Is the functor of divided powers a weakly monoidal functor?

Question

Let $R$ be a commutative ring with identity $e$. For every $R$-module $M$ the algebra of divided powers $D(M)$ is defined as follows. The generators of $D(M)$ are the symbolls $m^{(k)}$ for every $m\in M$ and every non-negative integers $k$. Defininig relations between the generators are given by

$m^{(0)} = e$

$m^{(k)}m^{(l)} = \binom{k+l}{l}m^{(k+l)}$

$(m_1+m_2)^{(l)} = \sum_{k=0}^l m_1^{(k)}m_2^{(l-k)}$

$(am)^{(k)}=a^k m^{(k)}$.

The algebra $D(M)$ is graded with the degree function given by $deg(m^{(k)})= k$.

Then $D$ is a functor from the category of $R$-modules to the category of graded $R$-algebras in the obvious way. By passing to the $r$-homogeneous component of $D(M)$ we get the endofunctor $D_r$ on the category of $R$-modules.

Then it seems functor $D_r$ can be given a structure of a weakly monoidal functor as follows.

The unit transformation $\eta\colon R\to D_r(R)$ is defined by $\eta(a)=a^re^{(r)}$.

The multiplication transformation $\tau\colon D_r(M)\otimes D_r(N)\to D_r(M\otimes N)$ is defined by

$$m_1^{(k_1)}\dots m_t^{(k_t)} \otimes n_1^{(l_1)}\dots n_s^{(l_s)} \mapsto \sum_{v} \prod_{i=1}^t \prod_{j=1}^s (m_i\otimes n_j)^{(v_{ij})},$$

where summation is over matrices $v\in M_{t,s}(\mathbb{N}_0)$ such that the sum of rows of $v$ is $(l_1,\dots,l_s)$ and the sum of collumns of $v$ is $(k_1,\dots,k_t)$.

Is there any reference where such monoidal structure on $D_r$ is considered?

Answer 1

This is a well-known result and, apart from terminology, should be found in Roby, Norbert Lois polynômes multiplicatives universelles. (French. English summary) C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 19, A869–A871.

Note that this becomes most natural if one interprets your $D_r(M)$ (usually denoted $\Gamma^r(M)$ and the map $M\to \Gamma^r(M)$ (given by $m\mapsto m^{(r)}$ as the universal homogeneous polynomial map of degree $n$. From this it is quite formal to conclude that there is a map $\Gamma^r(M)\bigotimes \Gamma^r(N)\to \Gamma^r(M\bigotimes N)$ characterised by $m^{(r)}\otimes n^{(r)}\mapsto (m\otimes n)^{(r)}$ (and commutation with scalar extension).

Addendum: I don't have access currently to Roby's article so let me give the argument. We have a map $M\times N\to \Gamma^r(M\bigotimes N)$ given by $(m,n) \mapsto (m\otimes n)^{(r)}$. This commutes with extension of scalars and is hence a polynomial map in Roby's sense. It is also bihomogeneous of degree $r$ and hence gives a linear map $\Gamma^r(M)\bigotimes \Gamma^r(N)\to \Gamma^r(M\bigotimes N)$.

Applied to the case when $A=M=N$, where $A$ is an algebra we get a map $\Gamma^r(A)\bigotimes \Gamma^r(A)\to \Gamma^r(A\bigotimes A)$ which composed with the multiplication map $A\bigotimes A\to A$ gives the algebra structure that Roby is considering.