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Bundles of Principal Parts

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

Author

Affiliation

Fei Ye

QCC-CUNY

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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.