Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,493 questions
2
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Related to fractional ideals
$K$ a field, $A\subset K$ a subring, $M\subset K$ an $A$-submodule. Define
$$(A:_{K}M):= \lbrace s\in K|sM\subset A\rbrace$$
Then it is easy to see that
$$M\subset A\Longleftrightarrow A\subset (A:_{...
- 2,857
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votes
3
answers
972
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Is Krull dimension non-increasing along ring epimorphisms?
Let $f \colon R \to S$ be an epimorphism of commutative rings, where $R$ and $S$ are integral domains. Suppose that $\mathfrak{p} \subset S$ is a prime such that $f^{-1}(\mathfrak{p}) = 0$. Does it ...
- 7,328
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votes
1
answer
579
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Is there a clean definition of the residue map in Milnor K-theory?
If K is a field, v a discrete valuation, and k the residue field, there is a residue map $\partial: K^M_n(K) \to K^M_{n - 1}(k)$. All the definitions I have seen for this map involve two pages of ...
- 467
3
votes
1
answer
475
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Can the factorisation of (p) in a number field K be described by the minimal polynomial of a primitive element?
Let $K$ be a number field, with ring of integers $O_K$, and let $\alpha\in O_K$ be a primitive element for the extension $K/Q$, with minimal monic polynomial $f(x)\in Z[x]$. If $p$ is a prime number, ...
- 2,406
9
votes
3
answers
1k
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Does "finitely presented" mean "always finitely presented", considered in general
I'm wondering about the question
"If we have a finitely presented __, is it necessarily finitely presented with respect to any finite generating set for it?"
I know this is true for groups and ...
- 2,585
12
votes
3
answers
4k
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Is being finitely generated a local property?
I am trying ot figure out a proof of the following fact, that I believe is true, but it seems to me that something is lacking.
Suppose we have commutative, unitary rings $A,B$ and a (unit preserving) ...
- 261
5
votes
0
answers
210
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Kahler differentials and the m-adic filtration
Let $A$ be a comm. local $k$-algebra with max. ideal $m$. Let $gr(-)$ be the associated graded algebra or module for the $m$-adic filtration. The canonical derivation $d:A \to \Omega^1_A$ satisfies $...
- 1,038
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vote
2
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364
views
Rig of fractions, including zero denominators
For some integral domain $R$, one forms the field of fractions $R^*$ by considering (equivalence classes of) formal pairs {$r/s : r \in R, s\in R\backslash 0$} and defining $+$ and $*$ as you'd expect ...
- 1,872
3
votes
0
answers
591
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Algebraic description of double vector bundles.
It is well known, by Serre-Swan theorem, that given a compact manifold M there is an equivalence of categories between the category of vector bundles over M and the category of finitely generated ...
- 51
6
votes
1
answer
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Nagata's bizzare examples
Hi,
due to Nagata and his clever and bizzare examples I'm unsure in this:
1) Is there a regular ring of infinite Krull dimension?
2) Is it true that: Regular ring of finite Krull dimension = ...
- 203
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votes
4
answers
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For which $c$ is $\mathbb{Z}[\sqrt{c}]$ a unique factorization domain? a Euclidean domain?
Let $c$ be an integer, not necessarily positive and $|c|$ not a square. Let $\mathbb{Z}[\sqrt{c}]$ be the set of complex numbers $$a+b\sqrt{c}, \quad a, b\in \mathbb{Z},$$
which form a subring of the ...
- 744
7
votes
4
answers
2k
views
commuting matrices
I have a set $\{A_1, A_2, .. A_k\}$ of $n$ by $n$ real matrices and I know that they are 'perturbated versions' of a set of commuting matrices : $\{P_1,..,P_k\}$, by perturbated versions I mean that I ...
- 71
3
votes
1
answer
374
views
Composition and intersection of residue fields
Let $A$ be a normal ring with quotient field $K$. Let $L/K$ be a finite separable extension.
Let $E_1/K$ and $E_2/K$ be extensions of $K$ contained in
$L$. Let $B_1$ (resp. $B_2$) be the normalization ...
- 1,673
27
votes
5
answers
14k
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Flat module and torsion-free module
All rings in this question are integral.
It is known that flat modules are torsion-free. Conversely, torsion-free modules over Prüfer domain (in particular, Dedekind domain) are flat, please see here. ...
- 951
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0
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451
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Proof of local structure theory for unramified morphisms [closed]
In Raynaud's "Anneaux locaux henseliens," a proof is given of the
following fact: Let $R \to S$ be a finite type morphism of rings, $\mathfrak{q}
\in \mathrm{Spec} S$, $\mathfrak{p} $ the inverse ...
- 25.6k
6
votes
1
answer
356
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Constructive Bezout cofactors in the ring of algebraic integers
We see the following in an answer to This Question : (mangled by me for my purposes, all errors my fault)
Dedekind stated (in 1871) that the ring of algebraic integers is a Bezout Domain. He calls it ...
- 30.1k
74
votes
3
answers
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Is there a "purely algebraic" proof of the finiteness of the class number?
The background is as follows: I have been whittling away at my commutative algebra notes (or, rather at commutative algebra itself, I suppose) recently for the occasion of a course I will be teaching ...
- 65.4k
5
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3
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769
views
Extension of scalars and support of a non-finitely generated module
This is a question about support of modules under extension of scalars.
Let $f \colon A \to B$ be a homomorphism of commutative rings (with unity), and let $M$ be a finitely generated $A$-module.
...
- 2,995
5
votes
1
answer
2k
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Calculating the normalization of an algebraic surface.
Suppose $X$ is a complex algebraic projective surface with one dimensional singular locus. For example consider the hypersurface $z^5=t^2(tx^2+y^3+t^3)$, whose singular locus is along the double line $...
- 51
3
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0
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277
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For which dg-algebras is a dg-module with bounded cohomology quasi-isomorphic to a bounded dg-module?
Hello!
Let S be a commutative local Noetherian base ring and A be a dg-S-algebra.
Suppose M is a bounded above dg-A-module with bounded cohomology. Does there exist a bounded dg-A-module isomorphic ...
- 2,756
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3
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Is the tensor product of regular rings still regular
An imprecise version of the question is that when $A$ and $B$ are regular rings, is $A \otimes B$ also regular? Please allow me to put more restrictions, here I am only interested in the case when $A$ ...
- 1,160
4
votes
4
answers
2k
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Why are principal ideal domains and Dedekind domains prominent, but I always seem to see Noetherian rings rather than Noetherian domains?
It seems to me that these 3 algebraic systems are closely related, but it always seems to be Noetherian rings rather than Noetherian domains appearing, and conversely I rarely seem to see principal ...
- 4,351
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votes
3
answers
2k
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How to define the orientation of a vector space over an arbitrary field?
I know the construction of the Hodge star operator in the context of (pseudo-)euclidean real vector spaces. Apart from the scalar product it involves a orientation of the vector space, which one has ...
- 2,399
3
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0
answers
240
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Flatness of a homogeneous quasi-coherent sheaf on the formal plane from flatness on lines through the origin?
(This is a follow-up to a previous question; as often happens, I didn't include enough hypotheses, so I'm asking a new, more-likely-to-be-true question).
Consider the formal plane $\operatorname{Spec}...
- 44.7k
1
vote
1
answer
268
views
Flatness on the formal plane from flatness on lines through the origin?
Consider the formal plane $\operatorname{Spec}\mathbb C[[t,h]]$, and let $\mathcal F$ be a quasi-coherent sheaf. Now assume that $\mathcal F$ is flat over infinitely many lines through the origin of ...
- 44.7k
11
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1
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Can ⨁_I A be isomorphic to ∏_I A for infinite I?
Suppose $A$ is a non-zero ring (say commutative unital) and $I$ is an infinite set. Can it happen that there is an isomorphism of $A$-modules $\bigoplus_{i\in I}A\cong \prod_{i\in I}A$?
The obvious ...
3
votes
1
answer
928
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How exotic can DVRs be in the ring of rational functions over a local field?
Suppose that $R$ is a complete DVR with field of fractions $K$, uniformiser $\pi$ and residue field $k$.
Let $B$ be a subring of the ring $K(t)$ of rational functions over $K$. Moreover assume that $...
- 3,545
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0
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197
views
Existence of flat models of a smooth finite type algebra over $R((t))$
Let $k$ be a field, $R$ a $k$-algebra (of finite type if necessary),
$B$ an algebra of finite type over ring of the formal Laurent series $R((t))$, which is smooth.
Up to this generality, can one ...
- 51
12
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1
answer
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intersection of ideals in a commutative ring vs their product
This question was inspired by this one. Given two ideals $A,B$ in a finitely generated commutative ring $R$. Is it possible to decide whether $A\cap B=AB$? Here $R$ is given by generators and ...
user6976
65
votes
4
answers
22k
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When is the product of two ideals equal to their intersection?
Consider a ring $A$ and an affine scheme $X=\operatorname{Spec}A$ . Given two ideals $I$ and $J$ and their associated subschemes $V(I)$ and $V(J)$, we know that the intersection $I\cap J$ corresponds ...
- 914
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4
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1k
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What formal properties should resolution of singularities have?
If I were going to propose a new construction as a "replacement for resolution of singularities", what properties would my replacement have to have? [I am going to do no such thing -- this is purely ...
- 5,846
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1
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183
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Projectively splitting module
Is there a name for such class of modules $M$ such that $M\rightarrow N\rightarrow 0$ splits for every $N$?
- 2,857
5
votes
2
answers
3k
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Zariski topology and compact \paracompact space?
Does the Zariski topology on a ring (not commutative in common) form a compact or paracompact space and why?
- 53
4
votes
2
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493
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Explicit Bézout cofactors
$\DeclareMathOperator\lcm{lcm}$This is a rather severe revision of a question I asked recently. We know over the integers that $\gcd(a^2,b^2)=\gcd(a,b)^2$. We might prove this via unique factorization....
- 30.1k
6
votes
1
answer
762
views
Nontrivial criteria for polynomials to have no common zeros?
When we work in $C[x_1,x_2,...,x_n]$,here $C$ denotes the complex field, we know that when polynomials $f_1,f_2,...,f_k$ have no common zeros, then there exists polynomials $g_1,...,g_k$ , such that ...
- 1,528
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Commutative ring Notes by M. Artin
In 1966, Professor Michael Artin gave a course for first-year graduate students at MIT on commutative algebra. In that course he covered many classical topics, (the Spectrum of a commutative ring, ...
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2
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2k
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Iterated calculation of determinants
Given a $4 \times 4$ matrix $S$ over a commutative ring $R$. I want to consider it as a $2\times 2$ matrix over $M_2(R)$. Lets say $S=\left(\begin{array}{cc} A&B \\\ C&D\end{array}\right)$ ...
- 11.1k
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4
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Reference for tensor products of fields
Does anybody know a reference for basic properties of tensor products of (finite) algebraic extensions of fields?
Ideally, I would like a description of $L \otimes_k K$ for arbitrary finite ...
- 467
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1
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251
views
What is a certain cartesian product of algebras?
Suppose $F$ is a field and $A$ the $F$-algebra $F[X]\times_{(F\times F)} F$ given by the missing corner of a cartesian square in $F$-algebras
\begin{equation}
F~\xrightarrow{\Delta} ~F\times F~ \...
- 1
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votes
3
answers
1k
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Can the I-fold direct product be free?
Let $A$ be a non-zero commutative ring with unit, $I$ a infinite set.
Can $\prod_{i\in I}A$ be free as an $A$-module?
I found when $A$ is a field or is isomophic to $\mathbb{Z}/m\mathbb{Z}$, then ...
- 268
3
votes
2
answers
810
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What is the divisibility theory for Bezout Domains?
There are many facts about integer gcds which can be proved by appealing to unique prime factorization (up to sign). for example $\gcd(a^2,b^2)=\gcd(a,b)^2$. One way to get the machinery (if that is ...
- 30.1k
29
votes
3
answers
7k
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Non finitely-generated subalgebra of a finitely-generated algebra
Ok, I feel a little bit ashamed by my question.
This afternoon in the train, I looked for a counter-example:
— $k$ a field
— $A$ a finitely generated $k$-algebra
— $B$ a $k$-subalgebra of $A$ that ...
3
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2
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How to calculate Tor(R/I, R/J) ??
How can I prove that $\text{Tor}_1(R/I,R/J) = (I \cap J)/IJ$, where $R$ is a ring and $I, J$ ideals.
Moreover, if we suppose $R=I+J$, how do I prove that $\text{Tor}_1(R/I,R/J)=0$?
Ps: No, this is ...
- 33
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votes
2
answers
1k
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Geometric motivation for the Stanley-Reisner correspondence
The Stanley-Reisner ring of an abstract simplicial complex $\Delta$ on the vertex set $\{1,...,n\}$ is the $k$-algebra
$$
k[X_1,...,X_n]/I_\Delta
$$
where $I_\Delta$ is the ideal generated by the $X_{...
- 2,782
12
votes
2
answers
799
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Normal Macaulayfications
Given any reduced excellent scheme $X$, there exists a Macaulayfication $\pi : Y \to X$. In other words, there exists a proper birational map $\pi$ from a Cohen-Macaulay scheme $Y$ to $X$. These ...
- 20.5k
8
votes
2
answers
2k
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Can we deduce that two rings $R_1$ and $R_2$ are isomorphic if their polynomial ring are isomorphic?
Given two rings $R_1$ and $R_2$ (with or without identity: it's not specified). If $R_1[x]$ is isomorphic to $R_2[y]$ (No such requirement that the isomorphism sends the constant terms to constant ...
- 1,079
2
votes
0
answers
261
views
On a characterization of the symbolic square of prime ideals in polynomial rings
If $R=k[x_1,...,x_n]$ is a polynomial ring in $n$ indeterminates over a field $k$ of characteristic $0$, there is a characterization of the symbolic square of a prime ideal $P$ (the $n$-th symbolic ...
- 805
15
votes
1
answer
649
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Primes that must occur in every composition series for a given module
Let $M$ be a finitely generated module over the commutative noetherian ring $R$. Let ${\cal C}(M)$ be the set of all primes $P$ in $R$ such that $R/P$ appears as a quotient in every composition ...
- 23k
4
votes
0
answers
338
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What to call the following variant of tame ramification
Suppose that $R \subseteq S$ is a generically separable extension of 1-dimensional normal domains (you can assume that $R$ is local if you'd like) of equal-characteristic $p > 0$ (for simplicity, ...
- 20.5k
4
votes
2
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354
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"un-nil-ifying" ideals via deformation
This is perhaps a naïve question; given an irreducible scheme $X$, is there a general procedure to find a flat family $Y \to T$ such that over some point $t_0$ we have $Y \times_T {t_0} \cong X$ but ...
- 467