Reflexive homology and involutive Hochschild homology as equivariant Loday constructions

Description

This paper can be classified as categorical algebraic topology. It doesn’t look like topology, though. The object of study are rings of different kinds. The paper is structured in a way that special rings are studied first and after that the theorems are generalised.

We are interested in Loday construction of rings. (a Loday construction seems to be a simplicial complex created from the ring or, more generally, a G-Tambara functor). We notice that under certain circumstances (we have a so-called anti-involution and 2 is invertible) the homotopy group (which is the generalisation of the fundamental group btw.) of the Loday construction is equivalent to two homology theories. It is later shown by counterexample that the special circumstances can’t be ignored.

The theorem is repeated in different generalities. First we study the homology of commutative rings, then commutative k-algebras, then an associative ring, and finally we look at constructions relative to a commutative ring.

There is a lot of categoric homology terminology going on, so the paper is quite hard to read.

The paper is structured as follows:

1. Introduction Describes the main theorems
2. Equivariant Loday constructions Gives a description of Equivariant Loday construction of G-Tambara functors/ but omits a detailed explanation of Loday constructions and G-Tambara functors. (possibly to keep the paper under 20 pages)
3. Basic results about fixed-point Tambara functors We define Fixed-point Tambara functors (again, involutions are not explained). $C_2$-Mackey functors seem to be of great importance but they too haven’t been defined. The basic results seem quite technical. I assume they are used later
4. Working relative to a commutative ground ring The last generalisation made in the introduction is that of a relative Loday construction. Here we define what we mean with relative.
5. Identifying ${\mathcal{L}}_{S^{\sigma }}^{C_2}({\underline{R}}^{\text{fix}})$ We start by identifying a very easy example
6. Relating ${\mathcal{L}}_{S^{\sigma }}^{C_2}({\underline{R}}^{\text{fix}})$ to reflexive homology We show that in the above example the reflexive homology is equivalent to the homotopy of the Loday construction. Then the example is generalised
7. Involutive Hochschild homology as a Loday construction We do the same for the Hochschild homology.
8. A counterexample in characteristic two We show that the restrictions we placed earlier must be there
9. The case of assiciate $C_2$-Green functors I guess that here more generalisations are taking place.