## Arpon Raksit - Hochschild homology and the derived de Rham complex revisited

Tags: 
#higher-algebra/derived #projects/notes/seminars  

Refs: 
[HH](Unsorted/HH.md)
[derived de Rham cohomology](Notes%20-%20The%20cotangent%20complex%20and%20derived%20de%20Rham%20cohomology.md)

> Reference: Arpon Raksit - Hochschild homology and the derived de Rham complex revisited. 
> <https://www.youtube.com/watch?v=E84gVDm1kvM>

- The [Unsorted/algebraic de Rham cohomology](Unsorted/algebraic%20de%20Rham%20cohomology.md) is used to define [derived de Rham cohomology](Notes%20-%20The%20cotangent%20complex%20and%20derived%20de%20Rham%20cohomology.md) :

![](attachments/image_2021-05-06-11-15-55.png)

- Can get a derived version: take a [nonabelian derived functor](nonabelian%20derived%20functor.md), i.e. take a [simplicial resolution](simplicial%20resolution) by simplicial polynomial algebras and apply the functor to the resolution.
  - Equivalently a left [Kan extension](Kan%20extension.md)?
- Define $\Ld \Omega^1 _{A/R}$ to be the [cotangent complex](cotangent%20complex.md) and take derived exterior powers for the other degrees.
- Derived [Hodge filtration](Hodge%20filtration.md) may not be complete, so take completion: take cone (cofiber?) in derived category and take a hocolim.

![](attachments/image_2021-05-06-11-20-04.png)

- Fact: de Rham complex has a universal property, initial (strictly, so odd elements square to zero) commutative [differential graded algebras](differential%20graded%20algebras) receiving a map $A\to X^0$, so the initial way to turn an algebra into a DGA.
  Does the derived version have a similar property?

- [Unsorted/HH](Unsorted/HH.md) is defined as $\HH(A/R) \da A^{\tensor_R S^1}$ the $S^1$ tensoring, take [homotopy fixed points](homotopy%20fixed%20points.md) to get $\HC^-$, [cyclic homology](cyclic%20homology).
- Associated graded of $\HH$ recovers derived de Rham:

![](attachments/image_2021-05-06-11-25-44.png)

  - Why does this happen? $\HH(A/R)$ is the initial [simplicial algebra](simplicial%20algebra.md) with an $S^1\dash$action receiving a map from $A$.
  - Analogies
    - [simplicial ring](simplicial%20ring) $\mapstofrom$ [CDGAs](differential%20graded%20algebras)
    - $S^1$ action $\mapstofrom$ the differential

- Can take homotopy groups of $\HH$????