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Grothendieck is arguably the most brilliant mathematician of the 20th century, with his influence felt the most in algebraic geometry, which he transformed. Some time ago the story used to be told was that Grothendieck put it on a rigorous footing after the Italian School starting in the 19-th century (Castelnuovo) gradually neglected rigor (Enriques), and ended up with results which no one could tell were true or not in 1940-50s (Severi). The actual story is much more nuanced, but more to the point I eventually learned, much to my surprise, that Zariski, who was originally part of the Italian School, left it and published a classical book, which was already up to the modern standards of rigor, back in 1935. There is an insightful discussion of the Italian School and its demise in this MO thread , where one of the commenters quipped: "Grothendieck didn't make algebraic geometry rigorous. Weil and Zariski made algebraic geometry rigorous".

I know of Grothendieck's technical innovations (like schemes, motives and topoi), and his achievements (like proving the Weil conjectures), but it is hard to put them into context. What was his role "structurally"? To give an analogy, in set theory we have Cantor exploring a new territory, but running into logical troubles (roughly like the Italian School), and then Zermelo making the set theory rigorous (roughly like Weil and Zariski). The next towering figure is perhaps Gödel, and he transformed set theory by going to a "meta" level, because some deeper problems, like continuum hypothesis or the status of axiom of choice, could not be resolved in Zermelo's framework. Similarly, Weierstrass's analysis was already rigorous but some major issues (like Lebesgue integration) could not be advanced without going to the next level of abstraction. My impression is that Grothendieck's role was similar to that, but the subject is so technical that for a non-expert it's hard to move beyond vague impressions and analogies.

Is the analogy correct? Can somebody explain in accessible terms what was still lacking after "Weil and Zariski made algebraic geometry rigorous", and why Grothendieck's contribution allowed such rapid progress after him? Why did AG need AG?

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    $\begingroup$ If you say that "Grothendieck was ARGUABLY the most brilliant mathematician" of 20-th century, then you probably know the answer to your question:-) Foundations of algebraic geometry is his most famous work. $\endgroup$ – Alexandre Eremenko Jan 13 '15 at 22:39
  • $\begingroup$ Weil and Zariski did the foundations apparently, Grothendieck's role is somewhat mysterious to me. Especially given that motives and topoi have spread beyond AG for reasons I'd like to understand. $\endgroup$ – Conifold Jan 13 '15 at 23:47
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    $\begingroup$ For Zariski and Weil, algebraic geometry was carried out over fields (Weil had a particular interest in not assuming the field is algebraically closed). The basic objects of interest were still close to those of earlier periods, such as (on the algebraic side) reduced finitely generated algebras over a field. Grothendieck relaxed this to allow arbitrary commutative rings in that role instead, including rings with nonzero nilpotent elements. $\endgroup$ – KCd Jan 14 '15 at 9:24
  • $\begingroup$ All this sounds more like Grothendieck created the area joining/identifying disparate pieces. No Grothendieck, no algebraic geometry to wonder why he was needed... [yes, I am a complete ignorant in this, just an amused bystander]. $\endgroup$ – vonbrand Mar 1 '16 at 0:17
  • $\begingroup$ +1 @AlexandreEremenko. $\endgroup$ – tttbase Jul 17 '16 at 3:57
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Just prior to Grothendieck's entry to the subject, Weil had gotten terrific results in number theory by algebro-geometric arguments, and pointed the way to far more, but some of his methods went beyond existing rigorous foundations. He aimed to supply new foundations adequate to his ideas. Around the same time Zariski and van der Waerden were also generalizing the foundations of algebraic geometry, and many others were introducing various innovations.

In particular the "Weil conjectures" suggested that topological methods of cohomology applied to algebraic geometry might have huge consequences for number theory -- but neither Weil nor anyone else in say 1954 saw how to actually do that.

Serre believed more strongly than Weil himself did that such a cohomology could actually be created. He recruited Grothendieck to the project.

Grothendieck gave two huge simplifications.

First simplification: cohomology in 1954 had numerous alternative forms in topology, geometry, and analysis. Believe me, you do not want the details, but it could be defined by covering spaces, or differential forms, or Cech covers, or simplicial decomposition, or, injective resolutions, or flabby resolutions, or bar resolutions, and much more. These had numerous complicated relations to one another. Grothendieck invented the now-current ideas of Abelian category and derived functor so that essentially all these cohomology theories used one simple kind of resolution: the injective.

The other resolutions are all still used today for calculations but Grothendieck's framework is the standard and the others are organized around it. It has led to creating very many more alternative cohomologies.

Second simplification: schemes. By mid 20th century leading algebraic geometers all saw that algebraic geometry needed a notion of algebraic space more general than spaces defined by polynomial equations over the complex numbers. Many people gave examples, led by Weil and notably Serre. In these, algebraic spaces could have coordinate functions much more general than polynomials with complex coefficients. As KConrad notes they all stayed fairly close to taking coordinates in fields, generally in algebraically closed fields.

This was a serious obstacle to Weil's hopes for algebraic geometry as a tool in number theory, since the integers do not form a field at all. The rational numbers form a field but it is as far from algebraically closed as you can be -- if an integer polynomial with lead coefficient 1 has no integer roots, then it has no rational roots either, but of course it has as many complex roots as its degree.

Grothendieck defined "schemes" so that every ring is the coordinate ring of a scheme. Rings that seem to be purely algebraic abstract entities still have geometry meaning n this theory. Some schemes by themselves have little apparent geometric sense, but they have eminently geometric relations to one another. You do not want the details but I give you Deligne's summary of it: "if the decision to let every commutative ring define a scheme gives standing to bizarre schemes, allowing it gives a category of schemes with nice properties" (Deligne, "Quelques idees maıtresses de l’œuvre de A. Grothendieck" 1998, p. 13 (his emphasis) see http://www.ihes.fr/~landais/smf_sem-cong_3_11-19.pdf).

You can define schemes without knowing what polynomials are, let alone knowing what "algebraically closed" means. In lightly over-simplified terms, you only need to know the commutative, associative, and distributive laws so that you can learn what a commutative ring is, and you can define a scheme.

That does not make scheme theory entirely easy. In fact it makes scheme theory very general so that there are vast numbers of important special cases to learn. In practice virtually all algebraic geometers who do not want to just use polynomials on the complex numbers, use scheme theory.

Schemes are important even for geometry over the complex numbers. Ancient Greek geometers already debated whether or not a circle tangent to a line meets it in a "bigger" point than when two lines cross. In scheme theory the answer is simply "yes." With schemes a point of tangency includes an infinitesimal neighborhood of an ordinary point. These schemes are not just cute but has a lot of uses in studying singularities of spaces, and degeneracies in families of spaces, and group actions on spaces, and they use the non-zero nilpotents that KConrad mentions.

These two ideas, derived functor cohomology, and schemes, let Grothendieck pick up another idea of Serre's ("unramified" covers, on which "isotrivial" fiber spaces become trivial) and turn it into "etale cohomology of schemes," which with Deligne's help fulfilled the goal of a cohomological proof for the Weil conjectures. Etale cohomology has also turned out to be the right setting for much more geometry and number theory, and it has spun off many variants to do much more geometry. Etale cohomology also led Grothendieck to create toposes, which have spun off into many other quite unrelated uses.

Grothendieck originally saw the "etale topos" of a given scheme as an expression of that scheme as a "generalized topological space." It was a category organizing all the basic data on the space, and Grothendieck came to see it as a mathematical world like the universe of sets, where you can do analogues of all of standard mathematics but it is intrinsically adapted to that one geometric scheme.

"Motives" for Grothendieck were a further radical simplification of cohomology that would give the same useful information with even less apparatus -- but that theory would depend on new basic theorems that have not yet been proved today. There has been a lot of work on this but so far as I can see the ideas are very far from taking any stable form.

It is entirely impossible now to do justice to Grothendieck's contribution in a few words. I will enter these few words in hopes of helping you see what he did.

Much of this answer is now expanded as http://www.ams.org/publications/journals/notices/201603/rnoti-p256.pdf

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    $\begingroup$ Vow, great answer, thanks! Couple of questions. Were schemes etc. also necessary for more subtle problems even over fields? How do motives and topoi fit into this? $\endgroup$ – Conifold Jan 14 '15 at 23:24
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    $\begingroup$ @Conifold I have added these. $\endgroup$ – Colin McLarty Jan 15 '15 at 14:42
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    $\begingroup$ Dear @ColinMcLarty: you write "Ancient Greek geometers already debated whether or not a circle tangent to a line meets it in a "bigger" point than when two lines cross." This is enormously intriguing. Could you maybe give a pointer towards a possible source? I would love to know more about this. $\endgroup$ – RP_ Feb 7 '16 at 1:14
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    $\begingroup$ @René I don't recall what I read on this, but start with references to Democritus and Proclus in mathforum.org/kb/… $\endgroup$ – Colin McLarty Feb 8 '16 at 11:58
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    $\begingroup$ @René (and anyone else interested, of course) Days after seeing this answer, I saw horn angles mentioned in a short article, From Mixed Angles to Infinitesimals by Bair and Henry. Unfortunately (while I did have access), all sources (such as this one) I can find are behind a paywall. While there was not much history in the article, they did point to a webpage that seems interesting (search for "history of horn angles" on page). $\endgroup$ – pjs36 Feb 2 '18 at 1:17

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