Questions tagged [divided-powers]

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What comes next in the sequence "symmetric algebras, exterior algebras, divided power algebras, ..."?

This question was posed by A Rock and a Hard Place in this discussion, where they mentioned the isomorphisms \begin{align*} \mathrm{L}\,\mathrm{Sym}^n_R(M[1]) &\cong (\mathrm{L}\,{\...
4
votes
0answers
145 views

divided powers of a deformation class

Let $A$ be a (unital, associative) $k$-algebra where $k$ is a field. Given a flat deformation of $A$ one gets the deformation class $h$ in the second Hochschild cohomology $HH^2(A)$. Suppose $k$ has ...
9
votes
2answers
440 views

Divided power algebra is artinian as a module over the polynomial ring

I already asked this on math.stackexchange.com, but did not receive much responses. I hope this is also appropriate for mathoverflow. In the paper Homological algebra on a complete intersection, with ...
6
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0answers
228 views

Universal property of $A_{\mathrm{cris}}/p^n$

It is well known that the ring $A_{\mathrm{cris}}$ of Fontaine is the universal $p$-adically complete divided power thickening of $\mathcal{O}_{\mathbb{C}_p}$ over $\mathbb{Z}_p$; in fact, this is one ...
4
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0answers
196 views

Explicit description of graded (counital) cofree cocommutative coalgebras

Let $k$ be a field of characteristic $p \neq 2$, and $V = \oplus V_{n}$ be a graded vector space over $k$. Then, can one compute the graded (counital) cofree cocommutative coalgebra $C(V)$ ...
2
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1answer
408 views

On the divided power ring over the integers

Consider the divided-power ring $A := \mathbb Z \langle x_1, \ldots, x_n \rangle$ consisting of $\mathbb Z$-linear combinations of divided-power monomials of the form $x_1^{(a_1)} \cdots x_n^{(a_n)}$; ...
11
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0answers
290 views

Subrings of invariants in divided power algebras

I am wondering to what extent the functors "ring of invariants under a group action $G$" and "divided power envelope with respect to a $G$-stable ideal" commute. To be precise, let $R$ be a ...