Schemes 18: Immersions

Schemes 18: Immersions

by Richard E Borcherds

Formal & Physical Sciences Mathematics PBMGeometryPBMWAlgebraic geometry
🎙 Richard E Borcherds 👥 82K 📅 July 13, 2020 ⏱ 24 min 👁 3K 📄 lecture 🧭 2026-08-17
Available in: English (current) Français

Keywords

open immersionclosed immersionschemequasi-finitefinite typequasi-compactreduced schemeintersection

Summary

This lecture is part of an online algebraic geometry course on schemes, based on Chapter II of Hartshorne’s ‘Algebraic Geometry’. The topic is immersions of schemes. The lecturer first defines open immersions, which correspond to open subsets of the underlying topological space, and gives examples such as the inclusion of the affine line minus a point into the affine line, and the inclusion of the affine line into projective space. He then discusses finiteness properties of open immersions: they are locally of finite type, but not necessarily finite, quasi-finite, or quasi-compact. In the Noetherian case, they are quasi-compact and quasi-finite, but in general they may fail these properties, as illustrated by an example involving infinite-dimensional affine space. The lecture then introduces closed immersions, which locally look like the spectrum of a quotient ring. Closed immersions are finite, and they correspond to closed subschemes, but they are not determined solely by the image set; there can be many closed subschemes with the same underlying closed set. The concept of a reduced closed subscheme associated to a closed subset is discussed, and an example shows that the intersection of two reduced closed subschemes need not be reduced. The lecture concludes by mentioning the theorem of Grothendieck (often called Zariski’s Main Theorem) that quasi-finite morphisms can be factored as an open immersion followed by a finite morphism.

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Critical Evaluation

Value of the Information & Strength of the Argument

The lecture provides a clear and rigorous exposition of open and closed immersions in scheme theory. The value lies in the precise definitions, the discussion of finiteness properties, and the illustrative examples, including counterexamples in the non-Noetherian case. The argumentation is solid: each concept is motivated, and the properties are either proved or justified with clear reasoning. The lecturer also highlights subtle points, such as the difference between closed immersions and closed subsets, and the non-reducedness of intersections of reduced subschemes.

Scientific Rigor, Source Quality, Title Accuracy

The lecture is based on the standard textbook ‘Algebraic Geometry’ by Robin Hartshorne, which is a reliable source. The presentation is rigorous and mathematically accurate. The title ‘Schemes 18: Immersions’ accurately reflects the content. The lecturer is a renowned mathematician, adding to the credibility. No external sources are cited in the video, but the reliance on Hartshorne is explicit.

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Title / Content Match

The title accurately reflects the content, which focuses on open and closed immersions in the context of schemes.

Quality & Reliability

9/10

Lecture by a renowned mathematician (Fields medalist) based on a standard textbook (Hartshorne). The content is rigorous, definitions and examples are precise, and the presentation is clear. The video is part of a well-structured course.

Key Moments

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

Cited Sources

  • Algebraic Geometry — The course is based on Chapter II of this book by Robin Hartshorne.

Contribution & Novelties

This lecture provides a clear and rigorous introduction to open and closed immersions in scheme theory, with a focus on their finiteness properties and examples. It clarifies the distinction between open and closed immersions and highlights subtle points such as the non-uniqueness of closed subschemes with the same underlying set. The lecture is particularly valuable for its counterexample in infinite-dimensional affine space and the discussion of Zariski’s Main Theorem.

Pour aller plus loin :

107 words

Radar Profile

The radar profile shows high scores across all dimensions, indicating a lecture that is information-dense, technically rigorous, and highly reliable. The balance between quantity and quality of information is excellent, making it a valuable resource for advanced students.

Reliability 9/10