It appears that there are more definitions of cyclic homology than there are people working on cyclic homology. As a newcomer, this confuses me to no end. I've written a list of definitions that some Google searches have given me, although I am confident that there are many more. My question is essentially: what is the relation between all of them?
- If $A$ is an associative algebra over a ring $k$, then it has a Hochschild complex $C_*(A)$ with differential $b$, whose homology is called Hochschild homology. If one instead quotients out by the cyclic permutation on the level of complexes, one obtains cyclic homology $HC_*(A)$ of $A$. If one moreover has an ideal $I$ of $A$, we can define relative cyclic homology $HC_*(A,I)$ to be the homology groups of the complex $\operatorname{ker}(C_*(A) \to C_*(A / I))$.
- One can also define Hochschild homology of an associative $k$-algebra $A$ to be $\operatorname{Tor}_*^{A \otimes A^{\text{op}}}(A,A)$.
- If $A$ is an associative algebra over a ring $k$, one can instead define an additional operator $B$ on $C_*(A)$, forming a double complex. The homology of the total complex should be cyclic homology. From this point of view, it appears that one can work more generally with bimodules over $A$.
- Letting $A$ again be an associative algebra over a ring $k$, one can also define $CC_*(A)$ to be the complex $C_*(A)[[u^{\pm 1}]] / u C_*(A)[[u]]$ with operator $b + u B$. The homology of this complex is cyclic homology. In addition, negative cyclic homology is obtained from the complex $C_*(A)[[u]]$ with differential $b + uB$, and periodic cyclic homology is obtained from $C_*(A)[[u^{\pm 1}]]$ with operator $b + uB$.
- In addition, I've read somewhere that we can view cyclic homology as being a derived functor on $(b,B)$-complexes, but I do not understand what that means.
- If $A$ is an associative, unital ring, one can consider the simplicial object formed by $A^{\otimes n}$, and the geometric realisation should somehow be Hochschild homology of $A$. This admits an $S^1$-action, and cyclic homology is defined by taking homotopy orbits, while negative cyclic homology is defined via homotopy fixed points.
- Connes has defined a cyclic homology theory in terms of what is called cyclic vector spaces. Cyclic homology can then be computed as certain Ext-groups. I have also been told that there is a categorical way of defining cyclic homology in terms of cyclic objects. I do not know if this is the same thing.
- Charles Weibel has defined cyclic homology for schemes in terms of hypercohomology, so this in particular applies to the case of commutative rings and algebras.
- In a derived setting, Hochschild homology can be understood as the cohomology of a certain free loop space object, and cyclic homology is the corresponding equivariant cohomology.
- For C*-algebras, as far as I can tell the definitions can carry over, but I vaguely understand that there are many choices for the tensor product, which complicates matters. In addition, Wikipedia tells me that cyclic homology (in whatever form) of C*-algebras is degenerate as it cannot take into account the topological structure, and there are many variants to incorporate this topology. Mentioned there are entire cyclic homology due to Connes, analytic cyclic homology due to Meyer, asymptotic cyclic homology and local cyclic homology due to michael Puschnigg.
- Then there's topological Hochschild homology and topological cyclic homology, and it seems one could make equally long lists for them.