$\def\sO{\mathcal{O}} \def\d{\mathrm{d}}$In ringed spaces theory, there is a notion of “conormal sheaf of an immersion” (mainly used in scheme theory), whereas in smooth manifold theory, there is the notion of “conormal bundle of an embedded submanifold.” I am interesting in understanding the relation between these two, which is the lemma below. Before stating it, we need definitions.

A morphism of ringed spaces $i:Z\to X$ is said to be a immersion if it is a homeomorphism onto a locally closed subset of $X$ and $i^{-1}\mathcal{O}_Z\to\mathcal{O}_X$ is onto, and $i$ is said to be a closed immersion if additionally $i(Z)$ is closed in $X$. Given a smooth manifold $N$, denote $C_N^\infty$ to the sheaf of real-valued smooth functions on $N$. An embedding of smooth manifolds $S\to M$ is the same as an immersion of ringed spaces $(S,C^\infty_S)\to(M,C^\infty_M)$. Given an immersion of ringed spaces $f:Z\to X$, one can define the conormal sheaf of $Z$ in $X$, denoted $\mathcal{C}_{Z/X}$, in the following way: let $\mathcal{I}=\operatorname{Ker}(\mathcal{O}_X\to i_*\mathcal{O}_Z)$ be the ideal sheaf associated with the immersion. Then $\mathcal{C}_{Z/X}=i^{-1}(\mathcal{I}/\mathcal{I}^2)=i^*(\mathcal{I}/\mathcal{I}^2)=i^*\mathcal{I}$. (to see all these equalities one must use that $\mathcal{O}_Z\cong i^{-1}(\mathcal{O}_X/\mathcal{I})$.)

On the other hand, if $i:S\to M$ is an embedding of smooth manifolds, the conormal bundle of $S$ in $M$ is defined to be the subbundle of $T^*M|_S$: $$ N^* S=\left\{(q, \eta) \in T^* M: q \in S,\left.\eta\right|_{T_q S} \equiv 0\right\}. $$ (See [Lee, Exercise 22-13].) It's not difficult to see that we have a canonical s.e.s. of bundles over $S$ $$ 0\to TS\to TM|_S\to (N^*S)^*\to 0 $$ (this requires some work, but it's not what I am interesting on here). That is, $(N^*S)^*$ is canonically isomorphic to the normal bundle of $S$ in $M$, $NS:=TM|_S/TS$. This justifies the terminology of “conormal bundle.”

Given a vector bundle $E$ over $N$, denote $\Sigma_E$ to the sheaf of sections over $N$ (which is a $C^\infty_N$-module). This is the statement I am interested in:

Lemma. Let $i:S\to M$ be an embedding of smooth manifolds. Then $\mathcal{C}_{S/M}\cong \Sigma_{N^*S}$ as $C^\infty_S$-modules.

In the case $S=\{p\}$, the lemma reads $\mathcal{I}_p/\mathcal{I}_p^2\cong T_p^*M$, as real vector spaces (this last iso is [Lee, Exercise 11-4] and is discussed in MSE.)

My first question is:

1. Is the lemma to be found somewhere in the literature? Also, in differential geometry, where does the notion of the conormal bundle pop up?

(I don't really know much of differential geometry literature; all I know is what I learnt from Lee's book.)

My second question is about the proof I worked out of the lemma:

2. Is the following argument sound?

I didn't check all the details, but it looks fine to me.

Proof of the lemma

We will use the following proposition of [Lee]:

The ideal $\mathcal{I}\subset C^\infty_M$ is the ideal sheaf of functions that vanish on $S$. Define a morphism of sheaves $\mathcal{I}\to i_*\Sigma_{TM|_S}$ sending $f$ to $\d f|_S$. Leibniz's rule guarantees that this morphism is $C_M^\infty$-linear. By the proposition, the map $f\mapsto \d f|_S$ is actually a map $\mathcal{I}\to i_*\Sigma_{N^*S}$. On the other hand, Leibniz's rule for the differential implies that $\mathcal{I}\to i_*\Sigma_{N^*S}$ annihilates $\mathcal{I}^2$. Hence, we get a morphism $\mathcal{I}/\mathcal{I}^2\to i_*\Sigma_{N^*S}$. The inverse image-direct image adjunction gives a map $i^{-1}(\mathcal{I}/\mathcal{I}^2)\to \Sigma_{N^*S}$ of $C^\infty_S$-modules. We claim that the latter is an isomorphism.

We must see that $\mathcal{I}_p/\mathcal{I}_p^2\to\Sigma_{N^*S,p}$ is an isomorphism for all $p\in S$. First we see that it is onto. This is the same as seeing that $\mathcal{I}_p\to\Sigma_{N^*S,p}$ is onto. Pick local coordinates $x^1,\dots,x^m$ of $M$ around $p$ such that $x^1,\dots,x^s$ are local coordinates for $S$ (i.e., in this coordinates, locally, $S$ is the set $\{x^{s+1}=0,\dots,x^m=0\}$). We have that $(\d x^1,\dots,\d x^m)|_S$ is a local frame of $T^*M|_S$ and that $(\d x^{s+1},\dots,\d x^m)|_S$ is a local frame of $N^*S$. Hence, any section $\sigma$ of $N^*S$ around $p$ writes locally as $\sigma=g_{s+1}\d x^{s+1}+\dots+g_m\d x^m$. Defining $f=g_{s+1} x^{s+1}+\dots+g_m x^m$ gives a local section of $\mathcal{I}$ such that $\d f|_S=\sigma$. This shows surjectivity of $\mathcal{I}_p/\mathcal{I}_p^2\to\Sigma_{N^*S,p}$. To see injectivity, we must see that if a germ $f_p\in\mathcal{I}_p$ is sent to zero in $\Sigma_{N^*S,p}$ then it must be $f_p\in\mathcal{I}_p^2$. For this, invoking this version of Hadamard's lemma, we can write $f=\sum_{i=s+1}^m g_ix^i$ locally around $p$. Hence, if $\d f|_S=\sum_{i=s+1}^m g_i\d x^i$ vanishes, we get that the $g_i$'s vanish on $S$, whence $f_p\in\mathcal{I}_p^2$. $\square$

Bibliography

[Lee] John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer.