Coherent sheaves on projective space have finite resolutions by twists

Every coherent sheaf on $\PP^n$ is built from finitely many twists $\OO(-d)$, in a resolution of length at most $n$.Suggest a slogan

Every coherent $\mcf$ on $\PP^n_k$ admits a finite resolution \[ 0 \to \mce_m \to \cdots \to \mce_1 \to \mce_0 \to \mcf \to 0 \] with each $\mce_j$ a finite direct sum of twists $\OO(-d)$, and one may take $m \leq n$.

Serre’s theorem A supplies the surjection $\mce_0 \surjects \mcf$; iterating it on the kernel gives a possibly infinite resolution, and the Hilbert syzygy theorem for $k[x_0,\dots,x_n]$ is what truncates it. So the statement is a graded-module theorem read through $\tilde{\wait\,}$, and its length bound is the global dimension $n+1$ of the polynomial ring, dropping by one on passage to $\Proj$.

This is the standard device for reducing any additive invariant of coherent sheaves to the twists. Cohomology, the Euler characteristic, and the Hilbert polynomial are all computed this way: break the resolution into short exact sequences, apply the known values on $\OO(d)$, and alternate signs. It is also why $K_0(\PP^n) \cong \ZZ^{n+1}$, generated by the classes of the twists.

How do you resolve a coherent sheaf on $\PP^n$?

How long can the resolution be taken to be?

What is this used for?

Dependencies

T-MODSERRESerre's theorems A and Buses
D-CB9XSThe twisting sheaves $\OO(n)$, and $\Pic(\PP^n) = \ZZ$uses

Backlinks

D-COHDEVISSAGEDévissage of coherent sheavesrelated-to

What links to this

Sheaves on projective space