Commutative algebra 61: Examples of regular local rings

Value of the Information & Strength of the Argument

The lecture provides valuable insights into subtle behaviors of regular local rings, especially in characteristic p, which are often overlooked. The argumentation is rigorous, with clear step-by-step explanations and concrete examples. The speaker effectively demonstrates why the naive notion of regularity fails over non-algebraically closed fields and introduces the concept of geometric regularity as a remedy. The number theory example illustrates the importance of normalization in achieving regularity.

Scientific Rigor, Source Quality, Title Accuracy

The lecture is based on Eisenbud’s textbook ‘Commutative algebra with a view toward algebraic geometry’, a standard reference. The mathematical content is presented with precision and correct definitions. The title accurately reflects the content. No external sources are cited beyond the textbook, but the lecture is self-contained and rigorous.

The title accurately reflects the content, which focuses on examples of regular local rings, including non-geometrically regular cases and number theory examples.

9/10

Lecture by a renowned mathematician, based on a standard textbook (Eisenbud), with rigorous definitions and examples. The content is mathematically sound and well-structured.

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

• 00:00Introduction: goal to give examples of regular local rings, especially in characteristic p.
• 01:00First example: curve x^p + y^p = a over non-algebraically closed field, maximal ideal generated by y, regular local ring.
• 03:00Over algebraic closure, ring becomes non-reduced and non-regular.
• 05:00Simplification to zero-dimensional case x^p - a, illustrating the phenomenon.
• 07:00Discussion of tensor product of fields: in characteristic zero it is a product of fields, in characteristic p it can have nilpotents.
• 10:00Introduction of geometrically regular rings and smoothness as fixes.
• 12:00Second example: Z[√-3] is not regular at (2, 1+√-3), but its normalization Z[(1+√-3)/2] is regular.
• 15:00Explanation of why Z[√-3] is not regular: m/m^2 has dimension 2.
• 17:00Normalization is regular: it is a Dedekind domain, localizations are DVRs.
• 19:00Warning about inverse image of prime ideals: ideal 2 in Z[(1+√-3)/2] pulls back to (2, 1+√-3) in Z[√-3].

• Commutative algebra with a view toward algebraic geometry — The course follows this book by David Eisenbud.

• Commutative algebra with a view toward algebraic geometry — The lecture follows this textbook, which covers regular local rings and related concepts.