Questions tagged [ac.commutative-algebra]

Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.

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Sections of non-reduced schemes

Let $X$ be an affine, irreducible (complex), generically reduced, scheme containing an embedded point, say at $x \in X$. Suppose further dimension of $X$ is strictly positive (can assume to be one ...
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37 views

Automorphisms of $\mathbb{C}[x, y, z]$ over $\mathbb C[x]$

What are the automorphisms of $\mathbb{C}[x, y, z]$ fixing $\mathbb{C}[x]$? I.e. those automorphisms $\phi:\mathbb{C}[x, y, z]\to\mathbb{C}[x, y, z]$ s.t. $\phi(x) = x$. I am interested in complete ...
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88 views

Varieties with everywhere good reduction isomorphic over every completion

Let $R$ be the ring of integers in a number field. Let $X$ and $Y$ be smooth and proper schemes over $R$. For a maximal ideal $\mathfrak{m}\subset R$ denote the localization of $R$ at $\mathfrak{m}$ ...
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73 views

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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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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128 views

Finitely generated commutative rings with the same profinite completion

Let $R_1$ and $R_2$ be two finitely generated commutative rings. Assume that their profinite completions are isomorphic: $\widehat{R_1}\cong \widehat{R_2}$. Suppose that $R_1$ is a domain. Does ...
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1answer
101 views

Categorical Kähler differentials and the Leibniz rule

From nlab, the module of Kähler differentials over some category $\mathcal{C}$ is the free functor: $$\Omega: \mathcal{C} \to \mathsf{Mod_{\mathcal{C}}}$$ left-adjoint to the (forgetful) embedding: $$...
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1answer
129 views

When is the module of Kahler differentials free?

As the title says, when is the module of Kahler differentials a free module? In particular, are there known conditions or criterions that could be met that ensures that it will be free? For example, ...
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57 views

A field with non zero characteristic is finite? [closed]

a field with non zero characteristic is finite?? If it is false, does somebody know contraexamples?
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199 views

Rings $R$ such that every [regular] square matrix with entries in $R$ is equivalent to an upper triangular matrix

Let $\text{M}_n(R)$ be the ring of $n$-by-$n$ matrices with entries in a commutative unital ring $R$. Theorem III in C.R. Yohe, Triangular and Diagonal Forms for Matrices over Commutative ...
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141 views

Question about polynomials in $\mathbb{C}[x, y, z]$

What can be said about pairs of polynomials $P, Q\in\mathbb{C}[x, y, z]$, such that $\frac{\partial P}{\partial y}\frac{\partial Q}{\partial z} - \frac{\partial P}{\partial z}\frac{\partial Q}{\...
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130 views

Degrees of syzygies of points in $\mathbb P^2$

Let $X$ be a collection of points in $\mathbb P^2$ over the complex numbers. Let $I_X$ be the defining ideal. I am interested in knowing when: The syzygies of $I_X$ contains no linear forms. Since ...
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Gröbner basis and integer programming

I was studying about grobner basis and observed one application of it in integer programming which is pretty much amazing but tougher than available methods like branch bound. Then what is the benefit ...
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79 views

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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Solving Problems of product Series [closed]

Is there any general method to solve various types of Product Series Problems including the pi product forms?
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Can the conductor of a local, unramified, Cohen-Macaulay domain ever be contained in a parameter ideal?

Let $(R, \mathfrak m)$ be a local Cohen-Macaulay domain which is analytically unramified (i.e. the $\mathfrak m$-adic completion of $R$ is reduced). Let $\bar R$ be the integral closure of $R$ in the ...
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On inequality between number of generators of ideals

Let $(R, \mathfrak m,k )$ be a regular local ring of dimension $3$ with infinite residue field $k$. Let $I$ be an $\mathfrak m$-primary ideal such that for every ideal $J$ containing $I$, it holds ...
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Certain endomorphisms of $\mathbb{C}(x,y)$

Let $f: (x,y) \mapsto (p,q)$ be a $\mathbb{C}$-algebra endomorphism of $\mathbb{C}(x,y)$ satisfying the following two conditions: (i) $\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in \mathbb{C}-\{0\}$. (...
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191 views

Ideals generated by regular sequences

In Vasconcelos' paper (Ideals generated by R-sequences), he proved If $R$ is a local ring, $I$ an ideal of finite projective dimension, and $I/I^2$ is a free $R/I$ module, then $I$ can be ...
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1answer
287 views

Are projective modules over a certain localised Laurent polynomial ring free?

Let $R=\mathbb{Z}[t^{\pm 1}]$ be the ring of Laurent polynomials, and let $S \subset R$ be the multiplicative subset generated by the polynomial $t-1$. I am interested in the ring $S^{-1}R=\mathbb{Z}[...
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1answer
148 views

Frobenius splitting for an excellent, non $F$-finite, $F$-pure hypersurface

Let $p$ be an odd prime. Let $k$ be a field of characteristic $p$ such that $[k:k^p]=\infty$ (i.e. $k$ is not $F$-finite ) . Also assume that $-1$ is not a square in $k$ . Consider the homogeneous ...
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Hilbert series of special linear sections of Grassmannian $Gr(2,n)$

Consider the Grassmannian $\operatorname{Gr}(2,n)$. I want to know Hilbert series of $H_1 \cap H_2 \dots \cap H_m \cap \operatorname{Gr}(2,n)$ in the Plücker embedding of $\operatorname{Gr}(2,n)$, ...
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102 views

Do Poincaré duality algebras need to be defined over a field?

I asked the below question here on MSE, but after some time and a bounty offering I have not received an answer. A graded commutative, connected $\mathbb{k}$-algebra $A$ is called a Poincaré duality ...
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255 views

Does the category of local rings with residue field $F$ have an initial object?

Let $F$ be a field. Does the category $C_F$ of local rings $R$ equipped with a surjective morphism $R\longrightarrow F$ have an initial object? This is, for instance, true if $F=\mathbb{F}_{p}$ for ...
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242 views

Algebraic analog of a geometric result

There is a famous topological result: Let $X$ be a smooth manifold of dimension $n$, $E$ be a vector bundle of rank $k > n$, then $E$ contains a trivial line bundle. So, I guess that (...
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2answers
250 views

Decomposing a polynomial ring into Specht Modules

Let $S_{\pi}$ where $\pi$ is an integer partition of $n$, denote the Specht module corresponding to $\pi$. I am trying to decompose the set of all homogeneous polynomials in $x_1,x_2,...,x_n$ ...
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1answer
117 views

Splitting of short exact sequence in the category of finitely generated modules over a commutative Noetherian ring

In the category of finitely generated modules over a commutative Noetherian ring, the splitting of a short exact sequence can be checked locally at the maximal ideals of the ring. One reference for ...
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143 views

Constructing non-trivial epimorphisms from commutative rings

As the title suggests, I wonder if anyone can share some techniques or references for constructing interesting epimorphisms from generic commutative rings. Generic is largely open to interpretation, ...
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1answer
52 views

Extension of Dedekind domains and their codifferent

Let $A\subset B$ be a finite extension of Dedekind domains. Let $0\neq b\in B$ and $0\neq a\in A$ such that $(a)=(b)\cap A$. In particular, we have $a=b\cdot c$ for some $c\in B$. Now for any $A$-...
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92 views

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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59 views

Does Kahler differential commute with completion

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}_{\hat{R}} \rightarrow \varprojlim \Omega^{1}_{...
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5answers
694 views

Computation of fraction field of formal series over the integers

What is the fraction field $K$ of the domain $\mathbb Z[[X]]$? It is strictly smaller than the field of Laurent series $L=\operatorname {Frac}\mathbb Q[[X]]$, since $\sum_{i\geq 0}\frac {X^i}{i!}\in ...
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1answer
93 views

Geometric meaning of colocalization of modules?

Let $A$ be a commutative ring and $S\subset A$ a subset. A localization of $A$ at $S$ is defined as a ring morphsim $A\to A[S^{-1}]$ which is initial with respect to inverting $S$. Similarly, a ...
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1answer
129 views

What Stanley-Reisner rings are $\mathbb{Q}$-Gorenstein?

Let $\Delta$ be a simplicial complex and let $R$ be the associated Stanley-Reisner ring. We can characterize when $R$ is Cohen-Macaulay or when $R$ is Gorenstein in terms of the topology of $\Delta$ (...
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1answer
113 views

Adjunctions capturing duality between ideals and saturated monoids in a commutative ring?

Let $R$ be a commutative ring. A saturated monoid in $R$ is a multiplicative submonoid $S\subset R$ which is closed under divisors, i.e $xy\in S\implies x\in S$. This is the converse of the analogous ...
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intersection of left and right orthogonals of a module

$\DeclareMathOperator\Ext{Ext}$Let $(R, m)$ be a commutative Noetherian Gorenstein local ring and let $M$ be an $R$-module. Let ${^\perp M}$ and $M^\perp$ be respectively the left and the right ...
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46 views

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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1answer
690 views

What is a module over a Boolean ring?

Recall that a (unital) Boolean ring is a (unital) commutative ring $A$ where every element is idempotent; it follows that $A$ is of characteristic 2. There is an equivalence of categories between ...
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48 views

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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28 views

Do there exist characterizations of divisors of a particular element for all algebra structures on a vector space

Let $k$ be a field and $V$ a $k$-vector space. Let $M$ be the subset of $\operatorname{Hom}_k(V \otimes_{k} V,V)$ formed by all elements giving $V$ the structure of a commutative (associative) $k$-...
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160 views

Necessary condition to extend a morphism of schemes

Consider two schemes $X,Y$ over a locally noetherian scheme $S$. Let $p \in X$ and assume that $X$ is irreducible and not affine spectrum of a semilocal ring. We assume moreover we have a morphism $...
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1answer
149 views

Conjugacy in the quaternion group

Let $G$ be a non-commutative group, and suppose we are given two elements $x, y \in G$ which are conjugate, i.e. we know there exists some $z \in G$ such that $zxz^{-1} = y$. Can we find $z$ given $x$ ...
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56 views

zeros of a polynomial in three variables over finite field

Consider a homogeneous symmetric polynomial $f(x,y,z)=(x+y+z)((xy)^{q-1}+(yz)^{q-1}+(zx)^{q-1})+x^{2q-1}+y^{2q-1}+z^{2q-1}$ of degree $2q-1$, where $q$ is an odd prime power. Under what condition ...
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1answer
97 views

Decide whether there are “linear” relations between quadrics

Let $k$ be an algebraically closed field of characteristic $0$. For a homogeneous ideal $I=(q_1,\dots, q_k)\subset k[x_0,\dots,x_n]$ generated by quadrics, is there a method to decide whether the ...
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1answer
196 views

Examples of noetherian local rings $R$ such that $K'_0(R)$ is not isomorphic to $\mathbb Z$

Does there exist a simple example of a commutative noetherian local ring $R$ such that $K'_0(R) = K_0(\mbox{Mod-}R)$ (by $\mbox{Mod-}R$ I mean the abelian category of finitely generated $R$-modules) ...
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89 views

Dimension of the socle of the first local cohomology module

Let $M$ be a graded $\mathbb{C}[z_0,\dots,z_n]$-module. Using local duality one can show that $$ \dim_\mathbb{C} (\text{soc} H_\mathfrak{m}^1(M))_k = \beta_{n,k+n+1}(M). $$ Here $H_\mathfrak{m}^1(M)$ ...
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59 views

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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1answer
120 views

Generic Galois alteration of an arithmetic model with semistable special fiber

Let $R$ be a DVR of char $0$ and $S=Spec(R)$. Let $X\longrightarrow S$ be a proper flat morphism. Assume $X$ is integral. De Jong's Theorem 8.2 in his paper Smoothness, semi-stability and alterations ...
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1answer
202 views

Commutative ring $R$ with no nontrivial idempotents, with a localization $R_r$ with infinitely many idempotents

I am looking for a commutative ring $R$ with $1$ such that $R$ has no idempotents and there exists $r\in R$ such that the localization ring $R_r$ has infinitely many idempotents.
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1answer
192 views

Is there a finite extension with a non-trivial class group of any PID?

Let $R$ be a PID with infinitely many prime ideals. Does there always exist a finite extension $R\subset R'$ with $R'$ being a Dedekind domain with a non-trivial class group?

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