Sheaf Cohomology in Algebraic Geometry¶

Job: coherent cohomology $H^i(X,\mathcal{F})$ via Čech and derived functors — Serre duality and classical Riemann–Roch as working tools on curves/projective space.

Core¶

Coherent and quasi-coherent sheaves¶

On a scheme $X$: an $\mathcal{O}_X$-module $\mathcal{F}$ is quasi-coherent if $X$ has an affine cover $U_i=\operatorname{Spec} A_i$ with $\mathcal{F}|_{U_i}\cong\widetilde{M_i}$ for $A_i$-modules $M_i$. Coherent (under locally Noetherian hyp.): locally $\widetilde{M}$ with $M$ f.g., and kernels of maps between finite free sheaves behave well (equiv. f.p. on affines when $A$ Noetherian).

Class	Affine $\operatorname{Spec} A$	Stability
$\operatorname{QCoh}$	all $A$-modules	kernels, cokernels, extensions; $f_*$ for qcqs
$\operatorname{Coh}$	f.g. modules ($A$ Noeth.)	same under proper pushforward (Grothendieck)
Locally free rank $r$	projective f.g. rank $r$	vector bundles

Serre’s $\mathcal{O}(n)$. On $\mathbb{P}^n_A$, twisting sheaves $\mathcal{O}(n)$; every coherent $\mathcal{F}$ fits $\bigoplus_i\mathcal{O}(n_i)^{\oplus m_i}\twoheadrightarrow\mathcal{F}$ for $n_i\ll 0$ (or dual formulation with surjections from $\mathcal{O}(-N)$).

Global sections and derived functors¶

$\Gamma(X,-):\operatorname{Ab}(X)\to\mathbf{Ab}$ (sheaves of abelian groups) is left exact. Sheaf cohomology: $$ H^i(X,\mathcal{F}) := R^i\Gamma(X,\mathcal{F}), $$ computed via injective (or flasque / Godement) resolutions. Long exact sequence from $0\to\mathcal{F}'\to\mathcal{F}\to\mathcal{F}''\to 0$.

Vanishing. If $X$ is Noetherian of dimension $d$ (topological), $H^i(X,\mathcal{F})=0$ for $i>d$ (Grothendieck). For $X$ affine, $H^i(X,\mathcal{F})=0$ for $i>0$ and $\mathcal{F}$ quasi-coherent (Serre).

Čech cohomology¶

For open cover $\mathfrak{U}=\{U_i\}$ and sheaf $\mathcal{F}$, Čech complex $\check{C}^\bullet(\mathfrak{U},\mathcal{F})$ with $$ \check{C}^p = \prod_{i_0<\cdots<i_p}\mathcal{F}(U_{i_0}\cap\cdots\cap U_{i_p}), $$ differential alternating restrictions. $\check{H}^p(\mathfrak{U},\mathcal{F})=H^p(\check{C}^\bullet)$.

Theorem (comparison). For $X$ separated, $\mathcal{F}$ quasi-coherent, and $\mathfrak{U}$ affine open cover, $\check{H}^p(\mathfrak{U},\mathcal{F})\cong H^p(X,\mathcal{F})$. On $\mathbb{P}^n$, standard affine cover $D_+(x_i)$ computes all coherent cohomology in practice.

Object	Computation
$H^0(\mathbb{P}^n_k,\mathcal{O}(m))$	homogeneous polynomials degree $m$ ($m\ge 0$); $0$ if $m<0$
$H^n(\mathbb{P}^n_k,\mathcal{O}(-n-1))$	$k$ (generator $x_0^{-1}\cdots x_n^{-1}$ in Čech)
$H^i(\mathbb{P}^n,\mathcal{O}(m))$	vanishes unless $i=0$ or $i=n$ (Serre)

Higher direct images¶

For $f:X\to Y$, $R^if_*\mathcal{F}$ sheafifies $U\mapsto H^i(f^{-1}U,\mathcal{F})$. Leray spectral sequence $E_2^{p,q}=H^p(Y,R^qf_*\mathcal{F})\Rightarrow H^{p+q}(X,\mathcal{F})$. Proper + coherent $\Rightarrow$ $R^if_*$ coherent (Noetherian hyp.).

Serre duality (pointer)¶

Theorem (Serre duality, smooth projective). $X/k$ smooth projective pure dim $n$, $\omega_X=\det\Omega_{X/k}$. For coherent $\mathcal{F}$, $$ H^i(X,\mathcal{F})^\vee \cong \operatorname{Ext}^{n-i}_X(\mathcal{F},\omega_X), $$ and if $\mathcal{F}$ locally free, $\operatorname{Ext}^{n-i}(\mathcal{F},\omega_X)\cong H^{n-i}(X,\mathcal{F}^\vee\otimes\omega_X)$. In particular $H^n(X,\omega_X)\cong k$ (trace).

Curves. Genus $g=\dim H^1(X,\mathcal{O}_X)=\dim H^0(X,\omega_X)$. Duality: $H^1(L)^\vee\cong H^0(\omega\otimes L^{-1})$ for line bundles $L$.

Grothendieck duality upgrades this to derived categories and proper morphisms — derived categories.

Riemann–Roch (classical)¶

Theorem (RR for curves). $X$ smooth projective curve genus $g$ over $k$ alg. closed, $D$ divisor (or line bundle $L=\mathcal{O}(D)$): $$ \dim H^0(X,L) - \dim H^1(X,L) = \deg L + 1 - g. $$ With Serre duality: $\ell(D)-\ell(K-D)=\deg D+1-g$.

Special case	Conclusion
$\deg L>2g-2$	$H^1(L)=0$; $\dim H^0=\deg L+1-g$
$L=\mathcal{O}$	$\chi(\mathcal{O})=1-g$
Canonical $K$	$\deg K=2g-2$; $\ell(K)=g$

Hirzebruch–RR / GRR (pointer). For higher dimension, $\chi(\mathcal{F})=\int_X \operatorname{ch}(\mathcal{F})\operatorname{td}(T_X)$; Grothendieck–RR in $K$-theory for proper morphisms.

Euler characteristic. $\chi(\mathcal{F})=\sum_i (-1)^i\dim H^i(X,\mathcal{F})$; additive in exact sequences; constant in flat proper families (upper semicontinuity of $h^i$, constancy of $\chi$).

See it¶

Canonical embedding of non-hyperelliptic curves. For $g\ge 3$ non-hyperelliptic smooth projective curve, $|K|$ is very ample and embeds $X\hookrightarrow\mathbb{P}^{g-1}$ as a degree $2g-2$ curve. RR + duality compute $\dim|K|=g-1$ and that $H^0(K)$ separates points/tangents precisely when no $g^1_2$ (hyperelliptic linear system). Petri’s theorem and moduli of curves use this cohomological package constantly.

Cohomology and base change. Flat proper $f:X\to S$, coherent $\mathcal{F}$ flat over $S$: for $s\in S$, the map $R^if_*\mathcal{F}\otimes\kappa(s)\to H^i(X_s,\mathcal{F}_s)$ is iso under cohomological flatness / vanishing of adjacent $R^{i\pm1}$ — failure modes organize degeneration of Hodge numbers in families.

Check¶

Compute $H^1(\mathbb{P}^1_k,\mathcal{O})$ and $H^1(\mathbb{P}^1_k,\mathcal{O}(-2))$ via Čech on $\{U_0,U_1\}$.

$U_i=D_+(x_i)\cong\mathbb{A}^1$, $U_0\cap U_1=\operatorname{Spec} k[t,t^{-1}]$. $\check{C}^0=\mathcal{O}(U_0)\times\mathcal{O}(U_1)$, $\check{C}^1=\mathcal{O}(U_0\cap U_1)$. For $\mathcal{O}$: sections polynomials in $t$ and in $t^{-1}$; difference on overlap — cokernel of $(f,g)\mapsto f-g$ is $0$ (Laurent with only nonnegative/nonpositive parts glue). For $\mathcal{O}(-2)$: transition multiplies by $t^{2}$ (conventions); $H^1\cong k\cdot t^{-1}$ one-dimensional, dual to $H^0(\mathcal{O})\cong k$ by Serre.

Why does RR not determine $\dim H^0$ alone for intermediate degrees?

Only $\chi(L)=\ell(D)-\ell(K-D)$ is topological/degree-theoretic. Both terms can be positive (special divisors, Brill–Noether); Clifford’s theorem bounds $\ell(D)\le\deg D/2+1$ for special $D$.

Pitfalls¶

Čech depends on cover until comparison theorems apply; bad covers give wrong groups.
Coherent $\neq$ quasi-coherent: infinite products of structure sheaves, $j_!\mathcal{O}_U$ on non-closed opens often fail coherence.
GAGA: for proper $X/\mathbb{C}$, analytic and algebraic coherent cohomology agree; open varieties need care.
Base field not alg. closed: dimensions after extension; geometric genus vs arithmetic genus.

Refs¶

Hartshorne, Algebraic Geometry (III)
Mumford, Abelian Varieties (RR applications)
The Stacks Project, Cohomology of Schemes
Huybrechts, Fourier-Mukai Transforms in Algebraic Geometry (derived upgrade)

Next¶

Foundations · Schemes · AG index · Homological algebra · Derived categories · Algebraic topology · Prismatic cohomology — the p-adic cohomology theory unifying crystalline / de Rham / étale

Class	Affine \(\operatorname{Spec} A\)	Stability
\(\operatorname{QCoh}\)	all \(A\)-modules	kernels, cokernels, extensions; \(f_*\) for qcqs
\(\operatorname{Coh}\)	f.g. modules (\(A\) Noeth.)	same under proper pushforward (Grothendieck)
Locally free rank \(r\)	projective f.g. rank \(r\)	vector bundles

Object	Computation
\(H^0(\mathbb{P}^n_k,\mathcal{O}(m))\)	homogeneous polynomials degree \(m\) (\(m\ge 0\)); \(0\) if \(m<0\)
\(H^n(\mathbb{P}^n_k,\mathcal{O}(-n-1))\)	\(k\) (generator \(x_0^{-1}\cdots x_n^{-1}\) in Čech)
\(H^i(\mathbb{P}^n,\mathcal{O}(m))\)	vanishes unless \(i=0\) or \(i=n\) (Serre)

Special case	Conclusion
\(\deg L>2g-2\)	\(H^1(L)=0\); \(\dim H^0=\deg L+1-g\)
\(L=\mathcal{O}\)	\(\chi(\mathcal{O})=1-g\)
Canonical \(K\)	\(\deg K=2g-2\); \(\ell(K)=g\)