As Wikipedia says:
In Grothendieck's retrospective Récoltes et Semailles, he identified twelve of his contributions which he believed qualified as "great ideas". In chronological order, they are:
- Topological tensor products and nuclear spaces.
- "Continuous" and "discrete" duality (derived categories, "six operations").
- Yoga of the Grothendieck–Riemann–Roch theorem (K-theory, relation with intersection theory).
- Schemes.
- Topoi.
- Étale cohomology and l-adic cohomology.
- Motives and the motivic Galois group (Grothendieck ⊗-categories).
- Crystals and crystalline cohomology, yoga of "de Rham coefficients", "Hodge coefficients", …
- "Topological algebra": ∞-stacks, derivators; cohomological formalism of topoi as inspiration for a new homotopical algebra.
- Tame topology.
- Yoga of anabelian algebraic geometry, Galois–Teichmüller theory.
- "Schematic" or "arithmetic" point of view for regular polyhedra and regular configurations of all kinds.
What are some modern, concise expository texts on these topics that students can use to learn the great ideas of Grothendieck? EGA, SGA, Les Dérivateurs, La Longue Marche, and so on don't count, because they are French, overwhelming, and maybe a bit outdated (I'm not sure) – anyway, it's not realistic for a student to go through them in their free time in addition to the courses they take for their degree. If you want to give a textbook, make sure it's as short as possible.
Please only one topic with expository text per answer.
Let me start by giving two examples of the kind of answers I am expecting:
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Leinster's An informal introduction to topos theory is an amazing introduction to the basic ideas surrounding topoi (and their connections to logic and geometry). And it is concise! This is idea 5.
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Milne's lecture notes on étale cohomology. Not as short as Leinster's paper, but better (for a student) than 1000 pages of SGA 4 and 4 1⁄2. This is idea 6.
Best Answer
A good roadmap for FGA (topic 4, with glimpses on topic 6) is
Fantechi, Barbara; Göttsche, Lothar; Illusie, Luc; Kleiman, Steven L.; Nitsure, Nitin; Vistoli, Angelo, Fundamental algebraic geometry: Grothendieck’s FGA explained, Mathematical Surveys and Monographs 123. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3541-6/hbk). x, 339 p. (2005). ZBL1085.14001.