Skip to main content

All Questions

Filter by
Sorted by
Tagged with
0 votes
0 answers
292 views

regular locus of an affine domain

Let $A$ be an affine domain over a field $k$ (need not be algebraically closed). Let $\mathfrak{p}$ be a prime ideal of $A$, such that $A_{\mathfrak{p}}$ is a regular local ring. Does there always ...
sagnik chakraborty's user avatar
0 votes
0 answers
109 views

Vanishing of the module of differentials of a extension of perfect fields

Let $L|F$ be a extension of perfect fields of characteristic $p$, $\phi_F:F \to F_{\phi}$, $\phi_L:L \to L_{\phi}$ the Frobenius isomorphisms ($F_{\phi}=F$ but considered as $F$-algebra via $\phi_F$). ...
user58841's user avatar
0 votes
0 answers
79 views

Stable analytic manifold under simple action

For an integer $m > 1$, let us define the action $$ f: X_i \to (1+X_i)^{m} - 1 $$ on $C[[X_1,...,X_N]]$, where $C$ is the complex number field. Consider the analytic manifold $V(I)$ defined by the ...
Pierre's user avatar
  • 1
0 votes
0 answers
105 views

$\Gamma_Z(\widetilde M)\cong\widetilde{ \Gamma_Z(M)}$

Let $R$ be a Noetherian ring and let $M$ is an $R$-module. Consider the associated affine scheme $(\text{Spec R},\mathcal{O}_{\text{Spec R}})$ and Suppose $Z\subset X$ is a closed subset of $\text{...
user49402's user avatar
0 votes
0 answers
182 views

Zariski open set of linear forms

Let $I$ be a graded homogeneous ideal over $k[x_1, ... ,x_n]$ and $h$ a linear form, let $H$ be the corresponding hyperplane and $I_H$ the restriction of $I$ to $H$. I am looking for a Zariski open ...
A.B.'s user avatar
  • 73
0 votes
0 answers
230 views

A question about the unbounded derived category of the polynomial ring in infinitely many variables

In this moment I am trying to understand the derived category of the polynomial ring in infinitely many variables over a field $k$, $R=k[x_{1},x_{2},\dots]$ and I am wonder if it is true that $Hom_{D(...
Manuel's user avatar
  • 11
0 votes
0 answers
350 views

Reference: A nowhere vanishing section of a vector bundle is locally split

Well-known fact: If $(A, \mathfrak{m})$ is a local Noetherian ring, $E$ is a finitely generated free $A$-module, and $e\in E$ is an element not contained in $\mathfrak{m}E$, then $E/eA$ is also a ...
Charles Staats's user avatar
0 votes
0 answers
136 views

Monoid action on an uncountably infinite set

The action of a monoid on a finite set is equivalent to a finite state machine, however I would like a categorical way to think about an uncountably infinite state machine (a state transition system?)....
smolloy's user avatar
  • 101
0 votes
1 answer
119 views

Colon operation after adjoint variables

Let $R$ be a commutative Noetherian ring and $M$ a finitely generated $R$-module. Let $I$ an ideal of $R$. We have $$0:_MI = \cap_x(0:_Mx),$$ where $x$ runs a set of generators of $I$. Now set $S = ...
Pham Hung Quy's user avatar
0 votes
0 answers
68 views

Let $(R, m)$ be noetherian local, $\dim(R)=1$. Show $CH^1(R)=\mathbb{Z}/(\gcd([k_i, R/m]))$, where $k_i$ are residue fields of normalization

Let $R$ be a $1$-dimensional noetherian local domain. Then we have that $CH^1(R)=\mathbb{Z}/(\gcd([k_i, k]))$ where the $k_i$ are residue fields of the normalization and $k$ is the residue field of $R$...
Pax's user avatar
  • 841
0 votes
0 answers
138 views

Profinite Local Ring inside Polynomial Ring

This is a "technical" question that I came across in my research. Let $A = \textbf{Z}_{p}[\![t_1, \cdots, t_a ]\!]<z_1, \cdots, z_b>$ be the $(p, t_1, \cdots, t_a)$-adic completion of the ...
david's user avatar
  • 61
0 votes
0 answers
355 views

Cubic field and the corresponding cubic binary form

I am currently reading about binary cubic forms and cubic number fields (mainly about using binary cubic forms with integer coefficients to parametrize orders in the cubic field) and I thought it ...
Heidi's user avatar
  • 21
0 votes
0 answers
320 views

Invariants of the Determinant Form

Consider a form of degree $r$ in $n$, that is, a homogeneous polynomial $$f(x_1, \ldots, x_n)=\sum_{i_1+\ldots i_n=r}\alpha_{i_1 ... i_n}x_1^{i_1} ... x_n^{i_n} $$ After the linear change of ...
zacarias's user avatar
  • 801
0 votes
0 answers
72 views

Decomposition results for locally commutative semigroups

Every finite abelian group is the direct product of its cyclic groups of prime order, and every commutative monoid divides a product of its cyclic submonoids. Could these results generalized to ...
StefanH's user avatar
  • 798
0 votes
1 answer
368 views

The closure of an effective Cartier divisor in a special situation

I am studying first order deformations and a natural question arises. Situation: Let $X_1$ be a scheme. $\pi: X_1 \to {\rm Spec}~ k[t]/(t^2)$ is a flat morphism of finite type, where $k$ is an ...
Yang Zhou's user avatar
0 votes
0 answers
548 views

Fitting ideal sheaves and determinant bundles

I am working on a problem in algebraic geometry which comes down to a fact in commutative algebra that I am hoping is well-known. Suppose $F$ is a coherent sheaf on a smooth variety $S$, and that the ...
Jack Huizenga's user avatar
0 votes
0 answers
52 views

Cohen-Macaulayness of inseparable isogeny k-algebras

Let $R$ and $S$ be 2 associated, commutative, and unita $k$-algebras where $k$ is an algebraically closed field of characteristic $p$. We call these algebras inseparable isogeny or $F$-isomorphism if ...
NN guest's user avatar
  • 127
0 votes
0 answers
181 views

A question on binary polynomials

This is probably a well-known result but I was not able to find a reference on my search. My question concerns general polynomials $f(x,y) \in \mathbb{Z}[x,y]$ such that $f$ cannot be written as a ...
Stanley Yao Xiao's user avatar
0 votes
0 answers
255 views

Image of critical points

Let $K$ be a field of characteristic $0$, $f:K^n\rightarrow K^n$ be an algebraic function, that is, $n$ polynomial functions in $n$ variables. Let $S$ be the set of critical points of $f$. If $K=\...
loup blanc's user avatar
  • 3,741
0 votes
0 answers
259 views

Ring algebraically closed in its completion.

First I would like to be clear about the definition, which I am having trouble finding. What does: The local ring $A$ is algebraically closed in $B\supset A$. (e.g. for $B:=\hat{A}$, the completion ...
O.R.'s user avatar
  • 807
0 votes
0 answers
93 views

Ring of even characteristic.

Is possible to choose three units $u,v,w$ of a ring $R$ (not containing a field) with even characteristic such that $u+v+w=0$. Thanks in advance.
Joaquín Moraga's user avatar
0 votes
0 answers
99 views

Example of a ring whose minimals are annihilators of idempotents?

I'm looking for examples† of rings with the property that for each $P={\rm Ann}_R(a)\in{\rm Min}(R)$ then $a\in R$ is idempotent (ie $a^2=a$) † other than domains!
QED's user avatar
  • 189
0 votes
1 answer
322 views

Height unmixed ideal

Suppose $R$ is a regular local ring and $I$ is a non-zero ideal such that $I$ is a radical ideal and $I$ is height unmixed. Suppose $J$ is any radical ideal contained in $I$ and with the same height ...
messi's user avatar
  • 3
0 votes
0 answers
123 views

Irreducibility of superelliptic curves

Let $k$ be an algebraically closed field of characteristic zero, let $a,d$ be integers, and let $f\in k[x]$ be a separable polynomial of degree $d$. Question: a) Is the affine plane curve $y^a=f(x)$ ...
Robert's user avatar
  • 23
0 votes
0 answers
166 views

The intersection complex and the Cohen-Macaulay property

Let $\Delta:Y\rightarrow X$ a closed immersion of $k$-schemes of finite type and equidimensionnal. We assume that $\Delta^{*}[-d]IC_{X}=IC_{Y}$, if $X$ is Cohen-Macaulay, does it imply that $Y$ is ...
prochet's user avatar
  • 3,472
0 votes
0 answers
355 views

Can we find a Groebner Basis?

I would like to ask the following. Given only the leading terms of an ideal $I$, namely the set $LT(I)$, is it possible to find a Groebner Basis of $I$? If not always, then when is it possible? We ...
Sln's user avatar
  • 1
0 votes
1 answer
468 views

Finite extensions of residue fields of Henselian DVRs

Let $K$ be an Henselian discrete valuation field such that its completion is separable over $K$. Let $F$ be its infinite residue field. Is it true that a finite extension of $F$ is a simple extension ...
Jana's user avatar
  • 2,032
0 votes
0 answers
245 views

Notation Problem, Fixed Rings and Fields

I am trying to make sense of the notation and certain sets in two articles by Annick Valibouze whose results I'm using for my bachelor's thesis, I hope it's relevant enough to merit an answer. In one ...
Erik Vesterlund's user avatar
0 votes
0 answers
355 views

abstract algebra for component wise operations on "vectors" or what it might be called

I have a quite tough problem to solve and need an algebra that allows to "vectors" following operations: - multiplication between two vectors are componentwise that means v=(v1, v2, v3,...) multiplied ...
al-Hwarizmi's user avatar
0 votes
0 answers
244 views

Properties of Gorenstein ideal

Fix an integer $k>4$. For any integer $r>0$, denote by $S_{r}:=\mathbb{C}[X_0,X_1,X_2,X_3]_{r}$ the vector space of degree $r$ polynomials in $X_i$ with coefficients in $\mathbb{C}$. Let $W$ be ...
Naga Venkata's user avatar
  • 1,040
0 votes
0 answers
87 views

Algorithm for computing basis of zero dimensional ring?

If given a zero dimensional ring over a field, for example, a polynomial ring $A=k[x_1,\ldots,x_n]/(f_1,\ldots,f_n)$ such that $A$ is 0-dimensional, is there an algorithm to compute a monomial basis ...
minimax's user avatar
  • 1,157
0 votes
0 answers
235 views

Power of ideals and exact sequences

Hello, I'm reading about analytic sheaves and I've a problem to understand something that's related with commutative algebra: Let $\mathfrak{a}\subset R$ an ideal and $M$ an $R$-module. Then, $\...
Pedro Montero's user avatar
0 votes
0 answers
315 views

Definitions for Oddness

In the thread Even Xor Odd Infinities I defined odd models of Modular Arithmetic (MA) as models satisfying the axioms of MA and two first order statements. Even XOR Odd Infinities? $O1) \forall x(x=0 ...
Russell Easterly's user avatar
0 votes
0 answers
383 views

Pseudo-cauchy sequence and valuation

Let $k$ be a field and $x$ is transcendental over $k$. Can we construct a pseudo-cauchy sequence $(a_{i})$ convergent to $x$ with each $a_{i}$ is algebraic over $k$ and $k(a_{i})\subseteq k(a_{i + 1})$...
Rajnish's user avatar
  • 173
0 votes
0 answers
178 views

The transcendence degree of the algebras of invariants

Let $V_n,V_m$ be the vector $\mathbb{C}$-spaces of the binary forms of degrees $n,m$ considered as usual $SL_2$-modules. Let $I_{n,m}=\mathbb{C}[V_n \oplus V_m]^{SL_2}$ and $C_{n,m}=\mathbb{C}[...
Melania's user avatar
  • 301
0 votes
1 answer
177 views

Laurent series with analytic coefficients

Let $A=H(D(0,1))$ the ring of holomorphic functions on the open unity disc. I consider the function $f$: $$f (t)=\sum f_{i}t^{i} \in A[[t]]$$ I suppose that the $t$-adic valuation of it is less or ...
prochet's user avatar
  • 3,472
0 votes
0 answers
381 views

Completion of commutative rings.

Assume that $(R,\mathfrak{m})$ is a commutative local ring of equal characteristic zero. So $R$ contains the field of rationals. The well known $\mathfrak{m}$-adic completion of $R$ provides a ...
Aurora's user avatar
  • 591
0 votes
0 answers
152 views

Kählerdifferentials and normal crossing divisors

Let $k$ be an algebraically closed field of arbitrary characteristic, $X$ a smooth surface over $k$, and $D_i \subset X$ be an regular, effective Divisor such that $D=\sum D_i$ has normal crossings ...
fschueller's user avatar
0 votes
0 answers
243 views

strict henselian and excellent henselian

Hello, everyone. I want to ask a problem about strict henselian ring. Let $A$ be a strict henselian DVR. Dose there exist subrings $A_{i}$ of $A$, such that $A=lim_{i} A_{i}$ and where $A_{i}$ are ...
kiseki's user avatar
  • 1,921
0 votes
0 answers
428 views

flat morphism between regular local rings

Suppose $f: A \rightarrow B$ is a local homomorphism of local rings. Assume that $A$ and $B$ are noetherian, regular and $\mathrm{Spec} B \rightarrow \mathrm{Spec} A$ is quasi-finite. Is is necessary ...
xuehang's user avatar
  • 153
0 votes
0 answers
346 views

Length of $\mathfrak{m}$-torsion module

Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring and $M$ is an $R$-module such that $\mathfrak{m}^tM=0$ for some non-negative integer $t$. Then the length of $M$ is finite. Is that right?...
minhtringuyen's user avatar
0 votes
0 answers
236 views

On vanishing orders of an ideal via the restriction

Let $Y$ be a submanifold of a complex manifold $X$, and $a$ be an ideal on $X$ which does not vanish along the entire $Y$. Consider a point $\xi$ on $Y$, there are the vanishing order $ord_{\xi}a$ ...
Zhengyu Hu's user avatar
0 votes
0 answers
87 views

Standard Notation for Monomial Orderings?

Is there a standard way to denote a particular lexicographic (resp. reverse lexicographic) monomial ordering using subscripts or superscripts? For example, I might want to refer to the lexicographic (...
stepanp21's user avatar
  • 326
0 votes
0 answers
212 views

Homomophism from Koszul complex to the original ring

In an article, I encounter an isomorphism relation as follows: Let S be a comm. ring, x an element in S. K[x,S] be corresponding Koszul complex. The article says "this is a classical isomorphism": $...
AlgRev's user avatar
  • 61
0 votes
0 answers
774 views

Discrete valuation rings.

Given an algebraically closed field $\mathbb F$ of characteristic $p$, let $\mathbb A$ be a discrete valuation ring of characteristic zero having $\mathbb F$ as its residue field ( it does exist, but ...
Angelo's user avatar
  • 1
0 votes
1 answer
583 views

Question about modules, quotient rings, and polynomial rings? [closed]

Consider an integer polynomial ring, $A = \mathbb{Z}[t]$, and a ring of fractions, $B = \mathbb{Z}[t, t^{-1}]$; obviously, $A$ is a subring of $B$. Now we consider two modules over $A$ and $B$, $M$ ...
Osiris's user avatar
  • 161
0 votes
0 answers
179 views

semigroup actions of groups on regular rooted trees

If $G$ is a group which has a semigroup action on a regular rooted tree via prefix-preserving, continuous transformations (I give the tree the path metric), what kinds of algebraic restrictions can we ...
dan's user avatar
  • 125
0 votes
0 answers
166 views

Can the zero-degree part of $M_f \otimes_{S_f} N_f$ be identified with $M_{(f)} \otimes_{S_{(f)}} N_{(f)}$?

The isomorphism ${(M \otimes _ {S} N)} _ {f} = M _ {f} \otimes _ {S _ {f}} N _ {f}$ is well-known. Here, $S$ is a graded ring, and $M,N$ are graded $S$ modules. Now, let $f$ be any homogeneous ...
Hiro's user avatar
  • 945
0 votes
0 answers
551 views

sub ring of algebra over subfield

Let $k$ be a field and $k[a]$ an algebric extension. If $A$ is a reduced commutative algebra over $k[a]$ and $B$ is a subring which is an algebra over $k$, then is the following true: if there exist ...
ventura's user avatar
0 votes
0 answers
237 views

resolution of singular points on plane curves and base change

Let $k$ be a field and $C/k$ be an affine plane curve over $k$, namely $C = \mathrm{Spec}(A)$ for some $A = k[x,y]/(f(x,y))$, here $f(x,y) \in k[x,y]$ is an irreducible polynomial. Let $B$ be the ...
user565739's user avatar
  • 1,109