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

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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
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2answers
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
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3answers
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
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1answer
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
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0answers
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
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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 ...
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1answer
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
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2answers
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
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1answer
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
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1answer
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
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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
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0answers
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
2answers
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
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0answers
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
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1answer
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
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2answers
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 $\{b_1,\dots,b_n\}$ ...
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0answers
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
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2answers
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
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1answer
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
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0answers
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
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1answer
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 ...
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0answers
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
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0answers
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
2answers
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
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1answer
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
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1answer
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
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1answer
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, ...
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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
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1answer
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
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1answer
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 ...
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0answers
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
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1answer
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
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0answers
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
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2answers
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
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0answers
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
3answers
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
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0answers
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
2answers
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
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0answers
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
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0answers
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
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3answers
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
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0answers
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
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1answer
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
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2answers
246 views

Sandpile group corresponding to Abelian group

How we can prove each finite Abelian group is the sandpile group for some graph ?
4
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0answers
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
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0answers
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
1answer
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
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1answer
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
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0answers
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
1answer
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 ...