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For an affine variety with coordinate ring $A$ and an $A$-module $M$, localization defines sections on principal open subsets by the displayed formula. Restrictions are the natural localization maps. Equivalently, a section is a family of stalk values which locally come from one fraction $m/f^r$. These data form a quasi-coherent sheaf, with global sections $M$ and stalk $M_{{\mathfrak{m}}_P}$ at $P$. Sheaf tensor products correspond to tensor products over $A$; pullback of a sheaf of modules corresponds to extension of scalars, and affine direct image to restriction of scalars.

For sheaves of abelian groups and a continuous map $f:X\to Y$, there is a natural bijection ${\mathrm{Hom}}_X(f^{-1}\mathcal{G},\mathcal{F})\cong {\mathrm{Hom}}_Y(\mathcal{G},f_{\ast }\mathcal{F})$. It comes from the universal property of an inverse image sheaf. The inverse image of abelian sheaves is different from the tensor-adjusted pullback of a sheaf of modules on a ringed space.

For a continuous map $f:X\to Y$, the inverse-image functor takes a sheaf on $Y$ to the sheafification of its local inverse-image presheaf on $X$. It is left adjoint to the direct image functor. For sheaves of modules on ringed spaces, extension of scalars gives the separate pullback of a sheaf of modules.

Write ${\mathbb{P}}^r={\mathbb{P}}_k^r$ over its ground field; the argument also works over any Noetherian affine base. Since $\iota$ is a closed immersion, $X$ is a noetherian scheme and a quasi-compact scheme. LetHere the pullback of a sheaf of modules makes $L$ an invertible sheaf, the restriction of the twisting sheaf on projective space. It is an ample line bundle.

By global generation by sufficiently positive twists, choose $n\ge 0$ such that $F(n)$ is a globally generated sheaf. Finitely many global sections suffice: at each point, finite generation of the stalk of the coherent sheaf $F(n)$ lets us select finitely many generating germs of global sections. Those sections generate on a neighbourhood, because their cokernel sheaf is a coherent sheaf with zero stalk at that point. A finite subcover, using that $X$ is a quasi-compact scheme, supplies finitely many global sections generating everywhere. This is finite global generation on a quasi-compact scheme, and gives a surjective sheaf morphism

Tensor with $(L^{\vee }{)}^{\otimes n}$, where $L^{\vee }$ is the dual of a sheaf. Here tensoring with an invertible sheaf is exact: locally the tensor product of sheaves is tensoring with a free module of rank one. Using the permitted identity $L\otimes L^{\vee }\cong {\mathcal{O}}_X$, we obtainA finite direct sum of sheaves of invertible sheaves is a locally free sheaf of finite rank, and it is a coherent sheaf on the noetherian scheme $X$. This constructs the required $G$. No regularity or smoothness assumption on $X$ is needed.

The exact sequence and associated modules. Put $A=k[x]$ and $\mathfrak{m}=(x)$. Evaluation at zero gives the short exact sequence of $A$-modulesThe affine module sheaf associated to $(x)$ is the ideal sheaf $\mathcal{J}$ of the closed point $p$, and the one associated to $A$ is the structure sheaf ${\mathcal{O}}_{{\mathbb{A}}_k^1}$. To identify the third term, on a principal open subset $D(g)$,which equals $\Gamma (i^{-1}D(g),{\mathcal{O}}_p)$. The restrictions also agree, so $\tilde{k}\cong i_{\ast }{\mathcal{O}}_p$. Exactness of localization proves that applying the affine module sheaf construction to the module sequence remains exact: on every stalk the sequence is its localization. ThusEach term is an affine module sheaf, hence a quasi-coherent sheaf. In this example each is also a coherent sheaf, since its defining module is finitely generated over the Noetherian ring $k[x]$.

The failure of left exactness. For an affine module sheaf $\tilde{M}$, the pullback of a sheaf of modules along $i$ corresponds to the tensor product of modules $k{\otimes }_{A}M$. In particular,The first vector space is nonzero, with basis the class of $x$. But $i^{\ast }j$ sends this class to the class of $x$ in $A/(x)$, which is zero. Thus $i^{\ast }j$ is the zero map $k\to k$, and is not injective. More explicitly, the pulled-back sequence iswhere exactness fails at its first copy of $k$. Thus $i^{\ast }$ is not a left-exact functor:Although $(x)$ is a free module over $A$ with one generator, its inclusion into $A$ becomes multiplication by $x$ after choosing its generator; the tensor product of modules with $A/(x)$ kills that map. This makes explicit why local freeness of $\mathcal{J}$ does not preserve this injection under the given pullback.

First form the inverse image sheaf ${\varphi }^{-1}\mathcal{H}$ by sheafifying the presheafwhere $U$ runs through open subsets of $X$. It is naturally a module over ${\varphi }^{-1}{\mathcal{O}}_X$, not yet over ${\mathcal{O}}_Y$. The structure morphism ${\varphi }^{-1}{\mathcal{O}}_X\to {\mathcal{O}}_Y$ allows extension of scalars, giving the pullback of a sheaf of modules:The tensor product of sheaves includes its own sheafification. At $Q\in Y$ the construction becomesThis adjustment of the coefficient ring distinguishes the pullback from the underlying inverse image sheaf.

The affine module sheaf construction makes the remaining statements explicit. For $A=k[V]$ and an $A$-module $M$, defineon the basis of principal open subsets. Restrictions are the natural localization maps. Equivalently, a section on an arbitrary open set is a family $s_P\in M_{{\mathfrak{m}}_P}$ which near each point is represented by a single fraction $m/f^r$, with $f$ nonvanishing throughout that neighbourhood. This locally representable family construction gives a sheaf; the usual localization gluing verifies its quasi-coherence. Its stalk is $M_{{\mathfrak{m}}_P}$ andThis last identification is precisely the standard module-localization gluing on the affine basis, not an assertion that arbitrary sheaves are determined by their global sections.

Now write $A=k[X]$, $B=k[Y]$ and $\alpha ={\varphi }^{\#}:A\to B$. Let $\mathcal{G}=\tilde{M}$, $\mathcal{H}=\tilde{N}$ for $A$-modules $M,N$, and let $\mathcal{F}=\tilde{L}$ for a $B$-module $L$. The affine versions of the three constructions areHere ${}_{A}L$ denotes restriction of scalars along $\alpha$, whereas the last associated sheaf lives on $Y$. To verify the tensor-product assertion, localize at every prime: both stalks are $M_{\mathfrak{m}}{\otimes }_{A_{\mathfrak{m}}}{N}_{\mathfrak{m}}$. For the direct image, ${\varphi }^{-1}D(a)=D(\alpha (a))$, andThese equalities identify the sheaves on a basis and respect restrictions. For the pullback of a sheaf of modules, its stalk at $Q$ is $B_{{\mathfrak{m}}_Q}{\otimes }_{A}N$, exactly the stalk of $\widetilde{B{\otimes }_{A}N}$. The local tensor maps are compatible, so these stalk identifications come from an actual sheaf isomorphism. Thus all three constructions are quasi-coherent in the stated affine setting.

Finally construct the natural projection formula for sheaves map by sending a local tensor $m\otimes b$ to $b(1\otimes m)$ in the pulled-back module, followed by direct image. In the affine description, the target ${\varphi }_{\ast }{\varphi }^{\ast }\tilde{M}$ is the sheaf associated with the $A$-module ${}_A(B{\otimes }_{A}M)$, while $\tilde{M}{\otimes }_{{\mathcal{O}}_X}{\varphi }_{\ast }{\mathcal{O}}_Y$ corresponds to $M{\otimes }_{A}B$. The mapsare inverse $A$-linear maps. Localizing them commutes with every restriction, provingNo flatness assumption on $A\to B$ or finiteness assumption on $M$ is needed.

First form the inverse image sheaf ${\varphi }^{-1}\mathcal{H}$ by applying sheafification toIt is a sheaf of modules over ${\varphi }^{-1}{\mathcal{O}}_Y$. The morphism of ringed spaces gives a ring map ${\varphi }^{-1}{\mathcal{O}}_Y\to {\mathcal{O}}_X$, so the pullback of a sheaf of modules isAt $x\in X$, writing $y=\varphi (x)$, its stalk is ${\mathcal{O}}_{X,x}{\otimes }_{{\mathcal{O}}_{Y,y}}{\mathcal{H}}_y$. The tensor construction makes the action of the structure sheaf explicit; taking the inverse image alone would not give the requested ${\mathcal{O}}_X$-module.

The tensor product of sheaves is the sheafification of the presheafwith restriction maps induced by those of $\mathcal{F}$, $\mathcal{G}$, and the structure sheaf of a scheme. Sending the germ of $f\otimes g$ to $f_P\otimes g_P$ gives a canonical mapIt is surjective because a finite sum of tensors of germs has representatives on one common neighborhood. It is injective because an equality in a tensor product of modules is witnessed by finitely many additive and balancing relations; all their germs, including the scalars in ${\mathcal{O}}_{X,P}$, can be represented and their equalities made valid on a common smaller neighborhood. Equivalently, a filtered colimit of modules commutes with these tensor products. Since sheafification preserves the stalk of a sheaf, this proves the asserted isomorphism.

For the pullback of a sheaf of modules, first form the inverse image sheaf ${\varphi }^{-1}\mathcal{F}$: sheafify $U\mapsto {\mathrm{colim}}_{V\supseteq \varphi (U)}\mathcal{F}(V)$. The morphism of sheaves of rings ${\varphi }^{-1}{\mathcal{O}}_Y\to {\mathcal{O}}_X$ then definesExactness of a sequence of sheaf of modules morphisms is checked on every stalk of a sheaf. Applying the right exactness of the tensor product of modules to the stalk sequence over ${\mathcal{O}}_{Y,\varphi (P)}$ gives the exact stalk sequence of the pullbacks. Thus ${\varphi }^{\ast }$ preserves sequences ending in zero, as required.

A quasi-coherent sheaf is a sheaf of modules locally of the form $\tilde{N}$ for a module $N$ over the coordinate ring of an affine neighborhood. On an affine variety $V$ with ring $B$, this associated sheaf is characterized on principal opens by $\tilde{N}(D(b))=N_b$, using localization of a module. Suppose $\mathcal{F}{|}_V\cong \tilde{N}$ and $U\subseteq {\varphi }^{-1}V$ is affine, with ring $C$. The induced algebra map $B\to C$ gives a canonical mapAt a point $P\in U$, both sides identify with ${\mathcal{O}}_{X,P}{\otimes }_{B}N$: on the right use ${\mathcal{F}}_{\varphi (P)}\cong {\mathcal{O}}_{Y,\varphi (P)}{\otimes }_{B}N$ and associativity of the tensor product of modules. The map is therefore an isomorphism on every stalk of a sheaf, hence an isomorphism of sheaves. Such $U$ cover $X$, proving that the pullback is a quasi-coherent sheaf.

Finally, on an affine open $V\subseteq Y$ with ring $B$, the presheaf defining $M_Y$ has, on $D(b)\subseteq V$, the valueConsequently $M_Y{|}_V\cong \widetilde{M{\otimes }_{R}B}$, so $M_Y$ is a quasi-coherent sheaf. The algebra map for $M_X$ is the composite $R\to \Gamma (Y,{\mathcal{O}}_Y)\to \Gamma (X,{\mathcal{O}}_X)$. Restriction of functions and the tensor-product construction give a canonical sheaf morphism ${\varphi }^{\ast }{M}_Y\to M_X$. At $P$, it is the associativity isomorphismIt is therefore a sheaf of modules isomorphism. In particular,

The two pairs of regular functionsdefine maps to the projective line on the loci where their respective coordinates do not vanish simultaneously. Those loci cover $Y$, since simultaneous failure would force $x=y=z=w=0$. On their overlap the equation $xy=zw$ says that the two projective points are equal. They therefore glue to a morphism

Let $[A:B]$ be the homogeneous coordinates on ${\mathbb{P}}_k^1$. The pullback of the hyperplane divisor $V(A)$ has local equation $x$ on the first chart and $z$ on the second. It is therefore exactly $D_Y=V(x,z)$. Compatibility of the pullback of a sheaf of modules with the line bundle associated to a divisor givesThis is the ruling morphism of the punctured three-dimensional affine quadric cone associated with $D_Y$.

An ${\mathcal{O}}_X$-module $\mathcal{F}$ is a quasi-coherent sheaf when every affine open $V=\mathrm{Spec}A\subseteq X$ has $\mathcal{F}{|}_V\cong \tilde{M}$ for some $A$-module $M$. For a morphism $f:X\to Y$, its direct image sheaf iswhile the pullback of a sheaf of modules is

If $i:Z\hookrightarrow X$ is a closed immersion of Noetherian schemes, then on an affine open $V=\mathrm{Spec}A\subseteq X$ one has $Z\cap V=\mathrm{Spec}(A/I)$. The restriction of $i_{\ast }{\mathcal{O}}_Z$ corresponds to the cyclic $A$-module $A/I$, so it is finitely generated. Hence $i_{\ast }{\mathcal{O}}_Z$ is a coherent sheaf; more generally this is the direct image of a coherent sheaf under a closed immersion.

Coherence need not survive an arbitrary pushforward. Letbe the open immersion. The sheaf ${\mathcal{O}}_{D(x)}$ is coherent, butis not a finitely generated $k[x]$-module. Therefore $j_{\ast }{\mathcal{O}}_{D(x)}$ is not coherent.

The tensor product of sheaves $\mathcal{F}{\otimes }_{{\mathcal{O}}_X}\mathcal{G}$ is the sheafification of the presheafEquivalently, its stalk at $x$ is ${\mathcal{F}}_x{\otimes }_{{\mathcal{O}}_{X,x}}{\mathcal{G}}_x$.

The direct image sheaf is defined on each open set $V\subseteq Y$ byThe pullback of a sheaf of modules iswhere $f^{-1}\mathcal{E}$ is the inverse image sheaf and the tensor product uses the structural morphism $f^{-1}{\mathcal{O}}_Y\to {\mathcal{O}}_X$ of the given ringed-space morphism.

For a morphism of ringed spaces, the pullback of a sheaf of modules is left adjoint to the direct image sheaf functor. The unit sends a local section $h$ to $1\otimes h$ after inverse image. On the structure sheaf it becomes the given ring homomorphism ${\mathcal{O}}_Y\to f_{\ast }{\mathcal{O}}_X$.

The restriction meant here is the pullback of a sheaf of modules to the closed subvariety, including tensoring with its structure sheaf. On an affine chart, tensoring $M$ with $A/I$ gives $M/IM$. This distinction matters because restricting only the underlying topological sheaf would retain the ambient ring action.