Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,493 questions
13
votes
4
answers
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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 ...
- 233
0
votes
1
answer
118
views
$S/I$-freeness of $I/I^2$ vs $I/I^{(2)}$, where $I$ is a radical ideal of regular local ring $S$
Let $I$ be a radical ideal of a regular local ring $S$. Put $R:=S/I$. Let $I^{(n)}$ be the $n$-th symbolic power of $I$. It is well-known that $I^n \subseteq I^{(n)}$.
Is it true that $I/I^2$ is $R$-...
- 6,262
4
votes
0
answers
128
views
Length of dual module
It is well known that, given a commutative ring $R$ and an $R$-module $M$, the dual module $M^\vee = \operatorname{Hom}_R(M, R)$ does not always satisfy $M^\vee \cong M \ (1)$, and not even $M^{\vee \...
- 223
3
votes
1
answer
185
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Uniformly closed ideals of smooth/real analytic functions
Consider $U\subseteq \mathbb{R}^n$ an open subset and denote by $R$ either the algebra of real-valued smooth or real analytic functions on $U$. In either case suppose that $R$ is equipped with the ...
- 21.5k
3
votes
1
answer
200
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Size of the Groebner basis and change of coordinates
Given an ideal $I$ of $k[x,y]$, the size of the Groebner basis depends on the monomial ordering. For example, if
$$
I = \langle x^3y^4 , x^2 + y^2 \rangle,
$$
then the Groebner basis with the ...
- 63.9k
22
votes
8
answers
5k
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Axiomatic definition of integers
The real numbers can be axiomatically defined (up to isomorphism) as a Dedekind-complete ordered field.
What is a similar standard axiomatic definition of the integer numbers?
A commutative ordered ...
- 309
0
votes
0
answers
60
views
Symbolic polyhedron of a monomial ideal
$\DeclareMathOperator\maxAss{maxAss}\DeclareMathOperator\conv{conv}$Let $I$ be a non-zero monomial ideal and $P$ $\subseteq$ $\mathbb R_+ ^ {n+1}$ be its symbolic polyhedron: then
$$
\alpha(P)= \min \{...
- 12.9k
3
votes
0
answers
156
views
Taylor-Wiles systems for higher dimensional deformation rings
Let $R$ be a deformation ring and $M$ be a finitely generated $R$-module.
A strategy for proving the theorems $R=T$ is to associate with $(R,M)$ a Taylor-Wiles system denoted $(R_{Q},M_{Q})$. Here I'm ...
- 10.4k
2
votes
1
answer
363
views
Is the completed tensor product (over a complete dvr) of two reduced complete Noetherian local rings again reduced?
To be more specific, Let $\mathcal{O}$ be a finite extension of $\mathbb{Z}_{p}$. Let $A=\mathcal{O}[[X_{1},\ldots, X_{n}]]/\left( f_{1},\ldots,f_{r}\right) $ and $B=\mathcal{O}[[Y_{1},\ldots, Y_{m}]]/...
- 783
4
votes
1
answer
669
views
Coherent sheaves, Serre’s theorem and ext groups
Let $X$ be a smooth projective variety over an algebraically closed field $k$ (if necessary we assume that $\operatorname{ch}(k)=0$).
Let $O_X(1)$ be a very ample invertible sheaf on $X$.
Then, the ...
- 889
16
votes
4
answers
2k
views
Neusis constructions
Is there some simple description of which complex numbers are "constructible" with straightedge and compass and neusis?
See http://en.wikipedia.org/wiki/Constructible_number and http://en.wikipedia....
- 19.3k
1
vote
0
answers
91
views
generating set of polynomial ring
I am considering the polynomials $P=P[x_1,x_2,\ldots,x_n]$ with coefficients in a ring $R$. Consider a subset $S=\{p_1,p_2,\ldots,p_k\}$ of $P$. There is a map $f\colon P[x_1,x_2,\ldots,x_k] \to P$ ...
- 63.9k
0
votes
0
answers
111
views
Totally isotropic space for bilinear pairing over ring
A duplicate of this:
Consider the following well-known inequality: Let $b$
be a non-degenerate symmetric bilinear pairing over a (finite-dimensional) $\mathbb F$-vector space $V$ and $W$
a totally ...
- 33.8k
1
vote
1
answer
105
views
Perturbing pole of Laurent polynomial/series in a single summand
I am working with the ring of Laurent polynomials $\mathbb{F}[X,X^{-1}]$ over $\mathbb{F}$ for some algebraically closed field $\mathbb{F}$ of any characteristic. I encountered a problem emerging from ...
- 21.5k
28
votes
6
answers
5k
views
Expressing $-\operatorname{adj}(A)$ as a polynomial in $A$?
Suppose $A\in R^{n\times n}$, where $R$ is a commutative ring. Let $p_i \in R$ be the coefficients of the characteristic polynomial of $A$: $\operatorname{det}(A-xI) = p_0 + p_1x + \dots + p_n x^n$.
I ...
- 1,446
2
votes
1
answer
318
views
Property of a commutative ring that is determined by the prime ideals of the ring
Robert Gilmer, in his paper "Commutative rings in which each prime ideal is principal", says:
Some well known theorems indicate that certain ideal-theoretic structure properties of a ...
- 63
2
votes
1
answer
121
views
Projective dimension and subrings
$\DeclareMathOperator\pd{pd}$Suppose that $R$ is a commutative ring and $R'$ is a subring of $R$ such that $R$ is a free $R'$-module of finite rank. Assume that both $R$ and $R'$ are regular local ...
- 12.9k
3
votes
0
answers
161
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Amalgamation of commutative subrings
Let $A$ and $B$ be commutative subrings of a non-commutative ring $X$.
Is there always a commutative ring $Y$ containing $A$ and $B$ preserving their intersection?
This is equivalent to ask if in the ...
- 39
8
votes
1
answer
639
views
A question on algebraic independence
Let $f_1,f_2,\ldots,f_n, g \in \mathbb{F}_q[x_1,...,x_m]$. Assume that $f_1,\ldots,f_n$ vanish at $0$, so that $\mathbb{F}_q[[f_1,...,f_n]]$ is a subring of $\mathbb{F}_q[[x_1,...,x_n]]$. Suppose that ...
- 28.7k
5
votes
2
answers
3k
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A direct proof that minimal primes are associated
It is a well-known theorem that, for a Noetherian ring $A$, the minimal primes of $A$ are among the associated primes of $A$; i.e., for every minimal prime $\mathfrak{p}$ of $A$, there is an element $...
1
vote
1
answer
229
views
Zero divisors in the boolean polynomial ring $\mathbb{F}_2[x_1,x_2,...,x_n]/(x_1^2-x_1,x_2^2-x_2,...x_n^2-x_n)$
Related to this question.
Let $n$ be positive integer and let $K$ be the boolean polynomial ring
$\mathbb{F}_2[x_1,x_2,...,x_n]/(x_1^2-x_1,x_2^2-x_2,...x_n^2-x_n)$.
For all $a$ in $K$ we have $a= -a$ ...
- 7,226
3
votes
1
answer
148
views
Formal étaleness along Henselian thickenings
Assume that $f:X\to Y$ is an étale map between smooth varieties and $(S,I)$ is a Henselian pair. Let $\alpha\in X(S/I)$. Can we say that the lifts of $\alpha$ to $X(S)$ are in bijection with the lifts ...
- 921
25
votes
6
answers
2k
views
What does the semiring of ideals of a ring R tell us about R?
Here is something I've wondered about since I was an undergraduate. Let $R$ be a ring (commutative, let's say, although the generalization to noncommutative rings is obvious). Ideals of $R$ can be ...
- 82.7k
6
votes
3
answers
1k
views
Complexity of solving systems of linear diophantine equations
It is "well known" that a matrix system $Ax=b$ where $A\in \Bbb Z^{m\times n}$, $x\in \Bbb Z^n,b\in\Bbb Z^m$ for some $m,n \in \Bbb N$, can be solved in polynomial time, using Smith/Hermite Normal ...
1
vote
0
answers
147
views
Gelfand's representation on matrices: construct maximal ideal in matrix algebra
I would like to see a constructive proof (some algorithm?) of the following statement:
Let $A_1, A_2, \dotsc ,A_k \in M_n(\mathbb C)$ be some commuting matrices, let $B$ be the commutative algebra (...
- 12.9k
0
votes
0
answers
109
views
Affine scheme over ring of meromorphic functions with finite poles on unit circle
I am looking into the set $S$ of meromorphic functions with a finite number of poles on the unit circle (i.e., rational functions with poles on the unit circle). I assume that any $h\in S$ has the ...
- 91
1
vote
0
answers
107
views
Computing simplicial resolution of rings
As the title says, I would like to ask how we can give "convient" simplicial resolutions of rings. In the category of modules this is often true: ifI have a ring $R$ and some ideal $I$ ...
- 4,418
1
vote
2
answers
558
views
Extension of the radical and radical of the extension of an ideal
If $A$ is a commutative ring, $I \subset A$ an ideal and $f:A \rightarrow B$ a ring homomorphism, then the extension of $I$, $I^e = \langle f(a): a \in I \rangle$ does not commute with the radical, I ...
- 1
1
vote
0
answers
63
views
Factorization of the symmetric function identity $E(t)=1/H(t)$ with the refined Euler characteristic polynomials of associahedra / Lagrange inversion
I've come across two matrix identities, flagged with daggers below, relating the two sets of elementary and complete homogeneous symmetric polynomials/functions via the two sets of refined Lah and ...
- 10.5k
8
votes
1
answer
1k
views
Why is $\operatorname{Spec}(\mathbb Z)$ supposed to lie over $\operatorname{Spec}(\mathbb F_1)$ rather than the other way around?
$\DeclareMathOperator\Spec{Spec}$I understand that one major motivation for the field with one element is supposed to be that there should be a map $\Spec(\mathbb Z) \to \Spec(\mathbb F_1)$, which has ...
- 32.5k
1
vote
0
answers
168
views
Does the symmetric algebra functor preserve inclusions?
Theorem: For any compact abelian group $G$, the homogeneous component $%
H^{2}\left( B_{G};%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
\right) $ of degree $2$ is naturally ...
- 1,367
2
votes
1
answer
138
views
height of sum of prime ideals
Suppose $R$ is a Cohen-Macaulay local ring and $P,Q$ are prime ideals in $R$. Let the height of $P$ and $Q$ be $m$ and $n$ respectively. Then is it true that the height of $P+Q$ is at most $m+n$?
- 38k
2
votes
1
answer
78
views
Is uniform dimension monotonic in quotients when there is a unique indecomposable injective?
The notion of uniform or Goldie dimension is something I’ve only seen discussed for categories of modules, but I believe the theory works the same way in any Grothendieck category $\mathcal C$. Recall ...
- 63.9k
6
votes
1
answer
367
views
Do groups of units change base nicely, assuming the fields are algebraically closed?
Let $K$ be an algebraically closed field. Let $X$ be an irreducible affine algebraic variety over $K$. Let $L/K$ be a field extension, where $L$ is also algebraically closed. Suppose the group of ...
- 148k
2
votes
1
answer
98
views
A perfect complex over a local Cohen--Macaulay ring whose canonical dual is concentrated in a single degree
Let $R$ be a complete local Cohen--Macaulay ring with dualizing module $\omega$. Let $M$ be a perfect complex over $R$. If the homology of $\mathbf R\text{Hom}_R(M,\omega)$ is concentrated in a ...
- 35.2k
0
votes
0
answers
329
views
Smooth morphisms under base change, Qing Liu's proposition 4.3.38
I have a concern about the first assertion in the proof of proposition 4.3.38 of Qing liu's "Algebraic Geometry and Arithmetic Curves". Referring to smooth morphisms, he says "The ...
- 6,367
3
votes
1
answer
224
views
Rings or algebras with many nilpotent elements and efficient computation
Crossposted from quantum.SE
where comment appears to suggest that solving modulo 2 might
be possible.
Searching the web for '"quantum computer" nilpotent'
returns many results, so maybe the ...
- 63.9k
4
votes
1
answer
173
views
If $\pi$ is a prime of a UFD $A$, is $\text{Spec }A$ a coproduct of $\text{Spec }A[\pi^{-1}]$ and $\text{Spec }A_{(\pi)}$ over $\text{Spec Frac }A$?
Let $A$ be a UFD (unique factorization domain) with fraction field $K$. Let $\pi\in A$ a prime. Let $A_{(\pi)}$ be the localization at the ideal $\pi$, and let $A[\pi^{-1}]$ be the localization w.r.t. ...
- 15.5k
5
votes
1
answer
191
views
Are module finite algebras over semiperfect rings again semiperfect?
Let $S$ be a Noetherian semiperfect ring (https://en.m.wikipedia.org/wiki/Perfect_ring). Let $R$ be a module finite associative $S$-algebra. Then, is $R$ also a semiperfect ring? (Clearly, $R$ is ...
- 35.2k
0
votes
0
answers
126
views
Nilpotent elements of $(\mathbb{Z}/n \mathbb{Z})[x_1,...,x_m]/\langle f_1(x_1,...,x_m),f_2(x_1,...,x_m),...f_k(x_1,...,x_m)\rangle$
This is generalization of the univariate case
and also related to open problem.
Let $n,k,m,B>1$ be positive integers and $f_1(x_1,...,x_m),f_2(x_1,...,x_m),...f_k(x_1,...,x_m)$ be polynomials with ...
- 25.4k
4
votes
0
answers
171
views
Nilpotent elements of $K=(\mathbb{Z}/n \mathbb{Z})[x]/f(x)$
This is related to an open problem.
Let $n$ be integer and $f(x)$ polynomial with integer coefficients and set $K=(\mathbb{Z}/n \mathbb{Z})[x]/f(x)$.
Let $S$ be the set of degree 2 nilpotent elements ...
- 63.9k
3
votes
1
answer
134
views
A question about minimal primes in the integral group ring of a finite abelian group
Let $G$ be a finite abelian group of order $n$ and let $R=\mathbb{Z}G$ denote the integral group ring of $G$. Let $R'$ denote the localization of $R$ with respect to the multiplicatively closed set ...
- 1
3
votes
1
answer
162
views
Question on the extension theorem from Humphreys' book on linear algebraic groups
On page 2 of Humphreys' book "Linear algebraic groups" he presents the "extension theorem". I will copy it below:
Extension theorem. Let $R/S$ be an integral extension, $K$ an ...
- 12.9k
0
votes
0
answers
117
views
A question on a system of quadratic polynomials
Consider the following system of quadratic polynomials $f_1,...,f_n \in \bar{\mathbb{F}}_2[x_1,....,x_n]$ :
$f_1 (\bar{x}) = x_1 + x_n^2 + q_1$
$f_i(\bar{x}) = x_i + q_i$ for $i \in \{2,...,n-1 \}$
$...
- 34.3k
1
vote
0
answers
166
views
Reference request showing that a very general Abelian variety $ A $ of genus $ g>1 $ has cyclic class group with ample generator
In Example of a $ \mathbb{Q} $-factorial, CM normal, projective, Mori dream space $ Z $ such that $ \operatorname{Cox}(Z) $ is integral and not CM I asked for an example of a Cohen Macaulay, normal, ...
- 912
31
votes
8
answers
21k
views
Reference book for commutative algebra
I'm looking for a good book in commutative algebra, so I ask here for some advice. My ideal book should be:
More comprehensive than Atiyah–Macdonald
More readable than Matsumura (maybe better ...
CommunityBot
- 1
2
votes
0
answers
73
views
From exact triangles in the stable category of maximal Cohen--Macaulay modules to short exact sequences
Let $R$ be a local Gorenstein ring. Let $\underline{\text{CM}}(R)$ be the stable category of maximal Cohen--Macaulay modules, it is known to carry a triangulated structure. My question is: If $M\to N\...
- 480
15
votes
1
answer
777
views
comparison of completion and Henselization in class field theory
Given a ring $R$ with maximal ideal $\mathfrak{m}$, we can form the localization $R_\mathfrak{m}$, the completion $\hat{R}_\mathfrak{m}$ or the Henselization $\hat{R}^h_\mathfrak{m}$ of $R$ with ...
- 781
0
votes
0
answers
71
views
"Approximating" ring of semi-invariants
I'm trying to calculate the semi-invariant ring for certain types of quivers. For a very brief introduction to semi-invariant rings of quiver please have a look at this wikipedia article at the ...
- 839
20
votes
2
answers
1k
views
Automorphisms of the hyperreals over the rationals and nontrivial automorphism groups
A classic result says the automorphism group of $\mathbb{R}$ (over $\mathbb{Q}$) is trivial. The proof is simple: every automorphism preserves squares, and hence fixes the positive reals, so it must ...
- 236k