Twisting sheaf on projective space ($\mathcal{O}(d)$)

The twisting sheaf $\mathcal{O}(d)$ on $\mathrm{Proj}k[x_0,\dots ,x_n]$ is associated with the shifted graded module $k[x_0,\dots ,x_n](d)$. For $d\ge 0$, its global sections are the homogeneous polynomials of degree $d$.

On projective space the twisting sheaf $\mathcal{O}(d)$ is the $d$th tensor power of $\mathcal{O}(1)$ for $d\ge 0$, and the corresponding dual power for $d<0$. Twisting another sheaf means tensoring it with this line bundle.

The local frame $X_i^m$ on $X_i\ne 0$ identifies a section with a homogeneous rational function of degree $m$ on the punctured affine cone. Homogenizing a local rational expression and cancelling common factors writes every nonzero section as a quotient of coprime homogeneous polynomials $F/G$ with $\mathrm{deg}F-\mathrm{deg}G=m$. Conversely, such a rational function regular on the cone gives local coefficients $r/X_i^m$, which glue with the twisting sheaf's transition functions of a vector bundle.

For $n\ge 1$, the top Čech cohomology of the standard affine cover of projective space is the degree-$m$ quotient of the Laurent ring by the sum of rings in which at least one variable has not been inverted. Its basis consists of Laurent monomials $X_0^{a_0}\cdots X_n^{a_n}$ with every $a_i<0$ and ${\sum }_i{a}_i=m$. Hence it vanishes for $m\ge -n$ and has dimension $\left(\genfrac{}{}{0ex}{}{-m-1}{n}\right)$ for $m\le -n-1$.

Over any field, the canonical bundle of projective space is $\mathcal{O}(-n-1)$. Indeed, dualizing the Euler sequence gives $0\to {\Omega }_{{\mathbf{P}}^n}^1\to \mathcal{O}(-1{)}^{\oplus (n+1)}\to \mathcal{O}\to 0$; taking determinants gives the result. Hence Serre duality identifies $H^i({\mathbf{P}}^n,\mathcal{O}(m))$ with the dual of $H^{n-i}({\mathbf{P}}^n,\mathcal{O}(-m-n-1))$.

On the standard charts of projective space, wedge the coordinate differentials in increasing order. The change $X_0/X_i=(X_i/X_0{)}^{-1}$ and $X_j/X_i=(X_j/X_0)/(X_i/X_0)$ gives the displayed Jacobian determinant. Multiplying the $i$th local generator by $(-1{)}^i$ removes this sign and yields the transition functions of the twisting sheaf on projective space with degree $-n-1$.

For a hyperplane $H\subset {\mathbf{P}}^n$ with inclusion $i$, multiplication by its defining linear homogeneous polynomial gives the short exact sequence of sheavesThe long exact sequence in sheaf cohomology relates twists in consecutive degrees and consecutive dimensions. Surjectivity of restriction on global homogeneous polynomials, together with eventual positive-twist vanishing, proves vanishing of intermediate cohomology by descending induction. The top-degree dimensions satisfy $h_n(m-1)=h_n(m)+h_{n-1}(m)$ for $n\ge 2$, where $h_r(m)=\dim H^r({\mathbf{P}}^r,\mathcal{O}(m))$.

The hyperplane exact sequence for twisting sheaves identifies consecutive top-degree groups whenever the hyperplane's preceding-degree group vanishes. Induction on dimension, starting with the surjective restriction of constants to a point on ${\mathbf{P}}^1$, propagates the degree-zero top vanishing down to $m=-n$. Its top-degree surjections also propagate vanishing to every positive twist. This proof does not apply the negative-twist global-section formula to ${\mathbf{P}}^0$.