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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_{i\geq 0} 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.

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.

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...

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_{i\geq 0} 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.

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)& = \operatorname{lim}_{k\to\infty }HC_{n+2k}(X) = \operatorname{colim}_{k\to\infty} HC^-_{n-2k}(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...

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...