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I was reading a solution of Exercise Ch I. 2.11 b) from Hartshorne's book that uses the following fact: if $S:=K[x_{0},\ldots,x_{n}]$ and $f_{1},\ldots,f_{m}\in S$ are linear polynomials then $\dim S/\langle f_{1},\ldots,f_{m}\rangle=n+1-m$. The proof begins with

"Since $S/\langle f_{1},\ldots,f_{m}\rangle\approx (S/\langle f_{1},\ldots,f_{m-1}\rangle) \, / \, \overline{\langle f_{m}\rangle}$ it suffices to prove the case when there is a single $f_{i}$, say $f$."

From this one constructs a homomorphism $S\to K[x_{1},\ldots,x_{n}]$ using $f$ in order to conclude that $S/\langle f\rangle \approx K[x_{1},\ldots,x_{n}]$.

I have some questions:

  1. Shouldn't it be $S/\langle f_{1},\ldots,f_{m}\rangle\approx (S/\langle f_{1},\ldots,f_{m-1}\rangle) \, / \, (\langle f_{1},\ldots,f_{m}\rangle/\langle f_{1},\ldots,f_{m-1}\rangle)= S/\langle f_{1},\ldots,f_{m-1}\rangle/\langle \, \overline{f_{m}} \, \rangle$ ?

  2. From this isomorphism, how does one conclude the induction step? I guess it would be an application of Theorem Ch I. 1.8A b) from Hartshorne's book, namely, $\mathrm{ht(p)}+\dim B/p=\dim B$ for a prime ideal $p$ and an integral domain which is a f.g. $k$-algebra. But do such hypotheses actually hold?

  3. Could this inductive process work to show that $\langle f_{1},\ldots,f_{m}\rangle$ is prime whenever the $f_{i}$'s are linear polynomials. If so, how? I see no reason as to why $\langle \, \overline{f_{m}} \, \rangle$ should be prime.

dwhydtea
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  • I think the result, as you've stated it, is false. I think you need to add the Rouché-Capelli-like condition that $\dim_K\operatorname{Span}{f_1-f_1(0),\cdots, f_m-f_m(0)}=\dim_K\operatorname{Span}{f_1,\cdots, f_m}$, call this number $u$, and then it's $n+1-u$ instead of $n+1-m$. – Sassatelli Giulio Dec 16 '24 at 06:45
  • This is kind of a suboptimal way to do this - you can do it all in one shot by a linear change of coordinates so that $f_i$ becomes $x_{n+1-i}$ and then all of these conclusions are trivial. See for example here. Does that solve your problem, or are you determined to answer these specific questions? (Also, your first numbered question looks like it's missing some parentheses and seems to be exactly what's written in the proof you're asking about.) – KReiser Dec 16 '24 at 07:21

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