Cohomology of Quasi Coherent Sheaves on Affine and Projective Schemes — Examples

Given: A field $k$, an integer $d$, the projective line $X={\mathbb{P}}_k^1$ with its twisting sheaves ${\mathcal{O}}_X(d)$; the Axiom of Choice is inherited from the cited suppliers.

Cohomology of the twists of projective space: for a commutative ring $A$ with $1$, an integer $n\ge 0$, the scheme ${\mathbb{P}}_A^n\cong \mathrm{Proj}A[x_0,\dots ,x_n]$ and every $d\in \mathbb{Z}$, one has $H^q({\mathbb{P}}_A^n,\mathcal{O}(d))=0$ unless $q=0$ or $q=n$; for $n>0$, $H^0({\mathbb{P}}_A^n,\mathcal{O}(d))\cong A[x_0,\dots ,x_n{]}_d$ when $d\ge 0$ and $H^0=0$ when $d<0$, while $H^n({\mathbb{P}}_A^n,\mathcal{O}(d))$ is the free $A$-module on the Laurent monomials $x_0^{e_0}\cdots x_n^{e_n}$ with $e_i<0$ for all $i$ and ${\sum }_i{e}_i=d$, so, for $n>0$, it is nonzero precisely when $A\mathrm{\ne }0$ and $d\le -n-1$. (Cohomology of O(d) on projective space, The polynomial ring $R[x_i:i\in I]$ as finitely supported coefficient families on monomials, Nonnegatively graded rings and modules, homogeneous elements, and twists, Relative projective space from standard charts, Twisting sheaf on Proj)

Finiteness and Euler characteristic: for a field $k$, a scheme proper over $k$ and a coherent ${\mathcal{O}}_X$-module $\mathcal{F}$, each $H^q(X,\mathcal{F})$ is a finite-dimensional $k$-vector space, only finitely many are nonzero, and $\chi (X,\mathcal{F})={\sum }_{q\ge 0}(-1{)}^q{\dim }_k{H}^q(X,\mathcal{F})$ is a well-defined integer. (Euler characteristic of a coherent sheaf, Finite-dimensional coherent cohomology over a field, Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis)

Properness: projective space ${\mathbb{P}}_A^n$ is proper over $\mathrm{Spec}A$ for every commutative ring $A$ and every $n\ge 0$. (Finite-dimensional projective space is proper over every base, Proper morphisms)

Local Noetherianity and coherence: a field is a Noetherian ring, the polynomial ring $k[t]$ is Noetherian, the standard charts of ${\mathbb{P}}_k^1$ are spectra of polynomial rings in one variable, so ${\mathbb{P}}_k^1$ is a locally Noetherian scheme; on a locally Noetherian scheme a quasi-coherent module is coherent if and only if it is of finite type. (A field has only the zero ideal and itself, hence is Noetherian, If $R$ is Noetherian then $R[x_1,\dots ,x_n]$ is Noetherian for every $n\in \mathbb{N}$, Relative projective space from standard charts, Locally Noetherian and Noetherian schemes, Coherent sheaves on a locally Noetherian scheme, Coherent module sheaves, Finite type and finitely presented module sheaves)

The twisting sheaves of ${\mathbb{P}}_k^1$ are invertible. Indeed, put $S=k[x_0,x_1]$ with its total-degree grading. On each of the two standard charts $D_{+}(x_i)$, multiplication by $x_i^d$ identifies $S_{(x_i)}$ with $S(d{)}_{(x_i)}$: in $S[x_i^{-1}]$ the element $x_i^d$ is a unit for every integer $d$, and its inverse sends each degree-$d$ element to degree zero. These maps commute with localisation, so the chart description of the twisting sheaf gives ${\mathcal{O}}_X(d){\mathrm{\mid }}_{D_{+}(x_i)}\cong {\mathcal{O}}_{D_{+}(x_i)}$. The two charts cover $X$, proving local freeness of rank one, hence invertibility, quasi-coherence and finite type. Thus each ${\mathcal{O}}_X(d)$ is coherent by [F4]. (Invertible sheaves, Locally free sheaves of finite rank, Twisting sheaf on Proj, Finite type and finitely presented module sheaves, Coherent module sheaves)

Given: A field $k$; the projective line $X={\mathbb{P}}_k^1$ with the standard charts $U_0=D_{+}(x_0)$, $U_1=D_{+}(x_1)$ and the twisting sheaf ${\mathcal{O}}_X(-2)$; and the Axiom of Choice inherited from the cited suppliers.

Charts and affine cover (Relative projective space from standard charts, Projective space is Proj of a polynomial ring, Standard opens are affine): $X={\mathbb{P}}_k^1\cong \mathrm{Proj}k[x_0,x_1]$; the standard charts $U_i=D_{+}(x_i)$ are affine open subschemes forming a cover of $X$, and the intersection $U_0\cap U_1=D_{+}(x_0{x}_1)$ is again affine; the index set $\{0,1\}$ is ordered by $0<1$.

Separatedness (The relative projective-space diagonal is closed): the diagonal ${\mathrm{\Delta }}_{X\mathrm{/}k}$ is a closed immersion, so the structure morphism of $X$ over $\mathrm{Spec}k$ is separated.

Twists and their sections (Twisting sheaf on Proj, Twists of a quasi-coherent sheaf, Sections of a graded-module sheaf on a standard open): ${\mathcal{O}}_X(-2)$ is the associated sheaf $\widetilde{S(-2)}$ of the graded module $S(-2)$, $S=k[x_0,x_1]$, it is quasi-coherent, and for a homogeneous $f$ of positive degree $\mathrm{\Gamma }(D_{+}(f),{\mathcal{O}}_X(-2))=S(-2{)}_{(f)}$ is the degree-zero part of the homogeneous localisation; for $f=x_0{x}_1$ this module has as $k$-basis the Laurent monomials $x_0^{e_0}{x}_1^{e_1}$ with $e_0+e_1=-2$, and the element $1\mathrm{/}(x_0{x}_1)$ is the member $e=(-1,-1)$.

Ordered Čech complex (Ordered Čech cochain complex of a cover, Fixed-cover Čech cohomology): for an open cover indexed by a linearly ordered set one has $C^p={\prod }_{i_0<\cdots <i_p}\mathcal{F}(U_{i_0}\cap \cdots \cap U_{i_p})$ with the alternating Čech differential; for the two-member cover $U_0,U_1$ this gives $C^0=\mathcal{F}(U_0)\oplus \mathcal{F}(U_1)$, $C^1=\mathcal{F}(U_0\cap U_1)$ and $C^p=0$ for $p\ge 2$, with ${\delta }^0(s_0,s_1)=(s_1-s_0){\mathrm{\mid }}_{U_0\cap U_1}$.

Monomial decomposition of the Čech complex (Laurent-monomial decomposition of the projective Cech complex): for $X={\mathbb{P}}_A^n$ with the ordered standard cover and ${\mathcal{O}}_X(d)$ one has $C^{\bullet }(\mathcal{U},{\mathcal{O}}_X(d))={⨁}_e{K}^{\bullet }(e)$, the sum over $e\in {\mathbb{Z}}^{n+1}$ with ${\sum }_i{e}_i=d$, where $K^p(e)$ is free on basis elements $x_{\sigma }^e$ indexed by the $(p+1)$-element subsets $\sigma \supseteq N(e)$, $N(e)=\{i:e_i<0\}$; if $N(e)=\mathrm{\varnothing }$ then $H^0(K^{\bullet }(e))=A$ and all higher cohomology vanishes, if $N(e)=\{0,\dots ,n\}$ then $H^n(K^{\bullet }(e))=A$ and the other groups vanish, and if $N(e)$ is nonempty and proper then $K^{\bullet }(e)$ is contractible; cohomology of the total complex is the direct sum of the cohomologies of the summands.

Čech comparison (Cech cohomology computes quasi-coherent cohomology on a separated scheme): for a quasi-compact separated scheme $X$ with a finite affine open cover $U_0,\dots ,U_r$ and a quasi-coherent ${\mathcal{O}}_X$-module $\mathcal{F}$, the canonical comparison ${\check{H}}^q(\mathcal{U},\mathcal{F})\to H^q(X,\mathcal{F})$ is an isomorphism for every $q\ge 0$.

Top cohomology of projective twists (Top cohomology of projective twists): for $n\ge 1$ the group $H^n({\mathbb{P}}_A^n,\mathcal{O}(d))$ is the free $A$-module on the Laurent monomials $x_0^{e_0}\cdots x_n^{e_n}$ with all $e_i<0$ and ${\sum }_i{e}_i=d$, and it is zero for $d>-n-1$; for $A=k$ a field, $n=1$ and $d=-2$ there is exactly one such monomial, $x_0^{-1}{x}_1^{-1}$, so $H^1({\mathbb{P}}_k^1,\mathcal{O}(-2))$ is free of rank one over $k$.

The Axiom of Choice (The Axiom of Choice): every family of nonempty sets has a choice function, inherited here from [F6] and [F7].

Given: A field $k$, a nonzero homogeneous cubic $f\in k[x_0,x_1,x_2]$, the closed subscheme $C=V_{+}(f)\subseteq {\mathbb{P}}_k^2$ with its structure sheaf ${\mathcal{O}}_C$, and the Axiom of Choice inherited from the cited suppliers.

The hypersurface sequence (Hypersurface cohomology sequence): for a commutative ring $A$ with $1$, $n\ge 0$, a homogeneous $f$ of degree $d>0$ and $X=V_{+}(f)\subseteq {\mathbb{P}}_A^n$ with closed immersion $i$, such that each dehomogenisation $f\mathrm{/}{x}_i^d$ is a nonzerodivisor of the chart ring $B_{(x_i)}$, $B=A[x_0,\dots ,x_n]$ (automatic for $A=k$ a field and $f\mathrm{\ne }0$), the sequence $0\to {\mathcal{O}}_{{\mathbb{P}}^n}(-d)\stackrel{\cdot f}{\to }{\mathcal{O}}_{{\mathbb{P}}^n}\stackrel{i^{\mathrm{♯}}}{\to }{i}_{\ast }{\mathcal{O}}_X\to 0$ is exact, the long exact sequence of sheaf cohomology reads $\cdots \to H^q({\mathbb{P}}^n,\mathcal{O}(-d))\to H^q({\mathbb{P}}^n,\mathcal{O})\to H^q(X,{\mathcal{O}}_X)\to H^{q+1}({\mathbb{P}}^n,\mathcal{O}(-d))\to \cdots$ with $H^q({\mathbb{P}}^n,i_{\ast }{\mathcal{O}}_X)\cong H^q(X,{\mathcal{O}}_X)$, the connecting maps give $H^q(X,{\mathcal{O}}_X)\cong H^{q+1}({\mathbb{P}}^n,\mathcal{O}(-d))$ for every $q\ge 1$, and in degree zero the sequence $0\to H^0({\mathbb{P}}^n,\mathcal{O}(-d))\to A\to H^0(X,{\mathcal{O}}_X)\to H^1({\mathbb{P}}^n,\mathcal{O}(-d))\to 0$ is exact, with $H^0({\mathbb{P}}^n,\mathcal{O}(-d))=0$ for $n\ge 1$ and $H^1({\mathbb{P}}^n,\mathcal{O}(-d))=0$ for $n\mathrm{\ne }1$.

Cohomology of twists on ${\mathbb{P}}^2$ (Cohomology of O(d) on projective space): for every commutative ring $A$, every $n\ge 0$ and every $d\in \mathbb{Z}$ one has $H^q({\mathbb{P}}_A^n,\mathcal{O}(d))=0$ unless $q=0$ or $q=n$; for $n=2$ and $A=k$ a field, $H^0({\mathbb{P}}^2,\mathcal{O})\cong k$, $H^2({\mathbb{P}}^2,\mathcal{O}(-3))$ is free on the negative triples summing to $-3$, namely on the single monomial $x_0^{-1}{x}_1^{-1}{x}_2^{-1}$, and $H^2({\mathbb{P}}^2,\mathcal{O})=0$ because $0>-n-1=-3$.

The Axiom of Choice (The Axiom of Choice): every family of nonempty sets has a choice function, inherited here from [F1] and [F2].

Given: The Axiom of Choice as inherited, a field $k$, an integer $n\ge 0$, the projective space $X={\mathbb{P}}_k^n$ with its standard embedding and twisting sheaves ${\mathcal{O}}_X(d)$.

Conventions: with the standard embedding of the statement the twisting sheaf ${\mathcal{O}}_X(1)$ is invertible, every twist $\mathcal{G}(m)=\mathcal{G}\otimes {\mathcal{O}}_X(1{)}^{\otimes m}$ of a coherent $\mathcal{G}$ is coherent, and $h_{\mathcal{G}}(m)={\dim }_k{H}^0(X,\mathcal{G}(m))$ and $P_{\mathcal{G}}(m)=\chi (X,\mathcal{G}(m))$ are defined for every $m\in \mathbb{Z}$. (Hilbert function and Euler characteristic on a projective scheme, Relative projective space from standard charts, Twisting sheaf on Proj, Invertible sheaves, Coherent module sheaves, Sheaf cohomology as right derived global sections, Euler characteristic of a coherent sheaf)

Cohomology of the twists: for every commutative ring with $1$ in place of $k$ and all $n\ge 0$, $d\in \mathbb{Z}$, $H^q(X,{\mathcal{O}}_X(d))=0$ unless $q=0$ or $q=n$; if $n>0$ then $H^0(X,{\mathcal{O}}_X(d))\cong k[x_0,\dots ,x_n{]}_d$ for $d\ge 0$ and $H^0(X,{\mathcal{O}}_X(d))=0$ for $d<0$, where $k[x_0,\dots ,x_n{]}_d$ is the degree-$d$ graded piece (Nonnegatively graded rings and modules, homogeneous elements, and twists, The polynomial ring $R[x_i:i\in I]$ as finitely supported coefficient families on monomials); and $H^n(X,{\mathcal{O}}_X(d))$ is the free $k$-module on the Laurent monomials $x_0^{e_0}\cdots x_n^{e_n}$ with $e_i<0$ for all $i$ and ${\sum }_i{e}_i=d$, so, for $n>0$ and a nonzero coefficient ring, it is nonzero precisely when $d\le -n-1$. For $n=0$ one has $X=\mathrm{Spec}k$ and $H^0(X,{\mathcal{O}}_X(d))\cong k$ for every $d\in \mathbb{Z}$, with all higher groups zero. (Cohomology of O(d) on projective space)

Counting multi-indices: the monomial $k$-basis of the degree-$m$ piece $k[x_0,\dots ,x_n{]}_m$ is indexed by the multi-indices $(e_0,\dots ,e_n)\in {\mathbb{N}}^{n+1}$ with ${\sum }_i{e}_i=m$, the monomials $x_0^{e_0}\cdots x_n^{e_n}$, by the uniqueness of the expansion of a polynomial (Monomials, coefficients, degree in each variable and total degree in $F[x_1,\dots ,x_n]$, The polynomial ring $R[x_i:i\in I]$ as finitely supported coefficient families on monomials). The number of $(n+1)$-tuples of nonnegative integers with sum $N\ge 0$ equals the number of compositions of $N+n+1$ into exactly $n+1$ positive parts, via $(f_i)\mapsto (f_i+1)$, and by Compositions of $n$ into $k$ positive parts are counted by $\left(\genfrac{}{}{0px}{}{n-1}{k-1}\right)$ this number is $\left(\genfrac{}{}{0px}{}{N+n}{n}\right)$ (with $\left(\genfrac{}{}{0px}{}{a}{b}\right)$ the count of The set $[A{]}^k$ of $k$-element subsets and the binomial coefficient $\left(\genfrac{}{}{0px}{}{n}{k}\right):=\mid [n{]}^k\mid$, and the value $0$ when $n+1$ parts exceed $N+n+1$). In particular the degree-$m$ piece of $k[x_0,\dots ,x_n]$ has dimension $\left(\genfrac{}{}{0px}{}{m+n}{n}\right)$ for every $m\ge 0$, and the set $\{e\in {\mathbb{Z}}^{n+1}:e_i<0,{\sum }_i{e}_i=d\}$ has $\left(\genfrac{}{}{0px}{}{-d-1}{n}\right)$ elements for every $d\le -n-1$. (cor-compositions-with-k-parts-are-counted-by-binomial-coefficients)

The product formula: for integers $0\le b\le a$ the identity $\left(\genfrac{}{}{0px}{}{a}{b}\right)\cdot b!\cdot (a-b)!=a!$ holds in $\mathbb{N}$, so $\left(\genfrac{}{}{0px}{}{a}{b}\right)=\frac{a(a-1)\cdots (a-b+1)}{b!}$; hence for $m\ge 0$ $\left(\genfrac{}{}{0px}{}{m+n}{n}\right)=\frac{(m+n)(m+n-1)\cdots (m+1)}{n!},$ which is the value at $t=m$ of the polynomial $\left(\genfrac{}{}{0px}{}{t+n}{n}\right)$ of the statement, and for $m\le -n-1$ $(-1{)}^n\left(\genfrac{}{}{0px}{}{-m-1}{n}\right)=(-1{)}^n\frac{(-m-1)(-m-2)\cdots (-m-n)}{n!}=\frac{(m+n)(m+n-1)\cdots (m+1)}{n!},$ the last equality because each factor $m+j$ for $1\le j\le n$ is the negative of $-m-j$; if $-n\le m\le -1$ then one of the factors $m+n,\dots ,m+1$ is $0$, so the polynomial value vanishes. ($\left(\genfrac{}{}{0px}{}{n}{k}\right)k!(n-k)!=n!$ for $k\le n$; hence $\left(\genfrac{}{}{0px}{}{n}{k}\right)k!=n^{\underset{‾}{k}}$, the quotient $n!\mathrm{/}(k!(n-k)!)$ is a natural number, and $\left(\genfrac{}{}{0px}{}{n}{k}\right)=\left(\genfrac{}{}{0px}{}{n}{n-k}\right)$, The factorial $n!$ and the falling factorial $n^{\underset{‾}{k}}$, defined by recursion in $\mathbb{N}$, The set $[A{]}^k$ of $k$-element subsets and the binomial coefficient $\left(\genfrac{}{}{0px}{}{n}{k}\right):=\mid [n{]}^k\mid$)

The Axiom of Choice is the choice principle named in the statement, inherited from the cohomology computation and the counting corollary cited above. (The Axiom of Choice)