every irreducible component of $Y \cap H$ has dimension $r-1$

Question

I am trying to show that "...every irreducible component of $Y \cap H$ has dimension $r-1$...", where the complete problem comes in the image, I have several doubts about the proof of this theorem

(I took this from http://sertoz.bilkent.edu.tr/courses/math591/2016/solutions.pdf , pag 1).

I know that this question is already posted on this site but my question is different.

(1) why $\bar f=\bar f_1 \dots \bar f_s$? Is this because $B=k[x_1,..., x_n]/p$ is a unique factorization domain?

(2) Why $k[x_1,..., x_n]/[(f_i)+p]$ is isomorphic to $B/(\bar f_i)$?

Thank you.

Answer 1

For 1), no $B$ may not be a UFD. In a Noetherian domain, every element can be written as a finite product of irreducible elements (irreducible does not mean prime, which is what will happen in a UFD).

For 2), What is your confusion? Going modulo $(f_i)+p$ is same as first going modulo $p$ (which gets you $B$) and then going modulo the image of $f_i$ which is $\overline{f_i}$ in $B$.