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I have been reading (in nLab) that "a typical Grothendieck proof consists of a long series of trivial steps where “nothing seems to happen, and yet at the end a highly non-trivial theorem is there.” I wonder if there is any example of Grothendieck's style of proof that an undergraduate can understand.

Thanks in advance.

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    $\begingroup$ Maybe a bit related: mathoverflow.net/questions/59071/… $\endgroup$ – Hans Lundmark Mar 5 '17 at 19:51
  • $\begingroup$ This proof somehow reminds me of Grothendieck. You define a weird topology on the integers. It's easy to check that it's a topology, and suddenly it follows that there are infinitely many primes: en.wikipedia.org/wiki/… $\endgroup$ – Flounderer Mar 6 '17 at 3:20
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    $\begingroup$ Also a bit related: The importance of EGA and SGA for students of today. $\endgroup$ – Markus Scheuer Mar 6 '17 at 8:21
  • $\begingroup$ @MarkusScheuer EGA and SGA are like a bomb: very powerful, but you need to be prepared to handle them, or you risk to harm yourself. $\endgroup$ – RandomGuy Mar 6 '17 at 10:09
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    $\begingroup$ @RandomGuy: Your answer (+1) brought me to this MO post and the highest voted answer there nicely affirms your opinion, I think. $\endgroup$ – Markus Scheuer Mar 6 '17 at 10:16
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It is very difficult to answer this question because Grothendieck himself wrote and designed in a masterly way extremely abstract theories, very hard to follow not just for an undergraduate or a graduate, but also for a professional mathematician working in the same fields. You can fully appreciate the depth of his mathematical thought by means of the image he gave of what proving a theorem means. He says, there are two ways. One is by brute force and the other is the conceptual way. It is like to open a nut. You can either break its shell with an act of force, but you can on the other hand place the nut in a container of water, or a solvent, and do basically nothing but wait until the time is mature and the nut opens up by itself, "naturally" (this is a key idea in AG thought). This is the idea of Mathematics that AG has pursued: instead of solving problems by tricks, more or less fortuitous stroke of genius, you have to work conceptually on the theories to understand how you can generalize them in order to solve a problem that in that theory appears impossible, and in the more generalized version of it will appear trivial. This is the difference between "tricky" mathematics and conceptual Mathematics.

AG proofs for the same reasons are extremely clear and neat to understand and read. There is an old saying among students of Algebraic Geometry which explains it all: if you don't understand it in Hartshorne, read it in Grothendieck. His mathematical style is so rich and clear that is almost impossible not to understand his arguments. If I may suggest something within the reach of an undergrad (although a good one of course), I would suggest section 0 of the EGA (French version), where he just introduces some concepts of Commutative Algebra, but in his style. Of course this is nothing to learn how AG worked and thought, but at least you can see how clear his proofs of some classic material were (compare it for example to some well-known book in Commutative Algebra to see the difference) and you can start to have an idea of how his proof style was. For a more mature student, I would suggest to read through most of his EGA and at least some part of SGA. Also his autobiography contains some wonderful material about how he thouught of Mathematics.

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