0
votes
1answer
43 views

Preparation for a graduate commutative algebra course based on Eisenbud

I am an undergraduate with two semesters of algebra(groups,rings, Galois theory, etc) under my belt and I am planning on going through Atiyah and MacDonald's book over the summer. Is this sufficient ...
2
votes
1answer
41 views

Flat modules on Stacks Project.

I have been reading through a bit of the material from the Stacks project, and there is a statement that I cannot make sense of. Lemma 10.36.19(7) states: Let $R$ be a ring. (7) Suppose ...
4
votes
2answers
97 views

Zorn's Lemma and Injective Modules

In my study of injective modules over commutative rings, i noticed that Zorn's Lemma is often employed in the proofs. Here are three examples: 1) Baer's Criterion 2) the characterization of injective ...
12
votes
5answers
463 views

Favourite applications of the Nakayama Lemma

Inspired by a recent question on the nilradical of an absolutely flat ring, what are some of your favourite applications of the Nakayama Lemma? It would be good if you outlined a proof for the result ...
8
votes
1answer
179 views

Construction(s) of new integral domains from “old ones”

Given an integral domain $D$, there are several ways how to construct a new integral domain related to D. For example, one can consider a ring of polynomials/formal power series/formal Laurent series ...
2
votes
2answers
128 views

Primary ideals confusion with definition

In a commutative ring, can I say An ideal $\mathfrak q$ in a ring $A$ is primary if $\mathfrak q \neq A $ and if $ xy \in \mathfrak q \Rightarrow $ either $ x \in \mathfrak q$ or $y^n \in \mathfrak ...
6
votes
1answer
181 views

Is there any deep connection between algebraic topology and homological algebra on rings?

There is a deep connection between algebraic topology and homological algebra on groups. A group $G$ can be interpreted as the fundamental group of a covering space $Y \rightarrow X$. (Co)Homology ...
7
votes
2answers
350 views

Classgroup of $\mathbb{Q}(\sqrt{2},\sqrt{-13})$

How would you compute the classgroup of the biquadratic number field $\mathbb{Q}(\sqrt{2},\sqrt{-13})$? I would prefer a method as "from scratch" as possible. Please avoid, if possible, quoting ...
4
votes
4answers
157 views

Why Does Finitely Generated Mean A Different Thing For Algebras?

I've always wondered why finitely generated modules are of form $$M=Ra_1+\dots+Ra_n$$ while finitely generated algebras have form $$R=k[a_1,\dots, a_n]$$ and finite algebras have form ...
44
votes
5answers
1k views

How does one give a mathematical talk?

Sometime tomorrow morning I will be presenting a mathematics talk on something related to commutative algebra. The people present there will probably be two mathematicians (an algebraic geometer and a ...
2
votes
1answer
207 views

How much connection is there between Commutative Algebra and Algebraic Topology?

How much connection is there between Commutative Algebra and Algebraic Topology? I am looking for general highlights, not complex details.
8
votes
2answers
379 views

Suggestions for further topics in Commutative Algebra

I am currently taking a semester long course in Commutative Algebra. We have covered a lot of dimension theory, and today finished proving Zariski's Main Theorem, which was the professor's original ...
25
votes
5answers
2k views

Why should I care about adjoint functors

I am comfortable with the definition of adjoint functors. I have done a few exercises proving that certain pairs of functors are adjoint (tensor and hom, sheafification and forgetful, direct image and ...