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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?

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as $\mathsf{GL}(V)$ modules, $V$ a vector space, $D^r(V)^*\cong S^r(V^∗)$ ($S$ is the symmetric power, $^*$ is the Hopf algebra dual). This is true in more generality, see the wiki article for divided power, so you can pass information about the symmetric power functor over to the divided power functor. –  Matthew Towers Mar 27 '11 at 13:28
Good point, it makes things to work for free $R$-modules, and at the moment I am not sure that $\tau$ is well-defined for all $R$-modules $M$ and $N$. –  Ivan Yudin Mar 27 '11 at 16:11

1 Answer 1

up vote 12 down vote accepted

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.

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I know about the article of Roby, but I didn't found the construction there. It seems, the univresal property give a map in the opposite direction. –  Ivan Yudin Mar 27 '11 at 15:50
Thank you! Sorry, I overlooked the full name of the paper and mixed it up with another paper of Roby. If he defines a natural algebra structure on $\Gamma^r(A)$ for every algebra $A$, it should be possible to recover a monoidal structure on $\Gammma^r$ by considering the product on $\Gamma^r(T(M\oplus N))$. –  Ivan Yudin Mar 28 '11 at 0:09
Indeed, having a product on $\Gamma^r(A)$ functorial in the algebra $A$ is equivalent to a weak monoidal structure. –  Torsten Ekedahl Mar 28 '11 at 6:39

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