# Polynomial maps of finite-dimensional vector spaces

I'm currently reading "General linear and functor cohomology over finite fields" by Franjou, Friedlander, Scorichenko and Suslin. There they define a polynomial map $$T : V \rightarrow W$$ of finite-dimensional space to be a morphism of the affine schemes $$\text{Spec}(S^*(V^\#)) \rightarrow \text{Spec}(S^*(W^\#))$$ where $$S^*$$ is the symmetric algebra and $$-^{\#}$$ is the standard $$k$$-duality. Moreover they say that this is equivalent to an element in $$S^*(V^\#) \otimes W$$. It is this isomorphism that I don't see. I see that for a finite-dimensional vector spaces we have that $$\text{Hom}(V,W) \cong V^{\#} \otimes W$$ using tensor-hom adjunction, but I can't see to find a similar argument for the symmetric algebra in this case.

Thank you very much for any help!

$$\operatorname{Hom}(V,W)$$ refers to the $$linear$$ maps between vector spaces $$V,W$$. Although linear maps are always polynomial maps, polynomial maps needn't be linear maps.

In the case that $$W=k$$ is the base field, the ring of polynomial functions is the ring generated by $$V^\lor=\operatorname{Hom}(V,W)$$ in the set of all functions on $$V$$. This is the symmetric/polynomial algebra, $$S(V^\lor)$$.

If you take bases $$\{v_i\}$$ and $$\{w_j\}$$ for the vector spaces, then the maps $$V\to W$$ are equivalent to maps $$k^n\to k^m$$ (assuming $$V,W$$ finite dimensional). Now, a polynomial map $$T:k^n\to k^m$$ is just a map of the form

$$T:(x_1,...,x_n)\mapsto (f_1(x_1,...,x_n),...,f_m(x_1,...,x_n))$$

where all the $$f_i$$ are polynomials in $$n$$-variables.

Ok, now putting this back into abtract form, a polynomial function on $$V$$ is an element of $$S(V^\lor)$$ (in my notation) and an element of $$S(V^\lor)\otimes W$$ is a sum of things of the form $$f\otimes w$$ for $$f\in S(V^\lor)$$ and $$w\in W$$, for instance, $$\sum_{j=1}^m f_j\otimes w_j$$, which is just the abstract way to write the coordinate definition above.

• Thank you for the answer! I don't see precisely how one would get a polynomial map from $\sum_{j=1}^m f_j \otimes w_j$. I guess that one should see the $f_j$ as the respective coordinate functions? For example, what polynomial map does $f \otimes w$ correspond to for a single polynomial function $f$ on $V$ and some element $w \in W$? Mar 29 at 11:11
• $\sum_{j=1}^m f_j\otimes w_j:v\mapsto \sum_{j=1}^m f_j(v) w_j$ Mar 29 at 13:07
• Look at what you get if the $w_j$ are the standard basis of $W=k^m$. Mar 29 at 13:08