Let X be a smooth and projective variety. Hochschild homology and cohomology have a very simple definition in terms of Ext groups of the diagonal of X. The Hochschild-Kostant-Rosenberg (HKR) theorem then tells us that these coincide with sums of cohomology of differential forms or polyvector fields (one reference for this is the papers by Caldararu).

  • Is there a simple definition of negative cyclic homology for X?

  • Is there an HKR-type theorem which relates this to more understandable gadgets?

  • Same question for ordinary cyclic and periodic cyclic homology (and cohomology)?

  • $\begingroup$ One of these should end up being algebraic de Rham cohomology, I think? $\endgroup$ – Qiaochu Yuan Nov 17 '16 at 6:53
  • $\begingroup$ You might find some pointers in imperium.lenin.ru/~kaledin/tokyo/final.pdf, e.g. in lecture 2. $\endgroup$ – pbelmans Nov 17 '16 at 12:05
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    $\begingroup$ The other differential in the mixed complex can be identified with de Rham d, but the difficulty of this calculation depends precisely on how you define things. In particular, for varieties over $\mathbb{C}$, the periodic cyclic homology always agrees with the two periodization of cohomology. There is an early paper by Weibel "Cyclic homology for schemes" that asserts this in one framework. You can also look at Toen and Vezzosi "S^1 Equivariant simplicial algebras and de Rham theory" or Nadler Ben-Zvi (they have a few papers on related topics). $\endgroup$ – Daniel Pomerleano Nov 17 '16 at 12:28

There are conceptually simple definitions, but they require a more symmetric definition of Hochschild homology. The Hochschild homology of $X/k$ (with coefficients in $\mathcal O_X$) is the homology of the complex

$$ \mathbf R\Gamma(X\times X, \mathbf R\Delta_*\mathcal O_X\otimes^{\mathbf L}\mathbf R\Delta_*\mathcal O_X)). $$

Equivalently, in the language of derived algebraic geometry, this is the complex $\mathcal O(LX)$ where $LX=X\times^{h}_{X\times X}X$. The circle comes in because $S^1$ is the homotopy pushout $*\coprod^h_{*\amalg *}*$, so, formally, $LX=X^{S^1}$ is the free loop space of $X$ (it is a derived scheme whose underlying scheme is $X$). Now $S^1$ is also a group and it acts on itself, hence it acts on $LX$ and on the complex $\mathcal O(LX)$ computing $HH_*(X)$.

The complexes computing the cyclic, negative cyclic, and periodic cyclic homology of $X$ are respectively the homotopy orbits $\mathcal O(LX)_{hS^1}$, homotopy fixed points $\mathcal O(LX)^{hS^1}$, and Tate fixed points $\mathcal O(LX)^{tS^1}$ of this circle action. (For $G$ a compact Lie group acting on a spectrum or chain complex $E$, there is a norm map $(\Sigma^{\mathfrak g}E)_{hG}\to E^{hG}$ whose homotopy cofiber is by definition $E^{tG}$. In our case this cofiber sequence induces the usual long exact sequence relating these three homology theories.) I'm not sure what the cyclic cohomology of a $k$-scheme is. At least if $X$ is affine, it is computed by the dual of the complex $\mathcal O(LX)_{hS^1}$, i.e., by the complex of $S^1$-invariant maps $\mathcal O(LX) \to k$.

ETA: Hochschild cohomology of $X$ with coefficients in $\omega_X$ is computed by the complex $\omega(LX)$, which has an $S^1$-action. So perhaps one gets reasonable "cohomology" versions of the above theories by replacing $\mathcal O(LX)$ by $\omega(LX)$.

To relate this to the traditional definitions, one shows that there is an equivalence between $S^1$-equivariant chain complexes and mixed complexes, and that the $S^1$-action on $\mathcal O(LX)$ corresponds to Connes' operator $B$.

Here's yet another way to understand the $S^1$-action on Hochschild chains which applies to the noncommutative setting as well. The complex $\mathcal O(LX)$ can be identified with the Euler characteristic (=trace of the identity) of $D_{qcoh}(X)$ in the symmetric monoidal $\infty$-category of presentable dg-categories. It is a general fact that the Euler characteristic of any object comes with an $S^1$-action. From the point of view of the cobordism hypothesis, this $S^1$ is now the framed diffeomorphism group of the circle, which is the Euler characteristic of the universal dualizable object in $\operatorname{Bord}_1^{fr}$.

If $\mathbb Q\subset k$ and $X$ is smooth and affine, the relations with Kähler differentials are given by:

\begin{align*} HC_n(X) &= \Omega^n(X)/B^n(X) \oplus \bigoplus_{i\geq 1} H^{n-2i}_{dR}(X),\\ HC_n^-(X) &= Z^n(X) \times \prod_{i\geq 1} H^{n+2i}_{dR}(X),\\ HC_n^{per}(X)& = \prod_{i\in\mathbb Z} H^{n+2i}_{dR}(X). \end{align*}

(Reference: Loday's book, 3.4.12, 5.1.12). The formula for $HC^{per}_n(X)$ remains valid if $X$ is not affine (because it satisfies Mayer-Vietoris), but the first two become more complicated...

Update. Here are the global formulas, I hope I got the indices right:

\begin{align*} HC_n(X) &= \bigoplus_{-\dim(X)\leq i\leq n} H^{n-2i}_{Zar}(X, s_{\leq n-i}\Omega^*_X),\\ HC_n^-(X) &= \prod_{0\leq i\leq \dim(X)-n} H^{n+2i}_{Zar}(X, s_{\geq n+i}\Omega^*_X). \end{align*}

Here $H_{Zar}$ is Zariski hypercohomology, $s_{\leq k}$ and $s_{\geq k}$ denote the stupid truncations, and $\Omega^*_X$ is the de Rham complex.

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  • $\begingroup$ Thank you a lot for the very thoughtful answer. The kind of things I'm after are equations like the last three you wrote. Is there really no HKR-style way to describe HC and HC- for X projective? [for example, the fact that equation 3 is always satisfied is kind of bittersweet, since it relates HCper with the equally mysterious HC and HC- !] $\endgroup$ – Yosemite Sam Nov 19 '16 at 4:23
  • $\begingroup$ What I wrote there was confusing, sorry. What I meant is that $HC_n^{per}$ is periodic de Rham cohomology even if $X$ is not affine (I changed the formula). I'm sure there are explicit formulas for $HC_n(X)$ and $HC_n^-(X)$. They are given by explicit complexes depending only on the mixed complex $(\bigoplus_n\mathbf R\Gamma\Omega_X^n, d)$. $\endgroup$ – Marc Hoyois Nov 19 '16 at 15:27
  • $\begingroup$ nice! thanks a lot. I also hope you got the indices right. $\endgroup$ – Yosemite Sam Nov 19 '16 at 17:51

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