(4.1) Complexes in abelian categories
Reference:
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Ch. 1.
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 $X$).
A complex in ${\mathcal A}$ is a sequence of morphisms
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
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Thm. 1.2.3.
Definition 4.1
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 $i$.
Proposition 4.3
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
\[ \cdots h^i(A^\bullet ) \to h^i(B^\bullet ) \to h^i(C^\bullet ) \to h^{i+1}(A^\bullet ) \to \cdots . \]
Reference:
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Thm. 1.3.1.
We need a criterion which ensures that two morphisms between complexes induce the same maps on all cohomology objects. Reference:
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1.4.
Definition 4.4
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 4.5
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 4.6
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 $i$.
June 24,
2019
(4.2) 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 4.7
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 4.8
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.
(4.3) $\delta $-functors
Reference:
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2.1.
Let ${\mathcal A}$, ${\mathcal B}$ be abelian categories.
Definition 4.9
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 4.10
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 $i > 0$, 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 4.11
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 4.12
Let $(T^i)_i$ be a $\delta $-functor from ${\mathcal A}$ to ${\mathcal B}$ such that for every $i>0$, the functor $T^i$ is effaceable. Then $(T^i)_i$ is a universal $\delta $-functor.
Reference:
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Thm. 2.4.7, Ex. 2.4.5.
(4.4) Injective objects
Let ${\mathcal A}$ be an abelian category.
Definition 4.13
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 4.14
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 4.15
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.
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.
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.
(4.5) Right derived functors
June 26,
2019
Theorem 4.16
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:
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.
We have an isomorphism $F\cong R^0F$ of functors.
For $I$ injective, we have $R^i FI=0$ for all $i>0$.
The family $(R^i F)_i$ is a universal $\delta $-functor.
We call the $R^iF$ the right derived functors of $F$.
Definition 4.17
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 $i>0$.
Definition 4.18
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 4.19
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.
General reference:
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H
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Ch. III,
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Stacks
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Ch. 20, 29.
(4.6) 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 ${\mathscr F}$.
July 1,
2019
(4.7) Flasque sheaves
Definition 4.21
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.
Lemma 4.22
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.
Proposition 4.23
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 $i>0$.
Corollary 4.24
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.
(4.8) Grothendieck vanishing
Reference:
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III.2.
Lemma 4.25
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
\[ H^i(Y, {\mathscr F}) = H^i(X, \iota _*{\mathscr F}), \quad i\ge 0. \]
Theorem 4.26 (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
\[ H^i(X, {\mathscr F}) = 0\quad \text{for all}\ i > n. \]
July 3,
2019
Reference:
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III.4,
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Stacks
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01ED (and following sections); a classical reference is
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Go
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.
(4.9) 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
\[ C^p({\mathscr U}, {\mathscr F}) = \prod _{i_0< \cdots < i_p} \Gamma (U_{i_0\dots i_p}, {\mathscr F}) \]
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 ${\mathscr F}$.
Definition 4.27
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 $\check{H}^0({\mathscr U}, {\mathscr F}) = \Gamma (X, {\mathscr F}) = H^0(X, {\mathscr F})$.
(4.10) 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 $I$.
(4.11) Passing to refinements
Definition 4.28
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 ${\mathscr F}$.
Proposition 4.29
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 $\check{H}^1(X, {\mathscr F}”)$.)
(4.12) A sheaf version of the Čech complex
We define a sheaf version of the Čech complex as follows:
\[ {\mathscr C}^p({\mathscr U}, {\mathscr F}) = \prod _{\underline{i}=(i_0< \cdots < i_p)} j_{\underline{i}, *}({\mathscr F}_{|U_{\underline{i}}}), \]
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 4.30
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.
Proposition 4.31
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 $p>0$.
Proposition 4.32
For every $i$, there is a natural map $\check{H}^i({\mathscr U}, {\mathscr F})\to H^i(X, {\mathscr F})$. 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 ${\mathscr F}$.
Proposition 4.33
For $i=0, 1$, the natural map $\check{H}^i(X, {\mathscr F})\to H^i(X, {\mathscr F})$ is an isomorphism.
July 8,
2019
(4.13) Vanishing of cohomology of quasi-coherent sheaves on affine schemes
Theorem 4.34
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 $i > 0$.
From this theorem, if 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 more is true:
Theorem 4.35
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 $i > 0$.
This follows from the above using Cartan’s Theorem (see e.g.,
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Go
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II Thm. 5.9.2,
[
Stacks
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01EO):
Theorem 4.36
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 $i> 0$. Then
we have $H^i(U, {\mathscr F}) = 0$ for all $i>0$,
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 ${\mathscr B}$.
The natural homomorphisms $\check{H}^i(X,{\mathscr F})\to H^i(X, {\mathscr F})$ are isomorphisms for all $i\ge 0$.
For $X$ noetherian, there is another approach which relies on the following result (see
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III.3):
Proposition 4.37
Let $A$ be a noetherian ring, $X=\operatorname{Spec}A$, and let $I$ be an injective $A$-module. Then $\widetilde{I}$ is a flasque ${\mathscr O}_X$-module.
From either approach, we also obtain the following consequence (of course, the second approach again works only in the noetherian situation):
Theorem 4.38
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 $i\ge 0$.
Corollary 4.39
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 $i > n$.
July 10,
2019
(4.14) The cohomology of line bundles on projective space
References:
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III.5,
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Stacks
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01XS.
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 4.40
Let $A$ be a noetherian ring, $n\ge 1$, $S=[T_0, \dots , T_n]$, $X = \operatorname{Proj}(S) = \mathbb {P}^n_A$. Then
There is a natural isomorphism $S \cong \bigoplus _{d\in \mathbb {Z}} H^0(X, {\mathscr O}(d))$.
For $i\ne 0, n$ and all $d\in \mathbb {Z}$ we have $H^i(X, {\mathscr O}(d)) = 0$.
There is a natural isomorphism $H^n(X, {\mathscr O}(-n-1)) \cong A$.
For every $r$, there is a perfect pairing
\[ H^0(X, {\mathscr O}(r)) \times H^n(X, {\mathscr O}(-r-n-1)) \to H^n(X, {\mathscr O}(-n-1))\cong A, \]
i.e., a bilinear map which induces isomorphisms
\[ H^0(X, {\mathscr O}(r)) \cong H^n(X, {\mathscr O}(-r-n-1))^\vee \]
and
\[ H^0(X, {\mathscr O}(r))^\vee \cong H^n(X, {\mathscr O}(-r-n-1)) \]
(where $-^vee = \operatorname{Hom}_A(-, A)$ denotes the $A$-module dual).
(4.15) Finiteness of cohomology of coherent ${\mathscr O}_X$-modules on projective schemes
Definition 4.41
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.
Lemma 4.42
Let $X = \mathbb {P}^n_A$, and let ${\mathscr F}$ be a coherent ${\mathscr O}_X$-module. Then there exist integers $d_1, \dots , d_s$ and a surjective ${\mathscr O}_X$-module homomorphism
\[ \bigoplus _{i=1}^n {\mathscr O}(d_i)\twoheadrightarrow {\mathscr F}. \]
Theorem 4.43
Let $X$ 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.
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
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H
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III.8).
(4.16) The Theorem of Riemann–Roch revisited
Reference:
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H
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III.7, IV.1.
Recall the Theorem of Riemann–Roch that we stated above (Thm. 2.9). 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 4.44
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 4.26) 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 4.45
Let ${\mathscr L}$ be a line bundle on $X$. Then
\[ \chi ({\mathscr L}) = \deg ({\mathscr L}) + \chi ({\mathscr O}_X). \]
Now we can define the genus of $X$ as $g:= 1-\chi ({\mathscr O}_X) = \dim _k H^1(X, {\mathscr O}_X)$, and choose for $K$ a divisor with ${\mathscr O}(K) \cong \Omega ^1_{X/k}$.
From the above, we immediately get
Corollary 4.46 (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 will follow from the Serre duality theorem (which will be discussed in the sequel to this course, Algebraic Geometry 3).
Theorem 4.47 (Serre duality)
Let $X$ be a smooth projective curve as above. For every line bundle ${\mathscr L}$ on $X$, there is a natural isomorphism
\[ H^1(X, {\mathscr L}) \cong H^0(X, {\mathscr L}^{-1} \otimes \Omega ^1_{X/k})^\vee \]
of $k$-vector spaces (where $-^\vee $ denotes the dual $k$-vector space).
A similar statement holds for every locally free sheaf ${\mathscr L}$ (and the theorem can be vastly further generalized).
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