Tensor, symmetric, and exterior algebras of a module

(tensor algebra)

Let $A$ be a commutative ring and let $M$ be an $A$-module. The tensor algebra is the unital associative $A$-algebra \[ T^\bullet_A(M)\coloneqq\bigoplus_{n\ge0}T_A^n(M),\qquad T_A^0(M)=A,\qquad T_A^n(M)=M^{\otimes_A n}\quad(n>0), \] with multiplication given by concatenation of elementary tensors. For every unital associative $A$-algebra $B$, an $A$-linear map $M\to B$ extends uniquely to an $A$-algebra homomorphism $T_A^\bullet(M)\to B$.

(symmetric algebra)

The symmetric algebra $\Sym_A^\bullet(M)$ is the quotient of $T_A^\bullet(M)$ by the two-sided ideal generated by $m\otimes n-n\otimes m$, for $m,n\in M$. Its degree-$q$ component is the symmetric power $\Sym_A^q(M)$. For every commutative $A$-algebra $B$, an $A$-linear map $M\to B$ extends uniquely to an $A$-algebra homomorphism $\Sym_A^\bullet(M)\to B$.

(exterior algebra)

The exterior algebra $\bigwedge_A^\bullet M$ is the quotient of $T_A^\bullet(M)$ by the two-sided ideal generated by $m\otimes m$, for $m\in M$. Its degree-$q$ component is the exterior power $\bigwedge_A^q M$. For every unital associative $A$-algebra $B$, an $A$-linear map $\phi:M\to B$ with $\phi(m)^2=0$ for all $m\in M$ extends uniquely to an $A$-algebra homomorphism $\bigwedge_A^\bullet M\to B$.

For integers $m\ge1$ and $q\ge0$ and a free module $M$ with ordered basis $e_1,\ldots,e_m$, the degree-$q$ tensor, symmetric, and exterior powers have bases consisting of all length-$q$ words, the degree-$q$ monomials, and the increasing length-$q$ wedges, respectively. Their ranks are $m^q$, $\binom{m+q-1}{q}$, and $\binom m q$. In degree zero all three modules are $A$. For $M=0$, every positive-degree component is zero [@Har10a, Exercise II.5.16(a)].

(A commutative tensor algebra)

For $M=Ae$ free of rank one, $T_A^q(M)$ has basis $e^{\otimes q}$ and concatenation gives $e^{\otimes p}e^{\otimes q}=e^{\otimes(p+q)}$. Hence $T_A^\bullet(Ae)\cong A[t]$, with $e$ corresponding to $t$. Thus a tensor algebra need not be noncommutative.

For homogeneous $u\in\bigwedge_A^p M$ and $v\in\bigwedge_A^q M$, \[ u\wedge v=(-1)^{pq}v\wedge u. \] The defining square relations give $m\wedge n=-n\wedge m$ by expanding $(m+n)\wedge(m+n)=0$; moving $p$ factors past $q$ factors proves the formula, and linearity extends it to homogeneous sums.

(Tensor operations on sheaves)

For a sheaf $\mcf$ of modules on a ringed space $(X,\OO_X)$, its tensor, symmetric, and exterior algebras are the sheafifications of the presheaves assigning the corresponding $\OO_X(U)$-algebra to $\mcf(U)$ on each open set $U$. Their degree-$q$ components are denoted $T^q(\mcf)$, $S^q(\mcf)=\Sym^q(\mcf)$, and $\bigwedge^q\mcf$ [@Har10a, Exercise II.5.16].

If $\mcf$ is quasi-coherent and $\mcf|_{\Spec A}\cong\widetilde M$, then \[ (\bigwedge^q\mcf)|_{\Spec A}\cong\widetilde{\bigwedge_A^q M}, \] and the corresponding identities hold for tensor and symmetric powers. These identities follow from compatibility of tensor products and their defining quotients with localization [@Har10a, Propositions II.5.2 and II.5.7, Exercise II.5.16(e)].

For a locally free sheaf $\mcf$ of finite rank $m$, its determinant is $\det\mcf=\bigwedge^m\mcf$. For a smooth $m$-dimensional variety over a field $k$, this gives the canonical invertible sheaf $\omega_X=\bigwedge^m\Omega_{X/k}$ [@Har10a, Chapter II, §8]. For a locally free sheaf $\mce$ of finite rank on a scheme $X$, the projective bundle in the quotient convention is $\PP(\mce)=\operatorname{Proj}_X\Sym^\bullet\mce$ [@Har10a, Chapter II, §7].

Define $T^\bullet M$, $\Sym^\bullet M$, and $\Wedge^\bullet M$, and state their universal properties.

What are the ranks of $T^n$, $\Sym^n$, and $\Wedge^n$ of a free module of rank $m$?

What is the determinant of a locally free module of rank $m$?

Dependencies

D-DEFGRRNGGraded rings, homogeneous ideals, and the irrelevant idealuses
D-DEFTENSThe tensor product of modules, and extension of scalarsuses