symmetric monoidal (∞,1)-category of spectra
A divided power algebra is a commutative ring together with an ideal and a collection of operations which behave like operations of taking divided powers in power series.
A divided power algebra is a triple with
a commutative ring (with identity);
an ideal of ;
an indexed set of functions (of underlying sets);
where we additionally adopt the convention (which is usually not in ), and this data is required to satisfy the following conditions:
For each , we have .
For each and , we have
For each , each and , we have
For each and each , we have
For each and each , , we have
For a given , a divided power structure on is a making a divided power algebra.
If is an -algebra for a ring , we call it a divided power -algebra or PD--algebra.
Some sources include in the data rather than as convention. Some sources give the data as typing while including for as an axiom.
Genuine powers can be constructed in the expected way from the divided powers, and when is torsion free, the reverse is true:
If is a divided power algebra, then for every and (taking ).
It is true for and by definition. For , this follows by induction, since .
If is a commutative, torsion free ring with an ideal such that is an -th multiple for every and , then has a unique divided power structure, and it is given by .
The hypotheses imply the quotients are unique and well-defined, and any divided power structure on must be given by that formula. It’s straightforward to check the definition does give a divided power algebra.
So in the torsion free case, the divided power algebras are precisely of the motivating form. In positive characteristic, though, examples can be somewhat more exotic.
We can define the concept of divided power in any symmetric monoidal category.
Let be a symmetric monoidal category.
The divided power of an object is then defined as the equalizer of the following diagram:
where there is one arrow for every . We write for the natural transformation associated to , which is defined in the entry symmetric monoidal category.
The divided power is thus described by the following equalizer diagram:
In the category of modules over some commutative ring, the divided power of a module is equivalently descibed as the space of invariants under the action of on by permutation of the factors. Note that the symmetric power of an object in a symmetric monoidal category is described as the coequalizer of the permutations . In a category of modules, it is equivalently described as the space of coinvariants by the action of . We then have the relation which probably means that these powers can be interpreted as graded exponential modalities in a kind of graded differential linear logic. In characteristic 0, we have . Nondegenerated models of such a logic in a category of vector spaces would thus be only in the case where the field is of positive characteristic.
Divided power algebras were originally introduced in
Their theory was further developed in Pierre Berthelot‘s PhD thesis (in the context of crystalline cohomology), which was later published as:
Review:
Pierre Berthelot, Arthur Ogus, Notes on crystalline cohomology, Princeton Univ. Press 1978. vi+243, (ISBN:0-691-08218-9)
Aise Johan de Jong et al., The Stacks Project, Chapter 09PD.
Recent works on divided power algebras include:
Sacha Ikonicoff?, Divided power algebras over an operad, Glasgow Math. J. 62 (2020) 477-517, doi:10.1017/S0017089519000223, pdf
Sacha Ikonicoff?, Divided power algebras and distributive laws, 2021, doi:10.48550/arXiv.2104.11736, pdf
It is related to differential categories in:
On divided powers:
See also:
In relation to the sphere spectrum
Last revised on November 19, 2022 at 20:44:34. See the history of this page for a list of all contributions to it.