Training from the Source A Bit Block Cipher Ed , Ziegler G. Chaos, Fractals, Power Laws Analysis and Design Fundamentals and Recent Advances Ed , Eckmann B. Ed - Intelligent Computing Everywhere Theory and Applications Kinematic Axioms for Minkowski Space-Time Proceedings of the 8th International Workshop on Complex Structures An English Translation with Commentary Mathematical Methods in Natural Science An Integrated Approach Net Kick Start Ed , Baldwin M.
The First Hundred Years 2nd edition Editor - Drug-Membrane Interactions. Analysis, Drug Distribution, Modeling. Methods and Principles in Medicinal Chemistry Theory and Methodology A Guide for Beginners and the Truly Skeptical Structure and Ontology Ed - Learning Theory Ed , Goldenfeld P.
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Interaction of Time and Size from Macro to Nano. NET OS 1st edition Ed , Jain L. Ed - Intelligent Paradigms for Healthcare Enterprises 1st edition We proved regular local rings have finite global dimension. We proved that a localization of a ring of finite global dimension has finite global dimension.
We proved that a minimal resolution of the residue field of a Noetherian local ring has length at least the dimension of the cotangent space. This in turn was done by induction showing that in this case the maximal ideal of A has at least one nonzerodivisor and induction. The statement about the existence of a nonzerodivisor was done using a consideration of supports and associated primes. Lecture XIX In this lecture we discussed the existence of enough injectives in the category of modules over a ring.
Then we used this to define Ext functors using Hom -, - and injective resolutions in the second variable or projective resolutions in the first variable. Then we defined depth of a finite module over a Noetherian local ring in terms of regular sequences. Next, we defined embedded associated primes of M and embedded primes of R. A CM module M does not have any embedded associated primes. This give a straightforward way to make non-CM things.
A Noetherian domain of dimension 1 is CM. A complete intersection over k is CM. The quotient of a CM local ring by a regular sequence is CM. Interesting rings are often CM. At the end of the lecture I stated Serre's criterion for normality. Next, given a Noetherian local ring A, m , k the following are equivalent dim A is 1 and A is a normal domain A is regular of dimension 1 A is a discrete valuation ring, A is a valuation ring not a field A is a domain and m is principal and not zero, A is a PID and not a field there is a t in m such that every nonzero element of A can be uniquely written as unit t n.
We defined a valuation ring. We proved the implication 1 implies 2 using an argument of Kollar. We computed some examples of normalizations to show the utility of Serre's criterion as a way to determine when you are done. Lectures XXIII, XXIV, XXV We talk a little bit about sheaves on topological spaces, pushforward and pullback of sheaves, stalks of sheaves, sheafification, sheaves on bases, ringed spaces, morphisms of ringed spaces, locally ringed spaces, morphisms of ringed spaces, Spec A as a locally ringed space, and finally we defined schemes.
If you intend to continue with the next part of this course next semester, it would be great if you could read about sheaves and maybe do some exercises to familiarize yourself with that material. Sometimes an exercise may be too hard. Then just skip it and start with the next one. Reading materials: The Stacks project. Closely related material is listed here, but be aware that the material in the Stacks project is often more general than what we discussed in the lecture.
People call this book the "old Matsumura" to distinguish from the next item The book "Commutative ring theory" by Matsumura translated by Miles Reid. The book "Commutative algebra. With a view toward algebraic geometry" by David Eisenbud. I never read this myself, but I think this is a good choice to look at. It has a lot of stuff in it and it is a bit more wordy if you like that. The book "Introduction to commutative algebra" by Atiyah and Macdonald.
Many of those whose first introduction to commutative algebra comes from this book, swear by this book. I think this is a psychological effect bc I swear by the old Matsumura. It only takes a minute to sign up. Connect and share knowledge within a single location that is structured and easy to search.
Of course the question "what is a good commutative algebra book for an algebraic geometer" has been asked before, see A good commutative algebra book and Reference request: introduction to commutative algebra for example. Learning recently about intersection multiplicity of varieties and multiplicity of a singular point of a variety, i was led into thinking Serre - Local algebra might be a good buy. However, it is almost never mentioned in answers to the questions asked before.
So my question now is: is it a good idea to buy Serre's book if one wants to use it for algebraic geometry? Are there algebraic geometers here that have found it useful to have, in addition to more standard commutative algebra books? Sign up to join this community. The best answers are voted up and rise to the top.
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