega ) = H^{n-i}(X, {\mathscr F}^\vee \otimes _{{\mathscr O}_X}\omega ). \]

This approach also gives a result (in terms of the derived category) without requiring that \(X\) be Cohen-Macaulay, and also for (proper) morphisms \(f\colon X\to Y\) where \(Y\) is not necessarily the spectrum of a field. See  [ GW2 ] Chapter 25 for more on this and for further references.

As another application of Serre duality, we mention the following result. The first part is named the Lemma of Enriques-Severi-Zariski. To name an application, we note that this is one of the ingredients of the proof that every regular proper surface over a field is projective (see  [ GW2 ] Theorem 25.151).

The result still holds (but in Part (1) only for \(H^1\), as stated, while for \(X\) Cohen-Macaulay one has an analogous result for all \(H^i\), \(i{\lt}n\)) if the assumption that \(X\) be Cohen-Macaulay is replaced by the assumption that \(X\) is normal (i.e., for every non-empty affine open \(U\subseteq X\), the ring \(\Gamma (U, {\mathscr O}_X)\) is integrally closed in its field of fractions \(K(X)\)). On the other hand it is clear that the assumption that \(X\) has dimension at least \(2\) cannot be dropped.

Theorem 6.57

Let \(K\) be an (algebraically closed) field, let \(X\) be an integral Cohen-Macaulay (e.g., smooth) projective \(k\)-scheme and let \(\iota \colon X\to \mathbb {P}^n_k\) be a closed immersion of \(k\)-schemes.

Assume that \(\dim X \ge 2\).

  1. Fix \(d {\gt} 0\) and let \({\mathscr L}:= \iota ^*{\mathscr O}_{\mathbb {P}^n_k}(d)\). Let \({\mathscr F}\) be a coherent \({\mathscr O}_X\)-module. We write \({\mathscr F}(n):={\mathscr F}\otimes _{{\mathscr O}_X}{\mathscr L}^{\otimes n}\), \(n\in \mathbb {Z}\). Then for \(n \ll 0\), \(H^1(X, {\mathscr F}(n)) = 0\).

  2. Let \(f\in k[T_0, \dots , T_n]\) be a non-constant homogeneous polynomial, and let \(Z := V_+(f)\), a closed subscheme of \(\mathbb {P}^n_k\). Then \(X\cap Z\) is connected.

Sketch of proof

Part (1) follows from Serre duality (Theorem 6.56) and the vanishing result of Proposition 6.51. To prove Part (2), let \(d\) denote the degree of \(f\). We view \(Z\) as an effective Cartier divisor. Then \({\mathscr O}_{\mathbb {P}^n_k}(Z)\cong {\mathscr O}(d)\). We set \({\mathscr L}= \iota ^*{\mathscr O}(d)\) and \({\mathscr F}= {\mathscr O}_X\) and apply Part (1) to find \(n\) such that \(H^1(X, {\mathscr L}^{-n}) = 0\). Let \(Z_n\) be the Cartier divisor \(n\cdot Z\) (with associated line bundle \(\cong {\mathscr O}(dn)\)). We then have a short exact sequence

\[ 0\to {\mathscr L}^{-n}\to {\mathscr O}_X\to {\mathscr O}_{X\cap Z_n}\to 0, \]

where \(X\cap Z_n\) denotes the scheme-theoretic intersection of \(X\) and \(Z\), i.e., \({\mathscr O}_{X\cap Z_n} = \iota ^*{\mathscr O}_{Z_n}\). The underlying topological space of \(X\cap Z_n\) is independent of \(n\ge 1\) and equals the set-theoretic intersection \(X\cap Z\). It is therefore sufficient to show that the scheme \(X\cap Z_n\) is connected. But the above short exact sequence induces, in view of the vanishing of \(H^1(X, {\mathscr L}^{-n})\) given by Part (1), a surjective homomorphism \(\Gamma (X, {\mathscr O}_X)\to \Gamma (X, {\mathscr O}_{X\cap Z_n})\) of \(k\)-vector spaces. Since \(\Gamma (X, {\mathscr O}_X) = k\), it follows that \(\Gamma (X, {\mathscr O}_{X\cap Z_n}) = k\), as well, and in particular \(X\cap Z_n\) is connected. See also  [ GW2 ] Section (25.28), [ H ] Section III.7.