**0**

votes

**1**answer

138 views

### Depth of polynomial ring $S=\Bbb{R}[x_1,x_2,x_3,…,x_n,…]$

Consider the polynomial ring of countable variables with coefficients in the real numbers, i.e, $S=\Bbb{R}[x_1,x_2,x_3,...,x_n,...]$. My Question is about the depth of this ring.
Question: Could we ...

**0**

votes

**2**answers

141 views

### h^0 of a sheaf supported at a point

$C$ : a projective curve over an algebraically closed field, $I$ : the ideal sheaf of $\mathcal{O}_C$ defining a point $p\in C$.
Why is the following hold?
...

**4**

votes

**3**answers

171 views

### Subset of Spec(A) realized as inverse image of some Spec(B)

Let $A$ be a (commutative) ring and $U$ be an arbitrary subset of $\mathrm{Spec}(A)$. Do there exist a ring $B$ and a ring homomorphism $\varphi\colon A\to B$ such that ...

**1**

vote

**1**answer

155 views

### chain of prime ideals in polynomial ring $S=\Bbb{R}[x_1,x_2,…,x_n,…]$

Consider the polynomial ring of countable variables with coefficients in the real numbers, i.e, $S=\Bbb{R}[x_1,x_2,x_3,...,x_n,...]$. Is the following question true?
Question: Is there any maximal ...

**0**

votes

**0**answers

71 views

### finiteness of the homology of an augmented Koszul complex

Let $(A,m)$ be a local ring $x_1,\cdots,x_n$ elements of $m$ and $M$ a finite $A$-module. Let $K(x)$ be the Koszul complex associated with $x_1,\cdots,x_n$. We define a new complex by $K(x) \otimes ...

**7**

votes

**0**answers

156 views

### Is the generation of rings by their units a question in K-theory?

Susan's question When can number rings be spanned (as $\mathbb{Z}$-modules) by units? smells like an algebraic K-theory question in disguise. I'll reformulate the question first:
Given an integral ...

**-1**

votes

**1**answer

121 views

### Algorithm to find symmetric function given specialization

I have a symmetric function f(c1,c2,c3,c4,c5) which, when c1 < c2 < c3 < c4 < c5, has the form p1(c1)+p2(c2)+p3(c3)+p4(c4)+p5(c5), where the p_i's happen to be polynomials of degree ...

**3**

votes

**2**answers

186 views

### Colon property of Gorenstein rings

Let $(R, \mathfrak{m})$ be a Gorenstein local ring of characteistic $p>0$. Let $x_1,...,x_d$ be a system of parameters of $R$. Let $I$ be an $\mathfrak{m}$-primary ideal containg $(x_1,...,x_d)$. ...

**2**

votes

**1**answer

147 views

### Finite generation of a commutative algebra via its quotients

I have a commutative algebra $A$ over the complex numbers.
I know that $\mathbb{C}[t]$ is a subalgebra of $A$, and that for any natural number $n$, the quotient $A/(t-n)$ is finite dimensional (and ...

**1**

vote

**1**answer

139 views

### Obstruction map for local singularities via tangent (Andre-Quillen) cohomology

Let $R$ be a local singularity (for example $R=\mathbb{C}[[x_1, \ldots , x_n]]/I$) ring over $\mathbb{C}$. Let $\mathbb{L}_{R}$ be a cotangent complex of $R$, then one can define tangent ...

**5**

votes

**3**answers

481 views

### geometric interpretation of “Euclidean domain”

I cite from Wikipedia:
Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields
All of these ...

**2**

votes

**0**answers

109 views

### Flags of varieties

I was wondering if there is a generalization of flags in the following way: Suppose you have a series of inclusions of affine varieties $V_1\hookrightarrow V_2\hookrightarrow\cdots\hookrightarrow V_n$ ...

**4**

votes

**2**answers

417 views

### Faithful-flatness for maps of formal power series rings

Let $R$ be a ring (commutative with unit).Let $f_1,...,f_n\in R$ elements that generate the unit ideal. The map $R\to R_{f_1}\times ...\times R_{f_n}$ is faithfully flat, since this is just a Zariski ...

**2**

votes

**0**answers

59 views

### Hilbert Regularity in relation to degree of generators

Suppose we look at the $\mathbb {C}$ module $\mathbb{C}[x_1,\dots,x_n]=\oplus{S_i}$ where $S_i$ are the polynomials of degree $i$. Then we look at a subring $R$ (also a $\mathbb {C} $ module) ...

**5**

votes

**1**answer

225 views

### Smooth affine algebraic subgroups as complete intersections

Let's fix an algebraically closed field $k$ of arbitrary characteristic and a connected nonsingular affine algebraic $k$-group $G$. Under what conditions can I assume that a connected nonsingular ...

**6**

votes

**2**answers

230 views

### Can transcedence degree be defined for arbitrary ring homomorphism?

Fix a homomorphism $f:A\rightarrow B$.
Choose $\{b_1,\dots,b_n\}$, $\{b'_1,\dots,b'_m\}$ subsets of elements in $B$. Suppose that $B$ is algebraic over $f(A)[b_1,\dots,b_n]$ and $\{b_1,\dots,b_n\}$ ...

**1**

vote

**0**answers

139 views

### Is every Artinian local ring a quotient of a nilpotent algebra over a field?

This question arose from a conversation with a friend where we tried to classify all one-point-schemes. Apologies if it's a totally stupid question.
If $R$ is a Noetherian commutative ring with ...

**4**

votes

**2**answers

377 views

### Quotient of $Z[x_1,…,x_n]$ by a maximal ideal is a finite field [duplicate]

I am seeing the proof of the Ax-Groethendieck theorem from commutative algebra and I have a problem. How can I prove that if $x_1,...,x_n$ are complex numbers and $I$ is a maximal ideal of ...

**2**

votes

**1**answer

105 views

### Does this solution guarantee $det(A)=0$ where $A\in M(R)$? [closed]

Suppose $R$ is a commutative ring with identity $1$ and the following matrix equation holds:
$\begin{pmatrix} a_n & & \\
\vdots & \ddots & \\
a_1 ...

**4**

votes

**0**answers

151 views

### What is known about this short exact sequence in Lie algebra cohomology?

In its most general form, I look at the following. Let $g$ be a dg Lie algebra and $Z$ be its center. The sequence $ Z \to g \to g/Z $ gives me a short exact
$ 0\to C(g,Z) \to C(g,g) \to C(g,g/Z) \to ...

**9**

votes

**1**answer

201 views

### Bernstein-Sato polynomial (one variable)

Let $R = \mathbb{C}[x_1,...,x_n]$, $p \in R$. There exists a monic (of lowest degree) $b_p(x) \in \mathbb{C}[s]$ and a differential operator $D(s)$ such that
$$b_p(s) p^s = D(x)p^{s+1}.$$
The ...

**1**

vote

**0**answers

194 views

### What is your expectation of the depth?

Let $S=k[x_1,...,x_9]$ be a polynomial ring over field $k$. Set $q_1=(x_1,x_2,x_5,x_6)$, $q_2=(x_1,x_2,x_6,x_7)$, $q_3=(x_2,x_3,x_7,x_8)$, $q_4=(x_1,x_5,x_6,x_7)$, $q_5=(x_1,x_6,x_7,x_8)$, ...

**0**

votes

**0**answers

163 views

### Monomial ideals: isomorphism problem for commutative algebras?

Theorem 5.27 in Polytopes, Rings, and K-Theory (Bruns, Gubeladze - 2009 - Springer SMM) claims:
Let $K$ be a field and $I\!\unlhd\!K[x]= K[x_1,\ldots,x_n]$ and ...

**2**

votes

**2**answers

183 views

### Tychonoff spaces and ideals

Let $X$ be a tychonoff space and let $T$ be the set of all $f \in C(X)$ such that for any $g$ the equation $fg = 1$ has at most finitely many solutions. Under what conditions on $X$, the set $T$ is an ...

**2**

votes

**1**answer

227 views

### Geometric meaning of the positive part of graded ring

Let's say I have a complex projective variety $X\subseteq\mathbb P^n$ with homogeneous coordinate ring $S=\bigoplus_{d\ge 0} S_d$. The localization by some homogeneous $f\in S$ (of nonzero degree) ...

**0**

votes

**1**answer

197 views

### Projective bundles

Fix $n$ and let
$0\leftarrow \mathcal{F}\leftarrow \bigoplus \mathcal{O}_{\mathbb{P}^n}(a_i)\leftarrow \bigoplus \mathcal{O}_{\mathbb{P}^n}(b_i)\leftarrow \cdots$
be an exact sequence.
Then we can ...

**4**

votes

**1**answer

282 views

### Computing cotangent complex

I would like to know if one can compute all the cohomology sheaves of the cotangent complex of a subvariety of the affine space once a resolution of its ideal sheaf is given?
In my precise situation, ...

**0**

votes

**0**answers

127 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 ...

**1**

vote

**1**answer

313 views

### Geometry behind Rees algebra (deformation to the normal cone)

Let me start with the formal definition of Rees algebra. If $A$ is a commutative ring over some field $k$, $I \subset A$ is an ideal, then Rees algebra is by definition
$$
R=\oplus_{i \in \mathbb{Z}} ...

**1**

vote

**1**answer

64 views

### integral closure of parameter ideals

Let $(R, \mathfrak{m})$ be an excellent domain of dimension $d$. Let $\mathfrak{q} = (x_1,...,x_d)$ be a parameter ideal of $R$.
Question: Is it true that $(x_1,...,x_{d-1}):x_d$ is contained in the ...

**1**

vote

**0**answers

35 views

### Finite window transformations--input+output (pure algebra)

This q. presents a complete approach to my previous q.: Indecomposability of image transformations ...
Let $\ A\ B\ $ be finite sets of cardinality $\ > 1$. Let $\ D:=A\times B$, ...

**15**

votes

**1**answer

548 views

### When is $f(x_1, \dots, x_n)+c$ an irreducible polynomial for almost all constants $c$?

Let $f\in \mathbb C[x_1, \dots, x_n]$, $n\ge 1$, be a non-constant polynomial. Consider the polynomial $f+t\in \mathbb C[t, x_1,\dots, x_n]$. This is an irreducible polynomial in $\mathbb C(t)[x_1, ...

**5**

votes

**0**answers

182 views

### Do tuples of pairwise commuting matrices form a variety?

Let $A$ be a ring and consider the ring generated by $A$ and $kn^2$ indeterminates $X_{lij}$ (with $1 \leq l \leq k $ and $1 \leq i,j \leq n$). If $M_l$ is the matrix $( X_{lij} )_{i,j}$, then one can ...

**2**

votes

**2**answers

180 views

### Partition of $\mathbb{F}_2^n$?

Consider an undirected graph $G$ with $n$ nodes denoted by $i$, $i \in [n] = \{1,2,...,n\}$. Denote the set of neighbours of node $i$ in the graph by $N(i)$.
Given that there exists a set ...

**8**

votes

**0**answers

440 views

### Getting a bound via polynomial equations

When studying the existence problem of power residue deference sets, I came across the following system of polynomial equations over $\mathbb{C}$,
\begin{cases}
...

**3**

votes

**3**answers

161 views

### On matrices in linear forms with vanishing determinant

This is a cross-post from my original question at math.se. I decided to post here because it seems more difficult than I originally thought.
Let $R=\mathbb C[x_1,\ldots,x_r]$ be a polynomial ring. ...

**3**

votes

**0**answers

222 views

### PAC field : Algebraically closed field :: ? : Henselian local ring

I'm wondering if the following exists in the world as a definition. I'll use the word "pseudo-Henselian." I'll restrict to DVRs for simplicity.
I'd want to call a DVR $(R,\mathfrak{m})$ ...

**4**

votes

**2**answers

187 views

### Do taking a general hyperplane section and taking a colon ideal commute?

Let $I$ be an ideal and $f$ be an element of $R = \mathbb{C}[x_1,\ldots,x_n]$, where $\mathbb{C}$ is an algebraically closed field of characteristic $0$. Does
$$
(I+H):f = (I:f)+H
$$
hold for a ...

**3**

votes

**0**answers

143 views

### Comparing different Euclidean algorithms on a Euclidean domain

I have posted this question at stackexchange (502413), without responses until now.
In the papers by T. Motzkin: The Euclidean Algorithm, Bull. AMS 55, 1949, pp. 1142--1146 and P. Samuel: About ...

**3**

votes

**0**answers

150 views

### Dimension of ring completion wrt to a decreasing chain of ideals

Let $(R,\mathfrak{m})$ be a Noetherian local ring and let $(I_{n})_{n\in\mathbb{N}}$ be a decreasing chain of ideals in $R$ such that $\bigcap_{n \in \mathbb{N}} I_{n} = \{ 0 \}$. Then there is a ...

**4**

votes

**3**answers

307 views

### A question concerning the isomorphic type of continuous functions

let $S$ be the ring of all functions $f : \Bbb{R}\longrightarrow \Bbb{R}$ which are continuous outside a bounded open interval containing zero (depended on $f$). Is it possible to consider $S$ as ...

**2**

votes

**0**answers

87 views

### Indecomposability of image transformations (pure algebra). Open questions

W-transformations -- definitions
We will consider a class called finite window transformations $\ T:C^\mathbb Z\rightarrow C^\mathbb Z\ $ defined a paragraph below; $\ \mathbb Z\ $ is the ring of ...

**3**

votes

**1**answer

178 views

### rational singularities of a certain variety

Let $X$ be a variety in $\mathbb{C}^{10}$ defined by the ideal
$I=\left<xz'-x'z, y'(u+z)-y(u'+z'), t'(u-z)-t(u'-z')+xy'-x'y\right>$ of $\mathbb{C}[x,y,z,u,t,x',y',z',u',t']$.
Note that ...

**3**

votes

**2**answers

246 views

### Sandpile group corresponding to Abelian group

How we can prove each finite Abelian group is the sandpile group for some graph ?

**4**

votes

**0**answers

205 views

### Epimorphisms between external tensor products

Let $k$ be a field, $R,S$ commutative $k$-algebras. If $\mathsf{Mod}_{fp}$ denotes the category of finitely presented modules, the external tensor product $\mathsf{Mod}_{fp}(R) \otimes_k ...

**2**

votes

**0**answers

48 views

### different and discriminant for finite invariants

Let $k$ be an algebraically closed field.
Let $B$ a $k$-algebra of finite type, normal and Cohen-macaulay. Let $G$ a finite group acting on $B$. We assume that the order of $G$ is prime to the ...

**4**

votes

**1**answer

241 views

### quotient by finite group actions that are smooth

Let $X$ an affine normal scheme of finite type over a field $k$ of characteristic zero.
Let $G$ a finite group acting on $X$ and $Y=X/G=Spec(K[X]^{G})$.
We assume that ...

**2**

votes

**1**answer

286 views

### About the practice of Bernstein-Kushnirenko theorem

The following refers to
common roots of bivariate polynomial equations and, in particular to the quim's and auniket's comments.
The BKK theorem (cf. arXiv:0812.4688. Theorem 5.4) asserts that if we ...

**2**

votes

**0**answers

135 views

### Relation between Castelnuovo-Mumford regularity for coherent sheaves and modules

Let $S$ be the ring $\mathbb{C}[X_0,...,X_n]$. Let $X$ be a smooth projective scheme of the form $\mathrm{Proj}(S/I_X)$ for some ideal $I_X$. Let $C$ be a scheme associated to a Cartier divisor on ...

**5**

votes

**1**answer

415 views

### Uncountable Reduced ring $R$ with $R[x]$ has only a countable number of maximal left ideals

The question is following:
Is there an uncountable reduced ring (i.e., a ring with no non-zero nilpotent element) $R$ (with identity) such that
$R[x]$ has only a countable number of maximal ...