Alexander Grothendieck

Mathematics

The mathematical universe

The mathematical universe: the movement from isolated problem solving toward languages capable of reorganizing whole fields. Preface

The diagrams of the book

The commutative diagrams drawn for this edition, as the chapters print them.

The diagram of the snake lemma: two exact rows and three vertical maps.
Figure 27. The diagram of the snake lemma: two exact rows and three vertical maps.Chapter 21 · The Homological Turn
A morphism of sheaves: the maps on sections commute with restriction.
Figure 28. A morphism of sheaves: the maps on sections commute with restriction.Chapter 21 · The Homological Turn
A commutative diagram.
Figure 29. A commutative diagram.Chapter 22 · The Language of Functors
A functor preserves the composition of morphisms.
Figure 30. A functor preserves the composition of morphisms.Chapter 22 · The Language of Functors
A natural transformation viewed as a functor.
Figure 31. A natural transformation viewed as a functor.Chapter 22 · The Language of Functors
Adjoint functors.
Figure 32. Adjoint functors.Chapter 22 · The Language of Functors
A natural transformation between two functors.
Figure 33. A natural transformation between two functors.Chapter 22 · The Language of Functors
The index theorem and the Grothendieck–Riemann–Roch theorem: the square commutes.
Figure 37. The index theorem and the Grothendieck–Riemann–Roch theorem: the square commutes.Chapter 23 · Riemann–Roch Recast
A fibre product (pullback) square.
Figure 42. A fibre product (pullback) square.Chapter 27 · The Relative View
A pullback fibration.
Figure 43. A pullback fibration.Chapter 28 · Representability
The pullback of a fibre bundle.
Figure 49. The pullback of a fibre bundle.Chapter 32 · Descent
The constant sheaf on a space of two points.
Figure 50. The constant sheaf on a space of two points.Chapter 33 · The Étale Site
Absolute and relative Frobenius morphisms.
Figure 52. Absolute and relative Frobenius morphisms.Chapter 34 · A Machine for Weil
The universal property of the fibre product of schemes.
Figure 73. The universal property of the fibre product of schemes.Chapter 51 · The Classroom and the World
From varieties to schemes, algebraic spaces and stacks.
Figure 77. From varieties to schemes, algebraic spaces and stacks.Chapter 54 · The Long Silence

Where the book tells the mathematics

The chapters that cite Grothendieck’s mathematical writings or draw a diagram, with their sections.

Part III

Learning from First Principles

17Nancy

18The First Harvest

Part IV

A Language Begins

19São Paulo

20Kansas

21The Homological Turn

22The Language of Functors

23Riemann–Roch Recast

24Toward Geometry

Part V

The Relative World

25A New Foundation

26The Scheme

27The Relative View

28Representability

29The House at Bures

30Elements of a Future Geometry

Part VI

The Architecture of Cohomology

31The Seminar

32Descent

33The Étale Site

34A Machine for Weil

35Collective Intelligence

36Moscow, Absent

Part VII

The Summit and the Breach

38Motives

39The Standard Conjectures

41The Ethics of Departure

Part VIII

Survival

47Mathematics and Refusal

Part IX

Teaching at the Margin

50Beginning Again

51The Classroom and the World

53The Hidden Workshop

Part X

Reaping, Sowing, Pursuing

58Pursuing Stacks

59Dreams and Proofs

Part XII

After the Life

72The Rising Sea

Grothendieck’s writings · Mathematics