Is there a page online that lists where exactly all the algebraic theorems in Hartshorne appear in Eisenbud? I know that Eisenbud uses the term "codimension" instead of "height", but just Flipping through the chapters on dimension theory, I'm not finding it very easy to locate the particular theorems referenced by Hartshorne.
I'm trying to find the proofs for the dimension theory in the first section, specifically:
- Theorem 1.8A (M, Ch. $5$, $\S14$)
- Theorem 1.11A (A-M, p. $122$)
- Theorem 1.12A (M, p. $141$)
Here "M" refers to Matsumura's Commutative Algebra and "A-M" to Atiyah and MacDonald's Introduction To Commutative Algebra, which are the references given in Hartshorne.
Part (a) is Theorem A, $\S13.1$, p. $286$ in Eisenbud:
Theorem A. If $R$ is an affine domain over a field $k$, then $$ \dim(R) = \operatorname{tr. deg.}_k(R), $$ and this number is the length of every maximal chain of primes in $R$.
The proof is given on p. 289.
Part (b) is Corollary 13.4, p. 286:
Corollary 13.4. If $R$ is an affine domain, and $I \subseteq R$ is an ideal, then $$ \dim(I) + \DeclareMathOperator{\codim}{codim} \codim(R) = \dim(R) \, . $$
Chapter 10, p. 232 of Eisenbud. Here $R$ is a (commutative) noetherian ring.
Theorem 10.1 (First version of the Principal Ideal Theorem). If $x \in R$, and $P$ is minimal among primes of $R$ containing $x$, then $\codim(P) \leq 1$.
Chapter 10, p. 234 of Eisenbud. $R$ is assumed noetherian.
Corollary 10.6. A domain $R$ is factorial if every codimension $1$ prime of $R$ is principal.