The residue sequence 0 ⟶ 𝒪_X(D) ⟶ 𝒪_X(D + y) ⟶ κ(y)_y ⟶ 0 #
Let X be a Noetherian integral scheme whose codimension-one local rings are discrete valuation
rings, D a Weil divisor on X and y a codimension-one point. Near y a section f of
𝒪_X(D + y) has order at least -D(y) - 1 at y, so for a rational function g of order
D(y) + 1 at y the product g f lies in the local ring 𝒪_{X,y}, and f is a section of
𝒪_X(D) exactly when g f vanishes at y. Taking the residue of g f at y therefore defines
a morphism from 𝒪_X(D + y) to the skyscraper sheaf κ(y)_y with kernel 𝒪_X(D). When y is a
closed point, as every codimension-one point of a curve is, this morphism is an epimorphism and
gives a short exact sequence
0 ⟶ 𝒪_X(D) ⟶ 𝒪_X(D + y) ⟶ κ(y)_y ⟶ 0.
The morphism depends on the choice of g, through multiplication by a unit of κ(y); its kernel
does not. When y is closed its image is the whole skyscraper sheaf
(epi_toSkyscraperResidueField), whatever the choice of g.
Main declarations #
SchemeWeilDivisor.toSkyscraperResidueField g hg, the morphism𝒪_X(D + y) ⟶ κ(y)_y, described on sections bySchemeWeilDivisor.skyscraperResidueFieldEquiv_toSkyscraperResidueField_app;SchemeWeilDivisor.toSkyscraperResidueField_app_eq_zero_iff: its kernel on sections is𝒪_X(D);SchemeWeilDivisor.residueShortComplex g hg, the sequence𝒪_X(D) ⟶ 𝒪_X(D + y) ⟶ κ(y)_y, exact in the middle (residueShortComplex_exact), and short exact whenyis closed (residueShortComplex_shortExact), sincetoSkyscraperResidueFieldis then an epimorphism (epi_toSkyscraperResidueField).
Over a base field k, the long exact cohomology sequence of the residue sequence gives
SchemeWeilDivisor.finiteDimensional_cohomology_sheaf_add_ofPoint_iff: when[κ(y) : k]is finite,Hⁱ(X, 𝒪_X(D + y))is finite-dimensional exactly whenHⁱ(X, 𝒪_X(D))is, andSchemeWeilDivisor.eulerCharBelow_sheaf_add_ofPoint:χ(𝒪_X(D + y)) = χ(𝒪_X(D)) + [κ(y) : k], the induction step of the Riemann–Roch formulaχ(𝒪_X(D)) = deg D + χ(𝒪_X).
References #
- R. Hartshorne, Algebraic Geometry, IV, proof of Theorem 1.3.
- J.-P. Serre, Algebraic Groups and Class Fields, Chapter II, §3.
The morphism 𝒪_X(D + y) ⟶ κ(y)_y sending a section f near y to the residue at y of
g f, for a rational function g of order D(y) + 1 at y. Its kernel is 𝒪_X(D)
(toSkyscraperResidueField_app_eq_zero_iff), and it is an epimorphism when y is closed
(epi_toSkyscraperResidueField).
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- One or more equations did not get rendered due to their size.
Instances For
The morphism to the skyscraper sheaf on sections. Over an open subset U containing y,
toSkyscraperResidueField g hg sends a section t of 𝒪_X(D + y) to the residue at y of the
element r of the local ring at y which is g t as a rational function.
The kernel of 𝒪_X(D + y) ⟶ κ(y)_y is 𝒪_X(D), on sections: a section of
𝒪_X(D + y) is sent to zero exactly when it is a section of 𝒪_X(D).
The sequence 𝒪_X(D) ⟶ 𝒪_X(D + y) ⟶ κ(y)_y of 𝒪_X-modules. It is exact in the middle
(residueShortComplex_exact), and short exact when y is a closed point
(residueShortComplex_shortExact).
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- One or more equations did not get rendered due to their size.
Instances For
𝒪_X(D) is the kernel of 𝒪_X(D + y) ⟶ κ(y)_y.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The sequence 𝒪_X(D) ⟶ 𝒪_X(D + y) ⟶ κ(y)_y is exact in the middle.
𝒪_X(D + y) ⟶ κ(y)_y is an epimorphism when y is a closed point.
The residue sequence 0 ⟶ 𝒪_X(D) ⟶ 𝒪_X(D + y) ⟶ κ(y)_y ⟶ 0 at a closed codimension-one
point y, on a Noetherian integral scheme whose codimension-one local rings are discrete
valuation rings.
Adding a point preserves finite-dimensionality of cohomology. If y is a closed
codimension-one point with finite residue field over k, and Hⁱ(X, 𝒪_X(D)) is
finite-dimensional, then so is Hⁱ(X, 𝒪_X(D + y)).
Removing a point preserves finite-dimensionality of cohomology. If y is a closed
codimension-one point with finite residue field over k, and Hⁱ(X, 𝒪_X(D + y)) is
finite-dimensional, then so is Hⁱ(X, 𝒪_X(D)).
Finite-dimensionality of cohomology is unchanged by adding a point. For a closed
codimension-one point y with finite residue field over k, Hⁱ(X, 𝒪_X(D + y)) is
finite-dimensional exactly when Hⁱ(X, 𝒪_X(D)) is.
The Euler characteristic of 𝒪_X(D + y). If y is a closed codimension-one point with
residue degree [κ(y) : k] finite and nonzero, and H⁰(X, 𝒪_X(D)) and H¹(X, 𝒪_X(D)) are
finite-dimensional, then
χ(𝒪_X(D + y)) = χ(𝒪_X(D)) + [κ(y) : k],
where χ(M) = dim H⁰(X, M) - dim H¹(X, M) is the Euler characteristic truncated at degree 2
(the full Euler characteristic on a curve). No vanishing of H² is needed: the skyscraper sheaf
has no H¹, so the connecting map into H²(X, 𝒪_X(D)) vanishes.