Intersection Products on Smooth Projective Surfaces

Given: a closed immersion $i:Z\to X$ of schemes, an invertible ${\mathcal{O}}_X$-module $\mathcal{L}$, a quasi-coherent ${\mathcal{O}}_Z$-module $\mathcal{G}$, and the Axiom of Choice (The Axiom of Choice).

A closed immersion is a morphism whose underlying map is a homeomorphism onto a closed subset $Z\subseteq X$ and for which ${\mathcal{O}}_X\to i_{\ast }{\mathcal{O}}_Z$ is surjective (Closed immersions of schemes). In particular $i^{-1}(U)=U\cap Z$ for every open $U\subseteq X$, the assignment $U\mapsto U\cap Z$ is a surjection from the open subsets of $X$ onto the open subsets of $Z$, and the open neighbourhoods $U\cap Z$ of a point $z\in Z$, with $U$ an open neighbourhood of $z$ in $X$, are cofinal among the open neighbourhoods of $z$ in $Z$ (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison).

Direct image is precomposition: $(i_{\ast }\mathcal{F})(U)=\mathcal{F}(i^{-1}U)$ with the evident restrictions, and it is a sheaf when $\mathcal{F}$ is (Direct image of a sheaf along a continuous map, Direct image preserves sheaves and objectwise algebraic structure). The stalk at $z\in X$ is the filtered colimit over open neighbourhoods $U$ of $z$ (The stalk of a presheaf at a point).

Pullback: $i^{\ast }\mathcal{L}={\mathcal{O}}_Z{\otimes }_{i^{-1}{\mathcal{O}}_X}{i}^{-1}\mathcal{L}$ is an ${\mathcal{O}}_Z$-module (Pullback of a module along a morphism of ringed spaces); it is quasi-coherent when $\mathcal{L}$ is quasi-coherent (Scheme pullback preserves quasi-coherence), and invertible ${\mathcal{O}}_X$-modules are quasi-coherent (Invertible sheaves, Locally free sheaves of finite rank). For every point $z$ the stalks satisfy $(i^{-1}\mathcal{L}{)}_z\cong {\mathcal{L}}_{i(z)}={\mathcal{L}}_z$ and $(i^{-1}{\mathcal{O}}_X{)}_z\cong {\mathcal{O}}_{X,z}$ (The stalk of an inverse image sheaf is the stalk over the image point); the stalk of a tensor product of $\mathcal{O}$-modules on a ringed space is the tensor product of the stalks over the stalk of the ring (The stalk of a tensor product sheaf is the tensor product of the stalks).

Module identifications: for a homomorphism of commutative rings $R\to S$, a right $S$-module $N$ and a left $R$-module $M$ there is a natural isomorphism $N{\otimes }_{R}M\cong N{\otimes }_S(S{\otimes }_{R}M)$ (Change of rings: $N{\otimes }_{R}M\cong N{\otimes }_S(S{\otimes }_{R}M)$); tensor products over a commutative ring are associative (Associativity of tensor products for compatible bimodules); and $R{\otimes }_{R}N\cong N\cong N{\otimes }_{R}R$ for every $R$-module $N$ (The regular module is a tensor unit: $R{\otimes }_{R}N\cong N$ and $M{\otimes }_{R}R\cong M$).

A morphism of sheaves on a topological space is an isomorphism if and only if it is bijective on every stalk (A morphism of sheaves is an isomorphism exactly when it is an isomorphism on every stalk).

Tensor products of quasi-coherent modules are quasi-coherent (Tensor product preserves quasi-coherence); on a locally Noetherian scheme a quasi-coherent module is coherent if and only if it is of finite type, and coherence is a local condition on the scheme (Coherent sheaves on a locally Noetherian scheme, Coherent module sheaves).

Assume AC. For a quasi-coherent ${\mathcal{O}}_Z$-module $\mathcal{F}$ and every $q\ge 0$ there is a canonical isomorphism $H^q(Z,\mathcal{F})\cong H^q(X,i_{\ast }\mathcal{F})$; if $X$ is locally Noetherian and $\mathcal{F}$ coherent, then $i_{\ast }\mathcal{F}$ is coherent (Closed immersion preserves cohomology and coherent pushforward, Euler characteristic of a coherent sheaf). The Axiom of Choice is inherited from these suppliers; the change-of-rings identification of [F4] and the stalk computations below make no selection.

Given: a field $k$, a proper $k$-scheme $X$, a closed point $p\in X$ with residue field $\kappa (p)$, the corresponding closed immersion $i:\mathrm{Spec}\kappa (p)\to X$, an invertible ${\mathcal{O}}_X$-module $\mathcal{L}$, and the Axiom of Choice (The Axiom of Choice).

Closed immersions: $i$ is a closed immersion precisely when its underlying map is a homeomorphism onto a closed subset and ${\mathcal{O}}_X\to i_{\ast }{\mathcal{O}}_Z$ is surjective (Closed immersions of schemes). The scheme $Z:=\mathrm{Spec}\kappa (p)$ has exactly one point $q$, so its only open subsets are $\mathrm{\varnothing }$ and $Z$ (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison); the stalk of any sheaf $\mathcal{F}$ on $Z$ at $q$ is therefore , since $Z$ is the only neighbourhood of $q$ (The stalk of a presheaf at a point). The structure sheaf satisfies ${\mathcal{O}}_Z(Z)=\kappa (p)$, so ${\mathcal{O}}_Z$ is the one-point sheaf with value $\kappa (p)$, written $\kappa (p)$ (The localization construction extends to the structure sheaf on Spec A, The residue field at a point of an affine scheme).

Properness of $X$ over $k$ gives finite type, so every affine chart of $X$ is a spectrum of a Noetherian ring and $X$ is locally Noetherian (Locally Noetherian and Noetherian schemes); on a locally Noetherian scheme a quasi-coherent module is coherent if and only if it is of finite type, and $Z$, being the spectrum of a field, is locally Noetherian (Coherent sheaves on a locally Noetherian scheme, Coherent module sheaves). The structure sheaf ${\mathcal{O}}_Z=\kappa (p)$ is finite type and quasi-coherent, hence coherent on $Z$; consequently $i_{\ast }\kappa (p)$ is coherent on $X$ whenever $X$ is locally Noetherian (Closed immersion preserves cohomology and coherent pushforward).

Residue degrees: the closed point $p$ lies in an affine open $U=\mathrm{Spec}A\subseteq X$ with $A$ a finite-type $k$-algebra, and corresponds to a maximal ideal $\mathfrak{m}\subseteq A$ with $\kappa (p)\cong A\mathrm{/}\mathfrak{m}$; by A maximal ideal of an affine algebra has finite residue field over the base field the residue field $\kappa (p)$ is a finite extension of $k$, so the degree $[\kappa (p):k]={\dim }_k\kappa (p)$ of The degree $[K:F]={\dim }_{F}K$ of a finite field extension is a finite integer.

Pushforward cohomology and vanishing: for the quasi-coherent ${\mathcal{O}}_Z$-module $\kappa (p)$ there are isomorphisms $H^q(Z,\kappa (p))\cong H^q(X,i_{\ast }\kappa (p))$ for every $q\ge 0$ (Closed immersion preserves cohomology and coherent pushforward). The space $Z$ is a one-point Noetherian space of dimension $0$ (Chain dimension and the empty-space convention), so $H^q(Z,\mathcal{F})=0$ for every sheaf of abelian groups $\mathcal{F}$ on $Z$ and every $q>0$ (Grothendieck vanishing on a Noetherian space); and $H^0(Z,\kappa (p))\cong \mathrm{\Gamma }(Z,\kappa (p))={\mathcal{O}}_Z(Z)=\kappa (p)$ (Degree-zero sheaf cohomology is global sections, [F1]).

Stalks of pullback: for the morphism $i$ and the point $q\in Z$ one has $(i^{-1}\mathcal{L}{)}_q\cong {\mathcal{L}}_{i(q)}={\mathcal{L}}_p$ and $(i^{-1}{\mathcal{O}}_X{)}_q\cong {\mathcal{O}}_{X,p}$ (The stalk of an inverse image sheaf is the stalk over the image point); the pullback $i^{\ast }\mathcal{L}={\mathcal{O}}_Z{\otimes }_{i^{-1}{\mathcal{O}}_X}{i}^{-1}\mathcal{L}$ (Pullback of a module along a morphism of ringed spaces) therefore has stalk $(i^{\ast }\mathcal{L}{)}_q\cong {\mathcal{O}}_{Z,q}{\otimes }_{{\mathcal{O}}_{X,p}}{\mathcal{L}}_p$ (The stalk of a tensor product sheaf is the tensor product of the stalks). Since $\mathcal{L}$ is invertible, ${\mathcal{L}}_p\cong {\mathcal{O}}_{X,p}$ (Invertible sheaves; Locally free sheaves of finite rank), and ${\mathcal{O}}_{Z,q}{\otimes }_{{\mathcal{O}}_{X,p}}{\mathcal{O}}_{X,p}\cong {\mathcal{O}}_{Z,q}=\kappa (p)$ (The regular module is a tensor unit: $R{\otimes }_{R}N\cong N$ and $M{\otimes }_{R}R\cong M$); hence $(i^{\ast }\mathcal{L})(Z)=(i^{\ast }\mathcal{L}{)}_q\cong \kappa (p)={\mathcal{O}}_Z(Z)$ by [F1].

Projection formula: for the closed immersion $i$, the invertible module $\mathcal{L}$ and the quasi-coherent ${\mathcal{O}}_Z$-module $\kappa (p)$, the canonical map $\mathcal{L}\otimes i_{\ast }\kappa (p)\to i_{\ast }(i^{\ast }\mathcal{L}{\otimes }_{{\mathcal{O}}_Z}\kappa (p))$ is an isomorphism, and $H^q(X,\mathcal{L}\otimes i_{\ast }\kappa (p))\cong H^q(Z,i^{\ast }\mathcal{L}\otimes \kappa (p))$ for every $q\ge 0$ (Projection formula for a closed immersion and an invertible sheaf); moreover $i^{\ast }\mathcal{L}{\otimes }_{{\mathcal{O}}_Z}\kappa (p)\cong {\mathcal{O}}_Z{\otimes }_{{\mathcal{O}}_Z}\kappa (p)\cong \kappa (p)$ once $i^{\ast }\mathcal{L}\cong {\mathcal{O}}_Z$ ([F5], The regular module is a tensor unit: $R{\otimes }_{R}N\cong N$ and $M{\otimes }_{R}R\cong M$).

The Axiom of Choice is inherited from the Euler-characteristic, closed-immersion and pushforward suppliers cited in [F2]–[F6]; the computations below make no selection.

Given: a field $k$, an integral proper $k$-scheme $X$ of dimension one, an invertible ${\mathcal{O}}_X$-module $\mathcal{L}$, a coherent ${\mathcal{O}}_X$-module $\mathcal{F}$, the generic point $\xi$ of $X$, and the Axiom of Choice (The Axiom of Choice).

Set-up: since $X$ is proper over the field $k$ it is of finite type over $k$, and its affine charts are spectra of Noetherian rings, so $X$ is locally Noetherian and quasi-compact, hence Noetherian (Proper morphisms, Locally Noetherian and Noetherian schemes, Every algebra of finite type over a Noetherian ring is a Noetherian ring). On the locally Noetherian scheme $X$ a quasi-coherent module is coherent if and only if it is of finite type (Coherent sheaves on a locally Noetherian scheme, Coherent module sheaves), and $\chi (X,\mathcal{G})$ is defined for every coherent $\mathcal{G}$ (Euler characteristic of a coherent sheaf).

Generic point and rank: because $X$ is integral it has a unique generic point $\xi$, and $\kappa (\xi )={\mathcal{O}}_{X,\xi }$ is its function field; the stalk of a coherent module at $\xi$ is a finite-dimensional $\kappa (\xi )$-vector space, so $r={\dim }_{\kappa (\xi )}{\mathcal{F}}_{\xi }$ is a finite integer (Integral schemes, Chain dimension and the empty-space convention). The degree is defined by ${\mathrm{deg}}_X(\mathcal{L})=\chi (X,\mathcal{L})-\chi (X,{\mathcal{O}}_X)$ (Degree of an invertible sheaf on a proper one-dimensional scheme).

Exactness of twisting: an invertible module is locally free of rank one (Invertible sheaves, Locally free sheaves of finite rank), and tensoring with a locally free module is exact: exactness of a sequence of sheaves is stalkwise (A sequence of abelian sheaves is exact exactly when it is exact on every stalk), at each point the stalk of $\mathcal{L}$ is free of rank one, and tensoring modules over a ring by a free module preserves kernels and cokernels. Tensor products of quasi-coherent modules are quasi-coherent (Tensor product preserves quasi-coherence), and the tensor product of an invertible module with a coherent module is coherent: the question is local, and on an affine open trivialising $\mathcal{L}$ the unit isomorphism of The regular module is a tensor unit: $R{\otimes }_{R}N\cong N$ and $M{\otimes }_{R}R\cong M$ identifies $\mathcal{L}\otimes \mathcal{G}$ with $\mathcal{G}$ for a coherent $\mathcal{G}$, while coherence is a local condition (Coherent module sheaves, Coherent sheaves on a locally Noetherian scheme).

Euler characteristic is additive in short exact sequences of coherent modules on the proper $k$-scheme $X$: $\chi (X,\mathcal{F})=\chi (X,{\mathcal{F}}^{\mathrm{\prime }})+\chi (X,{\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }})$ (Euler characteristic is additive in short exact sequences).

For a short exact sequence $0\to {\mathcal{F}}^{\mathrm{\prime }}\to \mathcal{F}\to {\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}\to 0$ of coherent sheaves, the stalk sequence at $\xi$ is a short exact sequence of $\kappa (\xi )$-vector spaces (A sequence of abelian sheaves is exact exactly when it is exact on every stalk, Localisation of modules is exact), and dimension is additive on short exact sequences (Rank-nullity: ${\dim }_{F}V=\mathrm{nullity}T+\mathrm{rank}T$); hence $r(\mathcal{F})=r({\mathcal{F}}^{\mathrm{\prime }})+r({\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }})$.

Noetherian devissage (Noetherian devissage for coherent proper pushforward): let $\mathcal{P}$ be a property of coherent ${\mathcal{O}}_X$-modules which holds for the zero module and satisfies the two-of-three property in every short exact sequence of coherent modules. Suppose that for every integral closed subscheme $Z\subseteq X$ with generic point $\eta$ there is a coherent ${\mathcal{O}}_X$-module $\mathcal{G}$ with $\mathrm{Supp}(\mathcal{G})\subseteq Z$, whose stalk ${\mathcal{G}}_{\eta }$ is annihilated by ${\mathfrak{m}}_{\eta }$ and is one-dimensional over $\kappa (\eta )$, and for which $\mathcal{P}(\mathcal{G})$ holds. Then $\mathcal{P}$ holds for every coherent ${\mathcal{O}}_X$-module.

The closed points: for a closed point $p\in X$ the canonical morphism $i:\mathrm{Spec}\kappa (p)\to X$ is a closed immersion, $i_{\ast }\kappa (p)$ is a coherent ${\mathcal{O}}_X$-module with support $\{p\}$, and $\mathcal{L}\otimes i_{\ast }\kappa (p)\cong i_{\ast }\kappa (p)$; moreover $\chi (X,i_{\ast }\kappa (p))=[\kappa (p):k]$ (Euler characteristic of a closed point, and invariance under an invertible twist, Support of a module sheaf).

The integral closed subschemes of the one-dimensional Noetherian space $X$ are $X$ itself and the closed points: a proper irreducible closed subset of the integral one-dimensional Noetherian scheme $X$ has dimension zero (otherwise a length-one chain inside it extends to a length-two chain in $X$), and an integral zero-dimensional scheme consists of its single generic point, which is then closed (Chain dimension and the empty-space convention, Integral schemes).

The Axiom of Choice enters through the Euler-characteristic, additivity and devissage suppliers [L1]–[L4]; the tensor and stalk computations below make no selection.

Given: a field $k$, a proper $k$-scheme $C$ of dimension at most one, invertible ${\mathcal{O}}_C$-modules $\mathcal{L}$ and $\mathcal{M}$, and the Axiom of Choice (The Axiom of Choice).

Set-up: $C$ is proper over $k$, hence of finite type, so its affine charts are spectra of Noetherian rings and $C$ is locally Noetherian and quasi-compact, hence Noetherian (Proper morphisms, Locally Noetherian and Noetherian schemes, Every algebra of finite type over a Noetherian ring is a Noetherian ring). On the locally Noetherian $C$ a quasi-coherent module is coherent if and only if it is of finite type (Coherent sheaves on a locally Noetherian scheme, Coherent module sheaves), and $\chi (C,\mathcal{G})$ is defined for every coherent $\mathcal{G}$; in particular ${\mathrm{deg}}_C$ is defined on invertible modules (Euler characteristic of a coherent sheaf, Degree of an invertible sheaf on a proper one-dimensional scheme).

Exactness and coherence of twisting: tensoring with an invertible module is exact, and the tensor product of an invertible module with a coherent module is coherent; the unit isomorphism gives ${\mathcal{O}}_C\otimes \mathcal{G}\cong \mathcal{G}$ (Invertible sheaves, Locally free sheaves of finite rank, A sequence of abelian sheaves is exact exactly when it is exact on every stalk, Tensor product preserves quasi-coherence, The regular module is a tensor unit: $R{\otimes }_{R}N\cong N$ and $M{\otimes }_{R}R\cong M$, Coherent module sheaves).

Euler characteristic is additive in short exact sequences of coherent modules on the proper $k$-scheme $C$ (Euler characteristic is additive in short exact sequences).

Noetherian devissage (Noetherian devissage for coherent proper pushforward): a property of coherent ${\mathcal{O}}_C$-modules that holds for the zero module, satisfies the two-of-three property in every short exact sequence of coherent modules, and holds for a suitable witness on every integral closed subscheme of $C$, holds for every coherent ${\mathcal{O}}_C$-module. The witnesses are modules with support in the subscheme whose generic stalk is annihilated by the maximal ideal and is one-dimensional over the residue field.

Integral closed subschemes of $C$: an integral closed subscheme $Z\subseteq C$ is nonempty, reduced and irreducible; since $\dim C\le 1$, either $Z$ has dimension one, or $Z$ is a single closed point (Integral schemes, Chain dimension and the empty-space convention, Closed immersions of schemes). Write $i:Z\to C$ for the closed immersion and ${\xi }_Z$ for the generic point of $Z$; then ${\mathcal{O}}_{Z,{\xi }_Z}=\kappa ({\xi }_Z)$ is a field, the structure sheaf ${\mathcal{O}}_Z$ is coherent on the locally Noetherian $Z$, and $i_{\ast }{\mathcal{O}}_Z$ is coherent on $C$, with $(i_{\ast }{\mathcal{O}}_Z{)}_{{\xi }_Z}\cong \kappa ({\xi }_Z)$ annihilated by the maximal ideal of ${\mathcal{O}}_{C,{\xi }_Z}$ (Coherent module sheaves, Closed immersion preserves cohomology and coherent pushforward).

Projection formula and closed-immersion cohomology (Projection formula for a closed immersion and an invertible sheaf, Closed immersion preserves cohomology and coherent pushforward): for an invertible ${\mathcal{O}}_C$-module $\mathcal{N}$ and a quasi-coherent ${\mathcal{O}}_Z$-module $\mathcal{G}$ there are isomorphisms $\mathcal{N}\otimes i_{\ast }\mathcal{G}\cong i_{\ast }(i^{\ast }\mathcal{N}\otimes \mathcal{G})$ and $H^q(C,\mathcal{N}\otimes i_{\ast }\mathcal{G})\cong H^q(Z,i^{\ast }\mathcal{N}\otimes \mathcal{G})$ for all $q$; when $\mathcal{G}$ is coherent, these isomorphisms also give $\chi (C,\mathcal{N}\otimes i_{\ast }\mathcal{G})=\chi (Z,i^{\ast }\mathcal{N}\otimes \mathcal{G})$.

Closed points: for a closed point $p$ of $C$ with residue field $\kappa (p)$ the pushforward $i_{\ast }\kappa (p)$ is coherent, and $\mathcal{N}\otimes i_{\ast }\kappa (p)\cong i_{\ast }\kappa (p)$ for every invertible $\mathcal{N}$, so that $\chi (C,\mathcal{N}\otimes i_{\ast }\kappa (p))=\chi (C,i_{\ast }\kappa (p))$ (Euler characteristic of a closed point, and invariance under an invertible twist; Support of a module sheaf).

Integral curves: if $Z\subseteq C$ is an integral closed subscheme of dimension one, then $Z$ is an integral proper $k$-scheme of dimension one, so for invertible modules $\mathcal{N}$ on $Z$ and coherent $\mathcal{F}$ on $Z$ one has $\chi (Z,\mathcal{N}\otimes \mathcal{F})=r{\mathrm{deg}}_Z(\mathcal{N})+\chi (Z,\mathcal{F})$ with $r$ the rank of $\mathcal{F}$ at the generic point; for invertible $\mathcal{F}$ the rank is one, and $i^{\ast }\mathcal{L}$, $i^{\ast }\mathcal{M}$ are invertible on $Z$ (Twisting a coherent sheaf by an invertible sheaf on an integral proper curve, Degree of an invertible sheaf on a proper one-dimensional scheme).

Dual and tensor: ${\mathcal{L}}^{\vee }{\otimes }_{{\mathcal{O}}_C}\mathcal{L}\cong {\mathcal{O}}_C$ canonically, so ${\mathrm{deg}}_C({\mathcal{O}}_C)=0$ and statements about $\mathcal{L}\otimes {\mathcal{L}}^{\vee }$ may be read off from the additive identity of statement 1 (Dual of a line bundle is its tensor inverse, Degree of an invertible sheaf on a proper one-dimensional scheme).

The Axiom of Choice enters through the Euler-characteristic, additivity, devissage and integral-curve suppliers [F3]–[F8]; the tensor computations below make no selection.