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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3
votes
0answers
55 views

Algorithm to compute the largest submodule of two given modules, containing a common submodule

Let $M_2\supset N\subset M_1$ be finitely generated modules, say over a polynomial ring. Here the inclusion just means an injection. There exists a maximal $K$ extending $N$, contained in both $M_1$ ...
3
votes
1answer
120 views

Origin of the relations of Leavitt path algebras

I know the formal definition of Leavitt path algebras, but I want know why the relations defining Leavitt Path Algebras are defined in that way? what is special of this relations? My real hidden ...
5
votes
0answers
170 views

Finite generation of kernel of derivations

Let $A$ be a finitely generated regular $k$-algebra, $k$ algebraically closed of characteristic zero, elements $x_1,\dots,x_n\in A$, such that $dx_1,\dots,dx_n$ give rise to a trivialization of the ...
3
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0answers
123 views

Mod p reduction of geometrically irreducible polynomials

Let $f\in \mathbb Z[t,x]$ be a polynomial of positive degree that is irreducible over $\overline{\mathbb Q}[t,x]$. Is it true that for all but finitely many primes $p$ the reduced polynomial $f_p\in \...
2
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0answers
24 views

Saturation of pure binomial homogeneous ideals

Let $2q$ monomials $m_1, m'_1, \cdots, m_q, m'_q$ of the polynomial ring ${\mathbb Z}[X_1, \cdots, X_n]$ such $\deg(m_i) = \deg(m'_i) := d_i$. I note $I$ the binomial ideal $I = \langle m_1 - m'_1, \...
2
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0answers
76 views

Infinitely many initial ideals for non-Artinian monomial orders?

Consider the polynomial ring $R=\mathbb Z[x_1,\ldots,x_n]$ and an ideal $I\subset R$. Let $<$ be a monomial order, i.e. a total order on the set of monomials in $R$ such that for any monomials $a$, ...
1
vote
1answer
139 views

Lifts of smooth algebras

Let $(R, I)$ be a Henselian pair, with $I$ a finitely generated ideal. We know that for any smooth $R/I$-algebra $A_0$, there exists a smooth $R$-algebra $A$ such that $A/I\simeq A_0$. We also know ...
23
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0answers
355 views

On the definition of regular (non-noetherian, commutative) rings

All rings are commutative with unit. A ring $R$ is called regular if it satisfies (Reg) Every finitely generated ideal of $R$ has finite projective dimension. Clearly this gives the usual ...
0
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0answers
69 views

tensor and hom complex of complexes of graded modules

If $R$ is a commutative ring, $^\cdot,Y^\cdot$ are complexes of $R$ modules,then it is natural to define $X^\cdot \otimes ^\cdot Y^\cdot$ by the n-th component $\coprod_{i+j=n}X^i\otimes_R Y^j$ with ...
6
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0answers
87 views

Contractible affine surfaces of log Kodaira dimension 2

The first examples of contractible smooth affine algebraic surfaces (over the complex numbers) of log Kodaira dimension 2 were constructed in a famous paper of Ramanujam https://www.jstor.org/stable/...
4
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200 views

Ultrapower of a field is purely transcendental

Let $F$ be a field, $I$ a set, and $U$ an ultrafilter on $I$. Is the ultrapower $\prod_U F$ a purely transcendental field extension of $F$? According to Chapter VII, Exercise 3.6 from Barnes, Mack "...
1
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1answer
124 views

Power series rings and the formal generic fibre

Let $S = K[[S_1,\ldots,S_n]]$ and consider $d$ elements \begin{equation*} f_1,\ldots,f_d \in S[[X_1,\ldots,X_d]] \end{equation*} and the prime ideal ${\frak P} \colon\!= (f_1,\ldots,f_d)$ generated ...
2
votes
1answer
99 views

Are integral extensions of a catenary ring still catenary?

A (commutative unitary) Noetherian ring $R$ of finite dimension is said to be catenary if for every prime ideal $\mathfrak{p}$ of $R$ one has $\mathrm{ht}(\mathfrak{p})+\mathrm{dim}(R/\mathfrak{p})=\...
3
votes
1answer
140 views

Complete local rings, automorphisms and approximation

Consider two local morphisms $f,g: B\rightarrow A$ of noetherian complete local rings and $f$ surjective. Does there exist an integer $n\in\mathbb{N}$, such that if $f=g \mod \mathfrak{m}_{A}^{n}$ ...
2
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0answers
117 views

Does a surjection between polynomial rings lose one Krull dimension per generator quotiented?

Let $k$ be a Noetherian commutative ring with unity (I am happy adding hypotheses, as I will ultimately want $k = \mathbb Z$) and $$\varphi\colon A = k[x_1,\ldots,x_n] \to k[y_1,\ldots,y_p] = B$$ a ...
3
votes
1answer
170 views

Are the ring of power series and the ring of germs of holomorphic functions catenary?

I am wondering if the following rings are catenary: If $k$ is a field, is the ring of formal power series $k[[X_1,\dots,X_n]]$ catenary? Is the ring of complex power series with a non-zero radius of ...
3
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0answers
149 views

What is wrong with this argument that $ \mathbb{A}^{2}_{k} $ is not cancellative in positive characteristic?

I have a question about a result of Abyankar, Heinzer, and Eakin, and a similar result in Russell. One of the results in the first paper is that if $ Y $ is a variety such that $ \mathbb{A}^{1}_{k} \...
3
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0answers
64 views

Ideals generalizing maximal ideals and ideals generated by regular sequences

Let $R$ be a local commutative Noetherian ring with maximal ideal $m$. My questions concern ideals $I \subseteq m$ of $R$ such that for any non-zero number $n \in \mathbb{N}$ the $R/I$-module $I^n/I^...
0
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0answers
165 views

Reconstructing almost known polynomial from a system of polynomials with common roots

We have $n$ algebraically independent degree $2$ homogeneous system of polynomials with $\mathbb Z$ coefficients in $n$ variables with exactly $t$ primitive (gcd of coefficients is $1$) integer roots ...
9
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2answers
489 views

Relations between homogeneous polynomials

Edit: The formulation of my question was incorrect, for several reasons. Here is what I hope to be the correct formulation: Let $\mathbb{P}$ be a projective space, and $V$ a general linear subspace ...
1
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0answers
131 views

Power series ring $R[[X_1,\ldots,X_d]]$ over a domain $R$

Let $R$ be a domain and \begin{align*} T \,\colon= R[[X_1,\ldots,X_d]]. \end{align*} Suppose that we have $d$ elements $f_1,\ldots,f_d \in T$ and let us consider an ideal $J$ of $T$ such that $(f_1,\...
8
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1answer
250 views

Equivalence of definitions of Cohen-Macaulay type

I know that the Cohen-Macaulay type has these two definitions: Let $(R,\mathfrak{m},k)$ be a Cohen-Macaulay (noetherian) local ring and $M$ a finite $R$-module of depth $t$. The number $r(M) = \dim_k ...
4
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0answers
94 views

Invariants of linear endomorphisms of tensor products

Let $V$ and $W$ be two finite dimensional vector spaces over an algebraically closed field $K$ of characteristic zero. Consider the coordinate ring $K[\mathrm{End}(V\otimes W)]$ of the affine space ...
6
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0answers
95 views

Standard reference/name for “initial ideals $\Leftrightarrow$ associated graded rings”

Let $R$ be commutative ring with a $\mathbb Z$-grading $\deg$ and let $I\subset R$ be an ideal. On one hand, we may consider the initial ideal $\mathrm{in}_{\deg}(I)$. That is the space spanned by ...
0
votes
1answer
144 views

Symmetric polynomials in two sets of variables

Suppose $f(x_1,...,x_m,y_1,...,y_n)$ is a polynomial with coefficients in some field which is invariant under permuting the $x$'s and the $y$'s. Then $f$ can be generated elementary functions $e_k(x_1,...
1
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1answer
147 views

For a holonomic $D_X$-module $M$, can $\operatorname{gr}M$ have embedded primes?

Let $M$ be a holonomic $D_X$-module. This means that the minimal primes in $\sqrt{\operatorname{Ann}(\operatorname{gr}M)}$ are $n=\dim X$ dimensional, for some (and any) good filtration on $M$. But ...
2
votes
0answers
121 views

Noether’s “set theoretic foundations” of algebra. Reference

In [C Mclarty] we read [Noether] project was to get abstract algebra away from thinking about operations on elements, such as addition or multiplication of elements in groups or rings. Her algebra ...
2
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1answer
203 views

Étale fibration for $K[[X_1,…,X_n]]$

Let us consider a formal power series ring $A_n \colon= K[[X_1,\ldots,X_n]]$ with $0 \ll n < \infty$ and we shall consider a prime ideal ${\frak P}$ of $A_n$ such that $1 < {\mathrm{ht}}({\frak ...
6
votes
2answers
270 views

Maximize $L^p$ norm over sphere

For $p \in \mathbb{R}$, consider the function $$F_p(\lambda_1, \dots, \lambda_n) = \lambda_1^p + \dots + \lambda_n^p.$$ My goal is to maximize this function under the constraints that $$ \lambda_1^2 +...
1
vote
1answer
146 views

Bound on number of proper ideals of norm equal to n

I have read in the paper by Einsiedler, Lindenstrauss, Michel and Venkatesh on Duke's Theorem the following bound that I don't understand: Let $d$ be a positive non-square interger and set let $K = \...
1
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0answers
164 views

Primes of the power series rings

Let $A_n \colon= K[[X_1,\ldots,X_n]]$ be a $n$-variable formal power series ring. By setting $X_n \mapsto 0$, we obtain a natural surjection \begin{equation*} \psi_{n,n-1} \colon A_n \...
4
votes
0answers
69 views

A presentation for a subalgebra

Let $K$ be a field, and let $I=(g_1,\ldots, g_r)$ be an ideal in $A:=K[X_1,\ldots ,X_n]$. Let $\{f_1,\ldots f_m\}$ be a subset of $A$, and let $B$ be the $K$-subalgebra of $A$ generated by $f_1,\...
3
votes
0answers
109 views

What is the $Ass(Ext^p_R(M,R))$?

Let $R$ be a Noetherian commutative local ring, $M$ a finitely generated $R$-module with $p=pd M<\infty$ (projective dimension of $M$). What is the relation between $Ass(Ext^p_R(M,R))$ and $Ass(M)$?...
3
votes
1answer
149 views

On a relation between the Hessian and the catalecticant matrix of a binary quartic form

I am currently working on a paper that requires using the theory of invariants of binary quartic forms. Playing around, I have found an interesting identity that gives the Hessian from the minors of ...
4
votes
0answers
54 views

Embedding the Mészáros subdivision algebra in an Orlik-Terao localization

The following is an open question (Question 4.1) from my paper $t$-Unique Reductions for Mészáros's Subdivision Algebra (published version in SIGMA 2018, and slightly updated preprint version with ...
1
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0answers
122 views

Algebraic generalization of Pascal's identity

Let $R$, $S$ be rings with identity. A map $f: R \times R \to S$ is said to be an a $R_S$-Pascal map if, for all $r_1, r_2 \in R$, the following relations are satisfied : $$\begin{align*} f(r_1-1_G, &...
4
votes
0answers
115 views

Injective resolution of the ring of entire functions

Let $R$ be the ring of entire functions on $\mathbb{C}$. I heard that the concrete value of the global dimension of the ring depends on continuum hypothesis. I would think that the injective dimension ...
1
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0answers
121 views

Infinite-dimensional representation theory of $K[x]$

Let $K$ be an algebraically closed field. The finite-dimensional representation theory of the polynomial algebra $K[x]$ is tame and completely understood, which I shall first summarise. It's ...
2
votes
0answers
56 views

Algorithm to find the minimal number of multiplications

Start with the $\mathbb{Q}$-vector subspace $V_0$ of the polynomial ring $Q[x_1,\ldots,x_n]$ spanned by $\{1,x_1,\ldots,x_n\}$. In each step, we can choose an element of the form $v_iv_i'$ for $v_i,...
4
votes
1answer
138 views

If a commutative graded algebra is free over a graded subalgebra, then must it have a graded basis?

Fix a field $\mathbf{k}$ and an $\mathbb{N}$-graded commutative $\mathbf{k}$-algebra $A = \bigoplus\limits_{n = 0}^{\infty} A_n$ of finite type. ("Finite type" means that each $A_n$ is a finite-...
0
votes
0answers
81 views

Reference request: J-P Serre, “Groupes finis d'automorphismes d'anneaux locaux réguliers”

Does anyone have, or know a link to, a copy of the paper named in the title? It is published in Colloq. d'Alg. École Norm. de Jeunes Filles, Paris (1967), 1-11. I do not have ready access to Serre's ...
3
votes
0answers
94 views

Flatness through parametrization

Let $A$ be a $\mathbb{C}$-algebra. Let $\phi:\mathbb{C}[X_1,...,X_n] \otimes_{\mathbb{C}} A \to \mathbb{C}[t] \otimes_{\mathbb{C}} A$ be a ring homomorphism, sending $X_i$ to say $f_i \in \mathbb{C}[t]...
6
votes
0answers
114 views

On the prime spectrum of $R[[X]]$ when the prime spectrum of $R$ is Noetherian

All rings below are commutative with unity. If $R$ has a.c.c. on radical ideals i.e. if $Spec R$ is Noetherian under Zariski topology, then so is $R[X]$, this is Theorem 2.5 in the following paper ...
4
votes
1answer
157 views

Graded Grothendieck Group and Hilbert Polynomial

I was wondering if any of the arguments from elementary dimension theory of local noetherian rings could be simplified with knowledge of the Grothendieck group. Let $A$ be a noetherian graded $K$-...
6
votes
1answer
348 views

Flatness of the integral closure

Let $R$ be a $p$-torsion free ring which is integrally closed in $R[1/p]$ and let $S$ be a finite etale extension of $R[1/p]$. Is it true that an integral closure $S^+$of $R$ in $S$ is flat over $...
4
votes
1answer
275 views

Is the integral closure of a valuation ring in a finite separable extension of its fraction field étale?

Let $K$ be a field endowed with a rank (height) one valuation with completion $\hat{K}$, which is not discrete. Let $R$ be the valuation ring of $K$. Let $L \subset \hat{K}$ be a separable finite ...
1
vote
0answers
89 views

Fine tuning the growth rate of the degrees of polynomials

Let $r$ be an integer with $r>1$. Suppose that if $k\geq 0$, then $p_{k}(x)$ is a polynomial with nonnegative integer coefficients with $p_{k}(0)=1$ but where $p_{k}\neq 1$. Suppose that $$\...
1
vote
0answers
122 views

Factorizations of etale morphisms

Let $f:X \rightarrow Y$ be a finitely presented separated etale morphism, with $Y$ quasicompact and quasiseparated. By Zariski’s main theorem, we can factor $f$ as $f= g \circ j$ with $j$ an open ...
23
votes
2answers
1k views

Is every commutative ring a limit of noetherian rings?

Edit of Feb. 14, 2019. After Laurent Moret-Bailly's accepted answer, only Questions 4 and 5 remain open. I don't care that much about Question 4, but I'm very curious about Question 5, which is Do ...
0
votes
0answers
54 views

Normality of certain subrings of polynomial rings in characteristic p

Let $k$ be an algebraically closed field of characteristic p. Let $Z\subset k[x_1,\cdots,x_n]$ be a graded $k$-subalgebra of a polynomial ring, such that for any $f\in Z,$ any divisor of $f$ (in $k[...