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I'm no expert but I think one of the big ones is Grothendieck's identification of affine schemes with the dual (categorical opposite) of commutative rings. That

I'm no expert but I think one of the big ones is Grothendieck's identification of affine schemes with the dual (categorical opposite) of commutative rings. That is, a "space" is just a commutative ring (of functions on it), but you think about it backwards. So a map of "spaces" A->B, is a map of commutative rings A<-B. John Baez has a bunch of TWF's about this stuff that I highly recommend. Also [1]. [1] https://qchu.wordpress.com/2009/11/24/spectra-of-rings-of-continuous-functions/ https://qchu.wordpress.com/2009/11/24/spectra-of-rings-of-co...

k0k0r0 4y ago
Yeah, that's somewhat correct but misses the intention. Back than classical algebraic geometry started to get extremely messy and mathematicians got lost in a maze. And would prove wrong theorems and stuff, a lot of notions they used were not (possible to be) properly defined. Redefinig the geometrical aspect of algebraic geometry in purely algebraic terms using rings clearified the previously used notions and provided a firm basis for the mathematics that already had been studied. Furthermore, it provided a way out of the algebraic geometry mess mathematicians had been doing and lead to completely new insights. Before, we mathematicians considered spaces like R^n or rather preferably C^n and then vanishing sets of polynomials in them, which form nice geometrical objects. E.g. The set of points (x,y) in R^2 such that x^2 + y^2 = 0, which gives a circe. At least if we only consider real values for x and y, but just that you know mostly people from algebraic geometry prefer the complex numbers C. Then, we asked questions about intersections of those, about what function one might define on them using algebraic terms, and how many straight lines they contain. The geometrical aspect of this provided a great intuition for this kind of mathematics, but it started to get really messy and mathematicians started to get stuff wrong. Then, Grothendieck came along and turned everything upside down. He stopped talking about for example the circe, which is described by the algebraic equation x^2+ y^2 = 0, as the geometric object. Instead, he said: from now on our geometric object of considerations is the set of prime ideals in the polynomial ring in two variables that contain the polynomial x^2 + y^2. And together with some more structure that's what is today known as an affine scheme. For the sake of this explaination its not really important what a prime ideal is, and you might want to look that up later, cause its actually a very simple thing. Also any further definitions would rather confuse any reader, that is not familiar with the subject. From now one the theory gets surprisingly abstract. For example, polynomials and polynomial rings quickly get replaced by arbitrary ring. The main point is: Now, the circle was replaced by an object that was defined in purely algebraic terms. From now on, in the field of algebraic geometry even the geometry was purely algebraic. On the first glance, that algebraic object seems to have nothing or little to do with the geometric object of the circle. And definitely the theory seems to be overkill, even when you have a look from the insight. However, grothendieck was suddenly able to explain a lot of the mess other mathematiciam where doing up till then, and even more, he proved theorems in algebraic geometry that where completely out of reach for any mathematicians before him. With Grothendiek in algebraic geometry a turning-point was reached. At that time people were hungry for answers and hungry for abstraction. They absorbed his theory as thirsty plants absorb long awaited rain. The nice properties of rings allowed to develop a massive machinery of abstract mathematics. That's what most of algebraic geometry had turned into after Grothendieck. I find its a beautiful theory and a lot of it is still very close to geometry. For anybody interested, there is this nice book about the huge field of toric varieties (a subfield of algebraic geometry) for example. Its from Cox, Little and Schenk. Even though it covers basically the whole subfield it also assumes no or very little familiarity with algebraic geometry from the reader. However, one needs to get through that abstract basics of algebraic geometry. You wont get far in algebraic geometry without it. Countless mathematician have tried and failed miserably, before Grothendieck came along with his fancy theory.
macrolocal 4y ago
Yep, though Grothendieck's perspective was more abstract. To him, affine schemes succinctly described coslice categories over a ring (with only localization maps). Adding equivalences back into the mix leads to his theory of stacks/descent. Adding more general localizations leads to his étale, fppf, fpqc results. His work has a more categorical than ring theoretic flavor.

k0k0r0 4y ago

Yeah, that's somewhat correct but misses the intention. Back than classical algebraic geometry started to get extremely messy and mathematicians got lost in a maze. And would prove wrong theorems and stuff, a lot of notions they used were not (possible to be) properly defined. Redefinig the geometrical aspect of algebraic geometry in purely algebraic terms using rings clearified the previously used notions and provided a firm basis for the mathematics that already had been studied. Furthermore, it provided a way out of the algebraic geometry mess mathematicians had been doing and lead to completely new insights. Before, we mathematicians considered spaces like R^n or rather preferably C^n and then vanishing sets of polynomials in them, which form nice geometrical objects. E.g. The set of points (x,y) in R^2 such that x^2 + y^2 = 0, which gives a circe. At least if we only consider real values for x and y, but just that you know mostly people from algebraic geometry prefer the complex numbers C. Then, we asked questions about intersections of those, about what function one might define on them using algebraic terms, and how many straight lines they contain. The geometrical aspect of this provided a great intuition for this kind of mathematics, but it started to get really messy and mathematicians started to get stuff wrong. Then, Grothendieck came along and turned everything upside down. He stopped talking about for example the circe, which is described by the algebraic equation x^2+ y^2 = 0, as the geometric object. Instead, he said: from now on our geometric object of considerations is the set of prime ideals in the polynomial ring in two variables that contain the polynomial x^2 + y^2. And together with some more structure that's what is today known as an affine scheme. For the sake of this explaination its not really important what a prime ideal is, and you might want to look that up later, cause its actually a very simple thing. Also any further definitions would rather confuse any reader, that is not familiar with the subject. From now one the theory gets surprisingly abstract. For example, polynomials and polynomial rings quickly get replaced by arbitrary ring. The main point is: Now, the circle was replaced by an object that was defined in purely algebraic terms. From now on, in the field of algebraic geometry even the geometry was purely algebraic. On the first glance, that algebraic object seems to have nothing or little to do with the geometric object of the circle. And definitely the theory seems to be overkill, even when you have a look from the insight. However, grothendieck was suddenly able to explain a lot of the mess other mathematiciam where doing up till then, and even more, he proved theorems in algebraic geometry that where completely out of reach for any mathematicians before him. With Grothendiek in algebraic geometry a turning-point was reached. At that time people were hungry for answers and hungry for abstraction. They absorbed his theory as thirsty plants absorb long awaited rain. The nice properties of rings allowed to develop a massive machinery of abstract mathematics. That's what most of algebraic geometry had turned into after Grothendieck. I find its a beautiful theory and a lot of it is still very close to geometry. For anybody interested, there is this nice book about the huge field of toric varieties (a subfield of algebraic geometry) for example. Its from Cox, Little and Schenk. Even though it covers basically the whole subfield it also assumes no or very little familiarity with algebraic geometry from the reader. However, one needs to get through that abstract basics of algebraic geometry. You wont get far in algebraic geometry without it. Countless mathematician have tried and failed miserably, before Grothendieck came along with his fancy theory.

macrolocal 4y ago

Yep, though Grothendieck's perspective was more abstract. To him, affine schemes succinctly described coslice categories over a ring (with only localization maps). Adding equivalences back into the mix leads to his theory of stacks/descent. Adding more general localizations leads to his étale, fppf, fpqc results. His work has a more categorical than ring theoretic flavor.