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Thanks. I have added some more hyperlinks (associative algebra, ring, ideal, …) . Your “operations” and “maps” should probably be linear maps. (?)
We ought to have an entry divided power, but unfortunately we don’t. Maybe you’d enjoy creating it?
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Added a reference to some informal discussion
for definiteness, I have changed
… a collection of maps
to
..an indexed set of functions (of underlying sets)
Also I added more hyperlinks to the references (author names, DOI-s, publisher pages,…)
Finally, I fixed the typesetting of the indices: due to an Instiki speciality, the source code nm gets rendered as “nm”. One needs a whitespace n m to tell Instiki that these are two distinct variables such as to obtain the desired italicized “nm”.
In the definition of a divided power R-algebra, the base ring R is never used. Is there a reason to define this for A an R-algebra rather than just for A a (possibly noncommutative) ring?
The definition given doesn’t agree with the stacks project, and the hypotheses on the data of the maps in the PD-structure doesn’t typecheck. I’m not sure what is meant, else I’d fix it.
I hadn’t noticed that either; IIRC you’re just supposed to have x,y∈I? Oh I see that’s been changed already.
Another thing to note (after making that correction) is that I is a commutative rng, since you can swap x and y in γ2(x+y)=γ2(x)+γ1(x)γ1(y)+γ2(y).
This meshes with the fact I thought when I saw divided power structures, they were on commutative rings with an ideal. Should we narrow the definition to A being commutative, or is there an application with noncommutative A?
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I’ve gone ahead and made the R-algebra case a secondary definition, and changed the definition to take A to be a commutative ring; feel free to adjust it if the noncommutative case really is of interest.
I think that the properties and constructions only really work well when I is in the center of A, so I expect substantial savings by making this change.
The definition is still a bit odd. in https://stacks.math.columbia.edu/tag/07GL, we see that the maps γn are meant to be endomorphisms (as sets) of the ideal I. But in the nLab page they are maps I→A. I’m not sure about the difference between a divided power algebra (as at the nLab) and a divided power structure (as at the Stacks Project), which only applies in the case of a divided power ring. Can someone sort this out?
It looks like item 3 in our entry forces γ to take values in I after all…
Maybe this is meant to be following the Wikipedia entry (here) that also insists on the codomain of γn being A.
But, yeah, it looks odd. I suggest to follow the StacksProject, make the codomain I and also fix the clauses 1 and 2 to have x∈I instead of x∈A.
Ah, the point is whether the indexing starts at 0 or 1, since γ0=1 indeed can’t take values in I unless I=A.
To me that seems like one should have maps γn:I→I for n≥1 and then just define γ0 by convention to be 1.
Don’t succeed to put all my edit because of crazy errors… That’s ok now, it was the error of putting a tikzcd diagram into $$
Added the reference Luis Narváez Macarro, Hasse-Schmidt derivations, divided powers and differential smoothness, 2009
Added that permutations σ:A⊗n→A⊗n are defined in the entry symmetric monoidal category.
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