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I recently came across divided power algebras here: http://amathew.wordpress.com/2012/05/27/lazards-theorem-ii/ It interests me because the free divided power algebra on one variable $x$, where $x^{(i)}$ models $x^i/i!$, seems like a good way of handling exponential and exponential-type series. (Divided power algebras that aren't generated by a single variable are more complicated to axiomatize, so I will stick to the one-variable case).

Quick summary: A system of divided powers for an element $x$ in an associative unital ring $R$ associates to each nonnegative integer $n$ a ring element $x^{(n)}$ with $x^{(0)} = 1$ and $x^{(1)} = x$ and satisfying the following condition for all $i,j \ge 0$:

$$x^{(i)}x^{(j)} = \binom{i + j}{i} x^{(i + j)}$$

As we can see, if $R$ is an algebra over the rationals, the obvious (and unique!) choice is to set $x^{(i)} = x^i/i!$ for all $i$.

Suppose I am interested in finding some $x^{[i]}$ that models $x^i/i$ instead of $x^i/i!$, i.e., I am interested in the kind of terms that appear in the expansions of logarithmic and inverse trig series. My best guess for the right analogue to consider is this: $x^{[i]}$ is defined for all positive integers $i$ with $x^{[1]} = x$ and satisfy the following condition. For each $i,j > 0$, write $ij/(i + j)$ as a reduced fraction $r/s$. Then, we must have:

$$rx^{[i]}x^{[j]} = sx^{[i + j]}$$

I'd like to know whether this structure has been studied in the past, and/or what names have been given to the structure in question. Also, I'd be interested if anybody has opinions on whether the above is a reasonable way of trying to model $x^i/i$.

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4 Answers 4

up vote 6 down vote accepted

First of all, let me remark that one normally speaks of a divided power structure on an ideal $I\subset R$ (rather than on the whole $R$), by asking the existence of divided powers only for elements in the ideal; the requirement is then that all divided powers but the $0$-th also lie in the ideal.

I think that your divided power structure would not be very useful – let me call it logarithmic divided power as Liz suggested. If you have the "usual" one on your ideal $I$ (let me denote it by $\gamma$ as in Berthelot&Ogus' book) you can immediately deduce a logarithmic one (call it $\lambda$) by setting $$ \lambda_i(x)=(i-1)!\gamma_i(x) $$ for all $x\in I$. Since $(i-1)!\in\mathbb{Z}$, this defines a unique logarithmic divided power structure on $I$.

But unfortunately, your logarithmic powers are weaker than usual ones: for instance, we know (see Berthelot&Ogus' book, page 3.2, Lemma 3.3) that if $V$ is a discrete valuation ring of mixed characteristic $(0,p)$ and uniformizer $(\pi)$, we can put a divided power structure on $(\pi^k)$ if and only if the absolute ramification index $e$ of $V$ verifies $e\leq k(p-1)$ (only the case $k=1$ is treated in the book, but computations are analogous for $k\geq 1$). If you want to stress that your logarithmic structure exist, you only need to insist that $v_\pi[\lambda_i(\pi^k)]\geq 1$ for all $i\geq 1$. Computing explicitly, this corresponds to $$ e\leq \frac{ik-1}{v_p(i)} $$ which, by monotonicity of $i/v_p(i)$, is satisfied as soon as $e\leq kp-1$. Since this is (in general) bigger than $k(p-1)$ you see that there are rings admitting one structure but not the other; since in crystalline cohomology and related topics one really needs exponentials (so, dividing out by $n!$), yours are too weak to be useful.

You started your question saying "Suppose I am interested in finding...": are you really?

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Filippo, thanks a lot for your answer. I don't know anything about crystalline cohomology, but I think I understand the rest of your answer. For the application that I had in mind, it isn't a disadvantage if the "logarithmic divided powers" are defined in a more diverse array of situations than the usual divided powers; that might even be an advantage. But I suspect that you may be right that the situations where logarithmic divided powers exist, and the usual ones don't, are not very interesting. –  Vipul Naik Jun 26 '12 at 16:07
Vipul, I think Laurent's answer is very enlightening, but as you say that you are not very familiar with crystalline cohomology I dare commenting a bit. The ring $B_{cris}$ you might have heard of is the divided power envelope of some ''universal'' creature; what Laurent observes is that one can, analogously, apply a whole bestiary of different divided power constructions to the universal creature to find many more rings which all look natural in $p$-adic Hodge theory, like $B_{cris}$. Your ''logarithmic divided power'' then corresponds to a (useful!) ring of functions with a growth condition. –  Filippo Alberto Edoardo Jun 27 '12 at 12:23

Vipul, I don't have an answer to your question, but I thought I would remind you why divided power algebras are so very natural (in contrast ...?), since that makes understanding the many variable case easy, at least if you are willing to have your algebras be commutative. Working over a field (not necessarily of characteristic zero), look at a polynomial algebra and give it a bialgebra structure by letting its generators be primitive. The dual algebra to the resulting coalgebra is the divided polynomial algebra. This makes sense and gives the right (commutative) definition even if you have infinitely many generators. Of course, you have to be careful about saying the dual is a bialgebra, but it is in zillions of graded situations in algebraic topology where there are finitely many generators of positive degree (even if the characteristic isn't $2$).

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Thank you for the insight, Professor May! –  Vipul Naik Jun 27 '12 at 18:47

One possible way to generalize divided powers is to model sequences $x^n/a_n$ where $a_{n+m}/(a_na_m)$ is integral (an integer, or at least integral in some sense). You can model these in the obvious way, like divided powers. Choosing $a_n = n!$ gives you divided powers, and choosing $a_n=p^n$ gives you a sequence which, from the point of view of $p$-adic analysis, is much more "regular" in its growth. You can come up with other examples.

As you may know, one uses divided powers to construct Fontaine's ring of periods $B_{cris}$ and using $a_n=p^n$ instead of $a_n=n!$ gives a ring $B_{max}$ that is similar in nature to $B_{cris}$ but much better behaved (see III.2 of Colmez' 1998 paper "Théorie d'Iwasawa des représentations de de Rham d'un corps local").

More generally, tweaking divided powers to get various convergence conditions is something that happens often in $p$-adic Hodge theory. See for instance 5.2.3 of Fontaine's "Le corps des périodes $p$-adiques" for one of many examples of custom made convergence conditions.

In the $p$-adic setting, I would say that your log series belongs to the "space of holomorphic functions on the $p$-adic open unit disk, having order of growth $\leq 1$". Using precise analytic conditions rather than "divided powers" seems the right way to study the analytic series that interest you.

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Why not call them logarithmic divided power algebras?

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