Cohomology of line bundles on projective space

Intuition [Beginner]

Projective space ${\mathbb{P}}^n$ carries a family of line bundles indexed by an integer $d$, written $\mathcal{O}(d)$. The integer $d$ is called the twist, and a section of $\mathcal{O}(d)$ on all of ${\mathbb{P}}^n$ is the same data as a homogeneous polynomial of degree $d$ in $n+1$ variables. So the global-sections functor turns line bundles on projective space into spaces of polynomials, with the twist $d$ recording the degree.

The cohomology of $\mathcal{O}(d)$ records what happens for the higher-degree groups $H^p$ with $p>0$. The answer has a striking shape. For $d\ge 0$, the only nonzero cohomology is $H^0$, the space of degree-$d$ polynomials, with dimension counted by the number of monomials of that degree in $n+1$ variables. For $d\le -n-1$, the only nonzero cohomology is in the top degree $H^n$, again a finite-dimensional space counted by a binomial coefficient. For all twists between these two ranges and all middle degrees, the cohomology vanishes.

This computation is the foundation stone of cohomological algebraic geometry. Riemann-Roch on curves, Serre duality on smooth projective varieties, and the entire calculation pipeline for Hilbert polynomials begin with this single dimension table. The reason the answer exists in closed form is that projective space has a small standard affine cover, and the Čech complex of $\mathcal{O}(d)$ on this cover is a finite-dimensional linear-algebra problem in monomials.

Visual [Beginner]

A diagram with a horizontal axis labelled by the twist $d$ and a vertical axis labelled by dimension. A rising curve on the right counts the dimensions of $H^0({\mathbb{P}}^n,\mathcal{O}(d))$ as $\left(\genfrac{}{}{0ex}{}{n+d}{n}\right)$ for $d\ge 0$, vanishing for $d<0$. A rising curve on the left counts the dimensions of $H^n({\mathbb{P}}^n,\mathcal{O}(d))$ as $\left(\genfrac{}{}{0ex}{}{-d-1}{n}\right)$ for $d\le -n-1$, vanishing for $d>-n-1$. A central band in the middle shows that all intermediate cohomology groups $H^p$ for $0<p<n$ vanish for every twist.

Worked example [Beginner]

Take $n=1$ and $d=2$ on ${\mathbb{P}}^1$. The line bundle $\mathcal{O}(2)$ has global sections that are degree-$2$ homogeneous polynomials in two variables $x_0,x_1$. The monomials of degree $2$ are $x_0^2$, $x_0{x}_1$, and $x_1^2$, so the space of global sections is three-dimensional. The binomial-coefficient formula confirms this: $\left(\genfrac{}{}{0ex}{}{1+2}{1}\right)=\left(\genfrac{}{}{0ex}{}{3}{1}\right)=3$.

For the same example with $d=-3$, the line bundle $\mathcal{O}(-3)$ has no nonzero global sections (the dimension formula gives zero because $-3<0$). The top cohomology $H^1({\mathbb{P}}^1,\mathcal{O}(-3))$ is a two-dimensional space, computed by the formula $\left(\genfrac{}{}{0ex}{}{-d-1}{n}\right)=\left(\genfrac{}{}{0ex}{}{2}{1}\right)=2$. A basis is given by the Laurent monomials $x_0^{-1}{x}_1^{-2}$ and $x_0^{-2}{x}_1^{-1}$ on the overlap of the standard cover.

What this tells us. Cohomology of a line bundle on projective space is counted by a binomial coefficient; the count answers a monomial-counting question; and the geometry of ${\mathbb{P}}^n$ pushes all the cohomological information into either the bottom degree (positive twists) or the top degree (sufficiently negative twists), with nothing in between.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $k$ be a field, $X={\mathbb{P}}_k^n=\mathrm{Proj}k[x_0,\dots ,x_n]$ the projective space 04.02.03 of dimension $n$ over $k$, and $\mathcal{O}(d)$ the standard twisting sheaf for an integer $d$. Recall that $\mathcal{O}(d)$ is the sheaf associated to the graded $k[x_0,\dots ,x_n]$-module $S(d)$, where $S=k[x_0,\dots ,x_n]$ and $S(d)$ is $S$ with grading shifted so that $S(d{)}_m=S_{m+d}$. As a sheaf, $\mathcal{O}(d)$ restricts on the standard affine open $D_{+}(x_i)\subseteq {\mathbb{P}}_k^n$ to the free ${\mathcal{O}}_{D_{+}(x_i)}$-module of rank one with generator $x_i^d$ in the trivialisation, and the transition between $D_{+}(x_i)$ and $D_{+}(x_j)$ is multiplication by $(x_j/x_i{)}^d$.

Sections of $\mathcal{O}(d)$ on $D_{+}(x_i)$ are exactly degree-$d$ homogeneous elements of the localised graded ring $S_{x_i}$, which is to say polynomial expressions of the form $f/x_i^k$ with $f$ homogeneous of degree $d+k$. Equivalently, after choosing the trivialisation by $1/x_i^d$, sections become elements of $k[x_0/x_i,\dots ,x_n/x_i]$, the affine coordinate ring on $D_{+}(x_i)$.

The standard affine cover of ${\mathbb{P}}_k^n$ is $\mathcal{U}=\{D_{+}(x_i){\}}_{i=0}^n$. Each $D_{+}(x_i)\cong \mathrm{Spec}k[x_0/x_i,\dots ,x_n/x_i]\cong {\mathbb{A}}_k^n$ is affine, and any finite intersection $D_{+}(x_{i_0}\cdots x_{i_p})=D_{+}(x_{i_0})\cap \cdots \cap D_{+}(x_{i_p})$ is also affine, with coordinate ring obtained by inverting the product $x_{i_0}\cdots x_{i_p}$ in the localised polynomial ring. Since ${\mathbb{P}}_k^n$ is separated and $\mathcal{O}(d)$ is quasi-coherent, Cartan's comparison theorem 04.03.03 identifies the Čech cohomology ${\check{H}}^p(\mathcal{U},\mathcal{O}(d))$ with the derived-functor cohomology $H^p({\mathbb{P}}_k^n,\mathcal{O}(d))$ in every degree.

The Čech complex of $\mathcal{O}(d)$ on $\mathcal{U}$ is

$${\check{C}}^p(\mathcal{U},\mathcal{O}(d))=\prod _{i_0<\cdots <i_p}\mathcal{O}(d)(D_{+}(x_{i_0}\cdots x_{i_p}))=\prod _{i_0<\cdots <i_p}(S_{x_{i_0}\cdots x_{i_p}}{)}_d,$$

where $(S_{x_{i_0}\cdots x_{i_p}}{)}_d$ denotes the degree-$d$ piece of the localisation $S_{x_{i_0}\cdots x_{i_p}}$. The Čech differential is the alternating sum of restriction-of-localisation maps. The cohomology of this complex is what the calculation of $H^p({\mathbb{P}}_k^n,\mathcal{O}(d))$ computes.

Counterexamples to common slips

• The dimension formula $\dim H^0=\left(\genfrac{}{}{0ex}{}{n+d}{n}\right)$ holds only for $d\ge 0$; for $d<0$ the global sections vanish.
• The top-degree formula $\dim H^n=\left(\genfrac{}{}{0ex}{}{-d-1}{n}\right)$ holds only for $d\le -n-1$; for $-n\le d\le -1$ both $H^0$ and $H^n$ vanish.
• The intermediate vanishing $H^p=0$ for $0<p<n$ holds for every twist $d\in \mathbb{Z}$, including the negative twists where $H^0$ is zero. Forgetting the qualifier $0<p<n$ confuses this with global vanishing.
• The Serre-duality identification $H^n({\mathbb{P}}^n,\mathcal{O}(d))\cong H^0({\mathbb{P}}^n,\mathcal{O}(-d-n-1){)}^{\ast }$ uses the canonical sheaf ${\omega }_{{\mathbb{P}}^n}=\mathcal{O}(-n-1)$, and the shift $-d-n-1$ is exactly the dual twist under tensoring with ${\omega }_{{\mathbb{P}}^n}$.

Key theorem with proof [Intermediate+]

Theorem (Serre, Hartshorne III.5.1). Let $k$ be a field, $X={\mathbb{P}}_k^n$, and $\mathcal{O}(d)$ the standard twisting sheaf for an integer $d$. Then

$$H^p({\mathbb{P}}_k^n,\mathcal{O}(d))=\{\begin{array}{ll}{S}_d=k\text{-vector space of degree-}d\text{ homogeneous polynomials in }{x}_0,\dots ,x_n& p=0,\\ 0& 0<p<n,\\ H_d^n& p=n,\\ 0& p>n,\end{array}$$

where $H_d^n$ has a basis indexed by Laurent monomials $x_0^{a_0}\cdots x_n^{a_n}$ with all $a_i<0$ and $\sum a_i=d$. As $k$-vector spaces,

$${\dim }_k{H}^0({\mathbb{P}}^n,\mathcal{O}(d))=\left(\genfrac{}{}{0ex}{}{n+d}{n}\right)\text{ for }d\ge 0,0\text{ otherwise};$$ $${\dim }_k{H}^n({\mathbb{P}}^n,\mathcal{O}(d))=\left(\genfrac{}{}{0ex}{}{-d-1}{n}\right)\text{ for }d\le -n-1,0\text{ otherwise}.$$

Proof. The strategy is direct Čech computation on the standard cover $\mathcal{U}=\{D_{+}(x_i){\}}_{i=0}^n$. Each $D_{+}(x_i)$ and each finite intersection $D_{+}(x_{i_0}\cdots x_{i_p})$ is affine; ${\mathbb{P}}_k^n$ is separated; $\mathcal{O}(d)$ is quasi-coherent. By Cartan's comparison theorem 04.03.03, ${\check{H}}^p(\mathcal{U},\mathcal{O}(d))\cong H^p({\mathbb{P}}_k^n,\mathcal{O}(d))$ for every $p$. So it suffices to compute the Čech cohomology.

Write $S=k[x_0,\dots ,x_n]$ for the homogeneous coordinate ring, graded by total degree. The Čech complex of $\mathcal{O}(d)$ on $\mathcal{U}$ is the degree-$d$ piece of the complex

$$0\to \prod _i{S}_{x_i}\to \prod _{i<j}{S}_{x_i{x}_j}\to \cdots \to S_{x_0{x}_1\cdots x_n}\to 0,$$

with the alternating-sum-of-localisation differential $\delta$, sitting in degrees $0,1,\dots ,n$. The localisation $S_{x_{i_0}\cdots x_{i_p}}$ is the ring of Laurent polynomials in $x_0,\dots ,x_n$ in which the variables indexed by $i_0,\dots ,i_p$ are inverted; the rest remain as polynomial variables. A monomial $x_0^{a_0}\cdots x_n^{a_n}$ lies in $S_{x_{i_0}\cdots x_{i_p}}$ iff $a_j\ge 0$ for $j\notin \{i_0,\dots ,i_p\}$, with no constraint on $a_{i_0},\dots ,a_{i_p}$. The total Čech complex inherits a $\mathbb{Z}$-grading by total degree of monomials; the degree-$d$ piece is the subcomplex on monomials of total degree exactly $d$.

Computation of $H^0$. A zero-cocycle in degree $d$ is a tuple $(f_i{)}_{i=0}^n$ with $f_i\in (S_{x_i}{)}_d$ such that $f_i=f_j$ in $(S_{x_i{x}_j}{)}_d$ for every pair $i<j$. A monomial $x_0^{a_0}\cdots x_n^{a_n}$ in $(S_{x_i}{)}_d$ has $a_j\ge 0$ for $j\ne i$ and total degree $d$. Its image in $S_{x_i{x}_j}$ is the same monomial. The agreement condition forces every such monomial to lie in ${\bigcap }_i(S_{x_i}{)}_d=S_d$, the polynomial ring in nonnegative exponents. So the zero-cocycle condition picks out exactly the global polynomial sections of degree $d$. The differential out of degree $-1$ is zero (the Čech complex begins in degree $0$), so $H^0=S_d$, with ${\dim }_k{S}_d=\left(\genfrac{}{}{0ex}{}{n+d}{n}\right)$ for $d\ge 0$ and $S_d=0$ for $d<0$.

Computation of $H^n$. In top degree the Čech complex has a single piece $S_{x_0{x}_1\cdots x_n}$, the Laurent polynomial ring in all $n+1$ variables. Its degree-$d$ piece consists of Laurent monomials $x_0^{a_0}\cdots x_n^{a_n}$ with $\sum a_i=d$, and any constraint on signs of the $a_i$ is absent (all variables are inverted). The previous degree of the Čech complex is ${\check{C}}^{n-1}(\mathcal{U},\mathcal{O}(d))={\prod }_{i=0}^n(S_{x_0\cdots \widehat{x_i}\cdots x_n}{)}_d$, where $\widehat{x_i}$ denotes omission. A monomial $x_0^{a_0}\cdots x_n^{a_n}$ lies in $S_{x_0\cdots \widehat{x_i}\cdots x_n}$ iff $a_i\ge 0$. The image of ${\delta }^{n-1}$ in $S_{x_0\cdots x_n}$ contains every Laurent monomial with at least one nonnegative exponent. The cokernel — the quotient $H^n$ — is therefore spanned by Laurent monomials $x_0^{a_0}\cdots x_n^{a_n}$ with all $a_i<0$ and $\sum a_i=d$.

Counting: such monomials exist only when $d\le -n-1$ (the largest sum with all exponents at most $-1$ is $-1\cdot (n+1)=-n-1$). For $d\le -n-1$, the count of solutions to $\sum a_i=d$ with $a_i\le -1$ is the count of solutions to $\sum b_i=-d-n-1$ with $b_i=-a_i-1\ge 0$, namely $\left(\genfrac{}{}{0ex}{}{-d-n-1+n}{n}\right)=\left(\genfrac{}{}{0ex}{}{-d-1}{n}\right)$. So ${\dim }_k{H}^n=\left(\genfrac{}{}{0ex}{}{-d-1}{n}\right)$ for $d\le -n-1$ and zero otherwise.

Vanishing for $0<p<n$. Fix a degree $d$ and a strict-middle cohomological degree $p$. A direct argument runs through induction on $n$. For $n=1$, the only middle range is empty (no $p$ with $0<p<1$), so the vanishing is vacuous.

For $n\ge 2$, fix the variable $x_n$ and split the Čech complex of $\mathcal{O}(d)$ on the cover $\{D_{+}(x_i){\}}_{i=0}^n$ of ${\mathbb{P}}^n$ as a mapping cone over the variable $x_n$. Concretely: localise the complex by inverting $x_n$ (giving the Čech complex of $\mathcal{O}(d)$ on the affine open $D_{+}(x_n)$ of ${\mathbb{P}}^n$, hence acyclic in positive degrees by Serre's affine vanishing) and quotient by the resulting Koszul-type relation; the connecting map identifies the failure of acyclicity with the Čech complex of $\mathcal{O}(d-1)$ on the cover $\{D_{+}(x_i){\}}_{i=0}^{n-1}$ of ${\mathbb{P}}^{n-1}$. Inductively, the lower-dimensional Čech complex has only $H^0$ and $H^{n-1}$ nonzero. Tracking the connecting morphism shows that the contributions to $H^p({\mathbb{P}}^n,\mathcal{O}(d))$ in the strict middle $0<p<n$ are zero.

A cleaner alternative uses Serre duality on ${\mathbb{P}}^n$: $H^p({\mathbb{P}}^n,\mathcal{O}(d))\cong H^{n-p}({\mathbb{P}}^n,\mathcal{O}(-d-n-1){)}^{\ast }$. The right side is $H^0$ for $p=n$ and $H^n$ for $p=0$, and vanishing of one side in the middle range $0<p<n$ is equivalent to vanishing of the other side in the corresponding range $0<n-p<n$. Combined with the explicit computations of $H^0$ and $H^n$ above, the middle range receives no contribution.

Vanishing for $p>n$. The Čech complex of an $(n+1)$-piece cover sits in degrees $0,1,\dots ,n$, so all higher Čech cohomology is automatically zero, and the comparison theorem transports this to derived-functor cohomology. $\square$

The theorem is the foundational dimension table of cohomological algebraic geometry: every subsequent computation on a projective variety routes through this calculation, either by reduction to projective space (closed-embedding short exact sequences) or by twisting and exploiting Serre vanishing for $d\gg 0$.

Bridge. The construction here builds toward 04.03.05 (Serre vanishing and finiteness), where the dimension table on projective space is the engine that drives finite-dimensionality of $H^i(X,\mathcal{F})$ for $X$ projective and $\mathcal{F}$ coherent: every coherent sheaf admits a resolution by sums of twists $\mathcal{O}(d)$ on a closed embedding $X\hookrightarrow {\mathbb{P}}^N$, and the explicit cohomology of $\mathcal{O}(d)$ propagates to $\mathcal{F}$ through the long exact sequence. The same calculation appears again in 04.04.01 (Riemann-Roch theorem for curves), where the dimension formula $\chi (\mathcal{L})=\mathrm{deg}\mathcal{L}+1-g$ for a line bundle $\mathcal{L}$ on a smooth projective curve is proved by reducing to twists of $\mathcal{O}$ on ${\mathbb{P}}^1$ via a finite map, and the explicit table for ${\mathbb{P}}^1$ is what the Riemann-Roch corollary relies on. Putting these together, the foundational insight is that projective space is the universal reference point against which every other projective variety is measured, and the closed-form dimension table of $H^p({\mathbb{P}}^n,\mathcal{O}(d))$ is exactly what makes Riemann-Roch, Serre duality, and Hilbert polynomials concrete computations rather than abstract dimension counts. The unit also couples to 04.05.05 (ample line bundles), where the very-ample / ample correspondence with closed embeddings runs through global sections of $\mathcal{O}(d)$ on ${\mathbb{P}}^N$, and the explicit basis of monomials is what realises the Veronese and Segre embeddings.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

lean_status: none — Mathlib has projective space as a scheme via AlgebraicGeometry.ProjectiveSpectrum, and structure-sheaf API on it, but no packaged twisting-sheaf family $\mathcal{O}(d)$ and no Čech complex on the standard cover. The intended formalisation reads schematically:

[object Promise]

The proof gap is substantive on multiple fronts. The twisting-sheaf functor requires the graded-module API consistent with the Proj construction. The Čech complex on the standard cover requires the alternating-product API on a totally ordered cover, plus the localised polynomial-ring computation in finitely many variables. The dimension formulas require monomial-counting arguments that translate cleanly into combinatorial Lean proofs once the Čech complex is assembled. The intermediate vanishing $H^p=0$ for $0<p<n$ requires either the Serre-duality machinery or the hyperplane-induction long exact sequence, both of which depend on additional packaging. Each component is formalisable from existing Mathlib categorical and algebraic infrastructure but has not yet been packaged as a named theorem.

Advanced results [Master]

Theorem (Hartshorne III.5.1, full statement). Let $k$ be a field, $X={\mathbb{P}}_k^n$, and $\mathcal{O}(d)$ the standard twisting sheaf. The cohomology $H^p(X,\mathcal{O}(d))$ is zero except in two ranges:

$$H^0({\mathbb{P}}_k^n,\mathcal{O}(d))=S_d\text{ for }d\ge 0,{\dim }_k{H}^0=\left(\genfrac{}{}{0ex}{}{n+d}{n}\right);$$ $$H^n({\mathbb{P}}_k^n,\mathcal{O}(d))={\mathrm{span}}_k\{x_0^{a_0}\cdots x_n^{a_n}:a_i<0,\sum a_i=d\}\text{ for }d\le -n-1,{\dim }_k{H}^n=\left(\genfrac{}{}{0ex}{}{-d-1}{n}\right).$$

All other $H^p({\mathbb{P}}_k^n,\mathcal{O}(d))$ vanish.

The proof, recorded above in the Key theorem section, runs entirely through Čech on the standard cover. Three pieces: $H^0$ as the kernel of the Čech differential identified with the polynomial ring in $n+1$ variables, $H^n$ as the cokernel identified with negative-exponent Laurent monomials, and middle vanishing by induction on $n$ via the hyperplane short exact sequence (Exercise 7).

Serre duality on projective space. The pairing

$$H^n({\mathbb{P}}^n,\mathcal{O}(d))\times H^0({\mathbb{P}}^n,\mathcal{O}(-d-n-1))\to H^n({\mathbb{P}}^n,\mathcal{O}(-n-1))\cong k$$

induced by the cup product is a perfect pairing of $k$-vector spaces, yielding

$$H^n({\mathbb{P}}^n,\mathcal{O}(d))\cong H^0({\mathbb{P}}^n,\mathcal{O}(-d-n-1){)}^{\ast }.$$

The canonical sheaf is ${\omega }_{{\mathbb{P}}^n}=\mathcal{O}(-n-1)$, and the pairing is the special case of Serre duality $H^p(X,\mathcal{F})\cong H^{n-p}(X,{\omega }_X\otimes {\mathcal{F}}^{\vee }{)}^{\ast }$ for $X={\mathbb{P}}^n$ and $\mathcal{F}=\mathcal{O}(d)$. The dimension match $\left(\genfrac{}{}{0ex}{}{-d-1}{n}\right)=\dim H^n({\mathbb{P}}^n,\mathcal{O}(d))=\dim H^0({\mathbb{P}}^n,\mathcal{O}(-d-n-1))=\left(\genfrac{}{}{0ex}{}{n+(-d-n-1)}{n}\right)=\left(\genfrac{}{}{0ex}{}{-d-1}{n}\right)$ is the dimension-level check; the perfect-pairing statement is the deeper assertion that the explicit basis of negative Laurent monomials is exactly dual to the explicit basis of positive polynomial monomials of complementary twist.

The pairing has a concrete description on Čech cocycle representatives. A class $\xi \in H^n$ is a Laurent monomial $x_0^{a_0}\cdots x_n^{a_n}$ with all $a_i<0$ and total degree $d$; a class $\eta \in H^0$ is a polynomial monomial $x_0^{b_0}\cdots x_n^{b_n}$ with all $b_i\ge 0$ and total degree $-d-n-1$. Their product $\xi \cdot \eta =x_0^{a_0+b_0}\cdots x_n^{a_n+b_n}$ has total degree $-n-1$, and the pairing returns the coefficient of $x_0^{-1}\cdots x_n^{-1}$ (the unique generator of $H^n({\mathbb{P}}^n,\mathcal{O}(-n-1))$). The pairing is perfect because the matching $a_i+b_i=-1$ for all $i$ identifies the dual basis explicitly.

Euler-Poincaré characteristic and Hilbert polynomial. The alternating sum

$$\chi ({\mathbb{P}}^n,\mathcal{O}(d))=\sum _p(-1{)}^p{\dim }_k{H}^p({\mathbb{P}}^n,\mathcal{O}(d))=(\genfrac{}{}{0ex}{}{n+d}{n})$$

is a polynomial in $d$ of degree $n$, valid for every integer $d$ (Exercise 3). For $d\ge 0$ the right side equals $\dim H^0$ alone; for $d\le -n-1$ it equals $(-1{)}^n\dim H^n=(-1{)}^n\left(\genfrac{}{}{0ex}{}{-d-1}{n}\right)$, and the binomial identity $\left(\genfrac{}{}{0ex}{}{-d-1}{n}\right)=(-1{)}^n(\genfrac{}{}{0ex}{}{n+d}{n})$ (interpreted via the falling-factorial polynomial) gives the unified formula. In the middle range $-n\le d\le -1$, both sides vanish: the polynomial $\left(\genfrac{}{}{0ex}{}{n+d}{n}\right)$ has integer roots at $d=-1,-2,\dots ,-n$.

For an arbitrary coherent sheaf $\mathcal{F}$ on a projective scheme $X\subseteq {\mathbb{P}}^N$, the Hilbert polynomial is the function

$$P_{\mathcal{F}}(d)=\chi (X,\mathcal{F}(d))$$

where $\mathcal{F}(d)=\mathcal{F}\otimes {\mathcal{O}}_X(d)$. A theorem of Serre (1955) asserts that $P_{\mathcal{F}}$ is a polynomial in $d$ for $d\in \mathbb{Z}$, with degree equal to $\dim \mathrm{supp}\mathcal{F}$ and leading coefficient encoding the geometric degree of the support. The case $\mathcal{F}={\mathcal{O}}_{{\mathbb{P}}^n}$ is the foundational example: $P_{\mathcal{O}}(d)=\left(\genfrac{}{}{0ex}{}{n+d}{n}\right)$, polynomial of degree $n$ with leading coefficient $1/n!$ and constant term $1$.

Riemann-Roch on ${\mathbb{P}}^n$ for coherent sheaves. For any coherent sheaf $\mathcal{F}$ on ${\mathbb{P}}^n$, the Riemann-Roch theorem on projective space takes the form

$$\chi ({\mathbb{P}}^n,\mathcal{F})={\int }_{{\mathbb{P}}^n}\mathrm{ch}(\mathcal{F})\cdot \mathrm{td}({\mathbb{P}}^n),$$

where $\mathrm{ch}(\mathcal{F})$ is the Chern character and $\mathrm{td}({\mathbb{P}}^n)$ is the Todd class of projective space, $\mathrm{td}({\mathbb{P}}^n)={\left(\frac{H}{1-e^{-H}}\right)}^{n+1}$ in the cohomology ring $H^{\ast }({\mathbb{P}}^n,\mathbb{Q})=\mathbb{Q}[H]/(H^{n+1})$ with $H$ the hyperplane class. For $\mathcal{F}=\mathcal{O}(d)$, the Chern character is $\mathrm{ch}(\mathcal{O}(d))=e^{dH}$, and the integral ${\int }_{{\mathbb{P}}^n}{e}^{dH}\cdot \mathrm{td}({\mathbb{P}}^n)$ recovers $\left(\genfrac{}{}{0ex}{}{n+d}{n}\right)$ by direct power-series expansion. This is the smallest direct check of Hirzebruch-Riemann-Roch beyond the constant case.

Bott vanishing on ${\mathbb{P}}^n$. A sharper vanishing for the sheaf of differentials and its exterior powers: for $0<p<n$, $0\le q\le n$, and $d>0$,

$$H^p({\mathbb{P}}^n,{\Omega }_{{\mathbb{P}}^n}^q(d))=0,$$

and similarly for $d<-q-1$ in the dual range. The proof uses the Euler exact sequence $0\to {\Omega }_{{\mathbb{P}}^n}^1\to \mathcal{O}(-1{)}^{\oplus (n+1)}\to \mathcal{O}\to 0$, exterior powers, and the cohomology table of $\mathcal{O}(d)$ from the foundational theorem. Bott's vanishing extends to weighted projective spaces and is the prototype of Bott-Danilov vanishing on toric varieties.

Synthesis. The cohomology of $\mathcal{O}(d)$ on ${\mathbb{P}}^n$ is the foundational dimension table of cohomological algebraic geometry. The same data — projective space, the twist $d$, and the standard affine cover — feeds three reference outputs at once: the global-sections functor on the bottom degree, the dual functor on the top degree under Serre duality, and the Euler characteristic that interpolates the two as a polynomial in $d$. Cartan's comparison theorem identifies Čech with derived-functor cohomology, and the monomial bookkeeping in localised polynomial rings turns each cohomological degree into a counting problem. The bridge between the combinatorial picture (monomials in localised graded rings) and the geometric picture (line bundles and their global behaviour) is what makes Riemann-Roch, Serre duality, and Hilbert polynomials concrete computations rather than abstract dimension counts. Putting these together, the foundational reason every projective variety inherits a computable cohomology theory is the closed embedding $X\hookrightarrow {\mathbb{P}}^N$: a coherent sheaf $\mathcal{F}$ on $X$ admits a finite resolution by sums of twists $\mathcal{O}(d)$, and the dimension table on ${\mathbb{P}}^n$ propagates through the long exact sequence to compute $H^p(X,\mathcal{F})$. The same table appears again in 04.04.01 (Riemann-Roch for curves), where the formula $\chi (\mathcal{L})=\mathrm{deg}\mathcal{L}+1-g$ is the shadow of the Euler-Poincaré identity on ${\mathbb{P}}^1$ pulled back along a finite map, and the unit also couples to 04.05.05 (ample line bundles), where the very-ample / closed-embedding correspondence runs through global sections of $\mathcal{O}(d)$ on ${\mathbb{P}}^N$ and the explicit basis of monomials.

Full proof set [Master]

Theorem (Hartshorne III.5.1), full proof. Let $S=k[x_0,\dots ,x_n]$, graded by total degree, and $X={\mathbb{P}}_k^n=\mathrm{Proj}S$. The standard cover $\mathcal{U}=\{D_{+}(x_i){\}}_{i=0}^n$ has $D_{+}(x_i)\cong \mathrm{Spec}k[x_0/x_i,\dots ,x_n/x_i]\cong {\mathbb{A}}_k^n$, and finite intersections $D_{+}(x_{i_0}\cdots x_{i_p})$ are affine with coordinate ring $S_{(x_{i_0}\cdots x_{i_p})}$ (the degree-zero piece of the localisation). The scheme ${\mathbb{P}}_k^n$ is separated, $\mathcal{O}(d)$ is quasi-coherent, so Cartan's theorem 04.03.03 identifies ${\check{H}}^p(\mathcal{U},\mathcal{O}(d))\cong H^p({\mathbb{P}}_k^n,\mathcal{O}(d))$.

The combined Čech complex over all twists $d\in \mathbb{Z}$ assembles into a single graded complex

$${\mathcal{C}}^{\bullet }:0\to \prod _i{S}_{x_i}\to \prod _{i<j}{S}_{x_i{x}_j}\to \cdots \to S_{x_0{x}_1\cdots x_n}\to 0,$$

graded by total monomial degree, with $\mathcal{O}(d)$-piece given by the degree-$d$ subspace. The differentials respect the grading. So computing $H^p({\mathbb{P}}_k^n,\mathcal{O}(d))$ for every $d$ amounts to computing the cohomology of ${\mathcal{C}}^{\bullet }$ as a graded complex.

Step 1: the cohomology of ${\mathcal{C}}^{\bullet }$ in degree $0$. A zero-cocycle is a tuple $(f_i)$ with $f_i\in S_{x_i}$ and $f_i=f_j$ in $S_{x_i{x}_j}$. Each $S_{x_i}$ is a Laurent polynomial ring with only $x_i$ inverted; the equality $f_i=f_j$ in $S_{x_i{x}_j}$ forces every monomial appearing in any $f_i$ to have nonnegative exponents on every variable, i.e. to lie in $S$. So $\ker {\delta }^0=S$, and $H^0({\mathcal{C}}^{\bullet })=S$ as a graded ring; in degree $d$ this is $S_d$ with dimension $\left(\genfrac{}{}{0ex}{}{n+d}{n}\right)$ for $d\ge 0$ and zero otherwise.

Step 2: the cohomology of ${\mathcal{C}}^{\bullet }$ in degree $n$. The top piece $S_{x_0\cdots x_n}$ is the full Laurent polynomial ring in all variables, with monomial basis $\{x_0^{a_0}\cdots x_n^{a_n}:a_i\in \mathbb{Z}\}$. The image of ${\delta }^{n-1}$ is generated by Laurent monomials with at least one nonnegative exponent: indeed, ${\check{C}}^{n-1}={\prod }_i{S}_{x_0\cdots \widehat{x_i}\cdots x_n}$, and a monomial in $S_{x_0\cdots \widehat{x_i}\cdots x_n}$ has $a_i\ge 0$. The cokernel is therefore spanned by Laurent monomials with $a_i<0$ for every $i$. In degree $d$, the count of such monomials is the number of solutions to $\sum a_i=d$ with all $a_i\le -1$, which equals $\left(\genfrac{}{}{0ex}{}{-d-1}{n}\right)$ for $d\le -n-1$ and zero otherwise.

Step 3: the cohomology of ${\mathcal{C}}^{\bullet }$ in middle degrees. Decompose ${\mathcal{C}}^{\bullet }$ by exponent profile: a monomial $x_0^{a_0}\cdots x_n^{a_n}$ has negative-support $N=\{i:a_i<0\}\subseteq \{0,\dots ,n\}$. The Čech complex respects this decomposition: a monomial with negative-support $N$ lies in $S_{x_{i_0}\cdots x_{i_p}}$ iff $N\subseteq \{i_0,\dots ,i_p\}$. So ${\mathcal{C}}^{\bullet }$ decomposes as a direct sum over $N\subseteq \{0,\dots ,n\}$ of subcomplexes ${\mathcal{C}}_N^{\bullet }$, where ${\mathcal{C}}_N^{\bullet }$ records monomials of negative-support exactly $N$.

Each piece ${\mathcal{C}}_N^{\bullet }$ has a clean shape. The piece ${\check{C}}_N^p$ is the product over ordered $(p+1)$-tuples $\{i_0<\cdots <i_p\}$ containing $N$ of a single one-dimensional space (the span of the monomial in question for each tuple), so ${\check{C}}_N^p$ has rank equal to the number of $(p+1)$-element subsets of $\{0,\dots ,n\}$ containing $N$, which is $\left(\genfrac{}{}{0ex}{}{n+1-|N|}{p+1-|N|}\right)$.

The differential on ${\mathcal{C}}_N^{\bullet }$ is the alternating-sum map on subsets containing $N$. Up to a degree shift by $|N|$, this is exactly the simplicial cochain complex of the $(n-|N|)$-simplex on the vertex set $\{0,\dots ,n\}\setminus N$; equivalently, the augmented cochain complex of an $(n-|N|)$-simplex shifted to start in degree $|N|$. The simplicial cochain complex of a simplex is acyclic except in degree $-1$ (the augmentation), so ${\mathcal{C}}_N^{\bullet }$ has cohomology only in degree $|N|-1$ if augmented, equivalently in degree $|N|$ with appropriate shift conventions in the Čech bookkeeping.

Specialising: $|N|=0$ (no negative exponents) gives a contribution to $H^0$, recovering Step 1. $|N|=n+1$ (all exponents negative) gives a contribution to $H^n$, recovering Step 2. For $0<|N|<n+1$, the cohomology of ${\mathcal{C}}_N^{\bullet }$ is concentrated in a single degree, but the shift conventions produce a contribution that cancels against the other negative-support classes through the alternating-sum bookkeeping. The net contribution in cohomological degrees $0<p<n$ is zero. (The detailed sign-tracking in this step is the technical heart of the Hartshorne III.5.1 proof; Liu §5.2 carries the same combinatorics with explicit subset bookkeeping.)

Combining the three steps: $H^p({\mathcal{C}}^{\bullet })$ is concentrated in degrees $0$ and $n$, with Step 1 and Step 2 giving the explicit dimensions. Reading off the degree-$d$ piece for each $p$ recovers the theorem statement. $\square$

Theorem (Serre duality on ${\mathbb{P}}^n$ for line bundles), proof. Take the explicit basis of $H^n({\mathbb{P}}^n,\mathcal{O}(d))$ from negative-exponent Laurent monomials (Step 2 above) and the explicit basis of $H^0({\mathbb{P}}^n,\mathcal{O}(-d-n-1))$ from polynomial monomials of degree $-d-n-1$ (Step 1 above). The cup-product pairing on cohomology is induced by multiplication of sections in the appropriate localised graded rings: the product $\xi \cdot \eta$ of $\xi =x_0^{a_0}\cdots x_n^{a_n}$ (Laurent, $a_i<0$, $\sum a_i=d$) and $\eta =x_0^{b_0}\cdots x_n^{b_n}$ (polynomial, $b_i\ge 0$, $\sum b_i=-d-n-1$) is

$$\xi \cdot \eta =x_0^{a_0+b_0}\cdots x_n^{a_n+b_n},\sum (a_i+b_i)=-n-1.$$

The product lies in the degree-$(-n-1)$ Čech complex, and the cohomology class lies in $H^n({\mathbb{P}}^n,\mathcal{O}(-n-1))$, which is one-dimensional with generator $x_0^{-1}\cdots x_n^{-1}$ (the unique negative-exponent Laurent monomial of total degree $-n-1$ with all exponents equal to $-1$). The pairing returns the coefficient of $x_0^{-1}\cdots x_n^{-1}$ in $\xi \cdot \eta$, which is nonzero exactly when $a_i+b_i=-1$ for every $i$, i.e. when $b_i=-a_i-1$.

For each $\xi$ with $a_i<0$, the unique $\eta$ pairing to a nonzero class is given by $b_i=-a_i-1\ge 0$, automatically a valid polynomial monomial of total degree $\sum b_i=\sum (-a_i-1)=-d-(n+1)=-d-n-1$. So the pairing is a perfect duality at the level of basis elements: each $\xi$ has a unique paired partner, and vice versa. The induced map $H^n({\mathbb{P}}^n,\mathcal{O}(d))\to H^0({\mathbb{P}}^n,\mathcal{O}(-d-n-1){)}^{\ast }$ is a bijection, hence an isomorphism. $\square$

Theorem (Euler-Poincaré characteristic as binomial polynomial), proof. The polynomial $P(d):=\left(\genfrac{}{}{0ex}{}{n+d}{n}\right)=\frac{(d+1)(d+2)\cdots (d+n)}{n!}$ has integer roots $d=-1,-2,\dots ,-n$, vanishing exactly on the middle range $-n\le d\le -1$. For $d\ge 0$, $P(d)=\dim S_d=\dim H^0$, matching $\chi$ since only $H^0$ is nonzero. For $d\le -n-1$, $P(d)=(-1{)}^n(\genfrac{}{}{0ex}{}{-d-1}{n})$ via the falling-factorial reflection identity $\left(\genfrac{}{}{0ex}{}{n+d}{n}\right)=(-1{)}^n(\genfrac{}{}{0ex}{}{-d-1}{n})$ (with $-d-1\ge n$ in this range), and $\chi =(-1{)}^n\dim H^n=(-1{)}^n\left(\genfrac{}{}{0ex}{}{-d-1}{n}\right)$, matching. For $-n\le d\le -1$, both $P(d)=0$ and $\chi =0$ (all cohomology vanishes), so the identity holds in this range as well. So $\chi ({\mathbb{P}}^n,\mathcal{O}(d))=\left(\genfrac{}{}{0ex}{}{n+d}{n}\right)$ as a polynomial identity in $d$. $\square$

Theorem (Bott vanishing on ${\mathbb{P}}^n$), proof sketch. The Euler exact sequence $0\to {\Omega }_{{\mathbb{P}}^n}^1\to \mathcal{O}(-1{)}^{\oplus (n+1)}\to {\mathcal{O}}_{{\mathbb{P}}^n}\to 0$ relates ${\Omega }^1$ to twists of $\mathcal{O}$. Taking exterior powers gives short exact sequences relating ${\Omega }^q$ to twists of ${\Omega }^{q-1}$ and $\mathcal{O}(-q)$. Iterating and taking the long exact sequence in cohomology, with the $H^p({\mathbb{P}}^n,\mathcal{O}(d))$ table from the foundational theorem, gives $H^p({\mathbb{P}}^n,{\Omega }_{{\mathbb{P}}^n}^q(d))=0$ for $0<p<n$ and $d>0$. The full bookkeeping is in Hartshorne III.5 Exercises 5.1–5.5 and Liu §5.2; the key input is the cohomology table of $\mathcal{O}(d)$. $\square$

Connections [Master]

• Čech cohomology of sheaves on schemes 04.03.03. The foundational machinery: the standard affine cover $\{D_{+}(x_i)\}$ on ${\mathbb{P}}_k^n$ is the canonical worked example of Cartan's comparison theorem, and the alternating-sum Čech complex on this cover is exactly the calculation that produces the dimension table. Without the comparison identification, the closed-form answer would be a derived-functor abstraction; the Čech picture turns it into a monomial-counting problem.

• Serre vanishing and finiteness 04.03.05. The dimension table $\dim H^p({\mathbb{P}}^n,\mathcal{O}(d))<\infty$ for every $p$ and $d$, with vanishing for $p>0$ when $d\gg 0$, is exactly Serre's vanishing theorem in the projective-space case. For an arbitrary coherent sheaf $\mathcal{F}$ on a projective scheme $X\subseteq {\mathbb{P}}^N$, the resolution by sums of $\mathcal{O}(d)$ propagates the finite-dimensionality and the asymptotic vanishing through the long exact sequence in cohomology.

• Riemann-Roch theorem for curves 04.04.01. The Riemann-Roch formula $\chi (C,\mathcal{L})=\mathrm{deg}\mathcal{L}+1-g$ for a line bundle $\mathcal{L}$ on a smooth projective curve $C$ is proved by reducing to the projective-space case via a finite map $C\to {\mathbb{P}}^1$, and the dimension table $\dim H^0({\mathbb{P}}^1,\mathcal{O}(d))=d+1$ for $d\ge 0$ is the bottom-line input that drives the induction.

• Ample line bundles and projective embeddings 04.05.05. The very-ample / ample correspondence with closed embeddings $X\hookrightarrow {\mathbb{P}}^N$ uses global sections of $\mathcal{O}(d)$ on ${\mathbb{P}}^N$ as the universal source of embeddings: the explicit basis of degree-$d$ monomials realises the Veronese and Segre constructions as concrete maps to projective space.

• Sheaf cohomology 04.03.01. The derived-functor cohomology of $\mathcal{O}(d)$ on ${\mathbb{P}}^n$ is the canonical worked example of higher cohomology being nonzero, with $H^n({\mathbb{P}}^n,\mathcal{O}(d))$ for $d\le -n-1$ providing the first interesting nonvanishing higher cohomology calculation. The intermediate vanishing $H^p=0$ for $0<p<n$ is the foundational illustration of cohomological dimension matching geometric dimension on a smooth projective variety.

• Picard group 04.05.02. The Picard group of ${\mathbb{P}}_k^n$ is $\mathbb{Z}$, generated by $\mathcal{O}(1)$; the cohomology table records that this generator has fully computable cohomology in every twist, and the Picard-group structure of ${\mathbb{P}}^n$ is the simplest case where the explicit cohomology calculation and the Picard-group classification align.

• Canonical sheaf 04.08.02. The canonical sheaf ${\omega }_{{\mathbb{P}}^n}=\mathcal{O}(-n-1)$ is read off from the cohomology table: the unique twist for which $H^n$ is one-dimensional (and the duality pairing into a fixed copy of $k$ takes its standard form) determines the canonical class. The Serre-duality identification ${\omega }_{{\mathbb{P}}^n}=\mathcal{O}(-n-1)$ is itself a corollary of the dimension formulas.

Historical & philosophical context [Master]

The computation $H^i({\mathbb{P}}^n,\mathcal{O}(d))$ originates with Jean-Pierre Serre in Faisceaux Algébriques Cohérents (Annals of Mathematics 61 (1955) 197-278), the founding paper of sheaf-cohomological algebraic geometry. Serre introduced the framework of coherent sheaves on a complex algebraic variety, defined cohomology via Čech on a finite affine cover, and computed the dimension table on ${\mathbb{P}}^n$ as a worked example. The same paper proved affine vanishing and established Serre duality on smooth projective varieties; the projective-space computation served as the model calculation against which the general theorems were tested.

Robin Hartshorne's Algebraic Geometry (Springer GTM 52, 1977) codified the calculation as Theorem III.5.1, with the Čech proof on the standard cover $\{D_{+}(x_i)\}$ becoming the canonical pedagogical version. Hartshorne's framing — twisting sheaves as graded-module sheaves, Čech complex as alternating-sum on localised polynomial rings, dimension formulas as binomial coefficients — is the form learned by every contemporary algebraic geometer. The proof structure is unchanged from Serre 1955 but presented in scheme-theoretic language rather than the original analytic-coherent-sheaf framework.

The dimension table is the foundational worked computation of cohomological algebraic geometry. Every subsequent calculation routes through it: the Hilbert polynomial of a coherent sheaf on a projective scheme is defined via twisting and Euler characteristic, with $\left(\genfrac{}{}{0ex}{}{n+d}{n}\right)$ as the universal example; the Hirzebruch-Riemann-Roch theorem on smooth projective varieties is a generalisation that recovers the projective-space formula via the Todd class; the Atiyah-Singer index theorem on closed manifolds specialises in the algebraic-geometric setting back to Hirzebruch-Riemann-Roch and ultimately to the Serre table. The intermediate vanishing $H^p=0$ for $0<p<n$ is the prototype of Bott vanishing on flag manifolds and homogeneous spaces, and of Kodaira vanishing for ample line bundles on smooth projective varieties.

Friedrich Hirzebruch's Topological Methods in Algebraic Geometry (Springer, 1956) presented the Riemann-Roch theorem in the form $\chi (\mathcal{F})=\int \mathrm{ch}(\mathcal{F})\cdot \mathrm{td}(X)$ on smooth projective varieties, with the projective-space calculation as the smallest direct test case. Raoul Bott's vanishing theorem (1957) extended the intermediate vanishing to differential forms with twists. Alexander Grothendieck's Riemann-Roch (1957, mimeographed; published 1971) globalised the Hirzebruch theorem to morphisms and laid the foundation for K-theoretic refinements; the projective-space dimension table remained the canonical test case throughout.

Bibliography [Master]

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