Update: have been reading Grothenieck, a mathematical portrait. A remarkable book. Recommend. One does need a serious acquitance with scheme theory & related fields to get most out of it.
One takeaway for me is that Grothenieck’s work was more evolutionary than revolutionary. Many ideas often associated with scheme theory were already pioneered by others, e.g. :
The idea of generic points and specializations(Andre Weil), lifting from characteristic p to zero (Weil & Zariski), definition of etale fundamental group & description in terms of complex fundamental group (Abhyankar, Zariski), the concept & need for the etale cohomology (Weil*), notion of etale maps & importance in cohomological computations (Serre), prime spectrum of a ring (Krull, many others), notion of scheme (Cartier, Serre, Chevalley, others. The name of schema is due to Chevalley iirc?), infinitesimals as key to deformations (Italian school—famously imprecise, but had many of the geometric ideas), jet & arc schemes (pioneered by John Nash Jr. - yes the Beautiful Mind John Nash), category theory & Yoneda lemma (invented and developed in great detail by Maclane-Eilenberg), locally ringed spaces (Cartan’s school), spectral sequences (invented by Leray), sheaf theory and sheaf cohomology (Leray, then introduced by Serre into algebraic geometry), injective/projective resolutions and abstract approach to cohomology (common technique in group cohomology).
More generally the philosophy of increased abstrraction and rising-sea style mathematics was common in the ‘French school’, famously as espounded by Nicolas Bourbaki.
One wouldn’t be wrong to say that, despite the scheme-theoretic language, 90% of the ideas in the standard algebraic geometry textbook of Hartshorne precede Grothendieck.
As pointed out clearly by Jean-Pierre Serre, the Grothendieckian style of mathematics wasn’t universally succesful as many mathematical problems & phenomena resist a general abstract framework.
[Jean-Pierre Serre favored the big tent esthetic of ‘belle choses’ (all things beautiful), appreciating the whole spectrum, the true diversity of mathematical phenomena, from the ultra-abstract and general to the super-specific and concrete.]
What then were the great contributions to Alexandre Grothendieck?
Although the abstraction has become increasingly dominant in modern mathematics, most famously pioneered by the French school of Bourbarki, Grothendieck was surely the more DAKKA Master of this movement in mathematics pushing the creation and utilization of towering abstractions to new heights.
Yet, in a sense, much of the impact of Grothendieck’s work was only felt many decades later, indeed much of its impact is perhaps yet to come.
To name just a few: the central role of scheme theory in the later breakthrus of arithmetic geometry (Mazur, Faltings, Langlands, most famously Wiles), (higher) stacks, anabelian geometry, galois-teichmuller theory, the elephant of topos theory. There are many other fields of which I must remain silent.
On the other hand, although Grothendieck envisioned topos theory he did not appreciate the (imho) very important logical & type-theoretic aspects of topos theory, which were pioneered by Lawvere, Joyal and (many) others. And although Grothendieck envisioned the centrality and importance of a very abstract homotopy theory very similar to the great influence and character of homotopy theory today, he was weirdly allergic for the simplicial techniques that are the bread-and-butter of modern homotopy theory. Indeed, simplicial techniques lie at the heart of Jacob Lurie’s work, surely the mathematician who most can lay the claim to be Grothendieck’s heir.
* indeed Weil’s conjectures were much more than a set of mere guesses but a precise set of conjectures, of which he proved important special cases, provided numerical evidence. Central to the conjectures was the realization & proof strategy that a conjectural cohomology theory -‘Weil cohomology’- would lead to the proof of these conjectures. Importantly, this involved a very precise description of conjectured properties the conjectured cohomology theory. Clearly circumscribing the need and use for a hypothetical mathematical object is an important contribution.
Update: have been reading Grothenieck, a mathematical portrait. A remarkable book. Recommend. One does need a serious acquitance with scheme theory & related fields to get most out of it.
One takeaway for me is that Grothenieck’s work was more evolutionary than revolutionary.
Many ideas often associated with scheme theory were already pioneered by others, e.g. :
The idea of generic points and specializations(Andre Weil), lifting from characteristic p to zero (Weil & Zariski), definition of etale fundamental group & description in terms of complex fundamental group (Abhyankar, Zariski), the concept & need for the etale cohomology (Weil*), notion of etale maps & importance in cohomological computations (Serre), prime spectrum of a ring (Krull, many others), notion of scheme (Cartier, Serre, Chevalley, others. The name of schema is due to Chevalley iirc?), infinitesimals as key to deformations (Italian school—famously imprecise, but had many of the geometric ideas), jet & arc schemes (pioneered by John Nash Jr. - yes the Beautiful Mind John Nash), category theory & Yoneda lemma (invented and developed in great detail by Maclane-Eilenberg), locally ringed spaces (Cartan’s school), spectral sequences (invented by Leray), sheaf theory and sheaf cohomology (Leray, then introduced by Serre into algebraic geometry), injective/projective resolutions and abstract approach to cohomology (common technique in group cohomology).
More generally the philosophy of increased abstrraction and rising-sea style mathematics was common in the ‘French school’, famously as espounded by Nicolas Bourbaki.
One wouldn’t be wrong to say that, despite the scheme-theoretic language, 90% of the ideas in the standard algebraic geometry textbook of Hartshorne precede Grothendieck.
As pointed out clearly by Jean-Pierre Serre, the Grothendieckian style of mathematics wasn’t universally succesful as many mathematical problems & phenomena resist a general abstract framework.
[Jean-Pierre Serre favored the big tent esthetic of ‘belle choses’ (all things beautiful), appreciating the whole spectrum, the true diversity of mathematical phenomena, from the ultra-abstract and general to the super-specific and concrete.]
What then were the great contributions to Alexandre Grothendieck?
Although the abstraction has become increasingly dominant in modern mathematics, most famously pioneered by the French school of Bourbarki, Grothendieck was surely the more DAKKA Master of this movement in mathematics pushing the creation and utilization of towering abstractions to new heights.
Yet, in a sense, much of the impact of Grothendieck’s work was only felt many decades later, indeed much of its impact is perhaps yet to come.
To name just a few: the central role of scheme theory in the later breakthrus of arithmetic geometry (Mazur, Faltings, Langlands, most famously Wiles), (higher) stacks, anabelian geometry, galois-teichmuller theory, the elephant of topos theory. There are many other fields of which I must remain silent.
On the other hand, although Grothendieck envisioned topos theory he did not appreciate the (imho) very important logical & type-theoretic aspects of topos theory, which were pioneered by Lawvere, Joyal and (many) others. And although Grothendieck envisioned the centrality and importance of a very abstract homotopy theory very similar to the great influence and character of homotopy theory today, he was weirdly allergic for the simplicial techniques that are the bread-and-butter of modern homotopy theory. Indeed, simplicial techniques lie at the heart of Jacob Lurie’s work, surely the mathematician who most can lay the claim to be Grothendieck’s heir.
* indeed Weil’s conjectures were much more than a set of mere guesses but a precise set of conjectures, of which he proved important special cases, provided numerical evidence. Central to the conjectures was the realization & proof strategy that a conjectural cohomology theory -‘Weil cohomology’- would lead to the proof of these conjectures. Importantly, this involved a very precise description of conjectured properties the conjectured cohomology theory. Clearly circumscribing the need and use for a hypothetical mathematical object is an important contribution.