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Some background, for those unfamiliar with this area. Algebraic geometry grew out of the study of solutions of systems of polynomial equations. This is natura

Some background, for those unfamiliar with this area. Algebraic geometry grew out of the study of solutions of systems of polynomial equations. This is naturally a question of geometry, which can be studied analytically (using tools from calculus, for example). If you are willing to consider solutions over the complex numbers, and not just the real numbers, you can study the question algebraically, as well. Once you take that step, you can ask what happens when you look at solutions over exotic objects such as finite fields or rational functions. Alexander Grothendieck pushed for a very general formulation of this algebraic approach, using the notion of "scheme". Together with Jean Dieudonne, he wrote an introduction to this topic, called EGA, which is French for "Elements of Algebraic Geometry". This is one of the foundational texts in algebraic geometry now. At some point, Grothendieck became disillusioned with both mathematics and academia, quit both, and isolated himself from most people. Not long before his death, he expressed the desire that all of his work cease to be published. He died 5 years ago now, and I'm not actually sure of the legal status of republishing his work, though people are determined to go ahead because of its great historical significance. While I think EGA has largely been superseded by newer sources, some of his later work (known as SGA) has not.

hackernewsname 7y ago
>While I think EGA has largely been superseded by newer sources, some of his later work (known as SGA) has not Can you elaborate what you mean by “superseded by”?
- QuesnayJr 7y ago
  I just mean that the content can be found from later sources, such as Hartshorne or the Stacks Project. For some parts of SGA, it is my impression that literally the only reference is SGA.
butterthebuddha 7y ago
I'd just like to point out that Grothendieck's notion of a scheme is simpler and more natural than other notions that were floating around at the same time and attempting to capture similar ideas. Why did mathematicians seek a general formulation of geometry (in fact, schemes assign a "natural" geometric interpretation to every commutative ring)? Well, the theory of schemes, has allowed number theoretic questions to be interpreted geometrically. An important result along these lines, that was proven only after the theory of schemes was made available, is an analogue of the Riemann hypothesis for finite fields. This result was famously proven by one of Grothendieck's students (Pierre Deligne).

hackernewsname 7y ago

>While I think EGA has largely been superseded by newer sources, some of his later work (known as SGA) has not Can you elaborate what you mean by “superseded by”?

QuesnayJr 7y ago
I just mean that the content can be found from later sources, such as Hartshorne or the Stacks Project. For some parts of SGA, it is my impression that literally the only reference is SGA.

butterthebuddha 7y ago

I'd just like to point out that Grothendieck's notion of a scheme is simpler and more natural than other notions that were floating around at the same time and attempting to capture similar ideas. Why did mathematicians seek a general formulation of geometry (in fact, schemes assign a "natural" geometric interpretation to every commutative ring)? Well, the theory of schemes, has allowed number theoretic questions to be interpreted geometrically. An important result along these lines, that was proven only after the theory of schemes was made available, is an analogue of the Riemann hypothesis for finite fields. This result was famously proven by one of Grothendieck's students (Pierre Deligne).