Tagged Questions
2
votes
0answers
43 views
Flatness over Jacobson ring
I need either a reference or a counter-example to the following statement. Let $A$ be a noetherian Jacobson ring (i.e. a noetherian ring where every prime ideal $\mathfrak{p} \subset A$ is an ...
1
vote
1answer
46 views
What's stronger: projective or locally free? flat or locally free?
maybe that's an idiot question, however I did not found anything related in the classical references. It's know that a finitely generated projective $A$-module $M$ is locally free ,since each ...
7
votes
1answer
77 views
Can we really understand $R$ by studying $R$-modules? [duplicate]
According to Algebra: Chapter 0, the category $R\operatorname{-Mod}$ reveals a lot about $R$. However after completing the first eight chapters, I still found no examples where this happens.
Can ...
3
votes
0answers
61 views
Die Relationen, welche zwischen den elementaren symmetrischen Functionen bestehen - Translation?
I am trying to find a translation of this paper either in English or French (preferably English).
I am not very optimistic, but i thought of asking in case somebody is more resourceful :)
3
votes
0answers
61 views
Access to Ribenboim's article on Hensel's lemma
In these pages I have seen a couple of references to Ribenboim's article:
Paulo Ribenboim, Equivalent forms of Hensel's lemma, Exposition. Math. 3 (1985), no. 1, 3-24.
Does anyone know of an ...
4
votes
2answers
175 views
Companion Lecture Notes to Atiyah-MacDonald?
Is there a set of lecture notes that follow Atiyah-MacDonald and expand on the dense passages, point out typos and so forth?
4
votes
2answers
129 views
Epic maps in the category of commutative rings with identity.
Here all rings are assumed to lie in the category $\cal C$ of commutative rings with identity, and ${\cal C} (\ R\ ,\ S\ )$ is the set of all ring homomorphisms $F$ from $R$ to $S$ for which ...
2
votes
2answers
190 views
Book recommendations for commutative algebra and algebraic number theory
Are there any books which teach commutative algebra and algebraic number theory at the same time. Many commutative algebra books contain few chapters on algebraic number theory at end. But I don't ...
7
votes
1answer
119 views
The Picard-Brauer short exact sequence
It seems to be a rather well understood fact that, given commutative rings $R,S$, and a homomorphism $R \to S$ there is a short exact sequence
$$\text{Pic}(R) \to \text{Pic}(S) \to F_0 \to ...
5
votes
1answer
52 views
A prime poset of ideals
Let $A$ be a ring (commutative unital), and $\mathcal I$ be a nonempty family of proper ideals of $A$.
I will say that $\mathcal I$ has property $\dagger$ if for any $\mathfrak a\in\mathcal I$ and ...
0
votes
1answer
82 views
$u\in R$ is a unit iff $u+x$ is a unit for all $x\in \mathcal{N}(R)$
Let $R$ be a commutative ring with identity and denote by $\mathcal N(R)$ its nilradical. It is known that an element $u\in R$ is a unit if and only if $u+x$ is a unit for all $x\in\mathcal N(R)$. In ...
2
votes
1answer
71 views
Describe invariant polynomials under action of commutative group of order eight.
I believe the question below should be fairly standard in invariant theory ; I hope someone more familiar with it than me can explain a bit more or point to a reference.
Let $F$ be polynomial field ...
1
vote
0answers
20 views
Classification of commutative Frobenius algebras
I would like to know if there is a pedagogical reference that explains the classification of all commutative Frobenius algebras.
3
votes
1answer
110 views
Direct limits and $\rm Hom$
I read that $\lim\limits_{\longleftarrow}\mathrm{Hom}(N_j,M)\cong\mathrm{Hom}(\lim\limits_{\longrightarrow}N_j,M)$. I was wondering if we can write $\lim\limits_{\longrightarrow}\mathrm{Hom}(N_j,M)$ ...
3
votes
2answers
162 views
Are projective modules over an artinian ring free?
Quoting a comment to this question:
By a theorem of Serre, if $R$ is a commutative artinian ring, every projective module [over $R$] is free. (The theorem states that for any commutative ...
0
votes
1answer
46 views
What is known about moduls $M = F^n$ over a ring $R$ where $F = R/I$ is a field
If $R$ is a ring and $I$ is an ideal of $R$, then $F = R/I$ is a homomorphic image of $R$, i.e. there is a homomorphism $f: R \rightarrow F$.
If you let $M = F^n$, and define $(\cdot): (R,M) ...
3
votes
2answers
103 views
What letter should I use to denote an ideal?
In commutative algebra, there seem to be two rather different notational conventions for ideals: either $I,J, \dots$ or $\mathfrak{a}, \mathfrak{b}, \dots$.
By itself, it is hardly surprising - after ...
3
votes
1answer
54 views
Reference Request: Finite Length Modules
Where is a good place to read about the properties of the length function on modules over a commutative ring (in particular, quotients of the ring)?
I'm looking mainly for basic properties.
I've ...
2
votes
1answer
90 views
Is the integral closure of $k[[t]]$ in a finite extension of $k((x))$ necessarily a free module?
In Milne Prop 2.29, it is said that the integral closure $B$ of a PID $A$ in a separable finite extension of its fraction field is a free $A$-module.
On the other hand, I have read here that if the ...
4
votes
4answers
137 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
...
5
votes
1answer
99 views
Original article on the Grothendieck group
Is there someone who knows the title of the original publication of Grothendieck on the construction of the Grothendieck group?
Thanks in advance.
0
votes
1answer
98 views
Irreducible polynomial over an algebraically closed field(2)
Suppose $k$ is algebraically closed field. And $p(x_1,\ldots,x_n)\in k[x_1,\ldots,x_n]$ is an irreducible polynomial.
I wonder to show $p(x_1,\ldots,x_n)+z\in \overline{k(z)}[x_1,\ldots,x_n]$ is ...
10
votes
4answers
272 views
Spectrum of $R[x]$
The spectrum of $\Bbb Z[x]$ is well known : a prime ideal of $\Bbb Z[x]$ is or $(Q, p)$, with $Q \in \Bbb Z[x]$ zero or irreducible modulo $p$, and $p$ prime or zero.
If I'm not mistaken, we have a ...
3
votes
1answer
324 views
A good commutative algebra book [duplicate]
Possible Duplicate:
Reference request: introduction to commutative algebra
I'm looking for a good book on commutative algebra covering most of (but not limited to) :
Basic Galois theory ...
4
votes
2answers
124 views
Proof that the $d$-th powers generate the $d$-th symmetric power of a vector space
Let $V$ be a $\mathbb{C}$-vector space of finite dimension. Denote its $d$-th symmetric power by $V^{\odot d}$. I am looking for a proof that $V^{\odot d}$ is generated by the elements $v^{\odot d}$ ...
4
votes
2answers
98 views
Does totally flat commutative ring imply all ideals are idempotent?
From reading Atiyah and MacDonald, I know of the result that a absolutely flat commutative ring has all principal ideals idempotent.
Reading around on math reference, I think that if a commutative ...
1
vote
1answer
129 views
Standard graded algebra
I am so sorry if you feel this kind of question is not appropriate for MS. But I hope you can sympathize with me, I tried to find the answer in all my books and even Google but I found nothing.
My ...
4
votes
1answer
156 views
Ideal as a projective module
I am sorry, this may not be a good question here. I am looking a good reference about when the ideal I of a given commutative ring R (local or may not be local) with identity is a projective module.
3
votes
2answers
264 views
Example of a non-Noetherian complete local ring
I was looking for an example of a non-Noetherian complete local commutative ring with $1$. I would appreciate if anyone can point to a reference.
15
votes
1answer
257 views
A space of ideals
Definition: Let $R$ be a commutative ring with 1. Endow the power set $2^R$ with the product topology. The ideal space $\mathcal{I}(R)$ is defined to be subset of $2^R$ consisting of ideals, ...
1
vote
1answer
121 views
Conceptualization of exterior powers of projective modules
Let $A$ be a commutative noetherian ring, and $P$ a projective $A$ module with $rank(P)=n$. I know that $\wedge^nP \simeq L$ for some rank 1 projective $A$-module, $L$; but I'm not sure of how to ...
4
votes
2answers
129 views
Characterization of primary ideals in a principal ideal domain?
On the commutative algebra wiki, a table of properties lists that "for a PID, the primary ideals coincide with the powers of prime ideals."
I played around with it, couldn't produce a proof, and have ...
11
votes
1answer
220 views
History of Commutative Algebra
There are books of the history of Algebraic Geometry, there are also papers about it (All had done by J.Dieudonné). But I could not find any book or paper about the history of Commutative Algebra. ...
2
votes
1answer
83 views
Free modules and the exactness of a sequence
When I read Thang Le's paper the coloured Jones polynomial and the A-polynomial of knots, it says in page 21 that:
Since $R=\mathbb{C}[t^{\pm1}]$ is a PID, and $C$ is free over $R$. So if we tensor ...
6
votes
2answers
528 views
Video lectures for Commutative Algebra
Are there any good video lectures for learning commutative algebra at level of Atiyah-Macdonald?
1
vote
0answers
189 views
What is a linear resolution?
Can anyone tell me where I may find an introduction to linear resolutions (of a $k[x_1,\ldots,x_n]$-module or ideal) including, of course, the standard definition of such a resolution, as well as its ...
2
votes
1answer
74 views
When is an affine algebra normal?
I was looking at results that give necessary (and possibly sufficient) conditions on an ideal $I$ in the ring $R=k[x_1,x_2,...,x_n]$ such that $R/I$ is normal. I don't know if there is a standard ...
4
votes
2answers
697 views
Prerequisites for Atiyah Macdonald
I am currently doing a one semester course on groups and rings where we have learned about (so far):
Definitions of groups, subgroups, cyclic and normal subgroups, the symmetric group, homomorphisms, ...
4
votes
1answer
99 views
For a ring of char $p$ where $p>0$ is a prime, what does $R^{1/p}$ mean?
If $R$ is a ring of characteristic $p\gt 0$, what does $R^{1/p}$ mean?
I am not sure how to search for it, since I don't know a name for it. From the notation, it seems to be a ring consisting of the ...
15
votes
2answers
508 views
Did Zariski really define the Zariski topology on the prime spectrum of a ring?
The question is not: “Did Zariski really define the Zariski topology?”
It is: “Did Zariski really define the Zariski topology on the prime spectrum of a ring?”
Here is the motivation. --- On page ...
2
votes
1answer
99 views
Regularity ascends from a Noetherian ring to a polynomial or power series ring over it
I am looking for a proof of the following statement:
A Noetherian ring $R$ is regular if and only if $R[x]$ is regular if and only if $R[[x]]$ is regular.
I am trying to understand the properties ...
1
vote
0answers
86 views
Hermite normal form and saturation
Recall that if $M$ is a submodule of $\mathbb{Z}^n$, then the saturation of $M$ (in $\mathbb{Q}$) is defined to be $\mathbb{Z}^n \cap (\mathbb{Q}\otimes_{\mathbb{Z}} M)$. According to an article of ...
19
votes
3answers
937 views
Reference request: introduction to commutative algebra
My goal is to pick up some commutative algebra, ultimately in order to be able to understand algebraic geometry texts like Hartshorne's. Three popular texts are Atiyah-Macdonald, Matsumura ...
2
votes
1answer
254 views
Max Noether's $AF + BG$ theorem
Wikipedia tells me about Max Noether's $AF + BG$ theorem but only gives one reference and one external link. I've had a look at the MathWorld link but it seems to be an entirely geometric formulation ...
