# Subschemes of the affine line over a domain

Let $$R$$ be a domain with affine spectrum $$S$$ and consider the scheme $$X=\mathbb A^1_R=\operatorname {Spec}R[T]$$ over $$S$$.
Let $$P\subset R[T]$$ be an ideal with $$P\cap R=0$$ and let $$Y\subset X$$ be the associated subscheme. The extension of $$P$$ to $$(\operatorname {Quot}(R))[T])$$ is principal, generated by some polynomial $$f$$ which we may assume has coefficients in $$R[T]$$. What is the relationship between $$Y$$, the subscheme $$V(f)\subset X$$ and (probably) inverse images of subschemes in $$S$$ ?
(I have asked this question three days ago on math stackexchange but got no answer nor comment)

Edit: I'm also interested in knowing if one can say more under the assumption that $$P$$ is prime.

• $V(f)$ is the union of $Y$ and a subscheme of $X$ which is the pull back of some subscheme from $S$. May 12, 2020 at 17:34
• @Mohan: Well of course, that's the obvious guess, as I hinted in my question. I'm asking for a detailed, rigorous proof. May 12, 2020 at 17:50
• Are you assuming $R$ is Noetherian? I was. May 12, 2020 at 17:52
• No, I was not assuming $R$ noetherian. But I would be happy to read a proof of your claim assuming noetherianness. May 12, 2020 at 20:01
• Then, use Krull's principal ideal theroem and look at the prime decomposition of $(f)$. Set theoretically this is the union of $Y$ and the pull back of a closed subset of $S$, since all minimal primes containing $f$ have height one. May 12, 2020 at 20:47