13
votes
2answers
248 views

Quotient of polynomials, PID but not Euclidean domain?

While trying to look up examples of PIDs that are not Euclidean domains, I found a statement (without reference) on the Euclidean domain page of Wikipedia that $$\mathbb{R}[X,Y]/(X^2+Y^2+1)$$ is ...
3
votes
2answers
166 views

What to study from Eisenbud's Commutative Algebra to prepare for Hartshorne's Algebraic Geometry?

I surveyed commutative algebra texts and found Eisenbud's "Commutative Algebra: With a View Toward Algebraic Geometry" to be the most accessible for me. The book outlines a first course in commutative ...
2
votes
1answer
88 views

The family of schemes $\operatorname{Spec} A[x]/(x^n)$

Consider the family $S_n:=\operatorname{Spec} A[x]/(x^n)$ of schemes, $A$ denoting any ring (which in our subject always means commutative and with identity). Is there some intuitive picture for ...
1
vote
1answer
64 views

Primary decomposition of $I = (x^2, y^2, xy)$

I want to find a primary decomposition of the ideal $$ I = (x^2,y^2,xy) \subset k[x,y]$$ where $k$ is a field. How to proceed? Are there algorithms to find such decompositions? Where can I find ...
1
vote
0answers
142 views

Generalization of Chinese Remainder Theorem to infinite ideals

I'm looking for any (obviously weaker) generalization of this famous theorem in the special case that the family of ideals is not finite.
0
votes
0answers
30 views

Suggest a good book or reference on graded modules over polynomial rings

I am looking for reference books or papers on graded modules over the polynomial ring $k[x_0, \ldots, x_n]$. Any good commutative algebra text like Eisenbud's Commutative Algebra already contains a ...
0
votes
1answer
51 views

Powers generate monomials

What is a reference in the literature for the following fact? Let $A$ be a commutative $\mathbb{Q}$-algebra. Then every monomial in $A$ of degree $n$ may be written as a linear combination of $n$th ...
1
vote
1answer
74 views

Commutative algebra with a geometric flavor

Does anybody know where can I find a book with topics similar to the ones in Atiyah's Introduction to commutative algebra, but with some sort of geometric motivation? Thanks!
0
votes
0answers
74 views

Readings for Noether

I'm studying the theory of Noether but I have only 4 pages of lecture notes with no details or examples. Are there any good lecture notes or chapters you know about? In my lectures the basics of ...
3
votes
0answers
57 views

Equations in the semiring of f.g. modules

Let $R$ be a commutative ring. Then we may consider the semiring $G(R)$ of isomorphism classes of finitely generated $R$-modules with $+ = $ direct sum, $* = $ tensor product, $0 = $ zero module, $1 = ...
0
votes
0answers
45 views

Proposition 5.23 from Atiyah-MacDonald.

Consider the following: Proposition $\bf \textit{$\textbf{5.23}.\,$}$ Let $A\subseteq B$ be integral domains, $B$ finitely generated over $A$. Let $v$ be a non-zero element of $B$. Then there ...
0
votes
1answer
78 views

$\biggl( \prod_{i \in \mathbb{N}}\mathbb{Z}/p^i\mathbb{Z}\biggr)\otimes_{\mathbb{Z}} \mathbb{Q}\ne 0$

I've found this claim $$\biggl( \prod_{i \in \mathbb{N}}\mathbb{Z}/p^i\mathbb{Z}\biggr)\otimes_{\mathbb{Z}} \mathbb{Q} \not\cong \prod_{i \in \mathbb{N}}\biggl( ...
2
votes
1answer
59 views

Good introduction to number theory that develops and/or makes heavy use of commutative ring theory and lattice theory?

I'd like to learn some number theory, since it provides a lot of motivation for commutative ring theory and even some motivation for lattice theory (at least, that's the impression I'm under). ...
7
votes
2answers
111 views

Techniques for showing an ideal in $k[x_1,\ldots,x_n]$ is prime

An affine variety $X$ over a field $k$ is irreducible if and only if its defining ideal $I(X)$ is prime (in this post we use the convention that varieties are not necessarily irreducible). Hence, it ...
0
votes
1answer
90 views

Solved exercises in commutative algebra

I'm looking for books or teaching material with solved exercises in commutative algebra, where can I find them ?
0
votes
0answers
77 views

Multivariable irreducible polynomials over finite fields

It is not difficult to prove the following result, and it seems that it should be already proved. I would appreciate it if someone offer me some reference to it. For any $f(x_1,\dots, x_n)=\sum ...
0
votes
0answers
102 views

Note or book on Examples of regular, Gorenstein, Cohen Macaulay, … rings

I need a good note or book with plenty of examples in commutative algebra and algebraic geometry which surveyed being regular, Gorenstein, Cohen Macaulay, .... Can you help? thanks.
3
votes
0answers
108 views

Infinitesimal thickening of a smooth closed subscheme

Let $A$ be a noetherian ring (if it is useful I can assume that $A$ is an algebra of essentially finite type over a field) and $I \subset A$ is an ideal s.t. $A/I$ is smooth. Is it true that extension ...
2
votes
0answers
44 views

Tor dimension in polynomial rings over Artin rings

I found this tricky problem in trying to understand some properties of local rings at non-smooth points of embedded curves. But this would be a very long story. So I make it short and I try to go ...
0
votes
1answer
111 views

Difference between Matsumura's Commutative Algebra and Commutative Ring Theory

I am a beginner in more advanced algebra and my question is very simple, I would like to know the difference between these books of the same author, Hideyuki Matsumura Commutative Ring Theory ...
3
votes
2answers
104 views

Module of $R$-valued functions on an infinite set is not countably generated

Let $R$ be an integral domain and $X$ be an infinite set. Let $R^X$ be the set of all functions $f: X \rightarrow R$, viewed as an $R$-module in the usual manner: for $\alpha \in R$, $\alpha f: x \in ...
2
votes
0answers
35 views

Question about the construction of the cotangent complex (S. Lichtenbaum's way)

I am trying to thoroughly understand the "old" construction of the cotangent complex. The first question I have is about the definition of an extension of degree two of a ring $B$ above a ring $A$ ...
3
votes
1answer
95 views

Counterexample for $A[[x, y]] = A[[x]][[y]]$

Maybe this is an idiot question, but I've heard that $A[[x, y]] = A[[x]][[y]]$ does not hold for $A$ an arbitrary commutative ring with identity, so I would like to know a counterexample, since the ...
3
votes
0answers
49 views

Endomorphism rings of MCM Modules

Let $k$ be a field (algebraically closed of characteristic not equal to two, if you like) and let $R = k[[t^2, t^{2n+1}]]$. It is well known $R$ has finite type and the MCM (maximal Cohen-Macaulay) ...
3
votes
0answers
51 views

“Localization” of a module at a family of elements

Let $x=(x_i)_{i \in I}$ be a family of elements of a commutative ring $R$. Typically $I$ is infinite. Let $M$ be an $R$-module. For every finite subset $E \subseteq I$ define $M_E = M$, and for ...
0
votes
1answer
93 views

Power series ring over a ring of integers

Let $K/\mathbb {Q}_p$ be a finite extension, $\mathcal{O} := \mathcal{O}_K$ the ring of integers of $K,$ $\frak p$ the maximal ideal of $\mathcal{O}$, and $\pi$ a uniformizer, i.e., $\frak{p} = ...
1
vote
1answer
77 views

Dedekind ring characterization via projective modules

I am looking for a book or course notes proving the following result: Let $R$ be an integral domain. Then $R$ is a Dedekind ring if and only if every submodule of a projective $R$-module is ...
3
votes
1answer
103 views

Some questions about Fitting ideals

Let $R$ be a ring and $M$ a finitely presented $R$-module. Given a free presentation $$ R^{\oplus m} \to R^{\oplus n} \to M \to 0 $$ we define $Fitt_k(M)$, the $k$-th Fitting ideal of $M$, to be the ...
0
votes
1answer
85 views

Poincaré series of an $A$-module

I'm studying Atiyah and Macdonald's "Introduction to Commutative Algebra" and I'm having some problems computing the Poincaré series of an $A$-module. Even the Example after $11.3$, which is dealt ...
1
vote
1answer
42 views

What is the name of this factor-algebra?

In the polynomial algebra $k[x_1,x_2,\ldots, x_n]$ consider an ideal $I$ generated by the polynomials of the form $x_i^k-x_i$, $i=1 \ldots n$ and $k=2,3,\ldots.$ Consider the quotient algebra ...
1
vote
1answer
67 views

Module of power series and change of rings

Let $R$ be a commutative ring with unit an $M$ an $R$-module. Define the formal power series over $M$ as $$M[[X]]=\{\sum_{n\ge 0}m_nX^n: m_n\in M\}$$ and make this an $R[[X]]$-module by ...
3
votes
1answer
164 views

When is the generic point of an integral noetherian scheme open (reference)?

Let $X$ be an integral noetherian scheme, let $\xi$ be its generic point. Then it is not so hard to show that $\{ \xi\}$ is open in $X$ if and only if $X$ is a finite set. In termes of algebra, it ...
4
votes
1answer
84 views

Proof that ideal of Plücker relations is a prime ideal

I am reading section 8.4 of Fulton's Young tableaux where he defines a certain ring as follows. Fix a complex vector space $E$ of dimension $m$ and integers $d_1,\ldots d_s$ such that $m \geq d_1 > ...
15
votes
1answer
176 views

Universal property of de Rham differential.

Suppose $A$ is a commutative algebra over a field $k$. It is well known that there is a module that generalizes the notion of differential $1$-forms. It is denoted $\Omega^1_{k}(A)$ and is called the ...
1
vote
1answer
101 views

a theorem on the dimension of finite algebras over a field (Hartshorne)

Robin Hartshorne in his Algebraic Geometry, Theorem 1.8A(b), p. 6, says that if $B$ is an integral domain which is a finitely generated $k$-algebra, $k$ a field, and $p$ a prime ideal of $B$, then ...
1
vote
1answer
83 views

Regular local ring result - reference request

Reference needed for the following result: Let $R$ be a regular local ring with maximal ideal $\mathfrak m$. If $A$ is a flat $R$-algebra and $A/(\mathfrak m)$ is a domain, then $A$ is a domain. ...
2
votes
0answers
113 views

(Finite) continued fractions over a general domain

I am looking for some literature (articles or books) where finite continued fractions over a general integral domains (that is, in a fraction field of that domain, but the "coefficients" are from the ...
2
votes
0answers
64 views

A criterion for finite generation of subalgebras of a polynomial ring

In her 1926 paper on invariant theory, Emmy Noether uses a certain "finiteness criterion" which I wish to translate and maybe find a more modern reference to. The original wording is: Ein ...
3
votes
2answers
77 views

A property of radical ideals

Let $A$ be a commutative ring with $1 \neq 0$. Theorem (Atiyah-MacDonald 1.13 (v)). Let $\mathfrak{a, b} \subseteq A$ be ideals. Then $\sqrt{\mathfrak{a + b}} = \mathfrak{\sqrt{\sqrt{a} + ...
7
votes
1answer
306 views

Maximal ideals in polynomial rings over a field

Let $K$ be an algebraically closed field and let $k$ be a subfield of $K$ such that the field extension $K \mid k$ is algebraic. Let $B$ be the polynomial ring $K [x_1, \ldots, x_n]$ and let $A$ be ...
5
votes
0answers
64 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
133 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
95 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 ...
4
votes
0answers
73 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
122 views

Access to Ribenboim's article on Hensel's lemma [closed]

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 ...
5
votes
2answers
315 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
146 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 ...
3
votes
2answers
388 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
176 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
66 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 ...