Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,497 questions
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On transforming pair of bivariate polynomials to pair of univariate polynomials by applying polynomial map
We know that a polynomial map $f(x,y), g(x,y)$ is polynomial automorphism if there exists polynomials $p(x,y)$ and $q(x,y)$ such that $f(p,q)$=x and $g(p,q)=y$. Jacobian conjecture tries to ...
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354
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A Characterization of Closed Ideals in $C^{\infty}(\mathbb{R}^n)$
The space $C^{\infty}(\mathbb{R}^n)$ can be turned into a topological ring using the Whitney topology. Whitney's Spectral Theorem says that the closure of an ideal in this ring is the ideal of all ...
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generalization Abhyankar's lemma
This question is related to a question I already asked on MO (smooth quotient out of a singular variety?), but I realized later that the hypotheses where not precise enough in my former question.
Let ...
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1
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Commuting Matrices and the Weak Nullstellensatz
In the Wikipedia article on Hilbert's Nullstensatz,
http://en.wikipedia.org/wiki/Hilbert%27s_Nullstellensatz
the following application of the Weak Nullstensatz is mentioned:
Commuting matrices
...
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2
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713
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How do you compute the space of lifts of an E-infinity map?
Let X, Y and B be $E_\infty$ spaces, and let $p: X \rightarrow Y$ and $f: B \rightarrow Y$ be $E_\infty$ maps. We can ask for the space of lifts of f across p, that is the space of $E_\infty$ maps $g:...
- 3,103
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0
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74
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self-cogenerator rings
Let $\mathbb{U}$ be a non-empty set (class) of objects of a
category $C$. An object $B$ in $C$ is said to be cogenerated by
$\mathbb{U}$ or $\mathbb{U}$-cogenerated if, for every pair of
distinct ...
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1
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455
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Iwasawa theory for Mazur's deformation ring R
The ideal class group $\mathrm{Cl}({\cal O}_K)$ and Mazur's deformation ring $R(\overline{\rho})$ for a number field $K$ are said to be similar to each other.
Let ${\Bbb Q}_{\infty}$ be the unique ...
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3
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1
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287
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I need to refind a reference on multigraded Hilbert series
I found a theorem about multigraded Hilbert series stated as follows:
Let $R$ be a Noetherian multigraded algebra $R:=\bigoplus_{j\in\mathbb{N}^m}{R_j}$ over $R_0=\mathbb{C}$. If $R$ is generated by $...
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1
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Canonical module of rees algebra
[Example 4.27, Integral Closure, Rees Algebras, Multiplicities, Algorithms] by Vasconcelos, says that if $I=(f_1,\ldots,f_g)$ is an ideal generated by a regular sequence with $g\ge 2$ then the ...
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108
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Structure of valuations on $\mathbb{F}_q(X,Y)$?
I'm looking to construct all valuations on $\mathbb{Q}(X,Y)$ extending the p-adic valuation on $\mathbb{Q}$ and understand their structural properties. In doing this, to obtain 3 dimensional valuation ...
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1
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1
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457
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Integrally closed
Is there any idea to prove that $F + xk[[x]]$ is not integrally closed when the field $k$ is a proper extension of the field $F.$
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Iwasawa invariants
Suppose $M$ is a finitely generated torsion $Z_p[[T]]$-module; the torsion comes from the $\mu$-invariant and the $\lambda$-invariant. Consider $M/(p)$ and $M[p]$ ($p$-torsion of $M$) which are $F_p[[...
- 1,209
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1
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Can a single DVR witness all specializations on a variety?
If $X$ is a noetherian scheme with points $x$ and $\xi$ so that $x$ is in the closure of $\{\xi\}$, then there exists a discrete valuation ring $V$ and a map $Spec(V)\to X$ sending the generic point ...
- 24.1k
4
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1
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Flat family of normal schemes over a normal base
Let $f \colon X \to Y$ be a flat morphism of schemes over $\mathbb{C}$. Suppose that $Y$ is normal and that the fibers over the closed points of $Y$ are all normal.
Can I say something about the ...
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2
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542
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Artin/Popescu approximation for (some) big rings
Fix a prime number $p$. Let $A = \overline{\mathbf{Z}_p}$ be the integral closure of the $p$-adic integers $\mathbf{Z}_p$ in some fixed algebraic closure of its fraction field, and let $B$ be the $p$-...
- 633
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Unibranch partial normalization
In a paper I recently read something about the "unibranch partial normalization" of a curve.
Say, $R$ is a local integral domain with maximal ideal $\mathfrak{m}$ and fraction field $K$. Is it ...
user78294
2
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2
answers
827
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Reduced varieties with no regular points?
Let $k$ be a field. Let $X$ be a reduced $k$-scheme of finite type. If $X$ is geometrically reduced, then it is a basic result that $X$ has a regular point (i.e. the local ring at that point is ...
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2
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2
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287
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If $M$ is a positively graded finitely generated module of dim 0, then why $R_{+}^{t}M=0$ for some $t\in \mathbb{N}$?
Let $M$ be a positively graded finitely generated module over a positively graded commutative ring $R$. Assume that $R_0$ is a local ring with maximal ideal $m_0$. Let $d$ be the Krull dimension of $M$...
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Does integral closure commute with pushforward
Suppose that $\pi : Y \to X$ is a proper birational morphism between normal varieties (schemes, whatever). Suppose that $I$ is an ideal sheaf on $Y$.
One can form $\pi_* I$ and construct an ideal ...
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2
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1
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155
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On conflicting descriptions for tor of a local cohomology group
Let $X$ be a smooth projective surface and $C$ a Cartier divisor on $X$. Denote by $\mathcal{H}^1_C(\mathcal{O}_X)$ the sheaf associated to the presheaf $U \mapsto H^1_{C \cap U}(\mathcal{O}_X|_U)$. ...
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When every module is a scalar extension?
Let $A \subseteq B$ be commutative noetherian domains.
Of course, if $M$ is an $A$-module, then $M \otimes_A B$ is a $B$-module.
I am curious to know if there exist additional conditions on $A$ and $B$...
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3
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1
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The global (weak) dimension of formal power series rings
Given a commutative ring $R$, what are relations between w.gldim$(R)$ and w.gldim$(R[[x]])$ (gldim$(R)$ and gldim$(R[[x]])$)?
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is every finitely n-presented (S^{-1})R-module a localization of a finitely n-presented R-module?
Let S be a multiplicative set in a ring R. We can see that every finitely generated $(S^{-1})R$-module is a localization of a finitely generated R-module.
Then, more generally, is every finitely n-...
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331
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Idempotent ideal in ring of continuous functions
Is there any equivalence conditions under which an ideal $I$ in ring of continuous functions be be an idempotent ideal?
- 1
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1
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637
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When is a local Artin C-algebra a subring of C[t]/t^n
Let $A$ be a local ring over $\mathbb{C}$, which moreover is a finite dimensional $\mathbb{C}$-vector space.
When is $A$ a subring of $\mathbb{C}[t]/t^n$?
What does the minimal ...
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481
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Is there some algorithm for calculating the least number of generators of an ideal in a polynomial ring?
My question comes from the context of algebraic geometry. By Krull's principal ideal theorem, the number of generators of the defining ideal of a variety gives an upper bound of its codimension.
...
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174
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if $ \lambda (I)= \dim R$, can one claim that $I$ is an $m$-primary ideal?
definition from Bruns-Herzog:
It is easy to see that if $I$ is a $m$-primary ideal of $R$ then $ \lambda (I)= \dim R$. I wonder if the converse is true:
if $ \lambda (I)= \dim R$, can one claim ...
- 1,355
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1
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534
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Checking flatness using radical ideals
Let $R$ be a commutative ring and $M$ a not necessary finitely presented $R$-module. I am looking for a prove or a counterexample to the following statement: $M$ is flat as an $R$-module if and only ...
- 73
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172
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Local cohomology commuting with fiber
Let $A$ be a nice commutative ring (say, $A=k[t_1,\ldots , t_n]$, ring of polynomials over an algebraically closed field $k$).
Let $M$ be an $A[x]$-module, which is finitely generated as an $A$-...
- 5,562
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203
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Smoothness of $A \to A[T]/(h)$
Let $A$ be a commutative noetherian domain of characteristic zero, $T$ an indeterminate, $h \in A[T]$, $B= A[T]/(h)$ and assume $B$ is also a domain.
When $B$ is (formally) smooth over $A$?; namely, ...
- 2,837
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0
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Closedness of the range of the distorsion of the multiplicative monoid of a number field
Let $H$ be a multiplicatively written monoid with identity $1_H$. An atom of $H$ is an element $x \in H \setminus H^\times$ such that $a \ne xy$ for all $x, y \in H \setminus H^\times$, where $H^\...
- 10.5k
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Change of grading used in the paper "The diagonal subring and the Cohen-Macaulay property of a multigraded ring" by Eero Hyry
I am currently reading the paper "The diagonal subring and the Cohen-Macaulay property of a multigraded ring" by Eero Hyry. I do not understand the following part in Lemma 1.1. here.
Let $T=\...
- 1,713
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4
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482
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analogue of a set with n binary operations
So a group is a type of structure with one binary operations that satisfies some list of axioms. A ring is a structure that has two binary operations that satisfy some list of axioms. Do there exist ...
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208
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elementary question on a completion of a ring
Let $k$ a field, and $k[\epsilon]=k[X]/(X^{2})$ , what is the completion of the ring $k[\epsilon][t]$ with respect to the ideal $(t^{2}+\epsilon)$?
- 3,472
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$\inf\{i\in \mathbb N \cup \{0\}\cup\infty\mid Ext^i_R(R/I,R)\neq 0\}=0 ?$
Let $R := k[x_1, \cdots, x_n, \cdots]/(x_1^1, \cdots , x^n_n, \cdots),$ where $k$ is a field. Set $I:=(x_1, \cdots, x_n, \cdots)$. the questions are:
Is $\inf\{i\in \mathbb N \cup \{0\}\cup \{\...
- 1,355
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1
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integral closure of m-primary ideals
I need help with this excercise
Let $k[X_1,\ldots,X_d]$ be the polynomial ring in $X_1,\ldots,X_d$ over a field $k$, and let $F_1,\ldots,F_m$ be forms of degree $n$. Assume that $(X_1,\ldots,X_d)=\...
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How to determine whether the following sum is nonzero for a given multivariate polynomial?
My research field is combinatorics. I am not very good at Algebra. So I want to ask for a given real multivariate polynomial $f(x_1,x_2,\cdots,x_n)$, is there any algebraic method to compute whether ...
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7
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1
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Semiring of vector bundles on $\mathbb{C}\mathbb{P}^1$
Consider the semiring
$$\mathbb{N}[H,H^{-1}]/(H^p+H^q = H^{p+q}+1)_{p,q \in \mathbb{Z}}.$$
Is it finitely presentable? Is there any simplification of the relations (except for $p \geq q \geq 0$)?
...
- 63.1k
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2
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416
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Computing toric ideals via saturation and Groebner bases of toric ideals
About a month ago I asked this question on math.stackexchange and unfortunately there was no response. Perhaps someone here knows the answer.
Let $A \in \mathbb{Z}^{m \times n}$ be a matrix of full ...
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264
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Are finitely presented modules finitely presentable? [closed]
Over a ring $R$ we have a notion of finitely presented module, namely:
Definition 1 A module $F$ is finitely presented if there are $m$, $n$ positive integers such that $R^m\to R^n\to F\to 0$ is ...
- 31
8
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1
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961
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Vanishing constant term in powers of a Laurent polynomial
This is motivated by idle curiosity. I recently learned a result of Duistermaat and Van Der Kallen in "Constant terms of powers of a Laurent polynomial" which says that:
If the constant term of $f^...
- 85.6k
4
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1
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Existence of Factor rings of UFDs which are UFDs
Suppose that $X=Spec(A)$ is an affine variety over an algebraically closed field $k$ which is normal and such that $Cl(X)=0$.
I am interested in hypersurfaces of $X$ which again satisfy this condition....
- 43
1
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1
answer
688
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Direct image of an ideal sheaf along a blow-up
Suppose that $I\subseteq\mathbb{C}[x_0,\ldots,x_n]$ is a saturated homogeneous ideal. Let $\mathcal{I}\subseteq\mathcal{O}_{\mathbb{P}^n}$ denote the corresponding coherent ideal sheaf, and then let $$...
user25585
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0
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94
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Is the tensor product of two commutative semiprime Q-algebras semiprime?
A ring is semiprime if it has no non-zero nilpotents. Let $Q$ denote the rational numbers and $A,B$ be a pair of commutative semiprime $Q$ algebras. Is $A\otimes_Q B$ semiprime? It is well known that ...
- 383
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Co-filtered and pro-finite manifolds, filtered algebras, and differential calculus on them
I've come across a lot of questions (and nice answers) on MO, concerning infinite-dimensional manifolds and differential calculus over them, but nothing suiting the simpler and special case I have in ...
- 2,527
7
votes
2
answers
1k
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any software to compute multivariable resultant?
Are there any software to computer resultant for a system of equations (more than 2) with more than 2 variables?
- 95
5
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1
answer
255
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computing the nonnegative part of a $\mathbb{Z}$-graded ring
Let $R = \bigoplus_{n \in \mathbb{Z}} R_n$ be a $\mathbb{Z}$-graded commutative ring with nonnegative part $R^+ = \bigoplus_{n \geq 0} R_n$ and nonpositive part $R^- = \bigoplus_{n \geq 0} R_{-n}$. By ...
- 211
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1
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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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1
answer
1k
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Conjugate Matrix
Let $A$ be a nilpotent square matrix, $J$ be the antidiagonal matrix with 1's on the secondary diagonal (i.e. $J^{2}=E$) and let $B=J A J.$ Suppose we conjugate the matrices $A,B$ by a matrix $...
- 301
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0
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132
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Local diffeomorphism (étale maps) in terms of infinitesimal tubular neighborhood?
In this MO question I asked for some help with several definitions of formally etale maps. The definitions I'm asking about are 'isomorphism of tangent spaces', i.e the square below is a pullback
$$\...
- 10.5k