# Questions tagged [ac.commutative-algebra]

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

4,519
questions

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62 views

### Minimal free resolution over arbitrary varieties

Over projective space, it is well-known that given a graded $S^\bullet$-module $M_\bullet$, where $S^\bullet = k[x_0, \dots, x_N]$, there is a unique minimal free resolution
$$
\cdots \to \...

**2**

votes

**1**answer

157 views

### Maximizing and minimizing the number of positive product $k$-subsets of an $n$-set

The question is simple but require some definitions. I came across resolving a certain inequality. If there is no closed answer is there a related sequence describing the situation?
Let
$$S\ :=\ \{X=...

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vote

**1**answer

136 views

### Flatness criterion for $I$-adic ring: $I$-torsion free

Let $R$ be an $I$-adically separated and complete valuation ring, with $I$ finitely generated.
It is used a few times in Bosch, Lectures on Formal and Rigid Geometry e.g. first lines of pg. 164, Cor. ...

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**0**answers

129 views

### Eichler orders in a certain quaternion algebra

Let us consider a totally real number field $K$ such that $[K \colon {\Bbb Q}] = {\mathrm{odd}}$. We shall consider the quaternion algebra $D$ over $K$ such that $D$ splits everywhere at finite places ...

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votes

**1**answer

122 views

### Construction of Jacobian Ideal

In Qing Liu's Algebraic Geometry and Arithmetic Curve, we have the following proposition(6.3.13):
Let $S$ be a locally Noetherian scheme. Let $X$, $Y$ be smooth schemes over $S$. Then any immersion $f:...

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**1**answer

124 views

### Localization at multivariate monic polynomials

Let $R$ be a ring and consider a monomial order $<$ on $R[X_1,\ldots,X_n]$. A nonzero polynomial $f \in R[X_1,\ldots,X_n]$ is said to be monic if its leading coefficient with respect to $<$ is $...

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**1**answer

131 views

### Structures of subgroups of a finite abelian p-group

$\newcommand\la{\langle}\newcommand\ra{\rangle}$Let $G=\mathbb{Z}/p^{i_1}\times\cdots\times\mathbb{Z}/p^{i_r}$ with $i_1\leq\ldots\leq i_r$ be a finite abelian $p$-group. Then there can be many ...

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vote

**1**answer

76 views

### Symbolic power of an ideal associated to non-singular algebraic set

Let $Z\subset \mathbb P^n$ be a reduced non-singular algebraic set and $I$ denote the saturated homogeneous ideal of $Z$. I have seen the following result without proof:
For all $ n\geq 1$, $I^{(n)}=(...

**2**

votes

**0**answers

77 views

### Finite locally free morphisms and degree

Let $f : X\to Y$ be a finite locally free morphism between integral normal schemes flat over a dvr $V$. Assume $f$ is of degree $d$ and factors through another finite locally free morphism of $V$-...

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99 views

### Generalization of the second Brauer-Thrall conjecture for arbitrary Artin algebras

Let $k$ be a field with infinite cardinality and $A$ a finite dimensional $k$-Algebra. The second Brauer-Thrall conjecture states the following: There are infinitely many natural numbers $n_1<n_2&...

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63 views

### Separable field extensions and base change

Suppose that there are field extensions
\begin{array}{ccc}
k & \longrightarrow & K \\
\downarrow & & \downarrow \\
L & \longrightarrow & M
\end{array}
where $M$ is generated by ...

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votes

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72 views

### Complete local Noetherian ring whose normalized dualizing complex has depth equal to dimension of the ring

Let $(R, \mathfrak m,k)$ be a complete Noetherian local ring. Let $D$ be a normalized dualizing complex of $R$. If $\text{depth}_R D=\dim R$, then must $R$ be Cohen-Macaulay (i.e. must $D$ be actually ...

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votes

**1**answer

121 views

### Open affine subscheme of a direct limit of smooth algebras

Let $R$ be (assumed to be commutative, Noetherian) a regular local ring. Let $A$ be a direct limit of $R$-smooth algebras, such that the transition maps are $R$-étale.
Let $U= Spec(B)$ be an affine ...

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vote

**1**answer

122 views

### Primary decomposition of huge ideals using M2/Singular

I used to ask similar questions in other communities, but so far never received any feedback.
Given four Hermitian $n\times n$ matrices $A_1,A_2,B_1,B_2$ together with the constraints $[A_i,B_j]=0$, I ...

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**1**answer

145 views

### Artin-Rees lemma for multiplicative subsets?

The classical Artin-Rees lemma tells the following. Let $R$ be a Noetherian commutative ring and $I\subset R$ be an ideal. Let $M$ be a finitely generated $R$-module and $N\subset M$ be a submodule. ...

**7**

votes

**1**answer

315 views

### On universally closed morphisms of reduced schemes

In this question I'd like to examine some properties of universally closed morphisms.
The question is self-contained. It can also be seen as a follow-up to this question.
Let $R$ be a discrete ...

**4**

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**1**answer

216 views

### Detecting closed immersions on fibers

Let $R$ be a dvr and $f : X\to S$ a universally closed morphism of $R$-schemes.
Assume $X$ and $S$ are $R$-flat and universally closed.
If the special fiber of $X\to S$ is a closed immersion, is $X\...

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**0**answers

138 views

### Integral extension and localization

Let ${\bf A} \subseteq {\bf B}$ be two rings such that ${\bf B}$ is integral over ${\bf A}$. If $S$ is a multiplicative subset of ${\bf B}$,
setting $\tilde{S}= S \cap {\bf A}$, can we say that the ...

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178 views

### Artin's “Versal Deformations and Algebraic stacks”: Question concerning proof of Theorem 3.3

I have been reading Artin's paper titled "Versal deformations and algebraic stacks" and am a bit confused about a statement he makes in the proof of Theorem 3.3, in the first few lines of pg....

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225 views

### Noetherian approximation and flatness

Let $R$ be a regular ring of finite type over an excellent domain, and $\{R_i, i\in I\}$ a directed system of $R$-algebras indexed by a directed set $I$, such that each of the maps $R\to R_i$ is ...

**2**

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**1**answer

134 views

### Coprime multivariate polynomials

Let ${\bf R}$ be a gcd domain, $n \geq 2$, $k \in \mathbb{N}^*$, and $f,g \in
{\bf R}[X_1,\ldots,X_n]$. Supposing that $f$ and $g$ are coprime in ${\bf R}[X_1,\ldots,X_n]$, that is, $\gcd(f,g)=1$, ...

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**1**answer

132 views

### Good prime ideals in tensor products of local rings

Let $L/K$ be a field extension.
Let $R,S$ be two local commutative $K$-algebras and let $\varphi : R \to S$ be a homomorphism of $K$-algebras, not assumed to be local. Let's call a prime ideal $\...

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104 views

### Integral face ring of the triangulation of a sphere

The integral face ring of a (finite) simplicial complex $K$ on $m$ vertices is the quotient ring
$$\mathbb{Z}[K]=\mathbb{Z}[v_1,...,v_m]/\mathcal{I}_K$$
where $\mathcal{I}_K$ is the ideal generated by ...

**2**

votes

**1**answer

122 views

### Is there a Banach algebra which $A^2$ is not dense in $A$ but $(A^{**})^2$ is dense in $A^{**}$?

Is there a Banach algebra which $A^2=\langle a_1a_2 ; a_1,a_2\in A\rangle$ is not dense in $A$ but $(A^{**})^2$ is dense in $A^{**}$?

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155 views

### What is this algebraic object (special case of a semigroup)?

Let $(M,*)$ be a finite semigroup. Further we demand the following:
Zero element: $\exists0\in M \forall m\in M:0*m=0=m*0$.
Left cancelation: $\forall m,n,n'\in M:0\neq m*n =m*n' \Rightarrow n=n'$.
...

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54 views

### Families of polynomials given by tuples of binary forms with finitely many reducible members

Let $G_1, \cdots, G_n \in \mathbb{Z}[x,y]$ be binary forms, and put $\mathbf{G} = (G_1, \cdots, G_n)$. Consider the family of monic polynomials
$$\displaystyle \mathcal{F}_\mathbf{G} = \{x^n + G_1(p,q)...

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votes

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249 views

### Ideals in Artinian Gorenstein local ring $(R,\mathfrak m)$ with $\mu(\mathfrak m)=2, \mathfrak m^2\ne 0$ and $\mathfrak m^3=0$

Let $(R,\mathfrak m,k)$ be an Artinian Gorenstein local ring such that $$\mu(\mathfrak m)=2, \quad\mathfrak m^2\ne 0,\quad\text{and}\quad \mathfrak m^3=0.$$
Then, is it true that every non-maximal ...

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74 views

### Are irreducible components of regularly embedded varieties regularly embedded?

Suppose I have a (reduced) subvariety $V \hookrightarrow X$ of a smooth variety $X$ such that $V$ is regularly embedded in $X$. (i.e. is locally cut out by a regular sequence of $\operatorname{codim}(...

**4**

votes

**1**answer

203 views

### Origin of Laguerre geometry?

Laguerre geometry is described as either the geometry of oriented lines and circles in the Euclidean plane, equipped with a certain unusual symmetry group (see https://en.wikipedia.org/wiki/...

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97 views

### derivation defined on $\mathbb{R}[sin(x),|x|]$

Given two algebraic independent elements, it is expected that the for subalgebra generated by these two elements, we can define derivations on the subalgebra by specify values on the two elements.
...

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votes

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138 views

### A criterion for rational singularities in mixed characteristic

Let $R$ be a mixed characteristic discrete valuation ring with perfect residue field and $f:X \to \mathrm{Spec}(R)$ a flat proper morphism.
If the generic fibre of $f$ is smooth and the special fibre ...

**8**

votes

**2**answers

608 views

### Do relatively prime polynomials $f$ and $g$ in $k[x,y]$ generate an ideal of finite codimension?

Let $k$ be a field and $f,g \in k[x,y]$ be two non-constant polynomials in two variables. Is it true that the following conditions are equivalent:
$f$ and $g$ are relatively prime in $k[x,y]$, in the ...

**1**

vote

**1**answer

218 views

### Product absolute value in rings of integers

Let $F$ be an algebraically closed field of characteristic $p$ equipped with a nonarchimedean dense absolute value $|\cdot|:F \rightarrow \mathbb{R}_{\ge 0}$ with respect to which $F$ is complete. Let ...

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votes

**3**answers

670 views

### What's the number of solutions of the quadratic equation $x_1^2+\dots+x_m^2=0$ over finite ring $\mathbb{Z}/p^n$?

I want to calculate the number of solutions to the quadratic equation $$x_1^2+\dots+x_m^2=0$$ where $m$ is odd (a given number) and $x_i\in\mathbb{Z}/p^n$ for a given prime number $p$ and a given ...

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**0**answers

143 views

### Unbounded derived Nakayama lemma

Let $R$ be a (commutative) local ring, which I don't assume to be noetherian. Let $m$ be its maximal ideal, and $k$ its residue field.
Let $X$ be a complex of $R$-modules with finitely generated ...

**5**

votes

**2**answers

244 views

### Tensor of finite-dimensional algebra over perfect field is semisimple

Let $K$ be a field and let $\Lambda_{1}$ and $\Lambda_{2}$ be two finite-dimensional $K$-algebras with Jacobson radicals $J_{1}$ and $J_{2}$ respectively. How to show or where can I find the proof of ...

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votes

**0**answers

119 views

### Is the category of topologically free $k[[h]]$-modules locally presentable?

$\newcommand{\colim}{\operatorname{colim}}$
Let $k$ be a field (of characteristic 0, say) and $M$ be a module over $R=k[[h]]$. Recall that the $h$-adic completion of $M$ is
$$
\hat M:=\lim M/h^nM,
$$
...

**15**

votes

**0**answers

347 views

### Reducible polynomials of the shape $f(t^2)$, where $f$ is irreducible

Let $f(x) \in \mathbb{Z}[x]$ be a monic, irreducible polynomial. What are necessary and sufficient conditions for $g(t) = f(t^2)$ to be reducible over $\mathbb{Q}$?
For instance, if $f(x) = x-1$ then $...

**3**

votes

**1**answer

178 views

### Interpolation of scheme-theoretic endomorphisms of closed fibers

Let $S$ be a scheme and $f : X\to S$ be an $S$-scheme. This question asks for examples of maps of sets $X(S) \to X(S)$ that do not come from an $S$-scheme endomorphism of $X$, but that, roughly, ...

**3**

votes

**1**answer

114 views

### Category of modules over internal monoid is abelian

I have asked the following question on MSE a few days ago, but without any success.
I am interested in proving the following statement:
Let $\mathcal{A}$ be a tensor category. Then the category of ...

**2**

votes

**1**answer

70 views

### Can the differentials in a minimal free resolution ever have a “long” row of $0$'s?

Assume just for sake of simplicity that $R = k[x_1 , \dots , x_n]$ is a standard graded polynomial ring over a field. If one considers the ideal
$$I = \left({x}_{1}{x}_{3},{x}_{2}^{2},{x}_{2}{x}_{3},{...

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vote

**1**answer

135 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 ...

**3**

votes

**1**answer

358 views

### Extending functors between K-algebras to schemes

Assume we have $K$ and $L$ (comm.) rings, and we have a functor $F$ from the category of $K$-Algebras to the category of $L$-Algebras (I work only with commutative rings). What conditions need to ...

**3**

votes

**1**answer

142 views

### Do there exist irreducible elements in this domain?

I asked this question on MSE. Here also I have the same motive in the question.
Let $D= \{\,a_1x^{r_1} + \cdots + a_n x^{r_n} \, \vert \, a_i \in \mathbb{C} \text{ for } i= 1,2,\dots,n \text{ and ...

**27**

votes

**1**answer

1k views

### Are large powers of polynomials linearly independent?

Let $P_1,\dots,P_k$ be polynomials over $\mathbf{C}$, no two of them being proportional.
Does there exist an integer $N$ such that $P_1^N,\dots,P_k^N$ are linearly independent?

**0**

votes

**1**answer

57 views

### Height of truncated system of parameter

Let $R$ be a Noetherian local ring of dimension $d$, and $a_1,\cdots,a_d$ is a system of parameter. I am wondering whether the following statement is true:
$\rm{ht}$$(a_1,\cdots,a_i)=i$ for all $i,1 \...

**2**

votes

**1**answer

112 views

### Reference to basic facts on non-Archimedean local fields

I need a reference to the following claims which, I believe, are correct and well known to experts (I am not one of them).
Let $K$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of ...

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vote

**1**answer

261 views

### Are there algebras over reals besides complex numbers, where identities, analoguous to $(-1)^i=e^{-\pi}$ and $i^i=e^{-\pi/2}$ hold?

Are there algebras over real numbers (with exponentiation), such that there is such $z$ that does not include components in $\mathbb{C}$ (or in a subset isomorphic to $\mathbb{C}$), for which $(-1)^z\...

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votes

**1**answer

121 views

### The minimal complexes and the direct limit

In my research I confronted with the following question. I do not have a counterexample, so I ask it here if anyone knows. I would like a positive answer, however a negative answer seems more likely ...

**3**

votes

**1**answer

138 views

### Are finite projective modules over $R[t]$ free when $R$ is DVR?

Let $R$ be a discrete valuation ring (DVR) and let $M$ be a projective module of finite type over the polynomial ring $R[t]$. Is $M$ free over $R[t]$?
As far as I understand, this should be a ...