Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
1,707
questions with no upvoted or accepted answers
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Betti numbers of a polynomial ideal after homogenization
Let $I \subset k[x_1,\ldots,x_n]$ be an ideal in a polinomial ring over a field $k$. Assume that $I$ is quasihomogeneous (that is, $I$ is not homogeneous with the usual grading, but it is homogeneous ...
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75
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Normal simple ring extensions
Let $R$ be a $k$-algebra, $k$ is an algebraically closed field of characteristic zero.
Assume that $R$ is an integral domain or a UFD (being a UFD makes things easier),
$a$ is an algebraic element ...
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92
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When do the kernels of module homomorphisms between rings whose kernels contain a given fixed ideal contain every prime ideal over it?
$\DeclareMathOperator{\Hom}{Hom}$All our rings are commutative with unity and, if necessary, we can suppose that they are actually polynomial rings over a field in finitely many variables where the ...
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30
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Making a generating set of a section of a graded polynomial $R$-module coming from a quotient into a basis of a quotient by higher degree polynomial
Denote the graded rings $R:=\mathbb{R}[x_{1},\dots x_{n}]$ and $S:=R[x_{0}]$ adding the homogenizing variable $x_{0}.$ Consider $h\in S$ a homogenous polynomial of degree $d$ with leading coefficient $...
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191
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Surjection from finite rank free $R$-module to finitely generated $R$-module and basis associated to generator set
Suppose the we have an epimorphism $s\colon M\to N,$ where $M$ is a free $R$-module of rank $r$ and $N$ is a finitely generated $R$-module, such that there exists a basis $B:=\{m_{1},\dots, m_{r}\}$ ...
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208
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Defining path on the prime spectrum
If $p$ and $q $ are two prime ideals of a commutative ring $R $ such that $p \subseteq q$, then we can easily define a continuous function (a path) $f$ from the unit interval $ [0,1]$ to the prime ...
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184
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Strongly graded rings
In Theorem 3.1 of Graded rings over arithmetical orders, the authors prove that for a strongly $\mathbb{Z}$-graded ring $R$, if $R_0$ is left and right Goldie and a maximal order in its (classical) ...
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72
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Initial Ideal have linear resolution
The ideal $J$ is generated by the set $B$ of all binomials and
$B$ forms a Grobner basis for $J$ with respect to the lexicographic order. Let $in(J)$ denote the initial ideal of $J$.
For non-negative
...
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81
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Defining cluster algebras of finite type $\mathrm{A}$ by generators and relations
Consider a cluster algebra of finite type $\mathrm{A}$. The set of all (distinct) cluster variables is of finite cardinality, denote it by $k$, for such algebra. Is it true that, for an arbitrary ...
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106
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Integer factorization given modular square root of 2
Let $N$ be composite. It is well-known that if $x^2 \equiv 1 \pmod N$, and $x \neq \pm 1 \pmod N$, then a factor of $N$ is easily found by computing gcd($N$, $x + 1$). I'm curious if there is a ...
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63
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Continous morphisms of a local field with conditions in positive characteristic
Let $P$ be a an irreducible polynomial of $k:=\mathbb F_q(T)$, $\Omega_P$ be the completion of an algebraic closure $\overline{k_P}$ of $k_P$, the completion of $k$ for the topology induced by the $P$-...
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70
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Gluing categorical limit over subgraphs
Let $C$ be a category, and $\Gamma$ a graph in $C$. Under good conditions it makes sense to talk about the limit $\lim \Gamma$ of $\Gamma$ in $C$.
Suppose $\Gamma$ is the union of two subgraphs $\...
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63
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Characteristic of ring completions
This may be a completely trivial question, but I haven’t seen it stated in any of the references I checked. Is the characteristic of a ring $R$ equal to that of its completions? This is true for the ...
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409
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Examples of almost Dedekind domains that are not Dedekind
All I know about almost Dedekind domains (which I have come to learn about only recently) is that they are integral domains whose localization at every prime is a discrete valuation ring. In other ...
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184
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Existence of integral extension of DVR satisfying some conditions
Let $X^{\prime}$ and $X$ be integral noether schemes over $\mathbb{C}$, and $p:X^{\prime}\rightarrow X$ be a surjective morphism.
Let $R$ be any discrete valuation ring over $\mathbb{C}$ with its ...
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200
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Is there a natural topology on $\mathbb{C}(t)[x_1,\ldots, x_n]$ with this property?
Is there a good topology on $A=\mathbb{C}(t)[x_1,\ldots, x_n]$ so that $A$ is a topological algebra with the following property:
For any $N>0$ and a polynomial $F\in\mathbb{C}[x_1,\ldots, x_n]$ ...
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79
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Completion of $K$-algebra of finite type with respect to the residue norm
Let $K$ be a non-archimedean field. For $n \in \mathbb{N}$ let
\begin{equation*}
T_n=\{ \sum_{\nu_1, \ldots, \nu_n \geq 0 } a_{\nu_1, \ldots, \nu_n }X_1^{\nu_1}\cdots X_n^{\nu_n} \in K[\![X_1,\ldots, ...
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60
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Local cohomology for infinitely generated modules
Let $R$ be a local (with maximal ideal $m$) commutative Gorenstein ring of dimension $d$.
Then for any $0 \leq i \leq d$ there are isomorphisms for the local cohomology:
$H_m^i(M) \cong D Ext_R^{d-i}(...
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253
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Almost ring theory and derivations
I don't understand the definition of $\boldsymbol{\Omega}_A$ in the context of almost rings. In Gabber and Ramero https://arxiv.org/pdf/math/0409584.pdf it is covered in 9.6.12. How is $\boldsymbol{\...
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173
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What are the irreps in this canonical action of $\operatorname{PGL}_2(F_q)$?
Consider the permutation action of $\operatorname{PGL}_2(\mathbb F_q)$ on $\mathbb P^1(\mathbb F_q)$ by fractional linear transformations. We can consider the associated (complex) representation of ...
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92
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What would be the quotient groups $U_{\mathrm{gen}}/U_{\mathrm{gen}}^{(n)}$ and $U_{\mathrm{gen}}^{(n)}/U_{\mathrm{gen}}^{(n+1)}$?
Let $K \supseteq \mathbb{Q}_p$ be a $p$-adic field with ring of integer $O$ and maximal ideal $m$. Let $O^*$ be the group of units in $O$.
Consider the group of units $U^{(0)}=U=O^*$ and $U^{(n)}=1+m^...
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309
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A composition of a simple extension and a separable extension is simple
Let $K/L/M$ be a tower of finite field extensions with $K/L$ separable and $L/M$ simple (in the sense of being generated by a single element). How does one show that $K/M$ is also simple?
I know that ...
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90
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Mod $N^2$ evaluation of a polynomial defined by first $N-1$ roots
Given a prime $N$ and integer $g$, where $g$ is able to generate the multiplicative subgroup $(\mathbb{Z}/N^2\mathbb{Z})^*$, I am interested in any results simplifying or evaluating $f\in (\mathbb{Z}/...
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126
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On the dual version of an isomorphism of spectral sequence term (from Cartan and Eilenberg)
I'm trying to take spectral sequences as a black box for application in commutative algebra and I admit that I haven't really gone through (or understand) all the proofs of all the isomorphisms ...
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88
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Automorphisms of the completion of a strict henselian local ring $R$ which come from automorphisms of $R$
Let $A\rightarrow R$ be a local homomorphism of Noetherian strict henselian local rings with completions $\hat{A},\hat{R}$.
Let $u\in R^\times, x\in R$ be such that there is a unique $\hat{A}$-linear ...
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221
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Non-zero divisors on an $I$-completely flat module
Let $A$ be a commutative ring (not necessarily Noetherian), $I=(f_1, f_2, \dots, f_n) \subseteq A$ a finitely generated ideal that is generated by a regular sequence. Let $M$ be an $A$-module, and ...
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145
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A series that is algebraic?
This question is a follow-up of question A series that is rational? . Let $k=\mathbb F_q(T)$. Can one prove (or disprove) that the series $\sum_{n\ge0}(1-TX^{q^n})Y^{q^n}\in k[[X,Y]]$ is algebraic ...
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154
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When every localization of the polynomial ring over a ring has finitely many idempotents
Let $R$ be a commutative ring such that every localization ring $R_r$ has finitely many idempotents for each non nilpotent element $r\in R$. Why dose every localization ring $R[x]_{f(x)}$ have ...
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98
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Notation question: bigraded direct sum of graded objects
In some work I'm doing I have two graded modules $M$ and $N$ (graded on $\mathbb Z$, say) and need to take, not the usual direct sum, but the bigraded sum consisting of all $M_p \oplus N_q$ (so graded ...
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70
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Constructive factorisation of null-homology map through acyclic complex
Let $f: C \rightarrow D$ be a maps of chain complexes on an idempotent complete additive category with all kernel or cokernel (or chain complexes on abelian category).
If $f$ induces a null map in ...
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88
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Commutative square of module of differential is cartesian?
Is it true that the following square is Cartesian? $\require{AMScd}$
\begin{CD}
R @>{d}>> \Omega^{1}_{R} \\
@VVV @VVV\\
\widehat{R} @>{ \widehat{d}}>> \Omega^{1}_{\widehat{R}}
\end{...
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65
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When minimal prime ideals are maximal with respect to not containing an element
Let $\{ P_i \}$ be the set of all minimal prime ideals of a commutative ring $R $. Is there any conditions on $R $ under which there exists an element $x\in R $ such that $P_i $ is an ideal of $R $ ...
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28
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Modified straightline complexity of almost square of sums
Assume every linear operation (such as inner product with constant vector) can be performed in one step and multiplication by variables (quadratic operation) can be performed in one step.
We know the ...
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160
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When localization is indecomposable
We know that if $R $ is a domain then any localization of $R $ at any multiplicative subset of $R $ is indecomposable, that is, has no non trivial idempotents. Now let $R $ be a commutative ring with ...
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120
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Self-intersecting irreducible real projective elliptic surface
I will say that a homogeneous polynomial $P(X)\in\mathbb{R}[X]$ ($X=(X_1,\ldots,X_n)$) is elliptic if its zero locus in $\mathbb{R}^n$ is $\{0\}$.
I will say that the zero locus of a homogeneous ...
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102
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Meaning and/or source of polynomials residually of the form $x^n(x-1)$ in Gabber's characterization of Henselian pairs?
Lemma 09XI in the stacks project includes a characterization (#5) by Gabber of Henselian pairs $(A,I)$: first, $I\leq \mathrm J(A)$ is contained in the Jacobson radical and second, every monic $f\in ...
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339
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Kähler differential of completion of algebra
Let $(R, \mathfrak{m}) $ be a local $k$- algebra and $\Omega^{1}_{R}$ denote the module of Kahler differential. Does the canonical map $ \Omega^{1}_{R} \otimes_{R} \hat{R} \rightarrow \Omega^{1}_{...
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On a structural decomposition of polynomials based on integral roots
Given an irreducible polynomial of structure $$f(x,y)=\sum_{\substack{i,j\in\{0,1,2\}\\i+j\leq3}}a_{i, j}x^iy^j\in\mathbb Z[x,y]$$ with $a_{2,1}a_{1,2}a_{1,1}a_{1,0}a_{0,1}a_{0,0}\neq0$ if $f(m,n)=0$ ...
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71
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Compatibility with multiplication of a cyclic order on a ring
I am copying my question from here: https://math.stackexchange.com/q/3233462/427611.
Is it correct that $\mathbb Z/3\mathbb Z$ and $\mathbb Z/4\mathbb Z$ are the only rings with three or more ...
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893
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Intersection of principal ideals
Let $x,y$ be nonzero elements in a commutative ring $R$. Is $(x)\cap (y)$ always finitely generated?
What if we further assume that $R$ is an integral domain? Can we construct non-Noetherian non-local ...
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95
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Degree reduction in decompositions of multivariate polynomials
Is the following statement true?
Let $m,n,d$ be natural numbers. Then there exists a natural number $D=D(d,m,n)$ with the following property: If a polynomial $P(x_1,\dots,x_n)$ of total degree $d$ ...
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152
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A structure on the groupoid of algebraic closures
Given a field $k$ let $\Omega(k)$ be the set of algebraic closures of $k$.
$\Omega(k)$ is obviously a groupoid. At each element $\bar{k}$ of $\Omega(k)$ we have its automorphism group over $k$, ...
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99
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Exactness of a certain sequence
Let $R$ be a commutative unitary ring and $I_1,..., I_n$ ideals in $R$. For each $p\in\{0,...,n-1\}$ consider the direct sums $\bigoplus_{i_0<...<i_p} I_{i_0}\cap...\cap I_{i_p}$ and define an $...
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224
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Categorical view of Hilbert’s Nullstellensatz, and Zariski topology
Let k be algebraic closed field. then $\mathbb{A}_n(k)$ as $\operatorname{Hom}(k_n,k)$ and $V(\alpha)$ as $\operatorname{Hom}(k_n/\alpha,k)$ which is true by using noether normalization theorem. so ...
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81
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Concerning $\mathbb{C}(s_1,s_2,s_3,y)=\mathbb{C}(x,y)$, where $s_1,s_2,s_3$ are symmetric
Perhaps the following question is not in the level of MO questions, but it has not received comments in MSE, so I ask it here also:
Let $\beta: \mathbb{C}[x,y] \to \mathbb{C}[x,y]$ be the involution ...
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87
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Characterizing subfields $\mathbb{C}(u,v) \subseteq \mathbb{C}(x,y)$ invariant under an involution
Let $\iota$ be an involution on $\mathbb{C}(x,y)$, namely, a $\mathbb{C}$-algebra automorphism of $\mathbb{C}(x,y)$ of order two.
Examples of involutions: $\alpha: (x,y) \mapsto (y,x)$, $\beta: (x,y) ...
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103
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formal smoothness and McQuillan formal schemes
Let $k$ be an algebraically closed field, $A\rightarrow B$ be a continuous map of weakly admissible topological $k$-local algebras.
We assume that it is formally smooth and topologically of finite ...
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60
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powers of linear forms in projections of complete intersections in codimension 3
Let $I\subset \mathbb{C}[x_0,x_1,x_2]=:A$ be a complete intersection, $I=(p_1,p_2,p_3)$, $p_i$ homogeneous all of the same degree d
for some $d>2$.
Let $l$ be a general linear form and let $J$ ...
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126
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Algebraic structures on graphs
There are many algebraic structures linked to graphs.
For example one can find zero divisor graphs $[1]$, $[2]$ and many other graphs.
Does there exist any survey paper which characterizes all the ...
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106
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How to obtain a linear basis from a Groebner basis?
Let $A$ be a finite dimensional algebra generated by $x_1, \ldots, x_n$ subject to certain relations $I_1, \ldots, I_m$. Could we obtained a linear basis $B$ consisting of monomials in $x_1, \ldots, ...