Arpon Raksit - Hochschild homology and the derived de Rham complex revisited
Tags: #higher-algebra/derived #projects/notes/seminars
Refs: HH derived de Rham cohomology
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 is used to define derived de Rham cohomology :
-
Can get a derived version: take a nonabelian derived functor, i.e. take a simplicial resolution by simplicial polynomial algebras and apply the functor to the resolution.
- Equivalently a left Kan extension?
- Define \(\mathop{\mathrm{{\mathbb{L} }}}\Omega^1 _{A/R}\) to be the cotangent complex and take derived exterior powers for the other degrees.
- Derived Hodge filtration may not be complete, so take completion: take cone (cofiber?) in derived category and take a hocolim.
-
Fact: de Rham complex has a universal property, initial (strictly, so odd elements square to zero) commutative differential graded algebras 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 is defined as \({\mathbb{H}}(A/R) \coloneqq A^{\otimes_R S^1}\) the \(S^1\) tensoring, take homotopy fixed points to get \({\operatorname{HC}}^-\), cyclic homology.
-
Associated graded of \({\mathbb{H}}\) recovers derived de Rham:
- Why does this happen? \({\mathbb{H}}(A/R)\) is the initial simplicial algebra with an \(S^1{\hbox{-}}\)action receiving a map from \(A\).
-
Analogies
- simplicial ring \(\rightleftharpoons\) CDGAs
- \(S^1\) action \(\rightleftharpoons\) the differential
- Can take homotopy groups of \({\mathbb{H}}\)????