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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,

An $A$-module $M$ is a flat module when the tensor functor $-{\otimes }_{A}M$ preserves injections, equivalently when it is exact.

Suppose first that $M$ is flat. For any nonzero $a\in A$, tensor the injection $A\stackrel{a}{\to }A$ with $M$. The resulting map $M\stackrel{a}{\to }M$ is injective, so $am=0$ implies $m=0$. Thus $M$ is a torsion-free module.

Conversely, suppose $M$ is torsion-free over the principal ideal domain $A$. Every finitely generated submodule of $M$ is a finitely generated torsion-free module over a PID, hence a finite free module and therefore flat. The module $M$ is the filtered colimit of these submodules. Tensor products commute with filtered colimits, and filtered colimits of modules preserve exact sequences, so $M$ is flat. This proves that a torsion-free module over a principal ideal domain is flat.

True. If $M$ is generated by $m_1,\dots ,m_r$, there is a surjection $R^r\twoheadrightarrow M$. The right exactness of the tensor product of modules gives a surjectionA finite direct sum of Noetherian modules is Noetherian, and a quotient of a Noetherian module is Noetherian. Hence $M{\otimes }_{R}N$ is Noetherian.

An $R$-module $M$ is Noetherian when every submodule is finitely generated, equivalently when every ascending chain of submodules stabilizes.

A free module is an $R$-module with a basis: every element has a unique expression as a finite linear combination of basis elements.

A flat module $M$ is one for which the tensor functor $-{\otimes }_{R}M$ is exact. Since tensor products are always right exact, it is equivalent to require that tensoring with $M$ preserve injections.

A projective module $P$ has the lifting property: for every surjection $E\twoheadrightarrow F$ and every map $P\to F$, there is a map $P\to E$ making the resulting triangle commute. Equivalently, $P$ is a direct summand of a free module.

For $R$-modules $M,N$, the tensor product of modules is an $R$-module $M{\otimes }_{R}N$ together with the balanced mapsuch that every balanced map $b:M\times N\to P$ factors through one unique $R$-linear map $\tilde{b}$:Equivalently,naturally in $P$. This is the universal property of the tensor product of modules.

The Tensor-hom adjunction is the natural isomorphismFor an R-module homomorphism $f$, it is given explicitly byConversely, an $R$-linear map $g:M\to {\mathrm{Hom}}_R(N,P)$ determines the balanced map $(m,n)\mapsto g(m)(n)$, so the universal property of the tensor product of modules givesThese formulas are inverse to each other because pure tensors generate the tensor product of modules.

The nth tensor power is the iterated tensor product of modules $V^{\otimes n}=V\otimes \cdots \otimes V$.

The Tor functor is the homology of a tensor product of modules with a free resolution: ${\mathrm{Tor}}_i^R(M,N)=H_i(F_{\bullet }{\otimes }_{R}N)$ for a free resolution $F_{\bullet }$ of $M$. It measures the failure of tensoring to preserve exact sequences. For a commutative ring, one may resolve either argument; the double complex of the two resolutions proves that both computations agree.