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

19
votes
0answers
183 views

The roots of the unity in commutative rings

For $i\in\{1,2\}$ let $A_i$ be a commutative ring with unity whose additive group is free and finitely-generated. Assume that $A_i$ is connected in the sense that $0$ and $1$ are unique solutions of ...
2
votes
1answer
91 views

$(x + y + z)…(x + y\omega_n^{n-1} + z\omega_n) = x^n + y^n + z^n - P(x,y,z)$ To find $P$

$$(x + y + z)(x + y\omega_n + z\omega_n^{n-1})(x + y\omega_n^2 + z\omega_n^{n-2})....(x + y\omega_n^{n-1} + z\omega_n) = x^n + y^n + z^n - P(x,y,z)$$ where $\omega_n$ is an nth root of unity. The ...
2
votes
0answers
55 views

Compute the closure of graph of function from complement of hypersurface in $\mathbb{A}^n$

I'm hoping someone can give me some tips to help speed up computation on the following problem: Suppose I have a map $G=(g_1/f,\dots,g_m/f):\mathbb{A}^n\setminus{V(f)}\to \mathbb{A}^m$. I'm ...
0
votes
0answers
78 views

gcd of polynomial values

Suppose that $f$ and $g$ are two coprime polynomials in $\mathbb Z[x]$. I'm interested in any sort of upper bound on $gcd(f(a),g(a))$, in terms of the integer $a$. Are there any results of this type?...
8
votes
1answer
365 views

Do commutative rings without unity have the IBN property?

Let $R$ be a commutative rng, i.e. a commutative ring without an identity element. Does $R$ still have the Invariant Basis Number (IBN) property? Recall that a ring is said to have the IBN ...
3
votes
0answers
127 views

Fraction fields of strict henselizations of DVRs

Let $A_{1},A_{2}$ be discrete valuation rings whose fraction fields are isomorphic. Let $A_{i}^{\mathrm{sh}}$ be the strict henselization of $A_{i}$, and let $K_{i}$ be the fraction field of $A_{i}^{\...
2
votes
1answer
44 views

Reflection-invariant monomial ideals and Alexander duality

First we give some definitions from Section 3 of the paper Monomials, Binomials, and Riemann-Roch by Manjunath and Sturmfels and then we restate a claim from that paper offered without proof. Finally ...
2
votes
1answer
61 views

Steinberg components of local deformation rings

Let $F=\mathbb Q_l$ and $E$ be a finite extension of $\mathbb Q_p$ with residue field $k \cong O_E/m_E$ ($p \neq l$). Let $\bar r: \Gamma_F=Gal(\bar F/F) \to GL_2(k)$ be a continuous representation. ...
3
votes
0answers
55 views

On ideals in Noetherian rings, isomorphic to the trace of some finitely generated module

Let $R$ be a Noetherian ring. For a finitely generated $R$-module $M$, let $tr_R(M):=Im(\tau_M)$, where $\tau_M:M\otimes Hom(M,R)\to R$ is the map defined as $\tau_M(m\otimes f)=f(m)$. Let $I$ be a ...
1
vote
0answers
39 views

Complete Intersection ideal for which $\mbox{In}_<(I)$ is not even Cohen-Macaulay

I would like to find a example of a graded ideal $I\subset k[x_1,\cdots,x_n]$ for which $I\mbox{In}_<(I)$ is not even Cohen-Macaulay (for some monomial order). I have tried to find such a ideal ...
7
votes
1answer
137 views

How should I think about the module of coinvariants of a $G$-module?

Let $G$ be a group, $M$ a $G$-module, then the group of coinvariants is the module $M_G := M/I_GM$, where $I_G$ is the kernel of the augmentation map $\epsilon : \mathbb{Z}G\rightarrow \mathbb{Z}$. ...
0
votes
0answers
110 views

Is $R$ finitely generated?

Let $A$ be a commutative ring with identity. Given two submodules $R,S$ of $A^n(n\in\Bbb N)$ and suppose $S$ is finitely generated, if there exists an isomorphism of $A$-modules $A^n/R\simeq A^n/S$, ...
1
vote
0answers
45 views

Persistence of saturation closure

Let $R$ be a ring. A closure operation $cl$ on a set of ideals $\mathcal{I}$ of $R$ is a set map $cl: \mathcal{I}\longrightarrow \mathcal{I}( I\mapsto I^{cl})$ satisfying the following conditions: (i)...
4
votes
0answers
76 views

Cohen-Macaulay local ring and its quotient by minimal prime ideal

Assume $R$ is a Noetherian local ring which is Cohen-Macaulay, must there exist an ideal $I$ of $R$ such its radical is a minimal prime and the quotient ring $R/I$ is still Cohen-Macaulay? If the ...
2
votes
0answers
108 views

Prime ideals being finitely-generated implies coherence?

Let $R$ be a non-noetherian local domain. Suppose that the following two conditions hold for $R$$\colon$ $(*)$$~\quad$An arbitrary prime ideal ${\frak P}$ of $R$ such that ${\mathrm{ht}}({\frak P}) &...
3
votes
0answers
94 views

Can one characterize power series that have polynomial inverses?

Going through my Commutative Algebra notes I found out that I don't know the answer to this question. Let $A$ be a commutative ring with unit and let $f(T):=\sum_{k=0}^\infty a_k T^k \in A[[T]]$ be a ...
0
votes
1answer
86 views

Valuation ring satisfying either a.c.c. or d.c.c. on prime ideals

If a commutative ring with unity has finite Krull dimension, then it satisfies a.c.c. and d.c.c. on prime ideals. The converse is not true in general, as can be seen from here An infinite dimensional ...
0
votes
0answers
92 views

Krull dimensions and regular sequences

I am trying to understand if a certain condition on quotient rings is sufficient for a sequence to be regular. Here is the setting: Let $\mathbb{C}[u_1,...,u_n]$ be the ring of regular functions on $\...
5
votes
0answers
165 views

Ideals of orders in number fields

Let $K$ be a number field with ring of integers $\mathcal{O}_K$. Let $\mathcal{O}$ be an order of $K$ and $A$ a proper ideal of $\mathcal{O}$, by which I mean that $\mathcal{O} = \{\alpha \in K : \...
2
votes
1answer
98 views

Transitivity of an invariant of finitely generated field extensions

For a finitely generated extension of fields $K/k$, let us define "$S_{K/k}$" to be the minimum of the degrees $[K:\ell]$ where $\ell/k$ ranges over the purely transcendental subextensions of $K$ with ...
7
votes
1answer
247 views

An infinite dimensional local domain whose chains of primes are finite

Does there exist a local domain of infinite dimension in which every chain of prime ideals is finite? Of course, such a ring must be neither noetherian nor catenary. (This question arose while ...
0
votes
2answers
140 views

Noetherian module, over Noetherian ring, which is isomorphic to its double dual [duplicate]

Let $M$ be a finitely generated module over a Noetherian ring $R$ such that $M$ is isomorphic with its double dual $M^{**}=Hom(Hom(M,R),R)$. Then is the natural map $j:M \to M^{**}$ defined as $j(m)(...
1
vote
0answers
77 views

A special family of prime ideals

I am looking for a commutative ring $R$ with identity which has a family $F:=\{p_i\}_{i\in A}$ of prime ideals such that for each $n\in \mathbb{N}$ there exists $m\in \mathbb{N}$ with $n\leqslant m$ ...
0
votes
0answers
51 views

counting solutions to a system of polynomial equations

Given a system of N polynomial equations for N complex variables, is there an efficient algorithm for counting the number of solutions (which one generically expects to be finite)? One can compute a ...
1
vote
1answer
45 views

Decomposing semihereditary rings

Let $R$ be a commutative semihereditary ring (i.e. every finitely generated ideal of $R$ is projective https://en.wikipedia.org/wiki/Hereditary_ring). Then is $R$ a finite direct product of Prufer ...
2
votes
1answer
68 views

Decomposing Noetherian hereditary rings of Krull dimension $1$ into product of hereditary domains (i.e. Dedekind domains)

Let $R$ be a commutative Noetherian hereditary ring (https://en.wikipedia.org/wiki/Hereditary_ring) of Krull dimension $1$. Then is it true that $R$ is a finite direct product of Dedekind domains ?
2
votes
1answer
170 views

Are quotient varieties local complete intersections?

Let $G$ be a reductive group acting on the smooth affine variety $X$ such that the stabilizers are finite. Is it true that the quotient $X/G$ is a local complete intersection (LCI)? In particular, is ...
1
vote
1answer
208 views

When does a subspace of the affine space form a regular sequence in a ring of regular functions?

Let $J$ be an ideal in the ring $\mathbb{C}[u] := \mathbb{C}[u_1,...,u_n]$ of regular functions on $\mathbb{C}^n$. Given a subspace $\mathbb{C}^k \subset \mathbb{C}^n$ and $I$ its ideal in $\mathbb{...
5
votes
1answer
112 views

On the linear factors of a polynomial obtained from the determinant of a matrix whose entries are related to Binomial expansion

Consider the polynomial ring $R=\mathbb C[x,y]$. Consider the matrix $A=\begin{pmatrix} x^5+y^5&5x^5&10x^5&10x^5&5x^5\\5y^5&x^5+y^5 &5x^5&10x^5&10x^5 \\10y^5&5y^5&...
2
votes
0answers
42 views

What is the ring of functions on the open unit disc with polynomially bounded Maclaurin coefficients called?

Let $R$ be the the set of complex-valued (analytic) functions $f$ on the open unit disc $\mathrm D:=\{z\in\Bbb C:|z|<1\}$ for which there exist constants $a_0$, $a_1$, ... in $\Bbb C$ and $n$ in $\...
2
votes
0answers
29 views

Numerator-cancellable Modules

I don't know why there is no investigation of the cancellability of quotients in category of modules. What I mean by cancellability of quotients in category of modules is the following : Let $R$ be a ...
2
votes
0answers
108 views

On finite dimensional commutative algebras and regular sequences

Let $\mathbb{C}[u]:= \mathbb{C}[u_1,...,u_n]$ the algebra of polynomial functions on $\mathbb{C}^n$. I consider two ideals $I$, and $J$, where $I$ is the ideal generated by the polynomials vanishing ...
14
votes
2answers
324 views

Ostrowski's Theorem for topological rings?

Ostrowski's theorem classifies all absolute values on a number field $K$. Questions: More generally, can one classify all Hausdorff topologies on $K$ making $K$ into a topological field? In ...
3
votes
1answer
210 views

Local ring of infinite dimension

Short version: Let $R$ be a commutative ring such that all chains of primes of $R$ with the same extremities have the same finite cardinality. Is $R$ locally finite-dimensional? Longer version: Let $...
4
votes
1answer
190 views

Resultants for compactly represented product form polynomials?

Typically computing resultnt of $n+1$ different $n+1$-variate homogeneous polynomials takes $O(poly(\prod_{i=1}^{d_{i}}))$ time where $d_i$ is degree of $i$th polynomial. In certain cases the ...
3
votes
0answers
112 views

derived symmetric powers of an ideal

Let $R$ be the polynomial algebra in $n$ variables over a field $F$ of characteristic $0$. Let $m$ be the ideal of the origin: $m=(x_1,...,x_n)$. We have a canonical map $Lsym^k(m)\to m^k$ from the ...
2
votes
0answers
70 views

On some finiteness properties of cohomological algebras of complex tori

Denote by $A := \mathbb{C}[u,u^{-1}]$, $u = (u_1,...,u_n)$, the algebra of polynomial functions on the complex torus $(\mathbb{C} \setminus \{ 0 \})^n$, which we consider as a $\mathbb{C}[u]$-module. ...
2
votes
0answers
99 views

Flatness of Endomorphism sheaf

Let $X$ be a nodal curve and $T$ be any scheme. Let $F$ be a flat family of torsion-free coherent sheaves on $X$ parametrized by $T$. Is there an example where $\mathcal{E}nd ~F$ is not flat over $T$? ...
8
votes
0answers
184 views

Counting exceptional divisors

Suppose that I blow up an ideal sheaf $J$ in $\mathbb A^2$ via a map $\pi : X \to \mathbb A^2$. I'd like to compute, from the ideal, how many exceptional divisors there are for $\pi$, and be able to ...
6
votes
0answers
121 views

Descending chain of subalgebras of $k[x,y]$

Let $k$ be a field of characteristic zero. Let $\{R_i\}_{i \in \mathbb{N}}$ be a descending chain of $k$-subalgebras of $k[x,y]$: $k[x,y]=:R_0 \supseteq R_1 \supseteq R_2 \supseteq \ldots$, such that ...
4
votes
0answers
73 views

Infinite-dimensional wild commutative algebras with non-trivial units

Let $A$ be an arbitrary (not necessarily finite-dimensional) associative algebra over an algebraically closed field $K$, and let $\mathrm{fin\,}A$ denote the category of finite-dimensional $A$-modules....
3
votes
0answers
186 views

Eudoxus real numbers

I recently remembered the eudoxus construction of the real numbers. Does anyone know what how the rational numbers $\mathbb Q$ can be characterised inside this construction? Clearification: The ...
1
vote
1answer
116 views

Regular local artinian k-algebra with residue field k is k

I am reading an article. There is a step in which I suspect that they use a "result" that "Let $A$ be a local artinian $k$-algebra with residue field $k$. If $A$ is regular then $A$ is nothing but $k$....
2
votes
1answer
106 views

Homomorphisms of ring extending nicely ideal intersections

Let $\varphi\!:\!S\to R$ be a homomorphisms of $K$-algebras for some field $K$. Let $\{a_{\lambda}\}_{\lambda}$ be a family of ideals of $S$. Is there some "natural" assumption on $\varphi$ to ...
7
votes
1answer
123 views

Reference for Kakutani result on power sum bases of symmetric functions

Numerical semigroups are additive submonoids $A$ of the natural numbers such that the greatest common divisor of all elements of $A$ is 1. The complement of a numerical semigroup in $\mathbb{N}$ is ...
12
votes
1answer
341 views

Is the identity function a unique multiplicative homeomorphism of $\mathbb N$?

Endow the set $\mathbb N$ of positive integers with the topology $\tau$ generated by the base consisting of arithmetic progressions $a+b\mathbb N_0$ where $\mathbb N_0=\{0\}\cup\mathbb N$, where $a,b\...
2
votes
1answer
134 views

difference between Cohen Macaulay and locally Cohen Macaulay curve

I am trying to understand the difference between Cohen Macaulay and Locally Cohen Macaulay curves. The stacks project https://stacks.math.columbia.edu/tag/02IN says that a scheme (a curve in ...
1
vote
1answer
113 views

Under which conditions on the homogeneous ideal $ I $, the quotient ring $ \mathbb{C} [X_0, \dots, X_n]/I $ is a regular ring?

If $ I $ is a homogeneous ideal of the ring of homogeneous polynomials $ \mathbb {C} [X_0, \dots, X_n] $ , under which conditions on the homogeneous ideal $ I $, and particularly on $ I_m $, the $m$ -...
3
votes
1answer
170 views

radical of a certain ideal of sixteen variable polynomial ring, generated by the entries of certain matrices

Consider the polynomial ring $R=\mathbb C[x_1,x_2,...,x_{16}]$, and set $$X=\begin{pmatrix} x_1 &x_2&x_3 &x_4\\ x_5&x_6& x_7&x_8\\x_9&x_{10}&x_{11}&x_{12}\\x_{13}&...
4
votes
1answer
119 views

Heights of contracted ideals

Let $R$ be a non-noetherian domain. $S$ be a multiplicatively closed subset of $R$. Let $S^{-1}R$ be a localisation of $R$, where all element of $S$ is invertible. Suppose we have an ideal $I$ of $R$. ...