Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
4,519
questions
2
votes
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
$$
\cdots \to \...
2
votes
1answer
157 views
Maximizing and minimizing the number of positive product $k$-subsets of an $n$-set
The question is simple but require some definitions. I came across resolving a certain inequality. If there is no closed answer is there a related sequence describing the situation?
Let
$$S\ :=\ \{X=...
1
vote
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
votes
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
votes
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
votes
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
votes
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
vote
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
votes
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
votes
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&...
1
vote
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 ...
0
votes
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 ...
0
votes
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
vote
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
votes
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
votes
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
votes
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\...
-1
votes
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 ...
3
votes
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
votes
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
votes
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
votes
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
votes
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 $A^{**}$?
1
vote
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: $\forall m,n,n'\in M:0\neq m*n =m*n' \Rightarrow n=n'$.
...
1
vote
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
$$\displaystyle \mathcal{F}_\mathbf{G} = \{x^n + G_1(p,q)...
4
votes
2answers
249 views
Ideals in Artinian Gorenstein local ring $(R,\mathfrak m)$ with $\mu(\mathfrak m)=2, \mathfrak m^2\ne 0$ and $\mathfrak m^3=0$
Let $(R,\mathfrak m,k)$ be an Artinian Gorenstein local ring such that $$\mu(\mathfrak m)=2, \quad\mathfrak m^2\ne 0,\quad\text{and}\quad \mathfrak m^3=0.$$
Then, is it true that every non-maximal ...
0
votes
0answers
74 views
Are irreducible components of regularly embedded varieties regularly embedded?
Suppose I have a (reduced) subvariety $V \hookrightarrow X$ of a smooth variety $X$ such that $V$ is regularly embedded in $X$. (i.e. is locally cut out by a regular sequence of $\operatorname{codim}(...
4
votes
1answer
203 views
Origin of Laguerre geometry?
Laguerre geometry is described as either the geometry of oriented lines and circles in the Euclidean plane, equipped with a certain unusual symmetry group (see https://en.wikipedia.org/wiki/...
-1
votes
0answers
97 views
derivation defined on $\mathbb{R}[sin(x),|x|]$
Given two algebraic independent elements, it is expected that the for subalgebra generated by these two elements, we can define derivations on the subalgebra by specify values on the two elements.
...
8
votes
0answers
138 views
A criterion for rational singularities in mixed characteristic
Let $R$ be a mixed characteristic discrete valuation ring with perfect residue field and $f:X \to \mathrm{Spec}(R)$ a flat proper morphism.
If the generic fibre of $f$ is smooth and the special fibre ...
8
votes
2answers
608 views
Do relatively prime polynomials $f$ and $g$ in $k[x,y]$ generate an ideal of finite codimension?
Let $k$ be a field and $f,g \in k[x,y]$ be two non-constant polynomials in two variables. Is it true that the following conditions are equivalent:
$f$ and $g$ are relatively prime in $k[x,y]$, in the ...
1
vote
1answer
218 views
Product absolute value in rings of integers
Let $F$ be an algebraically closed field of characteristic $p$ equipped with a nonarchimedean dense absolute value $|\cdot|:F \rightarrow \mathbb{R}_{\ge 0}$ with respect to which $F$ is complete. Let ...
9
votes
3answers
670 views
What's the number of solutions of the quadratic equation $x_1^2+\dots+x_m^2=0$ over finite ring $\mathbb{Z}/p^n$?
I want to calculate the number of solutions to the quadratic equation $$x_1^2+\dots+x_m^2=0$$ where $m$ is odd (a given number) and $x_i\in\mathbb{Z}/p^n$ for a given prime number $p$ and a given ...
5
votes
0answers
143 views
Unbounded derived Nakayama lemma
Let $R$ be a (commutative) local ring, which I don't assume to be noetherian. Let $m$ be its maximal ideal, and $k$ its residue field.
Let $X$ be a complex of $R$-modules with finitely generated ...
5
votes
2answers
244 views
Tensor of finite-dimensional algebra over perfect field is semisimple
Let $K$ be a field and let $\Lambda_{1}$ and $\Lambda_{2}$ be two finite-dimensional $K$-algebras with Jacobson radicals $J_{1}$ and $J_{2}$ respectively. How to show or where can I find the proof of ...
5
votes
0answers
119 views
Is the category of topologically free $k[[h]]$-modules locally presentable?
$\newcommand{\colim}{\operatorname{colim}}$
Let $k$ be a field (of characteristic 0, say) and $M$ be a module over $R=k[[h]]$. Recall that the $h$-adic completion of $M$ is
$$
\hat M:=\lim M/h^nM,
$$
...
15
votes
0answers
347 views
Reducible polynomials of the shape $f(t^2)$, where $f$ is irreducible
Let $f(x) \in \mathbb{Z}[x]$ be a monic, irreducible polynomial. What are necessary and sufficient conditions for $g(t) = f(t^2)$ to be reducible over $\mathbb{Q}$?
For instance, if $f(x) = x-1$ then $...
3
votes
1answer
178 views
Interpolation of scheme-theoretic endomorphisms of closed fibers
Let $S$ be a scheme and $f : X\to S$ be an $S$-scheme. This question asks for examples of maps of sets $X(S) \to X(S)$ that do not come from an $S$-scheme endomorphism of $X$, but that, roughly, ...
3
votes
1answer
114 views
Category of modules over internal monoid is abelian
I have asked the following question on MSE a few days ago, but without any success.
I am interested in proving the following statement:
Let $\mathcal{A}$ be a tensor category. Then the category of ...
2
votes
1answer
70 views
Can the differentials in a minimal free resolution ever have a “long” row of $0$'s?
Assume just for sake of simplicity that $R = k[x_1 , \dots , x_n]$ is a standard graded polynomial ring over a field. If one considers the ideal
$$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
votes
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
votes
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 ...
