Schemes 13: The functor of points

Value of the Information & Strength of the Argument

The lecture provides valuable insights into the foundational ideas of scheme theory, particularly the functor of points and the emphasis on morphisms. The argumentation is rigorous and well-structured, building from simple examples to more complex concepts. The lecturer clearly explains the motivation behind Grothendieck’s approach and demonstrates the power of the functorial perspective. The discussion of obstructions to representability, such as the two-point set example, is particularly illuminating and sets the stage for the introduction of stacks. The argumentation is solid and logically coherent, making it a valuable resource for students and researchers.

The title accurately reflects the content, which focuses on the functor of points and morphisms of schemes.

9/10

Lecture by a renowned mathematician, based on Hartshorne's standard textbook, with rigorous mathematical reasoning and clear explanations. No unsubstantiated claims; all concepts are well-established in algebraic geometry.

Markers derived by PSI from the transcript: the creator did not define chapters.

• 00:00Introduction to the lecture and overview of two themes: morphisms of schemes and functor of points.
• 01:00Explanation of why one should focus on morphisms of schemes rather than schemes themselves, using the example of affine n-space over a field.
• 03:00Introduction of schemes over a base scheme and the concept of families of schemes parameterized by a base.
• 05:00Example of an elliptic curve as a family over the affine plane, illustrating the idea of a family of curves.
• 07:00Discussion of the philosophy of defining properties of morphisms rather than schemes, with the example of proper maps and complete varieties.
• 09:00Introduction of the functor of points: definition of H_X(T) as morphisms from T to X, and its contravariance.
• 11:00Specialization to affine schemes and the covariant functor from rings to sets, with examples like the affine line.
• 13:00Discussion of the two questions: what functor does a scheme represent, and when is a functor representable by a scheme.
• 15:00Example of the spectrum of Z and the disjoint union of two copies, illustrating the sheaf condition for representability.
• 17:00Example of the affine line and elliptic curves as functors, and the notion of algebraic group schemes.
• 19:00Explanation that morphisms of schemes correspond to natural transformations of functors, and the embedding of schemes into functors.
• 21:00Discussion of the problem of classifying elliptic curves and the need for a moduli space, introducing the functor of families of elliptic curves.
• 23:00Obstruction to representability due to automorphisms, illustrated with the two-point set example and non-trivial families.
• 25:00Introduction of stacks as a solution to the automorphism problem, and mention of rigidification by adding level structure.
• 27:00Conclusion and transition to the next topic: properties of schemes and morphisms.

• Algebraic Geometry (Graduate Texts in Mathematics) — The course is based on chapter II of this book by Robin Hartshorne.

• Algebraic Geometry (Hartshorne) — The lecture follows the content of Chapter II of this standard textbook.