Takeshi Saito - 1/4 Singular Supports in Equal and Mixed Characteristics

This lecture by Takeshi Saito provides a rigorous and detailed introduction to the theory of singular supports in étale cohomology, a topic at the forefront of arithmetic geometry. The content is highly technical, assuming familiarity with derived categories, étale sheaves, and algebraic geometry. Saito’s exposition is clear and methodical, carefully defining concepts such as local acyclicity and micro-support before introducing the singular support. He emphasizes the non-triviality of the existence of a smallest micro-support, which is a key subtlety. The use of the Radon transform in Beilinson’s proof is explained, though the lecture is part of a series, so some details are deferred. The transition to mixed characteristics is well-motivated, highlighting the open problems and the introduction of the Frobenius-Witt cotangent bundle. The lecture is based on the speaker’s own research and lecture notes, which are provided in the description, adding to its credibility. However, as a live lecture, there are occasional informal exchanges and questions from the audience, which may disrupt the flow for some viewers. The video is not edited, so there are pauses and repetitions. Overall, the lecture is of high scientific quality, suitable for graduate students and researchers in the field. The title accurately reflects the content, and the lecture notes provide a valuable supplement. The main limitation is the lack of visual aids or slides in the video, which might make it harder to follow the mathematical arguments. Nevertheless, the depth and accuracy of the content make it a valuable resource for those already familiar with the basics.

The title accurately reflects the content: the lecture introduces singular supports in equal and mixed characteristics, as part of a series.

8/10

Lecture by a leading expert in arithmetic geometry, with detailed technical content and references to foundational work by Beilinson. The presentation is rigorous, but the video is a recording of a live lecture with occasional informal interactions and no post-production editing.

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

• 00:00Introduction and overview of the lecture series
• 02:00Definition of cotangent bundle and setting for equal characteristic
• 05:00Introduction of singular support and micro-support concept
• 10:00Discussion on the existence of the smallest micro-support
• 15:00Explanation of local acyclicity and its role
• 20:00Use of Radon transform in Beilinson's proof
• 30:00Transition to mixed characteristic case
• 40:00Introduction of Frobenius-Witt cotangent bundle
• 50:00Adaptation of Beilinson's argument for mixed characteristics

• Lecture notes by Takeshi Saito — The lecture notes accompanying this talk, providing detailed definitions and proofs.
• Carmin.tv — Platform hosting the video, offering additional features for the research community.

• Beilinson's paper on singular support — The foundational work on singular support, which the lecture builds upon.