This is probably a trivial question.


While reading the paper 


R. Elkik, Singularites rationnelles et deformations, *Invent. Math.* 47 Ž1978., 139147.


I came across the following short exact sequence. Consider a flat morphism $f:X\rightarrow S=Spec(R)$ of k-schemes of finite type, $X$ normal + CM, and pick a regular parameter $t\in R$. If $X\_t=X\times\_{S} Spec R/tR$ is the fiber over $t$, then multiplication by $t$ induces a short exact sequence $$0 \rightarrow \omega\_X \stackrel{t}{\rightarrow} \omega\_X \rightarrow \omega\_{X\_t} \rightarrow 0$$


Is it straightforward to derive such a sequence?


Thanks in advance for any suggestion.



Canonical sheaf of the fiber of a flat morphism
Sure.


Note first that we have a short exact sequence


$$
0 \to O\_X \xrightarrow{t} O\_X \to O\_{X\_t} \to 0
$$
If $D$ is the divisor corresponding to $t = 0$ on $X$, you can also view this as 
$$0 \to O\_X(-D) \to O\_X \to O\_D \to 0.$$


Anyways, now apply the functor $\mathcal{H}om\_{O\_X}(\bullet, \omega\_X)$. You get
$$
0 \to \mathcal{H}om\_{O\_X}(O\_{X,t}, \omega\_{X}) \to \mathcal{H}om\_{O\_X}(O\_X, \omega\_X) \xrightarrow{t} \mathcal{H}om\_{O\_X}(O\_X, \omega\_X) \to \mathcal{E}xt^1(O\_{X\_t}, \omega\_X) \to \mathcal{E}xt^1\_{O\_X}(O\_X, \omega\_X)
$$
Ok, the first term is zero (homing torsion into non-torsion), the last term is also zero since you are homing from a free module. Finally, $\mathcal{E}xt^1(O\_{X\_t}, \omega\_X) = \omega\_{X\_t}$ by basic properties of dualizing/canonical modules, and the result follows.


