The "restriction to the cover" map for an A-module M: it sends m to the
pair of its images in the localizations away from f and from g. This is the
first map of the Čech complex for the two-element cover {D(f), D(g)}.
Instances For
Sheaf condition (separation/injectivity): if f, g generate the unit ideal,
then the product of the two localization maps is injective. This is the H⁰
part of the Čech sheaf condition for {D(f), D(g)}.
Flatness of localization at a single element: A[f⁻¹] is A-flat.
This is essential for showing that taking cohomology commutes with localization.
f becomes a unit in A[(fg)⁻¹].
g becomes a unit in A[(fg)⁻¹].
The restriction A[f⁻¹] → A[(fg)⁻¹] induced by the universal property
of localization. This is one of the two Čech restriction maps for {D(f), D(g)}.
Instances For
The restriction A[g⁻¹] → A[(fg)⁻¹] induced by the universal property
of localization.
Instances For
The Čech differential d applied to an element of A coming from both
A[f⁻¹] and A[g⁻¹] gives zero. This expresses d ∘ d = 0 at the H⁰ level.
The Čech differential A[f⁻¹] × A[g⁻¹] → A[(fg)⁻¹], sending
(s, t) ↦ s|_{D(fg)} - t|_{D(fg)}. Its cokernel is H¹ for the two-element cover.
Instances For
The restriction map from A[f⁻¹] agrees with algebraMap A _ on elements
of A.
The restriction map from A[g⁻¹] agrees with algebraMap A _ on elements
of A.
Sending the formal inverse of f in A[f⁻¹] into A[(fg)⁻¹] and multiplying
by f yields 1.
Sending the formal inverse of g in A[g⁻¹] into A[(fg)⁻¹] and multiplying
by g yields 1.
Iterated version of cechMapFromF_invSelf_mul: pushing f⁻ⁿ into
A[(fg)⁻¹] and multiplying by fⁿ gives 1.
Iterated version of cechMapFromG_invSelf_mul: pushing g⁻ⁿ into
A[(fg)⁻¹] and multiplying by gⁿ gives 1.
Computation of the Čech restriction map from A[f⁻¹] on an element of the
form c / fⁿ, cleared by (fg)ⁿ: the result is c · gⁿ.
Computation of the Čech restriction map from A[g⁻¹] on an element of the
form c / gⁿ, cleared by (fg)ⁿ: the result is c · fⁿ.
Vanishing of Čech H¹ for the structure sheaf on an affine open:
if f, g generate the unit ideal of A, then the Čech differential
A[f⁻¹] × A[g⁻¹] → A[(fg)⁻¹] is surjective. Equivalently, H¹ of the
two-element cover by basic opens vanishes for the structure sheaf — the
algebraic origin of affine acyclicity.
H⁰(ℙ¹, O(n)) = 0 for n < 0: no global sections in negative degree.
H¹(ℙ¹, O(n)) = 0 for n ≥ 0: higher cohomology of non-negative twists
vanishes on ℙ¹.
Serre duality on ℙ¹, non-negative case: dim H¹(O(n)) = dim H⁰(O(-2 - n)).
For n ≥ 0 both sides vanish, confirming the duality. The canonical bundle on
ℙ¹ is O(-2), so O(n)∨ ⊗ K = O(-2 - n).
Serre duality on ℙ¹, negative case: dim H¹(O(n)) = dim H⁰(O(-2 - n)).
For n < 0, -2 - n may be ≥ 0 or = -1; both sub-cases give matching
dimensions.
Euler characteristic of O(n) on ℙ¹: χ(O(n)) = h⁰(O(n)) - h¹(O(n)) = n + 1.
This is the Riemann–Roch formula for line bundles on ℙ¹ (genus 0).