Conormal exact sequence Let $X$ be a smooth variety and $Y\subset X$ a smooth subvariety with ideal sheaf $\mathcal{I}_{Y/X}$.
For $n\geq 2$ is there an analogue of the exact sequence
$$0\rightarrow \mathcal{I}_{Y/X}/\mathcal{I}_{Y/X}^{2}\rightarrow\Omega_{X|Y}\rightarrow\Omega_Y\rightarrow 0$$
involving $\mathcal{I}_{Y/X}^n/\mathcal{I}_{Y/X}^{n+1}$?
 A: In general, when there is an exact sequence of vector bundles
$$0\to A\to B\to C\to 0,\label{1}\tag{1}$$
there is an associated long exact sequence
$$0\to S^kA\to S^{k-1}A\otimes B\to S^{k-2}A\otimes \wedge^2B\to \cdots\to \wedge^kB\to \wedge^k C\to 0\label{2}\tag{2}$$
for each $k\ge 1$, where $S^k$ is to take the $k$-th symmetric power. The map on the right is to take $\wedge^k$ of $B\to C$, while others are contraction of $A\to B$. See [Gr94, Lecture 4]. For example, when $k=3$, the second map from the left $$S^2A\otimes B\to A\otimes \wedge^2B$$ is give by $a_1\otimes a_2\otimes b+a_2\otimes a_1\otimes b\mapsto a_1\otimes (a_2\wedge b)+a_2\otimes(a_1\wedge b)$.
In particular, apply to the conormal exact sequence
$$0\to N_{Y|X}^*\to \Omega_{X|Y}\to \Omega_Y\to 0,$$
and use the fact that $S^k(N^*_{Y|X})=S^k(I_{Y}/I^2_{Y})=I_{Y}^k/I^{k+1}_{Y}$, one has the following exact sequence
$$0\to I_{Y}^k/I^{k+1}_{Y}\to I_{Y}^{k-1}/I^{k}_{Y}\otimes \Omega_{X|Y}\to I_{Y}^{k-2}/I^{k-1}_{Y}\otimes \wedge^2\Omega_{X|Y}\to \cdots\to \wedge^k\Omega_{X|Y}\to \wedge^k \Omega_Y\to 0.$$
Edited: As pointed out by (@Yai0Phah, there is another long exact sequence, called Koszul sequence, associated to \eqref{1} (which only need the flatness assumption of $A,B,C$):
$$0\to S^kA\to S^kB\to S^{k-1}B\otimes C\to \cdots \to B\otimes \wedge^{k-1}C\to \wedge^kC\to 0.$$
See the second exact sequence in [Ill71, I.1.4.3.1.7, p.110] and use the isomorphism of graded symmetric algebra to the graded divided power algebra. So you get another long exact sequence involving $I_{Y}^k/I^{k+1}_{Y}$ by applying to the conormal sequence.
[Gr94] Green, Mark L. Infinitesimal methods in Hodge theory. In Algebraic cycles and Hodge theory (Torino, 1993), Springer, Berlin, 1994.
[Il71] Illusie, Luc.
Complexe cotangent et déformations. I.
Lecture Notes in Mathematics, Vol. 239. Springer-Verlag, Berlin-New York, 1971. xv+355 pp.
