---- > [!theorem] Theorem. ([[on the sheaf cohomology of line bundles over projective space]]) > ~ ^theorem Let $S:=k[X_{0},\dots,X_{r}]$ be the [[polynomial 4|polynomial ring]] over a [[field]] $k$ in $r+1$ indeterminates. Let $\mathbb{P}^{r}:=\text{Proj }S$ denote rank-$r$ [[projective space]] over $k$, where $\text{Proj}$ denotes the [[proj construction]]. - [ ] [[locally free sheaf|line bundle]] notation Then: 1. **(Degree zero recovers $S$)** There exists an [[isomorphism]] of [[graded ring|graded rings]] $S \cong \bigoplus_{n \in \mathbb{Z}}H^{0}(\mathbb{P}^{r}, \mathcal{O}_{\mathbb{P}^{r}}(n)).$ That is, $H^{0}(\mathbb{P}^{r}, \mathcal{O}_{\mathbb{P}^{r}}(n))=\Gamma\big( \mathbb{P}^{r}, \mathcal{O}_{\mathbb{P}^{r}}(n) \big)$ is $S_{n}$, the [[homogeneous polynomial|homogeneous]] [[polynomial 4|polynomials]] of degree $n$.[^3] 2. **(Intermediate cohomology vanishes)** $H^{i}\big(\mathbb{P}^{r}, \mathcal{O}_{\mathbb{P}^{r}}(n)\big)$ for $0<i<r$.[^2] 3. **(Twisting the rank)** $H^{r}\big( \mathbb{P}^{r}, \mathcal{O}_{\mathbb{P}^{r}}(-r-1) \big) \cong k$ 4. **(Recovering the rest)** There is a [[perfect pairing]] $\begin{align} \overbrace{ H^{0}\big( \mathbb{P}^{r}, \mathcal{O}_{\mathbb{P}^{r}}(n) \big) }^{ S_{n} } &\times H^{r}\big( \mathbb{P}^{r}, \mathcal{O}_{\mathbb{P}^{r}} (-n-r-1) \big) \\ & \to \underbrace{ H^{r}\big( \mathbb{P}^{r}, \mathcal{O}_{\mathbb{P}^{r}} (-r-1) \big) }_{ \cong k } \end{align}$ of finite-dimensional $k$-[[vector space|vector spaces]]. (To remember, imagine that the arguments of $\mathcal{O}(\cdot)$ add, $n+(-n-r-1)=-r-1$) Put together, these allow us to compute the [[sheaf cohomology]] $H^{i}\big( \mathbb{P}^{r}, \mathcal{O}_{\mathbb{P}^{r}}(m)\big)$ given any $i,r,m$. Specifically, the only nonzero cases are $i=0$ and $i=r$; the former case just gives $S_{m}$ and in the latter case we find $H^{r}\big( \mathbb{P}^{r}, \mathcal{O}_{\mathbb{P}^{r}}(m)\big) \cong S_{-m-r-1}^{\vee}.$ Note that there is some built-in vanishing here, since there is no such thing as a homogeneous polynomial of negative degree: $S_{-1,-2,\dots}=0$. > [!specialization] > We have seen $(3)$ already, when computing $H^{1}({\mathbb{P}^{1}}, \mathcal{O}_{\mathbb{P}^{1}}(-2))$ — see the examples in [[Čech sheaf cohomology]]. ^specialization > [!proof]- Proof. ([[on the sheaf cohomology of line bundles over projective space]]) > ~ **Method.** We shall use [[Čech sheaf cohomology|Čech cohomology]] over the standard [[cover]] by [[affine scheme|principal affines]] $\mathscr{U}=\{ D_{+}(X_{0}), \dots, D_{+}(X_{r}) \}$. We will compute the [[(co)homology of a complex|cohomology]] [[group|groups]] all at once, by computing the cohomology of the [[quasicoherent sheaf|quasicoherent]] [[sheaf]] $\mathcal{F}=\bigoplus_{n \in \mathbb{Z}} \mathcal{O}_{\mathbb{P}^{r}}(n)$; this is okay because [[Čech sheaf cohomology|Čech cohomology]] commutes with [[direct sum of sheaves|direct sums]]. **Preliminaries.** Recall the [[lines bundles and transition functions|transition functions]] for $\mathcal{O}_{\mathbb{P}^{r}}(1)$ are $\frac{X_{i}}{X_{j}}$, by definition. So the transition functions for $\mathcal{O}_{\mathbb{P}^{r}}(m)$ are $\frac{X_{i}^{m}}{X_{j}^{m}}$ for $m \geq 0$, and the inverse of this for $m<0$. As notation, put $X=\mathbb{P}^{r}$. For $I \subset \{ 0,\dots,r \}$, write $U_{I}:=\bigcap_{i \in I}^{}U_{i}$. Note that $U_{I}=D_{+}\left( \underbrace{ \prod_{i \in I}^{}X_{i} }_{ =:X_{I} }\right).$ Remember how, in the motivating example computing $H^{*}(\mathbb{P}^{r},\mathcal{O}_{\mathbb{P}^{1}}(-2))$, to compute the codifferential we had to choose a trivialization and 'translate' everything into it? The following lemma yields a bookeeping device which allows us to work without worrying about which trivialization we are in. > [!proposition] A Laurent Lemma. > We may identify $\mathcal{O}_{\mathbb{P}^{r}}(m) (U_{I}) =S_{(X_{I})}$ with the vector space spanned by [[Laurent polynomial|Laurent polynomials]] of degree $m$ — i.e., monomials of the form $X_{0}^{a_{0}}\cdots X_{r}^{a_{r}}$ with $a_{i} \in \mathbb{Z}$, $\sum_{i=1}^{r}a_{i}=m$ — satisfying $a_{i} \geq 0$ if[^4] $i \not \in I$. > > > [!proof] Proof. > > > ^proof Thus (back to doing everything at once), $\Gamma(U_{I}, \mathcal{F})$, $\mathcal{F}=\bigoplus_{m \in \mathbb{Z}} \mathcal{O}_{\mathbb{P}^{r}}(m)$, may be identified with $S_{X_{I}}$:[^1] $\Gamma(U_{I}, \mathcal{F})=S_{X_{I}}.$ [^1]: This is earnest [[localization]], *not* [[proj construction|localization in degree zero]]: the assertion is that $\Gamma(U_{I}, \mathcal{F})$ is $S_{X_{I}}=\{ 1,X_{I}, X_{I}^{2},\dots \} ^{-1}k[X_{0},\dots,X_{r}]$. **The Čech complex.** We are now ready to write down the Čech complex: with the identifications discussed, it is $\prod_{0 \leq i_{0} \leq r} S_{X_{i_{0}}} \xrightarrow{d^{0}} \prod_{0 \leq i_{0} < i_{1} \leq r} S_{X_{i_{0}}X_{i_{1}}} \xrightarrow{d^{1}} \dots \xrightarrow{d^{r-2} }\prod_{0 \leq i \leq r}^{} S_{X_{0} \cdots \widehat{X_{i}} \cdots X_{r}} \xrightarrow{d^{r-1}} S_{X_{0}X_{1}\cdots X_{r}} \to 0.$ We can now prove the theorem. **0th Cohomology.** What is $H^{0}(\mathbb{P}^{r}, \mathcal{F})=\operatorname{ker }d^{0}$? First have to write down $d^{0}$; it is given by $d^{0}\big( (M_{i_{0}})_{0 \leq i_{0} \leq r} \big)= (M_{i_{1}}-M_{i_{0}})_{1 \leq i_{0}<i_{1} \leq r}.$ We don't have to do anything regarding restriction, because the Laurent Lemma above ensures that everything is being represented in the correct way. Specifically, we are viewing $M_{i_{1}}-M_{i_{0}} \in S_{X_{i_{0}}i_{1}} \subset S_{X_{0}\dots X_{r}}$. $M$ is in the kernel when $M_{i_{1}}=M_{i_{0}}$ for all $0 \leq i_{0}<i_{1}\leq r$. That is, $M$ is in all of the $S_{X_{i_{0}}}$; hence $\operatorname{ker }d^{0}=\bigcap_{0 \leq i_{0} \leq r}^{}S_{X_{i_{0}}} \subset S_{X_{0}\cdots X_{r}}.$ So we have to compute this intersection. Note that any homogeneous element of $S_{X_{0} \cdots X_{r}}$ can be written as[^4] $X_{0}^{\ell_{0}} \cdots X_{r}^{\ell_{r}} f(X_{0},\dots,X_{r}), f \in S=k[X_{0},\dots,X_{r}] \text{ not divisible by any }X_{i}.$ This lives in $S_{X_{i_{0}}}$ iff $\ell_{j} \geq 0$ for all $j \neq i_{0}$. It lives in the intersection precisely when all such exponents are nonnegative, i.e., is an element of $S$. Thus $H^{0}(\mathbb{P}^{r}, \mathcal{F})=S$ and this respects degrees: $H^{0}(\mathbb{P}^{r}, \mathcal{O}_{\mathbb{P}^{r}}(m))=S_{m}$. **Top cohomology.** Next we must compute $H^{r}\big( \mathbb{P}^{r}, \mathcal{F} \big)=\operatorname{coker}d^{r-1}$. Where $d^{r-1}$ is given by $d^{r-1}\big( (M_{i})_{0 \leq i \leq r} \big)= \sum_{k=0}^{r} (-1)^{k} M_{k} =\sum_{i=0}^{r} \pm M_{i} $ (we don't care about the signs, because it's just the image that matters). What is $\operatorname{im }d^{r-1}$? ---- #### [^2]: It follows from [[sheaf cohomology is well-behaved with respect to the dimension of Noetherian spaces]] that $H^{i}(\mathbb{P}^{r}, \mathcal{O}_{\mathbb{P}^{r}}(1))=0$ for all $i>r$. Takeaway: all we care about are the cases $i=0$ and $i=r$. [^3]: This result was promised in [[the imperfect dictionary to divisors]]. [^4]: Otherwise we would be 'dividing by zero'. [^5]: ----- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```