Exercise 9, § 5

(Polynomial laws) Denote by E and F two modules over the commutative ring k. By a polynomial law of E in F we understand, for each (associative commutative and unital) k-algebra L, a mapping $\varphi_L$ of $L \otimes_k E$ into $L \otimes_k F$ satisfying the condition: If L and R are two k-algebras and f is a unitary homomorphism of L into R, then

a) Let Q be a polynomial law of E into F, let $L = k[(T_i)_{i \in I}]$ be the polynomial algebra in a family of indeterminates $T_i$ over k and let $(x_i)_{i \in I}$ be a family of finite support of elements of E. Show that there exists a unique family of finite support $(y_\xi)_{\xi \in \mathbf{N}^{(I)}}$ of elements of F such that Show that if R is an algebra over k, then for every family $(a_i)_{i \in I}$ of elements of R we have

b) A family $(\varphi^j)_{j \in J}$ of polynomial laws of E into F is called summable if for every algebra L over k and every $u \in L \otimes E$, the family $\varphi^j_L(u)$ has finite support. Show that if $(\varphi^j)_{j \in J}$ is a summable family of polynomial laws of E into F, then there exists a unique polynomial law Q of E into F such that for every k-algebra L and every $u \in L \otimes E$ we have $\sum_i \varphi^i_L(u) = \varphi_L(u)$. This polynomial law is called the sum of the laws $\varphi^j$ and is denoted by $\sum_j \varphi^j$.

c) Let E, F, G be three k-modules, $\varphi$ a polynomial law of E into F and $\psi$ a polynomial law of F into G. Show that there exists a unique polynomial law $\eta$ of E into G such that for every k-algebra L we have $\eta_L = \psi_L \circ \varphi_L$. This law $\eta$ is called the composition of Q and $\psi$ and is denoted by $\psi \circ Q$.

d) Let p be an integer $\geq 0$. A polynomial law Q of E into F is said to be homogeneous of degree $p$ if for every algebra L over k we have $\varphi_L(au) = a^p \varphi_L(u)$ for any $a \in L$ and $u \in L \otimes E$. Show that if Q is homogeneous of degree p, then in formula (1) we have $y_\xi = 0$ for $|\xi| = \sum_i \xi(i) \neq p$. Determine all homogeneous polynomial laws of degrees 0 and 1.

e) With the same notation as in c), show that if Q and $\psi$ are homogeneous laws of degrees p and q respectively, then $\psi \circ Q$ is a homogeneous law of degree $pq$.

f) Let p be an integer $\geq 0$. For every commutative k-algebra L we denote by $\varepsilon_L$ the injection $a \mapsto aT$ of L into the polynomial algebra $L[T]$ and by $\beta_L^p$ the homomorphism of the L-module $L[T]$ into L which associates with each element $\sum_k a_k T^k \in L[T]$ the coefficient $a_p$. Let Q be a polynomial law of E into F. Show that the mappings $\varphi_L^p = (\beta_L^p \otimes \mathrm{Id}_L) \circ \varphi_{L[T]} \circ (\varepsilon_L \otimes \mathrm{Id}_L)$ form a polynomial law of E into F which is homogeneous of degree p. This polynomial law is called the homogeneous component of degree $p$ of the law Q. Show that the homogeneous components of Q form a summable family with sum $\varphi$.