# Definition of cotangent and conormal bundle

I have read the following definition of cotangent bundle:

Let $X$ be a $n$-dimensional smooth algebraic variety. For any $p\in X$ there exist a neighbourhood $U_{p}\subseteq X$ and functions (named local parameters) $u_{1},\ldots,u_{n}\in \mathcal{O}_{X}(U_{p})$ such that $$\mathfrak{M}_{q}=\langle u_{1}-u_{1}(q),\ldots,u_{n}-u_{n}(q)\rangle$$ for every $q\in U_{p}$, where $\mathfrak{M}_{q}$ is the maximal ideal of the local ring $\mathcal{O}_{X,q}$. Associating to each open subset $U_{p}$ the module $\Omega_{S/R}$ of relative differential forms of $S=\langle u_{1},\ldots,u_{n}\rangle$ over $R=\mathcal{O}_{X}(U_{p})$, gives rise to a locally free coherent sheaf. We define the cotangent bundle to be the associated vector bundle $\Omega_{X}$.

Now, suppose we have a closed immersion $Y\subset X$. We can find an open cover $X=\bigcup_{i\in I}U_{i}$ such that on each $U_{i}$, the closed subscheme $Y\cap U_{i}$ can be described as the zero locus of $r:=\mathrm{codim}(Y,X)$ regular functions $f_{1},\ldots,f_{r}\in\mathcal{O}_{X}(U_{i})$. The conormal bundle $\mathcal{N}^{*}_{Y/X}$ of $Y$ in $X$ is defined to be the kernel of the surjective morphism $$\Omega_{X}|_{Y}\rightarrow \Omega_{Y}$$ induced by the restriction of differentials.

My problem is that I don't understand this morphism and therefore I don't understand the definition of conormal bundle. What does "restriction of differentials" mean?

Any help (or correction in case I was misunderstanding something) would be appreciated.

• Some of this seems off to me. (1) $S \subseteq R$, so it seems like you would want to form $\Omega_{R/S}$. It seems to me like this is going to be $0$. (2) Are you sure you don't want to look at differentials with respect to the ground field? (3) Just as in differential geometry, whenever you have a morphism $f\colon Y \to X$ over $S$ you get a pullback map on forms $f^*\Omega_{X/S} \to \Omega_{Y/S}$, sometimes denoted $df$. If $f$ corresponds locally to a ring map $\phi\colon A \to B$ this map is $b \otimes da \mapsto b \cdot d(\phi(a))$. – Hoot Jun 22 '16 at 10:55
• @Hoot I want the sections of $\Omega_{X}$ in $U_{p}$ to be of the form $g_{1}du_{1}+\cdots+g_{n}du_{n}$ with $g_{1},\ldots, g_{n}\in \mathcal{O}_{X}(U_{p})$. Does it make sense? – Srinivasa Granujan Jun 22 '16 at 13:40
• You can do that by picking étale coordinates, sure, but I don't think it's relevant for this question about restriction. – Hoot Jun 22 '16 at 19:44