Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,493 questions
18
votes
4
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Flatness of normalization
Let $X$ be a noetherian integral scheme and let $f \colon X' \to X$ be the normalization morphism. It is known that, if non trivial, $f$ is never flat (see Liu, example 4.3.5).
What happens if we ...
- 3,704
5
votes
1
answer
220
views
When are these rings regular?
Let $R$ be a noetherian regular domain. Suppose that $a, b \in R$, with $b \neq 0$, and consider the ring $S:=R[\frac{a}{b}]=R[X]/(bX-a)$. Is $S$ regular? If this is not the case are there some ...
- 3,704
1
vote
0
answers
138
views
Bases of Ideals With no Monomials
Let $K$ be an algebraically closed field and $K[\underline{x}]$ its ring of polynomials in $n$ variables $x_1,\cdots, x_n$. Let $J\leq K[\underline{x}]$ be an ideal such that there are no monomials in ...
- 345
7
votes
3
answers
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Products of Ideal Sheaves and Union of irreducible Subvarieties
Assume I have a nonsingular, irreducible, algebraic variety $X$ and irreducible, nonsingular subvarieties $Z_1,\ldots,Z_k\subseteq X$. Let $\mathcal{I}_i$ be the ideal sheaf of $Z_i$ and $\mathcal{I}:=...
- 4,902
10
votes
1
answer
1k
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Formally smooth morphisms, the cotangent complex, and an extension of the conormal sequence
I'm reading Daniel Quillen's paper "Homology of commutative rings," in which he proves:
A finitely presented morphism of rings $A \to B$ is
Formally etale iff $L_{B/A}$ (this denotes the cotangent ...
- 25.6k
2
votes
0
answers
245
views
Is simplicity preserved under completion of the base ring?
Let $(A,\mathfrak{m})$ be a noetherian local ring and $R$ be an $A$-algebra, which is finitely generated generated as an $A$-module (module finite $A$-algebra). Let $\widehat{A}$ be the $\mathfrak{m}$-...
- 1,391
11
votes
1
answer
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Does completion commute with localization?
Suppose $A$ is a Noetherian (not necessarily local) ring and $\mathfrak{m}\subset A$ a maximal ideal. Then is it true that $$\hat{A}_{\hat{\mathfrak{m}}}=\widehat{A _{\mathfrak{m}}},$$ where hats ...
- 2,857
4
votes
1
answer
662
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Modules with flat duals
Let $R$ be a commutative ring, $M$ an $R$-module, $M^*=Hom_R(M,R)$ its dual. What are sufficient (and possibly necessary) conditions on $M$ that ensure that $M^*$ is flat? Is there a name for such ...
- 12.3k
37
votes
3
answers
3k
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What does it mean geometrically that an element in a domain is irreducible?
Consider a domain $A$ and a non-zero element $f\in A$. That element $f$ is prime if and only if the subscheme $V(f)\subset \operatorname{Spec}(A)$ is integral and this is a completely satisfactory ...
- 47.5k
24
votes
4
answers
4k
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Is there a Galois correspondence for ring extensions?
Given an ring extension of a (commutative with unit) ring, Is it possible to give a "good" notion of "degree of the extension"?. By "good", I am thinking in a degree which allow us, for instance, to ...
- 1,415
11
votes
1
answer
1k
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Do I need to know what an infinity-Gerstenhaber algebra is, and if so, what is it?
I am in the following situation. I have two (rather explicit and specific) dg commutative algebras $R,S$ over a field of characteristic $0$. In fact, $S$ is an $R$-algebra, in that I have a map $R \...
- 54.6k
15
votes
2
answers
2k
views
Is every poset the poset of prime ideals of a ring?
The answer to this question, as it is, is trivially false, for one necessary condition is the existence of maximal element(s), i.e., maximal ideals exist and are prime.
My question was inspired from ...
- 899
3
votes
1
answer
2k
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Multiplicity of a singular point
Let $X$ be a smooth projective complex variety. Assume that $Z$ is a subvariety. Let $T$ be a generic complete intersection of codimension $\dim Z-1$. Assume that $p$ is a point in $Z_T:=T\cap Z$. Is ...
- 2,444
3
votes
1
answer
268
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Universal catenarity and Laurent algebras
A Noetherian (commutative) ring $A$ is called universally catenary if every $A$-algebra of finite type is catenary. If one wants to know whether $A$ is universally catenary, then this definition ...
- 6,700
11
votes
1
answer
840
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Which cluster algebras are coordinate rings of double Bruhat cells?
Background
A uselessly vague paragraph follows. A cluster algebra is a commutative algebra $A$ with a distinguished set of generators called cluster variables. These cluster variables are grouped ...
- 13k
6
votes
1
answer
1k
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reference for p-local and p-complete integers
Can anyone suggest a good thorough reference for $p$-localization and $p$-completion of the integers? I'm an algebraic topologist who's found himself washed up without any intuition.
In particular, ...
- 499
2
votes
0
answers
506
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Zariski's Main theorem [closed]
Sir,
I am studying Zariski's Main theorem. May i know good source of problems related to Zariski's main theorem to understand it better?
Thanks in advance.
- 161
24
votes
6
answers
5k
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Pythagorean 5-tuples
What is the solution of the equation $x^2+y^2+z^2+t^2=w^2$ in polynomials over C ("Pythagorean 5-tuples")?
There are simple formulas describing Pythagorean n-tuples for n=3,4,6:
n=3. The formula ...
- 1,488
3
votes
0
answers
474
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Jacobson-Bourbaki correspondence
The Jacobson-Bourbaki correspondence induces the traditional, finite Galois correspondence by suitable restriction; I've been pondering two things: 1. Are there any (other) interesting applications of ...
3
votes
0
answers
473
views
Infinite Galois correspondence "according to Artin"
Ever since Artin's lectures on Galois Theory one knows how to set up and derive the usual Galois correspondence in the finite(-dimensional) case using just a bit of elementary Linear Algebra, and ...
13
votes
6
answers
7k
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Applications of commutative algebra
Hi. I'm preparing a thesis in commutative algebra, and when I say this to my friends they always ask me what are the applications to "real-world", and I don't know what to answer. This let me think ...
9
votes
2
answers
667
views
irreducibility of generic linear combination of polynomials?
I would be shocked if the following were not true, but I can't seem to see a proof.
Claim:
Let $R$ be an integral domain containing an AC (uncountable if you wish) field $k$. Let $a, b \in R$, and ...
- 525
10
votes
1
answer
3k
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Rings with all modules projective ?
Is there a classification of the commutative rings (with unit) such that each module over the ring is projective ?
- 16.2k
12
votes
2
answers
3k
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Vector bundles on affine scheme
I have already asked similar questions before, but now I realized that there a nice
general way to ask what I want. Namely let $X$ be a normal affine variety over
a field $k$. Assume first that $k$ is ...
- 7,037
11
votes
3
answers
2k
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When is a blow-up Cohen-Macaulay?
Let $X$ be a smooth projective variety and $Z$ a closed subscheme. Let $X'$ be the blow-up of $X$ with center $Z$.
Under what conditions on $Z$ is $X'$
Cohen-Macaulay?
In the case $Z$ is non-...
- 11.6k
5
votes
2
answers
491
views
Is independence meaningful for commutative $C^*$-algebras?
I don't know very much about spectral theory so probably the answer to my question has a basic reference which I would appreciate.
Let's say I have two self-adjoint operators on a Hilbert space and ...
- 2,243
1
vote
1
answer
272
views
Criteria for Preservation of a Module Structure under Extension of Scalars.
Let $A\to B$ be a morphism of (commutative) algebras and $M$ a $B$-module. The $A$-bilinear map $B\times M\to M$ given by $(b,m)\mapsto bm$ induces a surjective homomorphism $B\otimes_{A}M\to M$.
...
- 486
18
votes
2
answers
961
views
How to prove that $e^{zw}$ can not be written as a rational expression in functions in $z$ and in $w$?
Let $K$ be the field of fractions of
$\mathbb{C}[[z]]\otimes_{\mathbb{C}}\mathbb{C}[[w]]\subset \mathbb{C}[[z, w]].$ Given
a formal power series in $t, f\in \mathbb{C}[[t]],$ is there any simple ...
- 689
2
votes
2
answers
492
views
Model Theoretic Localization
This is a re-post on a previous question I asked. My first question was too vague to warrant detailed responses. Really, I have two specific questions to ask.
1) Let $\sigma = (A; \{0,1\}; +, \times)...
- 761
2
votes
1
answer
521
views
Kahler differentials of a hypersurface over a non-algebraically closed field
The following was recently on my algebraic geometry homework:
Let $k$ be an algebraically closed field, $f\in B=k[x_1,\ldots,x_n]$, and $A=B/(f)$. Show that $\Omega_{A/k}$ is locally free of rank $...
- 6,792
9
votes
3
answers
1k
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Quasi-compact maps in Number Theory
Can someone give me an example of a non-quasi-compact morphism of schemes which arises naturally in the field of Algebraic Number Theory?
- 761
6
votes
1
answer
970
views
Reflexive sheaves on singular surfaces
Let $S$ be a normal surface over an algebraically closed field $k$ and let
$s$ be a point of $S$. Let ${\mathcal F}$ be a reflexive sheaf on $S$ of generic rank $n$ . Consider the (derived) fiber of $...
- 7,037
9
votes
2
answers
1k
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Non-Standard Prime
Hello,
My question is about the non-standard models of the integers. If we add to the Peano's axioms $P$ of arithmetic the following axioms for a fixed constant $c$:
$c \neq 0$, $c \neq 1$, $c \neq 1+...
- 663
6
votes
2
answers
2k
views
When does the conormal bundle sequence split?
Let $X\subset \mathbb{P}^n$ be a smooth projective variety with ideal sheaf $I_X$. The conormal sequence is given by
$$
0\to I_X/I_X^2\to \Omega_{\mathbb{P}^n}|_X\to \Omega_{X}\to 0.
$$
For which ...
- 11.6k
7
votes
2
answers
512
views
Is the reduction of a flat, finite, surjective scheme over an integral base still flat?
Is the reduction $X_{red}$ of a flat, finite, surjective scheme $X$ over an integral base $S$ still flat?
I could possibly add that I am already aware we can assume the base $S$ to be local and ...
- 1,347
1
vote
2
answers
1k
views
maximal ideal in local subrings
Let $A,B$ be two local rings and put $\mathfrak{m}_A, \mathfrak{m}_B$ their maximal ideals. Now suppose that we have an injection $0 \to A \to B$ and put $\mathfrak{n} := A \cap \mathfrak{m}_B $. It ...
- 379
6
votes
2
answers
850
views
Decomposition of finite algebras over finite fields
Let $K$ be a number field, $Z_K$ its ring of integers, and $p$ a rational prime number. Then $A_p = Z_K/(p)$ is a finite ${\mathbb F}_p$-algebra. Using ideal arithmetic in $Z_K$ and the Chinese ...
- 32.6k
222
votes
8
answers
35k
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How to memorise (understand) Nakayama's lemma and its corollaries?
Nakayama's lemma is mentioned in the majority of books on algebraic geometry that treat varieties. So I think Ihave read the formulation of this lemma at least 20 times (and read the proof maybe ...
- 14.3k
7
votes
1
answer
800
views
Extensions of torsion modules
Given a regular local ring $R$ and an $R$-algebras $S$, which is torsion free and finitely generated (even free if needed) as an $R$-module.
Assume we have a nontrivial surjective map $f: M \...
- 1,391
3
votes
1
answer
546
views
Center of the category of $R$-algebras
Let $R$ be a ring (assumed to be commutative and with unit). What is the center of the category of $R$-algebras, i.e. $\text{Z}(R\text{-Alg})$? This is a commutative monoid. See this recent question ...
- 63.1k
7
votes
1
answer
747
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Non-normal domain with algebraically closed fraction field
I am looking for an integral domain $A$ with the following properties:
$A$ is not integrally closed
$A$ has a quotient field $K$ that is algebraically closed and that has characteristic 0
There is an ...
- 2,275
4
votes
0
answers
1k
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An example of a noetherian ring in which the integral closure of a finite extension of its field of fractions is not noetherian
Similar to another question I posted. Does anyone know of an example of a noetherian ring in which the integral closure of a finite extension of it's field of fractions is not noetherian.
- 113
3
votes
2
answers
667
views
Normality and rational singularities via Hilbert series
Let $A$ be a finitely generated ${\mathbb Z}_{\geq 0}$-graded algebra over a field without zero divisors;
assume that all graded components are finite-dimensional and that $Spec(A)$ is smooth
outside ...
- 7,037
15
votes
4
answers
6k
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how to determine whether an ideal is prime or not by an algorithm
Given polynomials $f_{1},\cdots,f_{n}\in \mathbb{C}[x_{1},\cdots,x_{m}]$, do we have an algorithm to determine whether the ideal $I=(f_{1},\cdots,f_{n})$ is prime ideal or not? Of course, we assume ...
- 1,528
3
votes
1
answer
320
views
Decision problem about the existence of solution for an integer matrix equation
Given $A,B,C \ $ integer matrices of dimensions $l \times m$, $l \times n$ and $l \times m$, we want to decide (algorithmically) about the existence of $X$ (unimodular) and $Y \ $ integer matrices ...
- 61
2
votes
1
answer
504
views
Zero-dimensional algebras of infinite vector space dimension
Consider an algebra $A$ over a field and suppose that $A$ is zero-dimensional as a ring. It is well-known that if, in addition, $A$ is finitely generated, it has a finite vector space dimension. ...
2
votes
2
answers
406
views
Extending a polynomial function from an open subset
I am a bit embarrassed to ask this question, but still: assume that I have
a finite morphism $\pi:X\to Y$ of affine algebraic varieties over a field (probably
finiteness is too strong an assumption, ...
- 7,037
8
votes
1
answer
289
views
Top degree local cohomology under action by a non-zerodivisor
Let $R$ be a noetherian commutative ring of dimension $n$, and let $M$ be a faithful finite $R$-module. Let $I$ be a proper ideal of $R$, and let $x\in I$ be a non-zerodivisor on $M$.
When does ...
- 19.6k
7
votes
2
answers
2k
views
An effective way to tell if the saturation of a homogeneous ideal is the irrelevant ideal
Let $\Bbbk$ be an algebraically closed field, let $R$ denote the graded ring $\Bbbk[x_0, \dotsc, x_N]$, and let $f_1, \dotsc, f_n \in R_m$ be nonconstant homogeneous polynomials. Then the common ...
- 7,338
2
votes
1
answer
456
views
Generic liftings of a regular sequence on the initial ideal
Hi everyone,
I've got a question about explicitly lifting regular sequences. Let $I$ be an ideal in a polynomial ring $S$ with some term order. We'll denote the initial ideal by $in(I)$. It is ...
- 103