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A line bundle trivial on each open set is determined by unit-valued transition functions on pairwise intersections. Their cocycle identities make the local trivial bundles glue; changing frames changes the cocycle by a coboundary. The tensor product of sheaves multiplies the transition functions, giving the displayed group isomorphism with Čech cohomology.

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.

Form the presheaf whose value on $U\subseteq X$ iswith restrictions induced from those of the two sheaves of modules and the structure sheaf. Its sheafification is the tensor product of sheaves $\mathcal{G}{\otimes }_{{\mathcal{O}}_X}\mathcal{H}$. Sheafification is needed because taking a tensor product on each open set need not itself satisfy the gluing axiom. The resulting sheaf represents bilinear sheaf morphisms, and localization gives the stalk formulaIndeed, sheafification preserves stalks, and filtered direct limits commute with the module tensor product.

Construct the tensor product of sheaves by first forming the presheaf $U\mapsto \mathcal{F}(U){\otimes }_{{\mathcal{O}}_X(U)}\mathcal{G}(U)$, with restriction maps induced by those of the two sheaves of modules, and then applying sheafification. The local module actions are compatible with restrictions and therefore give the sheaf an ${\mathcal{O}}_X$-module structure. Its stalks areIndeed, a finite collection of germs of sheaf sections can be represented on a common neighbourhood, and every finite tensor relation holds on a sufficiently small neighbourhood. Equivalently, this construction represents bilinear maps of sheaves of modules that are balanced over the structure sheaf. Sections of the resulting sheaf need not themselves be tensors of global sections: that is why the sheafification step matters.

The degree-one generation condition for Proj ensures that the opens $D_{+}(f)$ for $f\in S_1$ cover $X$. Put $B=S_f$ and $R=B_0$. Multiplication by $f^n$ is an $R$-linear bijection $R\to B_n$ for every integer $n$, because $f$ is an invertible degree-one element. The convention $S(n{)}_d=S_{n+d}$ therefore identifies $\widetilde{S(n)}$ on this chart with the free rank-one $R$-module $B_n$, generated by $f^n$. Hence every twisting sheaf ${\mathcal{O}}_X(n)$ is invertible, including negative twists.

For the tensor product of sheaves, consider the natural mapIt is an isomorphism by degree-one localization of a graded module. Explicitly, a homogeneous tensor $m_a\otimes n_{-a}$ of total degree zero is represented on the left by $f^{-a}{m}_a\otimes f^a{n}_{-a}$. Multiplying a tensor factor by a homogeneous element of $B$ gives the same result after using the tensor relation, since that element is a degree-zero coefficient times a power of $f$. This defines the inverse. Since localization commutes with tensor products, the target is $((M{\otimes }_{S}N{)}_f{)}_0$. These natural chart maps agree on overlaps, soThe degree-one hypothesis matters: on general Proj constructions, twisting sheaves need not be line bundles. The local trivializations are also recorded in Stacks Project, Section 27.10.

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,

For a line bundle $L$ on $X$, define the homomorphism associated to a line bundle on an abelian varietyThe Theorem of the square givesso ${\varphi }_L$ is a homomorphism. Pullback distributes over the tensor product of sheaves, and thereforeIterating the homomorphism law in $x$ gives

Suppose $L^{\otimes n}\in {\mathrm{Pic}}^0(X)$. Then ${\varphi }_L(nx)$ is trivial for every $x$. The multiplication-by-n morphism on an abelian variety is surjective, so ${\varphi }_L$ is trivial and $L\in {\mathrm{Pic}}^0(X)$. Thus the Néron-Severi groupis torsion-free.

Finally, put $M={\varphi }_L(x)$. For every $y$,which is trivial by the Theorem of the square. Hence

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.