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Cech cohomology of line bundles on the projective line

Statement

Assume the Axiom of Choice, inherited from sheaf cohomology.

For $d \in Z$ , let $O_{P^{1}} (d)$ have standard frames $e_{0}, e_{1}$ on $U_{0} = {x \neq 0}$ and $U_{1} = {y \neq 0}$ with $e_{1} = z^{d} e_{0}$ on the overlap, where $z = y / x$ and $w = 1 / z$ . Then $\dim H^{0} (P^{1}, O (d)) = \max (d + 1, 0), \dim H^{1} (P^{1}, O (d)) = \max (- d - 1, 0),$ and $H^{q} = 0$ for $q \geq 2$ . For $d \geq 0$ , restrictions to $U_{0}$ of global sections on $P^{1}$ , written in the frame $e_{0}$ , are precisely the polynomials of degree at most $d$ ; for $d \leq - 2$ , $H^{1}$ has Laurent basis $z^{d + 1}, z^{d + 2}, \dots, z^{- 1}$ .

Facts & Assumptions

Given: The Axiom of Choice (The Axiom of Choice), $P^{1}$ with the two standard affine charts, an integer $d$ , the line-bundle transition $e_{1} = z^{d} e_{0}$ , and sheaf cohomology with the supplied injective resolution datum.

[F1]

Under the assumed Axiom of Choice inherited from the definition of sheaf cohomology, the two-chart affine cover computes cohomology of a quasi-coherent sheaf on the separated scheme $P^{1}$ (Cech cohomology computes quasi-coherent cohomology on a separated scheme).

[F2]

The sheaf $O (d)$ is invertible, hence quasi-coherent, with the standard transition rule (Invertible sheaves).

Proof

1.1F1F2givenalgebra

The Cech complex for $U_{0}, U_{1}$ has $C^{0} = C [z] e_{0} \oplus C [w] e_{1}, C^{1} = C [z, z^{- 1}] e_{0},$ with differential $(f_{0}, f_{1}) \mapsto z^{d} f_{1} (1 / z) - f_{0} (z)$ . Its kernel consists of polynomials $f_{0} (z)$ that also lie in $z^{d} C [z^{- 1}]$ . If $d \geq 0$ , these are exactly the span of $1, z, \dots, z^{d}$ ; if $d < 0$ , the exponent ranges ${0, 1, \dots}$ and ${d, d - 1, \dots}$ are disjoint, so the kernel is zero. This proves the assertion for $H^{0}$ .

2.1F1step 1.1algebra∎

The degree-one cohomology is the Laurent-polynomial quotient $C [z, z^{- 1}] / (C [z] + z^{d} C [z^{- 1}]) .$ The first summand spans exponents $n \geq 0$ and the second spans $n \leq d$ . If $d \geq - 1$ , these ranges cover every integer, so the quotient is zero. If $d \leq - 2$ , the missing exponents are exactly $d + 1, d + 2, \dots, - 1$ , which gives the stated basis and dimension $- d - 1$ . The Cech complex has no degree above one, so $H^{q} = 0$ for $q \geq 2$ .

Depends on

Used by

Dependency tree · two levels

21 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Masaki Kashiwara, Representation theory and D-modules on flag varieties, Astérisque 173–174 (1989) (standard reference, not scraped)