Bundles of Principal Parts

A short introduction to bundles of principal parts (a.k.a jet bundles).

Fei YE https://yfei.page (QCC-CUNY)https://qcc.cuny.edu

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 \(o\).

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 \(\mathcal{P}_{X/S}^0(\mathcal{F})\cong\mathcal{F}.\)

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 \[ \begin{aligned} d^k: \mathcal{O}_X &\to \mathcal{O}_{X\times_SX}/\mathcal{J}^{k+1}=\mathcal{P}_{X/S}^k\\ a &\mapsto \overline{1\otimes_{f^{-1}\mathcal{O}_S} a} = 1\otimes_{f^{-1}\mathcal{O}_S} a + \mathcal{J}^{k+1}. \end{aligned} \]

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.

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 \(r\cdot {\dim X+k \choose k}\).

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 \[ \begin{array}{ccccccccc} 0 & \longrightarrow & S^{k}\left(f^{*} \Omega_{Y}^{1}\right) \otimes f^{*} \mathcal{E} & \longrightarrow & f^{*} \mathcal{P}_{Y}^{k}(\mathcal{E}) &\longrightarrow & f^{*} \mathcal{P}_{Y}^{k-1}(\mathcal{E}) & \longrightarrow & 0\\ & & \downarrow & & \downarrow & & \downarrow & &\\ 0 & \longrightarrow & S^{k}\left( \Omega_{X}^{1}\right) \otimes f^{*} \mathcal{E} & \longrightarrow & \mathcal{P}_{X}^{k}(f^{*} \mathcal{E}) &\longrightarrow & \mathcal{P}_{X}^{k-1}(f^{*} \mathcal{E}) & \longrightarrow & 0. \end{array} \]

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.

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.


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