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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1answer
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
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1answer
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=...
1
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1answer
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. ...
2
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0answers
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 ...
2
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1answer
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:...
2
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1answer
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 $...
2
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1answer
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 ...
1
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1answer
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
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0answers
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 $V$-...
4
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0answers
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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0answers
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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0answers
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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1answer
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 ...
1
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1answer
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 ...
6
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1answer
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
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1answer
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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1answer
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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0answers
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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0answers
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....
4
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0answers
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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1answer
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, $\gcd(f,g)=1$, ...
2
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1answer
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 $\...
4
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0answers
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
1answer
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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0answers
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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0answers
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)...
4
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2answers
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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0answers
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
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1answer
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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0answers
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. ...
8
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0answers
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
2answers
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
1answer
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 ...
9
votes
3answers
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 ...
5
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0answers
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
2answers
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 ...
5
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0answers
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
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0answers
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
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1answer
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
1answer
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
1answer
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},{...
1
vote
1answer
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
1answer
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
1answer
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
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1answer
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
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1answer
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
1answer
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 ...
1
vote
1answer
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\...
2
votes
1answer
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
1answer
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 ...

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