In 033Q we find defined what some sources call “Serre's conditions $\mathrm{S}_k$ and $\mathrm{R}_k$” (if you don't know what a scheme is, you can read the definition for a commutative Noetherian ring in 031P).

These conditions are used to characterize properties of Noetherian rings:

Lemma. Let $A$ be a commutative Noetherian ring. Then:

  1. $A$ is reduced $\Leftrightarrow$ $\mathrm{R}_0$, $\mathrm{S}_1$ hold. (031R)
  2. $A$ is normal $\Leftrightarrow$ $\mathrm{R}_1$, $\mathrm{S}_2$ hold. (031S)
  3. $A$ is Cohen-Macaulay $\Leftrightarrow$ $\mathrm{S}_k$ holds for all $k$. (0342)

Also, $\mathrm{S}_k$ can be defined for an $A$-module (see 031P), and then $\mathrm{S}_1$ has a characterization given in 031Q.

My questions are:

(1a) Who defined Serre's conditions the first time? (was Serre himself?)

(1b) Who chose the notation $\mathrm{S}_k$ and $\mathrm{R}_k$ for them? (It seems standard in all modern sources).

In EGA IV₂, conditions $\mathrm{S}_k$ and $\mathrm{R}_k$ are defined in §5.7 and §5.8, respectively. H. Matsumura also defines them in his Commutative Algebra (in (17.I)) and Commutative Ring Theory (in the remark before Theorem 23.8).

The second point of the lemma above is sometimes called “Serre's criterion for normality” (in EGA IV, this is (5.8.6), whereas in Matsumura's CA it's (17.I), Theorem 39, whose proof is “after EGA IV”). Right before stating Theorem 39, Matsumura says that Serre's criterion for normality is due to Krull(1931) in the case $A$ is a domain, and to Serre in the general case. However, there is no “Krull(1931)” in the book bibliography. For these reasons, my other questions are:

(2) Which paper is “Krull(1931)” that Matsumura quotes? A search for the author and year in google scholar shows three papers. However, after skimming through them I couldn't find the statement.

(3a) Where did Serre proved his criterion for normality?

(3b) Was Serre really the first one proving it? (For an arbitrary Noetherian ring.)



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