# Bundles of Principal Parts

Algebraic Geometry

Definitions and properties of bundles of principal parts (also known as jet bundles) and sheaves of differential operators will be studied in this post.

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August 25, 2019

## 1 The Intuition

Jet bundles are also known as bundles of principal part (more generally, modules of principal parts).

Roughly speaking, sections of a jet bundle, called jets, are operators that send sections to Taylor polynomials. For example, for a function $$f\in\mathcal{O}_{\mathbb{A}^1}$$, the second jet of $$f$$ at the origin $$o\in \mathbb{A}^1$$ is determined by $J_o^2 f= f(o)+Df(o)(x-o)+\frac{1}{2!}D^2f(o)(x-o)^2,$ where $$D$$ the is the differential operator $$\frac{\partial}{\partial x}$$.

Consider a function $$f\in\mathcal{O}_{\mathbb{A}^n}$$. For a vector $$\alpha=(\alpha_1,\dots, \alpha_m)$$ with $$m\leq n$$ and $$\alpha_i\in\mathbb{N}$$, we write $$|\alpha|=\alpha_1+\cdots+\alpha_m,$$ $$\alpha!=\alpha_1!\cdots\alpha_m!,$$ $$x^\alpha=x_1^{\alpha_1}\cdots x_m^{\alpha_m},$$ $D_\alpha(f)=\frac{1}{\alpha!}\frac{\partial^\alpha f}{\partial x^\alpha}:=\frac{1}{\alpha!}\frac{\partial^{|\alpha|} f}{(\partial x_1)^{\alpha_1}\cdots (\partial x_1)^{\alpha_1} },$ and $$\mathrm{d}x^\alpha=(\mathrm{d}x_1)^{\alpha_1}\cdots(\mathrm{d}x_m)^{\alpha_m}$$. Then the $$k$$-th jet of $$f$$ at the origin $$o$$ is $J_o^kf=\sum_{|\alpha|\leq k}D_{\alpha}(f)\mathrm{d}x^\alpha.$

On a smooth variety $$X$$, a jet $$J_o^k f$$ can be considered as an element in $$\mathcal{O}_{o,X}/\mathfrak{m}_o^{k+1}$$. The space of the image of $$J^k_o$$ is called the $$k$$-th jet space at $$o$$.

## 2 Definition of Sheaves of Principal Parts

Let $$f: X\to S$$ be a morphism between schemes $$S$$. Denote by $$\pi_1$$ and $$\pi_2$$ the projection of $$X\times_S X$$ on the first and second factor respectively. Let $$\Delta^{(k)}$$ be the $$k$$-th infinitesimal neighborhood of the diagonal, that is, $$\mathcal{O}_{\Delta^{(k)}}=\mathcal{O}_{X\times_S X}/\mathcal{J}^{k+1}$$, where $$\mathcal{J}$$ is the ideal sheaf of the diagonal $$\Delta\subset X\times_S X$$. Denote by $$\delta^{(k)}: \Delta^{(k)}\to X\times_SX$$ the immersion morphism. Set $$\delta=\delta^{(0)}$$, $$p=\delta^{(k)}\circ\pi_1$$ and $$q=\delta^{(k)}\circ\pi_2$$.

Definition 1 Given a $$\mathcal{O}_X$$-module $$\mathcal{F}$$ on $$X$$, the sheaf of $$k$$-th order principal part (or the $$n$$-jets of sections) of $$\mathcal{F}$$ over $$S$$ is defined as $\mathcal{P}_{X/S}^k(\mathcal{F}):=p_*(q^*\mathcal{F})={\pi_1}_*(\mathcal{O}_{X\times_SX}/\mathcal{J}^{k+1}\otimes_{\mathcal{O}_{X\times_SX}} \pi_2^*\mathcal{F}).$ In particular, $$\mathcal{P}_{X/S}^k:=\mathcal{P}_{X/S}^k(\mathcal{O}_X)={\pi_1}_*(\mathcal{O}_{\Delta^{(k)}})$$.

From the identity maps $$\pi_i\circ \delta: X \overset{\delta}{\to} \Delta\overset{\pi_i}{\to} X$$, we see that $$\mathcal{P}_{X/S}^0\cong\mathcal{O}_X$$, and $$\mathcal{P}_{X/S}^0(\mathcal{F})\cong\mathcal{F}.$$

## 3 Bimodule Structure on the Sheaf of Principal Parts

As the direct image of a $$\mathcal{O}_{X\times_S X}$$-module, the sheaf $$\mathcal{P}_{X/S}^k$$ has a natural left $$\mathcal{O}_X$$-module structure defined by the canonical map $c^k: \mathcal{O}_X\to \mathcal{P}_{X/S}^k={\pi_1}_*(\mathcal{O}_{\Delta^{(k)}}).$

Because $$\pi_1=\pi_2\circ s$$, $$\pi_2=\pi_1\circ s$$, and $$s\circ\delta =\delta$$, where $$s=\left(\pi_{2}, \pi_{1}\right)_S:X\times_S X \to X \times_S X$$ is the involutional automorphism, known as the canonical symmetry, the sheaf $$\mathcal{P}_{X/S}^k$$ also admits a right module structure derived from $$\pi_2$$ as follows $d^k: \mathcal{O}_X\overset{(\pi_2)_*}{\to} (\pi_2)_*\mathcal{O}_{\Delta^{(k)}}\overset{s_*}{\to} \mathcal{P}_{X/S}^k.$

Note that the $$s^*$$ is an identity map on $$\Delta$$ as $$s\circ s=\mathrm{id}_{X\times_SX}$$ and $$\delta\circ s=\delta$$.

There is an alternative way to see the bimodule structure of $$\mathcal{P}_{X/S}^k$$.

Using $$\pi_i\circ \delta \cong \mathrm{id}_X$$, we may identify $\mathcal{O}_{X\otimes_S X}=\mathcal{O}_X\otimes_{f^{-1}\mathcal{O}_S}\mathcal{O}_X$ and $$\mathcal{J}=\ker{m}$$, where $m: \mathcal{O}_X\otimes_{f^{-1}\mathcal{O}_S}\mathcal{O}_X \to \mathcal{O}_X$ is the natural multiplication map.

By slight abuse of notation, the left module structure is given by \begin{aligned} c^k: \mathcal{O}_X &\to \mathcal{O}_{X\times_SX}/\mathcal{J}^{k+1}=\mathcal{P}_{X/S}^k\\ a &\mapsto \overline{a\otimes_{f^{-1}\mathcal{O}_S} 1} = a\otimes_{f^{-1}\mathcal{O}_S} 1 + \mathcal{J}^{k+1} \end{aligned} and the right module structure is given by \begin{aligned} d^k: \mathcal{O}_X &\to \mathcal{O}_{X\times_SX}/\mathcal{J}^{k+1}=\mathcal{P}_{X/S}^k\\ a &\mapsto \overline{1\otimes_{f^{-1}\mathcal{O}_S} a} = 1\otimes_{f^{-1}\mathcal{O}_S} a + \mathcal{J}^{k+1}. \end{aligned}

## 4 Sheaves of Principal Parts as Tensor Products

Since the sheaf $$\mathcal{P}_{X/S}^k(\mathcal{F})$$ has a support contained in $$\Delta^{(k)}$$, using the canonical symmetry $$s$$, we may identify $\mathcal{P}_{X/S}^k(\mathcal{F})\cong \mathcal{P}_{X/S}^k \otimes_{\mathcal{O}_{X}}\mathcal{F},$ where in order to tensor over $$\mathcal{O}_X$$, $$\mathcal{P}_{X/S}^k$$ is considered as a right $$\mathcal{O}_X$$-module which is given by $$d^k:\mathcal{O}_X\to\mathcal{P}_{X/S}^k$$.

The bimodule structure on $$\mathcal{P}_{X/S}^k$$ induces a bimodule structure on $$\mathcal{P}_{X/S}^k(\mathcal{F})$$ via the tensor product. Locally, for $$a\in\mathcal{O}_X(U)$$, $$b\in\mathcal{P}_{X/S}^k(U)$$, and $$t\in\mathcal{F}(U)$$, the bimodule structure can be described by $a\cdot (b\otimes t) = (ab)\otimes t$ and $(b\otimes t)\cdot a =(b\cdot a)\otimes t = b\otimes (at) = (b d^k(a))\otimes t = (d^k(a)\cdot b)\otimes t.$

Using the tensor product interpretation of $$\mathcal{P}_{X/S}^k(\mathcal{F})$$, we define a map \begin{aligned} d_{\mathcal{F}}^k: \mathcal{F} & \to \mathcal{P}_{X/S}^k(\mathcal{F}) \cong \mathcal{P}_{X/S}^k \otimes_{\mathcal{O}_{X}}\mathcal{F} \\ t & \mapsto 1\otimes_{\mathcal{O}_{X}} t = \overline{1 \otimes_{f^{-1}\mathcal{O}_S} 1} \otimes_{\mathcal{O}_X} t \end{aligned} which is a homomorphism of sheaves of abelian groups. In general, if $$d^k_{\mathcal{F}}$$ is not an $$\mathcal{O}_X$$-module homomorphism unless the sheaf $$\mathcal{P}_{X/S}^k$$ is considered as a right $$\mathcal{O}_X$$-module. Indeed, we have $d^k_{\mathcal{F}}(ta) = 1\otimes (ta) = 1\otimes (at) = (1\otimes t)\cdot a = d^k_{\mathcal{F}}(t)\cdot a$ and $d^k_{\mathcal{F}}(at)=(1\cdot a)\otimes t=(d^k(a))\otimes t=d^k(a)\cdot(1\otimes t).$

Here, we note that $$d^k_{\mathcal{F}}$$ is $$\mathcal{O}_X$$-linear with respect to the right module structure on $$\mathcal{P}_{X/S}^k$$.

It’s clear that $$d_{\mathcal{F}}^k$$ lifts any global section $$g$$ of $$\mathcal{F}$$ to a global section $$d_{\mathcal{F}}^k\circ g$$ of $$\mathcal{P}_{X/S}^k(\mathcal{F})$$, where we identify a global section $$g$$ as a homomorphism $$g: \mathcal{O}_X\to \mathcal{F}$$.

For $$\mathcal{F} = L$$, a line bundle on $$X$$, over a local neighborhood $$U$$ of a point $$o$$ with local coordinates $$(x_1, x_2, \cdots, x_n)$$, the homomorphism $$d^k_L$$ sends a section $$t(x_1,x_2,\cdots, x_n)e$$ of $$L$$ to its truncated Taylor series $t(x_1,x_2,\cdots, x_n)e\mapsto \sum_{|\alpha|\leq k}\left( \frac{1}{\alpha !} \frac{\partial^\alpha t}{\partial x^\alpha}\mathrm{d}x^\alpha\otimes e\right),$ where $$\{e\}$$ is the basis (the local frame) of the line bundle $$L$$ over $$U$$ and $$\{\mathrm{d}x^\alpha\otimes e\mid |\alpha|\leq k\}$$ forms a basis for $$\mathcal{P}_{X/S}^k(L)$$.

The above description can be checked using the fundamental exact sequence in the next section.

## 5 Fundamental Exact Sequence

Suppose that $$X$$ is smooth over $$S$$ (more generally, $$\mathcal{J}$$ is locally generated by a regular sequences ) and $$\mathcal{F}$$ be a locally free sheaf on $$X$$. Then $Sym^k\Omega_X^1=Sym^k({p}_*(\mathcal{J}/\mathcal{J}^2))={p}_*(Sym^k(\mathcal{J}/\mathcal{J}^2))={p}_*(\mathcal{J}^k/\mathcal{J}^{k+1})$ and $$R^1{\pi_1}_*(\mathcal{J}^k/\mathcal{J}^{k+1})=0$$ because, $$\pi_1: \Delta\to X$$ is an isomorphism.

Proposition 1 The sheaves of principal parts on $$\mathcal{F}$$ on a smooth variety $$X$$ over $$S$$ fit in the following exact sequence $0\to Sym^k\Omega_X^1\otimes\mathcal{F}\to \mathcal{P}_{X/S}^k(\mathcal{F})\overset{\phi^k_{\mathcal{F}}}{\to} \mathcal{P}_{X/S}^{k-1}(\mathcal{F})\to 0$ is exact for each $$k\ge 1$$.

Proof. The proposition follows by applying $${\pi_1}_*(\cdot\otimes\pi_2^*\mathcal{F})$$ to the exact sequence $0\to \mathcal{J}^k/\mathcal{J}^{k+1}\to \mathcal{O}_{X\times_S X}/\mathcal{J}^{k+1}\to \mathcal{O}_{X\times_S X}/\mathcal{J}^k\to 0$ and identifying $$\mathcal{J}^k/\mathcal{J}^{k+1}\otimes\pi_1^*\mathcal{F}=\mathcal{J}^k/\mathcal{J}^{k+1}\otimes\pi_2^*\mathcal{F}$$ and $${\pi_1}_*(\mathcal{J}^k/\mathcal{J}^{k+1})=\mathcal{J}^k/\mathcal{J}^{k+1}$$ using the canonical symmetry $$s$$ and the fact that $$\mathcal{J}^k/\mathcal{J}^{k+1}$$ is supported on the diagonal $$\Delta$$. The proof is completed.

Note that $$d^{k-1}_{\mathcal{F}}=\phi^k_{\mathcal{F}}\circ d^{k-1}_{\mathcal{F}}: \mathcal{F}\to \mathcal{P}_{X/S}^{k-1}(\mathcal{F})$$.

By induction using the fundamental exact sequences, we obtain the following result.

Proposition 2 Suppose that $$X$$ is smooth over $$S$$ and $$\mathcal{F}$$ is locally free of rank $$r$$ on $$X$$. Then the $$\mathcal{O}_X$$-module $$\mathcal{P}_{X/S}^k(\mathcal{F})$$ is locally free of rank $$r\cdot {\dim X+k \choose k}$$.

## 6 Functorial Properties

The functor $$\mathcal{P}_{X/S}^k(\cdot)$$ is a covariant function from the category of $$\mathcal{O}_X-\mathrm{Mod}$$ to itself.

Let $$f: X\to Y$$ be a morphism of schemes over $$S$$. Then there is a morphism $f^*\mathcal{P}_{X/S}^k(\mathcal{F})\to \mathcal{P}_{X/S}^k(f^*\mathcal{F}).$

Proposition 3 Let $$f: X\to Y$$ be a morphism between smooth $$S$$-schemes and $$\mathcal{F}$$ a locally free sheaf on $$Y$$. Then the pullback map induces commutative diagram of exact sequences of left $$\mathcal{O}_X$$-modules $\begin{array}{ccccccccc} 0 & \longrightarrow & S^{k}\left(f^{*} \Omega_{Y}^{1}\right) \otimes f^{*} \mathcal{E} & \longrightarrow & f^{*} \mathcal{P}_{Y}^{k}(\mathcal{E}) &\longrightarrow & f^{*} \mathcal{P}_{Y}^{k-1}(\mathcal{E}) & \longrightarrow & 0\\ & & \downarrow & & \downarrow & & \downarrow & &\\ 0 & \longrightarrow & S^{k}\left( \Omega_{X}^{1}\right) \otimes f^{*} \mathcal{E} & \longrightarrow & \mathcal{P}_{X}^{k}(f^{*} \mathcal{E}) &\longrightarrow & \mathcal{P}_{X}^{k-1}(f^{*} \mathcal{E}) & \longrightarrow & 0. \end{array}$

## 7 Sheaves of Differential Operators

Definition 2 (EGA IV Definition (16.8.1)) A homomorphism $$D: \mathcal{F}\to \mathcal{G}$$ of sheaves of abelian groups is called a differential operator of order $$\leq k$$ if there exists a unique $$\mathcal{O}_X$$-module homomorphism $$u: \mathcal{P}_{X/S}^k(\mathcal{F})\to \mathcal{G}$$ such that $$D=u\circ d^k_{\mathcal{F}}$$, where the left $$\mathcal{O}_X$$-module structure on $$\mathcal{P}_{X/S}^k(\mathcal{F})$$ is taken for $$u$$.

Since the left and right module structures in general don’t agree on $$\mathcal{P}_{X/S}^k(\mathcal{F})$$, a differential operator of order $$\leq k$$ is not a $$\mathcal{O}_X$$-module homomorphism in general, but it is always a $$f^*\mathcal{O}_S$$-module homomorphism.

It is clear that the set of differential operators of order $$\le k$$ from $$\mathcal{F}$$ to $$\mathcal{G}$$ is an abelian group, denoted by $$\mathrm{Diff}_{X/S}^k(\mathcal{F}, \mathcal{G})$$.

Let $$D:\mathcal{F}\to \mathcal{G}$$ be a differential operator of order $$\le k$$. For any open set $$U$$, it’s clear that $$D|_U:\mathcal{F}|_U\to\mathcal{G}|_U$$ is a differential operator of order $$\le k$$. We define a sheaf of abelian groups $$\sDiff_{X/S}^k(\mathcal{F},\mathcal{G})$$ by $\sDiff_{X/S}^k(\mathcal{F},\mathcal{G})(U)= \sDiff_{U/S}^k(\mathcal{F}|_U,\mathcal{G}|_U)$

Proposition 4 (EGA IV Proposition (16.8.4)) There is a natural isomorphism of sheaves of abelian groups $\sHom_{\mathcal{O}_X}(\mathcal{P}_{X/S}^k(\mathcal{F}), \mathcal{G}) \cong \sDiff_{X/S}^k(\mathcal{F},\mathcal{G})$

Since $$\mathcal{P}_{X/S}^k(\mathcal{F})$$ has a bimodule structure, the sheaf $$\sDiff_{X/S}^k(\mathcal{F},\mathcal{G})$$ inherits a bimodule structure which are given locally as follows.

• The left module structure: $$(a\cdot D)(t)=a(D(t))$$, where $$a\in \mathcal{O}_X(U)$$ and $$t\in \mathcal{F}(U)$$.

• The right module structure: $$(D\cdot a)(t)=D(at)$$, where $$a\in \mathcal{O}_X(U)$$ and $$t\in \mathcal{F}(U)$$.

Proposition 5 (EGA IV Proposition (16.8.8)) Let $$\mathcal{F}$$ and $$\mathcal{G}$$ be $$\mathcal{O}_X$$-modules and $$D: \mathcal{F}\to \mathcal{G}$$ be a homeomorphism of $$f^*\mathcal{O}_S$$-modules. For a nonnegative integer $$k$$, the following are equivalent

1. the homomorphism $$D$$ is a differential operator of order $$\le k$$

2. for any section of $$\mathcal{O}_X$$ over an open set $$U$$, the homomorphism $$D_a=a\cdot D|_U-D|_U\cdot a: \mathcal{F}|_U\to \mathcal{G}|_U$$ is a differential operator of order $$\leq k-1$$.

We call the sheaf $\mathcal{D}_{X/S}^k(\mathcal{F})=\sHom_{\mathcal{O}_X}(\mathcal{P}_{X/S}^k(\mathcal{F}), \mathcal{F})\cong (\mathcal{P}_{X/S}^k(\mathcal{F}))^\vee\otimes_{\mathcal{O}_X}\mathcal{F}$ the sheaf of differential operators of order $$\le k$$ on $$\mathcal{F}$$. In particular, we write $\mathcal{D}_{X/S}^k=\mathcal{D}_{X/S}^k(\mathcal{O}_X)=\left(\mathcal{P}_{X/S}^k\right)^\vee.$

We note that $$\mathcal{D}_{\mathbb{A}^n/S}^k$$ is generated by the differential operators of order $$\le k$$ $D_\alpha=\frac{1}{\alpha!}\frac{\partial^\alpha }{\partial x^\alpha}, \qquad |\alpha|\le k$ (see the section The Intuition Introduction for notations).

Since $$\mathcal{P}_{X/S}^0(\mathcal{F})=\mathcal{F}$$, we see that $\mathcal{D}_{X/S}^0(\mathcal{F})\cong\sHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{F})\cong \mathcal{F}^\vee\otimes_{\mathcal{O}_X}\mathcal{F}.$

In particular, if $$\mathcal{F}=L$$ is a line bundle, then $\mathcal{D}_{X/S}^0(L)\cong\mathcal{O}_X.$

When $$\mathcal{F}$$ is locally free, we also have a short exact sequence for $$\mathcal{D}_{X/S}^k(\mathcal{F})$$ which is obtained by dualizing of the fundamental exact sequence for $$\mathcal{P}_{X/S}^k(\mathcal{F})$$.

For a line bundle $$L$$ on a smooth variety $$X$$ over $$S$$, the sheaf $$\mathcal{D}_{X/S}^k(L)$$ is locally free of rank $${\dim X+k\choose k}$$ and fits in the exact sequence $0\to \mathcal{D}_{X/S}^{k-1}(L)\to \mathcal{D}_{X/S}^k(L)\to Sym^k\mathcal{T}_X\to 0,$ where $$\mathcal{T}_X$$ is the tangent bundle of $$X$$.

Note that a $$\mathcal{O}_X$$-module homomorphism $$\psi: \mathcal{O}_X\to \mathcal{F}$$ determines an unique global section $$h=\psi(1)\in\Gamma(X, \mathcal{F})$$ and vise verse ($$\psi_h: \mathcal{O}_X\to \mathcal{F}$$ given by $$\psi_h(f)=fh$$.)

Given a global section $$g:\mathcal{O}_X\to L$$, we get a homomorphism of bundles \begin{aligned} j^k_{L}(g): \mathcal{D}_{X/S}^k(L)\cong\sHom_{\mathcal{O}_X}(\mathcal{P}_{X/S}^k(\mathcal{L}), L) &\to L\cong \sHom_{\mathcal{O}_X}(\mathcal{O}_X, L)\\ D=d_L^k\circ u_D &\mapsto D(g)=u_D\circ d_L^k\circ g(1). \end{aligned} which can be considered as the transpose of $$d^k_{L}(g)$$.

Locally, $$j^k_{L}(g)$$ is just the map which takes a differential operator $$D$$ of order $$\le k$$ to the function $$D(g)$$. It follows that $$j^k_{L}(g)$$ is zero precisely at the locus where $$g$$ vanishes to order greater than $$k$$.

Remark. Note that one may also define the $$k$$–th order differential operator. A 0th-order differential operator is defined to be 0. The $$k$$–th order differential operator (locally) $$D^k: M\to N$$ is a $$R$$–linear morphism such that for any $$g\in R$$,the map $$m\mapsto D(gm)-gD(m)$$ is a $$k-1$$–st differential operator, where $$M$$ and $$N$$ are $$R$$–modules. For more details, see for example The Stacks project, Section 09CH.