Today we’ll talk about the Hochschild (Co)Homology of schemes and compute it in the case the scheme is either the projective space or a smooth and proper curve. Throughout X will be a smooth projective scheme over a field k. We assume k is algebraically closed and of characteristic zero. To any scheme we can associate the bounded derived category of coherent sheaves on X, denote it by D(X).

Roughly speaking, the Hochshcild (Co)Homology of X is a collection of vector spaces arising as the derived endomorphisms of the identity functor on D(X), or in the homology case it arises from the derived tensor square of the identity functor (or equivalently the derived maps from the identity functor to the Serre functor). For this to make sense, we need to realize these supposed vector spaces as an extension or hom space in a k-linear category.

Since X is separated, we can consider the diagonal embedding \[ \Delta\colon X\to X \times X. \] Define \(\mathcal{O}_\Delta = \Delta_\ast\mathcal{O}_X\) and the Hochschild Cohomology groups as \[ HH^\ast(X) := \mathrm{Ext}^\ast_{X\times X}(\mathcal{O}_\Delta, \mathcal{O}_\Delta). \]

As for the Hochschild Homology groups, we have \[ HH_\ast(X) := \mathcal{O}_\Delta\otimes_{X\times X}^{\mathbf{L}}\mathcal{O}_\Delta. \]

We view the diagonal subscheme as defining the identity functor by using the formalism of Fourier-Mukai functors. The Fourier-Mukai kernel corresponding to the identity functor is \(\mathcal{O}_\Delta\).

Since we assume X is smooth, we can take a bounded, locally free resolution of \(\mathcal{O}_\Delta\), say \(\mathcal{F}^\cdot\to \mathcal{O}_\Delta\) is the resolution. Then we can equivalently define the Hochschild Cohomology groups as \begin{align*} HH^\ast(X) &= \mathrm{Ext}^\ast_{X\times X}(\Delta_\ast\mathcal{O}_X, \Delta_\ast\mathcal{O}_X) \\ &\cong \mathrm{Ext}^\ast_{X\times X}(\mathcal{F}^\cdot,\Delta_\ast\mathcal{O}_X) \\ &\cong \mathrm{Ext}^\ast_X(\Delta^\ast\mathcal{F}^\cdot,\mathcal{O}_X) \\ &\cong H^\ast(X;(\Delta^\ast\mathcal{F}^\cdot)^\vee). \end{align*}

Similarly, we have the isomorphism: \[ HH_\ast(X) = H^\ast(X;\Delta^\ast\mathcal{F}^\cdot). \]

Since the sheaf \( \Delta^\ast\mathcal{F}^\cdot\) computes the Hochschild (Co)Homology groups, we sometimes refer to \(\mathbf{L}\Delta^\ast\mathcal{O}_\Delta\) as the sheafy Hochschild complex.

There are two simple cases where these groups are relatively easy to compute from the definition: when X is a curve and when X is a projective space.

Let’s start by supposing we have a smooth projective curve C over k. Then the diagonal embedding is the inclusion of a Cartier divisor. We therefore have the exact sequence \[ 0\to\mathcal{O}_{C\times C}(-\Delta)\to \mathcal{O}_{C\times C}\to \Delta_\ast\mathcal{O}_C\to 0 \] where \(\mathcal{O}_{C\times C}(-\Delta)\) is just the ideal sheaf of \(\Delta(C)\). In particular, we have been gifted a free resolution of the diagonal \(\mathcal{O}_\Delta\). Since \(\Delta^\ast\mathcal{O}_{C\times C}(-\Delta)\cong \omega_C\), the restriction of this resolution to the diagonal is the complex \[ 0\to \omega_C\xrightarrow{0}\mathcal{O}_C\to 0\cong \omega_C[1]\oplus \mathcal{O}_C[0]. \]

We can now compute the Hochschild (Co)Homology groups. Let’s start with cohomology: \begin{align*} HH^\ast(C) &\cong \mathrm{Ext}^\ast(\omega_C[1]\oplus \mathcal{O}_C[0],\mathcal{O}_C) \\ &\cong H^\ast(C;\omega^\vee_C[-1]\oplus \mathcal{O}_C[0]) \\ \end{align*} and so \[ HH^\ast(C) = \begin{cases} H^0(\mathcal{O}_C) & \ast = 0 \\ H^0(\omega^\vee_C)\oplus H^1(\mathcal{O}_C) & \ast = 1 \\ H^1(\omega^\vee_C) & \ast = 2 \end{cases}. \]

As for Homology, we have \begin{align*} HH_\ast(C) &\cong H^\ast(C;\omega_C[1]\oplus \mathcal{O}_C) \\ &\cong H^\ast(\omega_C[1])\oplus H^\ast(\mathcal{O}_C) \end{align*} and so \[ HH_\ast(C) = \begin{cases} H^0(\omega_C) & \ast = -1 \\ H^0(\mathcal{O}_C)\oplus H^1(\omega_C) &\ast = 0 \\ H^1(\mathcal{O}_C) & \ast = 1 \end{cases}. \]

Let me take a moment to say that we have just verified the Hochschild-Kostant-Rosenberg isomorphism for a smooth projective scheme: \[ HH_n(X) = \bigoplus_{p-q=n}H^p(X;\Omega_X^q),\ HH^n(X) = \bigoplus_{p+q=n} H^p(X;\Lambda^q T_X) \] in the case X is a curve.

In the case X is the projective space \(\mathbb{P}^n\), we have a special free resolution of the diagonal given by Beilinson’s resolution: \[ 0\to\Omega^n(n)\boxtimes\mathcal{O}(-n)\to \cdots \to \Omega(1)\boxtimes\mathcal{O}(-1)\to \mathcal{O}_{\mathbb{P}^n\times\mathbb{P}^n} \] where the term in homological degree -i is \(\Omega^{i-1}(i-1)\boxtimes\mathcal{O}(1-i)\) and the notation \(\mathcal{E}\boxtimes\mathcal{F}\) means \(\pi_1^\ast\mathcal{E}\otimes\pi_2^\ast\mathcal{F}\). Upon restricting this resolution to the diagonal, we get a complex with vanishing differentials: \[ \bigoplus_{i=0}^n \Omega^i[i]. \]

The Hochschild Cohomology and Homology groups are now easily computed. The homology groups are: \[ HH_\ast(\mathbb{P}^n) = \bigoplus_{i=0}^n H^i(\Omega^i)[0]\cong \bigoplus_{i=0}^n k[0]. \] The cohomology groups are similar but with polyvector fields. We have now verified the HKR isomorphism in the case of projective spaces as well.