Symmetric algebra of a quasi-coherent module

Assume the Axiom of Choice, inherited from associated-sheaf existence and the affine quasi-coherent equivalence (The Axiom of Choice, The associated module sheaf exists, Affine quasi-coherent sheaves are modules). Let $X$ be a scheme and let $\mathcal{F}$ be a quasi-coherent ${\mathcal{O}}_X$-module (Quasi-coherent module on a scheme).

The affine models. Let $A$ be a commutative ring with $1$ and let $M$ be an $A$-module. The tensor algebra $T_A(M)={⨁}_{n\ge 0}{M}^{{\otimes }_{A}n},M^{{\otimes }_{A}0}=A,$ is the graded $A$-algebra whose multiplication on homogeneous pure tensors is concatenation (the case of a module over a field is Tensor algebra of a vector space). The symmetric algebra of $M$ is the quotient ${\mathrm{Sym}}_A(M)=T_A(M)\mathrm{/}I,$ where $I$ is the two-sided ideal generated by all elements $m\otimes m^{\mathrm{\prime }}-m^{\mathrm{\prime }}\otimes m$ with $m,m^{\mathrm{\prime }}\in M$. The relators are homogeneous of degree two, so $I$ is graded, and ${\mathrm{Sym}}_A(M)$ is a graded commutative $A$-algebra, generated as an $A$-algebra by its degree-one part $M$, with ${\mathrm{Sym}}_A^0(M)=A,{\mathrm{Sym}}_A^1(M)=M\mathrm{.}$ Its degree-$d$ piece is the quotient of $M^{{\otimes }_{A}d}$ by the relations $m_1\otimes \cdots \otimes m_d=m_{\sigma (1)}\otimes \cdots \otimes m_{\sigma (d)}$ for all permutations $\sigma$ of $\{1,\dots ,d\}$; for $M=A^r$ this is the polynomial algebra $A[T_1,\dots ,T_r]$ on $r$ degree-one generators.

The algebraic universal property follows directly from this quotient: an $A$-linear map $u:M\to D$ into a commutative $A$-algebra extends on a pure tensor to the product $u(m_1)\cdots u(m_n)$, and the commutator ideal maps to zero. This extension is unique because degree one generates the algebra. Consequently, for any ring map $A\to C$, maps from either $C{\otimes }_A{\mathrm{Sym}}_A(M)$ or ${\mathrm{Sym}}_C(C{\otimes }_{A}M)$ to a commutative $C$-algebra $D$ correspond to $A$-linear maps $M\to D$: a $C$-linear map out of $C{\otimes }_{A}M$ has the form $c\otimes m\mapsto cu(m)$. Applying these extensions to the degree-one maps in both directions gives mutually inverse algebra maps, since both composites fix degree one and scalars. These are graded and compatible with successive base changes by the same uniqueness. This proves the base-change assertion used in gluing.

Gluing the affine models. Let $U=\mathrm{Spec}A\subseteq X$ be an affine open subscheme with $\mathcal{F}{\mathrm{\mid }}_U\cong \tilde{M}$ for an $A$-module $M$; such opens cover $X$, because $\mathcal{F}$ is quasi-coherent, and $M$ is determined up to canonical isomorphism by $\mathcal{F}{\mathrm{\mid }}_U$ under the affine equivalence (Affine quasi-coherent sheaves are modules, Module sheaf on an affine scheme). Put ${\mathrm{Sym}}_U:=({\mathrm{Sym}}_A(M){)}^{\sim },$ an ${\mathcal{O}}_U$-module carrying a commutative ${\mathcal{O}}_U$-algebra structure (Modules on a ringed space). Let $V=\mathrm{Spec}C\subseteq U$ be a smaller affine open, so that $\mathcal{F}{\mathrm{\mid }}_V\cong \widetilde{(C{\otimes }_{A}M)}$ and, by naturality of the identification $(\tilde{N}){\mathrm{\mid }}_V\cong \widetilde{(C{\otimes }_{A}N)}$ (An associated sheaf restricts to an associated sheaf on an affine open), the canonical base-change isomorphism of graded $C$-algebras ${\mathrm{Sym}}_A(M){\otimes }_{A}C\cong {\mathrm{Sym}}_C(M{\otimes }_{A}C)$ induced by $M\to M{\otimes }_{A}C$ gives a canonical isomorphism of ${\mathcal{O}}_V$-algebras ${\theta }_{U,V}:{\mathrm{Sym}}_U{\mathrm{\mid }}_V\stackrel{\cong }{\to }{\mathrm{Sym}}_V\mathrm{.}$ Here ${\theta }_{U,U}=\mathrm{i}\mathrm{d}$ and ${\theta }_{U,W}={\theta }_{V,W}\circ {\theta }_{U,V}{\mathrm{\mid }}_W$ for affine $W\subseteq V\subseteq U$, because the base-change isomorphisms are compatible under composition of ring maps. On an overlap $U\cap U^{\mathrm{\prime }}$ the isomorphisms ${\theta }_{U^{\mathrm{\prime }},W}^{-1}\circ {\theta }_{U,W}$ for affine $W\subseteq U\cap U^{\mathrm{\prime }}$ agree on further affine opens by the cocycle identity, so they glue to an isomorphism ${\mathrm{Sym}}_U{\mathrm{\mid }}_{U\cap U^{\mathrm{\prime }}}\cong {\mathrm{Sym}}_{U^{\mathrm{\prime }}}{\mathrm{\mid }}_{U\cap U^{\mathrm{\prime }}}$, and the triple-overlap condition follows on affine refinements. Hence the ${\mathrm{Sym}}_U$ together with these identifications form a gluing datum on the cover by all such affine opens (A gluing datum for sheaves on an open cover), and the resulting glued sheaf of commutative ${\mathcal{O}}_X$-algebras $\mathrm{Sym}(\mathcal{F})$ is the symmetric algebra of $\mathcal{F}$; it is determined up to unique isomorphism by the affine models ${\mathrm{Sym}}_U$ (Compatible local sheaves glue uniquely up to unique isomorphism).

Grading and generation. The gluing is degreewise: writing ${\mathrm{Sym}}_A(M)={⨁}_{d\ge 0}{\mathrm{Sym}}_A^d(M)$, the degree-$d$ parts $({\mathrm{Sym}}_A^d(M){)}^{\sim }$ glue to sheaves ${\mathrm{Sym}}^d(\mathcal{F})$ with $\mathrm{Sym}(\mathcal{F})={⨁}_{d\ge 0}{\mathrm{Sym}}^d(\mathcal{F}),{\mathrm{Sym}}^0(\mathcal{F})={\mathcal{O}}_X,{\mathrm{Sym}}^1(\mathcal{F})=\mathcal{F},$ the direct sum being taken in the category of ${\mathcal{O}}_X$-modules; on an affine chart it corresponds under the equivalence $N\mapsto \tilde{N}$ to the direct sum of the modules ${\mathrm{Sym}}_A^d(M)$ (Affine quasi-coherent sheaves are modules), and degreewise gluing is legitimate because restriction isomorphisms are degree-preserving. The multiplication of the affine models induces multiplication maps ${\mathrm{Sym}}^d(\mathcal{F}){\otimes }_{{\mathcal{O}}_X}{\mathrm{Sym}}^e(\mathcal{F})\to {\mathrm{Sym}}^{d+e}(\mathcal{F})$, so $\mathrm{Sym}(\mathcal{F})$ is generated as an ${\mathcal{O}}_X$-algebra by its degree-one part $\mathcal{F}$, local sections of $\mathcal{F}$ commute in $\mathrm{Sym}(\mathcal{F})$, and the degree-$d$ piece is the quotient of the $d$-fold tensor power ${\mathcal{F}}^{\otimes d}$ (Tensor product of sheaves of modules) by the local relations $v_1\otimes \cdots \otimes v_d=v_{\sigma (1)}\otimes \cdots \otimes v_{\sigma (d)}$ for all permutations $\sigma$; the tensor powers are quasi-coherent with $({\mathcal{F}}^{\otimes d}){\mathrm{\mid }}_U\cong (M^{{\otimes }_{A}d}{)}^{\sim }$ by iteration of Tensor product preserves quasi-coherence. Equivalently, $\mathrm{Sym}(\mathcal{F})$ is obtained from the tensor-algebra presentation of $\mathcal{F}$ by imposing the local relations $v\otimes w=w\otimes v$.

Affine model. If $U=\mathrm{Spec}A\subseteq X$ is affine with $\mathcal{F}{\mathrm{\mid }}_U\cong \tilde{M}$, then $\mathrm{Sym}(\mathcal{F}){\mathrm{\mid }}_U\cong ({\mathrm{Sym}}_A(M){)}^{\sim },{\mathrm{Sym}}^d(\mathcal{F}){\mathrm{\mid }}_U\cong ({\mathrm{Sym}}_A^d(M){)}^{\sim },$ so restricting to the members of the covering family by all such affine opens shows that $\mathrm{Sym}(\mathcal{F})$ is a quasi-coherent graded ${\mathcal{O}}_X$-algebra (Quasi-coherent module on a scheme); in particular $\mathrm{Sym}({\mathcal{O}}_X^r)\cong {\mathcal{O}}_X[T_1,\dots ,T_r]$ with $\mathrm{deg}{T}_i=1$ and $\mathrm{Sym}(\mathcal{F})$ satisfies the universal property that every ${\mathcal{O}}_X$-linear map $\mathcal{F}\to \mathcal{A}$ into a sheaf of commutative ${\mathcal{O}}_X$-algebras extends uniquely to an ${\mathcal{O}}_X$-algebra map $\mathrm{Sym}(\mathcal{F})\to \mathcal{A}$.

Choice. The cover used is the family of all affine opens on which $\mathcal{F}$ is associated, which is determined by the data, and the identifications ${\theta }_{U,V}$ are canonical base-change isomorphisms, so no chart, module or isomorphism is selected: on each affine open one may use $M=\mathrm{\Gamma }(U,\mathcal{F})$ and the canonical comparison isomorphism of the affine equivalence. AC is inherited from the associated-sheaf machinery and from Affine quasi-coherent sheaves are modules, whose proof step 2.1 uses compactness of $\mathrm{Spec}A$ to obtain a finite principal cover. The algebraic quotient and the canonical gluing introduce no additional choice.