26
$\begingroup$

Reading "H. Matsumura - Commutative Ring theory" I had the impression that the definitions were all made to mean something in algebraic geometry afterwards. I wonder what was commutative algebra before the modern algebraic geometry (which ideas were developed, how sophisticated they were, etc). Moreover, whether the success of algebraic geometry had inspired the expansion of commutative algebra, and how much. Any older reference would also be interesting (not for studying, just to figure out). Thanks

$\endgroup$
  • 4
    $\begingroup$ If I'm not wrong, most, if not all the material in Matsumura's book was already motivated by pre-Grothendieck algebraic geometry. $\endgroup$ – YCor Nov 9 '16 at 3:53
  • 3
    $\begingroup$ I often wonder this myself! I like to think it would be incredibly useful for students to have some list that takes the definitions and theorems in a standard textbook (say, Matsumura) and indicates whether they were originally motivated by purely algebraic concerns, or more geometric considerations. That way one could abstract and sort 'how much' of either subject (CA vs. AG) was 'belongs' to that subject $\endgroup$ – Samantha Y Nov 9 '16 at 5:08
  • $\begingroup$ The short introduction of Matsumura's book ( math.hawaii.edu/~pavel/cmi/References/… ) includes historical context. $\endgroup$ – YCor Nov 9 '16 at 5:59
  • $\begingroup$ @João, I corrected your language a bit, I hope that you do not oppose to it. By the way, am I right that "how developed it was" should mean "how sophisticated/advanced it was", not "how it was developed"? $\endgroup$ – evgeny Nov 9 '16 at 9:48
  • $\begingroup$ The oldest reference is the first edition of "Modern Algebra" by B.L. van der Waerden. It is partially based on lectures by E. Artin and E. Noether. No references are mentioned. Here we are far from any "Modern Algebraic Geometry", at least explicitly ! $\endgroup$ – Al-Amrani Nov 9 '16 at 16:47
26
$\begingroup$

here is an excerpt from Zariski's talk at the Icm several decades ago:

"The arithmetic trend in algebraic geometry is not in itself a radical departure from the past. This trend goes back to Dedekind and Weber who have developed, in their classical memoir, an arithmetic theory of fields of algebraic functions of one variable. Abstract algebraic geometry is a direct continuation of the work of Dedekind and Weber, except that our chief object is the study of fields of algebraic functions of more than one variable. The work of Dedekind and Weber has been greatly facilitated by the previous development of classical ideal theory. Similarly, modern algebraic geometry has become a reality partly because of the previous development of the general theory of ideals. But here the similarity ends. Classical ideal theory strikes at the very core of the theory of algebraic functions of one variable, and there is in fact a striking parallelism between this theory and the theory of algebraic numbers. On the other hand, the general theory of ideals strikes at most of the foundations of algebraic geometry and falls short of the deeper problems which we face in the post- foundational stage. Furthermore, there is nothing in modern commutative algebra that can be regarded even remotely as a development parallel to the theory of algebraic function fields of more than one variable. This theory is after all itself a chapter of algebra, but it is a chapter about which modern algebraists knew very little. All our knowledge here comes from geometry. For all these reasons, it is undeniably true that the arithmetization of algebraic geometry represents a substantial advance of algebra itself. In helping geometry, modern algebra is helping itself above all. We maintain that abstract algebraic geometry is one of the best things that happened to commutative algebra in a long time."

$\endgroup$
  • 10
    $\begingroup$ This is from Zariski's 1950 ICM address entitled 'The Fundamental Ideas of Abstract Algebraic Geometry' and found, for instance, here: mathunion.org/ICM/ICM1950.2/Main/icm1950.2.0077.0089.ocr.pdf $\endgroup$ – Samantha Y Nov 9 '16 at 5:10
  • 1
    $\begingroup$ A variant of your answer could be: compare the books by Matsumura and Zariski-Samuel. One thing that happened inbetween is the development of cohomological methods. $\endgroup$ – ACL Nov 9 '16 at 14:33
  • $\begingroup$ @ACL, probably you know that Zariski Samuel discuss homological codimension and cohomological dimension (due essentially to Hilbert) via syzygies in their volume 2, chVII, and use the latter concept to prove that regular local rings are ufd's following Auslander and Buchsbaum, in appendix 7. I.e. I would suggest that basic cohomological methods and deep applications may be said to have been already included there, but I think your point has validity still in the subsequent ascendancy of those methods. $\endgroup$ – roy smith Nov 10 '16 at 23:56
  • $\begingroup$ Actually, I had just browsed at the table of contents of volume 2, and had not realized that. Thank you! $\endgroup$ – ACL Nov 11 '16 at 0:40

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.