# p-polar rings

What if one modified the definition of a commutative ring to only allow for $p$-fold multiplications instead of binary multiplications? This is the starting point of this project. There are various flavors of such $p$-polar rings, as I call them. The most basic one is the following:

Definition. Let $k$ be a ring and $p$ be a prime. A $p$-polar $k$-algebra is a $k$-module $R$ together with a map $$\mu\colon R \times \cdots \times R \to R \quad \text{(p factors)}$$ such that:

• $\mu$ is $k$-linear in each factor
• $\mu(x_1,\dots,x_p)$ is independent of the order of $x_1,\dots,x_p$, and
• the associativity axiom holds: $\mu(\mu(x_1, \dots,x_p ),x_{p+1}, \dots,x_{2p-1} )$ is independent of the order of $x_1,\dots,x_{2p-1}$.

Obviously, every $k$-algebra gives rise to a $p$-polar $k$-algebra (its $p$-polarization), but not every $p$-polar $k$-algebra arises in this way.

As part of current research, I have shown that any affine $p$-group scheme defined over a perfect field $k$ of characteristic $p$ is in fact defined on the category of $p$-polar $k$-algebras. In particular, cofree Hopf algebra constructions only depend on $p$-polarizations of rings. Furthermore, $p$-typical Witt vectors can be constructed for $p$-polar algebras. A preprint will be available shortly.

The aim of this project is to shed light on the more elementary properties of the category of $p$-polar algebras. Which notions from ring theory and from algebraic geometry can be generalized to the $p$-polar setting? Is $p$-polarization a kind of useful localization of the categories of rings or of schemes?

I have funding to employ an international Masters student from KTH as a research assistant for 6 months between now and August 2021 at 20% FTE (i.e. you will be spending 20% of your work week on this project) . The research work you will be doing cannot be used as course work or a Master’s thesis (but, a Master’s thesis could certainly build on it).

You will be working with me, in person or remotely as the situation permits, and we will have regular meetings, much like for thesis work. Your results are yours.

What I expect: You are excited about algebra, you like to pursue not only problems given to you but also your own explorations in mathematics, you can work independently to some extent, you have good knowledge of rings and modules, you optimally have had some exposure to abstract algebraic geometry (schemes and sheaves), and you have 20% of your time to spare without negatively affecting your course work.

What is in it for you? Besides being paid, training in doing research (on a basic level) and communicating that research with the possibility of a culmination in a publication.