ag.algebraic geometry - finite surjective l.c.i morphism is flat

Let $X,Y$ be locally Noetherian schemes. Let $f:Xto Y$ be a finite, surjective, and locally complete intersection morphism, i.e., locally it can be decomposed as regular immersion followed by a smooth morphism.
Recall: an immersion $Xto Y$ is called a regular immersion at a point $x$ if $mathcal{O}_{X,x}$ is isomorphic as $mathcal{O}_{Y,y}$-module to $mathcal{O}_{Y,y}$ modulo an ideal $I$ generated by a regular sequence of elements of $mathcal{O}_{Y,y}$.

Question: prove that $f$ is flat. In particular, $f$ will be a simultaneously open and closed morphism.

• Next reference request - Fundamental groups of noncompact surfaces
• Previous nt.number theory - What primes divide the discriminant of a polynomial?