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

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Extension of canonical surjection to a place of fraction field

Let $R$ be an integrally closed domain, $K$ its field of fractions, and $m$ a maximal ideal of $R$. By Chevalley theorem, the canonical surjection $R\to R/m$ extends in at least one way to a place ...
6
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1answer
307 views

Irreducibility of a polynomial

For $n\ge 1$, let $g(x_1,x_2,\ldots,x_n)$ be an irreducible homogeneous polynomial in $n$ variables over a field $k$ and $f(x)$ an irreducible polynomial of $k[x]$. Is $f(g(x_1,x_2,\ldots,x_n))$ ...
4
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0answers
167 views

Inversion, Koszul duality, combinatorics and geometry

According to this MO answer Koszul duality is related to operations on generating series; 1) multiplicative inversion for quadratic algebras, 2) compositional inversion for quadratic operads, 3) ...
3
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2answers
177 views

Divisibility of the degree of an extension by the degree of its residual field

Let $A$ be an integrally closed domain whose quotient field is $K$, $L$ be a finite Galois extension of $K$, and $B$ be the integral closure of $A$ in $L$. Let $M_A$ be a maximal ideal of $A$, and ...
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0answers
84 views

Moduli space of points - Gorenstein ideal

I've been working on algebraic covers, $\varphi\colon X\rightarrow Y$, ($\varphi_*\mathcal{O}_X$ is a locally free $\mathcal{O}_Y$-algebra of rank d). I'm more interested in the algebraic point of ...
4
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85 views

Multiplicity of $Ext^{d-t}(M,\omega_R)$, ($d=\dim R, t=\dim M$)

Let $R=\bigoplus_{i \geq 0} R_i$ be a Cohen-Macaulay graded ring ($R_0$ is a field and $R$ is generated by $R_1$) of dimension $d$ with canonical module $\omega_R$, and $M$ a graded Cohen-Macaulay ...
1
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1answer
116 views

Reducedness of a ring with prime nilradical

Let $A$ be a regular ring and $\mathfrak q$ be an ideal, such that $\sqrt{\mathfrak q}$ is prime. Further assume that $\mathfrak q$ is locally principal (i.e. $\mathfrak q$ is an irreducible divisor ...
6
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3answers
356 views

Irreducible/prime/indivisible elements

in what follows all the rings are commutative, nontrivial, with unit. Recall the following definitions: 1) $\pi\in A$ is prime if $(\pi)$ is a nonzero prime ideal 2) $\pi\in A$ is irreducible if ...
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62 views

ramification index generalized

I am trying to rewrite the theory of decomposition/inertia/ramification groups independently of the theory of Dedekind or valuation rings (I believe this has been done elsewhere, but I found only few ...
6
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2answers
574 views

Is being reduced a generic property of schemes?

(Naive formulation:) Let $X$ be an (irreducible) affine variety (over an algebraically closed field $k$) and $I$ be an ideal of the coordinate ring $R$ of $X$. Assume $Y = V(I)$ is equidimensional. ...
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213 views

What is a good introduction to cluster algebras from surfaces?

What is a good reference for cluster algebras from surfaces, with a view to their connection to Teichmuller theory? In my view, that means it should start off with unpunctured surfaces (and in ...
7
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3answers
389 views

Ring of differential operators of a quotient ring

All rings are assumed to have unity. Let $k$ be a field. Recall the definition of Grothendieck's ring of ($k$-linear) differential operators $D(R;k)$ of a commutative $k$-algebra $R$: ...
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1answer
102 views

question about a particular Polynomial ring [closed]

Let K be a field, let $T = K[X_1, X_2,...]$ be a polynomial ring, let $R=K[X_1^{2}, X_1X_2,..,X_i X_j,..]$, and let $L = Frac(R)$ = field of fractions of R. How can we prove that $R =T \cap L$ ?
7
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1answer
385 views

Some questions about the ring Z((x))

$\newcommand{\ZZ}{\mathbb{Z}}$ $\newcommand{\dim}{\text{dim }}$ Let me begin by apologizing for the length of this question, but I thought this might be interesting to some of you. This ring isn't ...
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0answers
108 views

Is the natural map $Pic A[M] \rightarrow Pic A[N]$ injective?

Let $A$ be a commutative ring. Let $M\subseteq N$ be an extension of positive seminormal monoids. Is the natural map $Pic A[M] \rightarrow Pic A[N]$ injective?
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1answer
59 views

betti-numbers of Gin(I), generic initial ideal of $I$

here in the paper Ideals with Stable Betti Numbers there is a theorem that I can't uderstand it, both in details (which highlighted) and sketch of the proof of (b): can you help please? ...
2
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1answer
193 views

finiteness dimension

$R$ is a local Noetherian ring. $f_I(M)$, the finiteness dimension of a module $M$ relative to $I$, is defined in ...
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310 views

Valuation of an ideal in a two-dimensional regular local ring

Let $f,g$ be two coprime elements in the ring $K[[x,y]]$, with $K$ a field. What is the smallest integer $n$ such that the inclusion of ideals $$(x^n)\subset (f,g)$$ holds in $K[[x,y]]$? Can we ...
4
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0answers
136 views

cohomology algebra of unordered configuration space with coefficients the finite fields

in the paper The cohomology algebra of unordered configuration spaces (Y. Félix, D. Tanré, J. London Math. Soc., 2005), Theorem 4: Let $M$ be an odd-dimensional, compact, closed, oriented manifold. ...
2
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1answer
173 views

When do we get free modules from Noether normalization

Let $X \subseteq \mathbb{P}_{\mathbb{C}}^n$ be an irreducible, projective, Cohen-Macaulay variety of dimension $k$. Let $L \subseteq \mathbb{P}_{\mathbb{C}}^n$ be a linear space of dimension $n-k-1$ ...
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0answers
100 views

Can you always find a regular sequence consisting of monomials?

Let $\mathbb{k}$ be a field, and let $S=\mathbb{k}[x_1,x_2,\ldots,x_n]$. Let $M$ be an $S$-module. A sequence $$f_1,f_2,\ldots,f_r$$ of polynomials in the maximal ideal $\langle x_1,\ldots,x_n\rangle$ ...
5
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1answer
289 views

automorphisms of local rings vs local change of coordinates

Let $R$ be a local (commutative, associative) ring over a field of zero characteristic. (My typical examples are $k[[x_1,..,x_p]]/I$, $k\{x_1,.,x_p\}/I$, $C^\infty(\Bbb{R}^p,0)$. If it helps one can ...
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110 views

On the computational complexity of the Hilbert polynomial of numerical semigroup rings

Let $(R, \mathfrak{m}) = k[[X^a, X^b, X^c]]$, $a<b<c$, $gcd(a, b, c) = 1$, be a semigroup ring. We have $R$ is a Cohen-Macaulay local ring of dimension one. It is well known that ...
3
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1answer
228 views

Non-commutative normalization

Let $A$ be a (non-commutative) associative algebra with 1. Assume that $A$ contains a cental subalgebra $Z$ such that a) $Z$ is a noetherian domain b) $A$ is a finitely generated module over $Z$. ...
7
votes
3answers
613 views

Completion of a local ring of a curve

Let $X$ be a smooth projective irreducible curve defined over an algebraically closed field $\mathbb{K}$ (of arbitrary characteristic), and let $p\in X$ be a closed point. Denote by $\mathcal{O}_p(X)$ ...
2
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1answer
191 views

Hochschild cohomology of commutative quotients

Notation: Let $k$ be a commutative local ring and let $HH^{i}(A,N)$ denote the $i^{th}$ Hochschild cohomology $k$-module of a $k$-algebra A with coefficients in an $(A,A)$-bi-module $N$. If ...
3
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1answer
184 views

Classical deformation of algebras

Given a complex manifold (or a smooth scheme) $X$, the classical (infinitesimal) deformation theory is parametrized by the first cohomology with coefficients in the tangent sheaf $H^1 (X, T_X)$. ...
0
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1answer
135 views

if $ \lambda (I)= \dim R$, can one claim that $I$ is an $m$-primary ideal?

definition from Bruns-Herzog: It is easy to see that if $I$ is a $m$-primary ideal of $R$ then $ \lambda (I)= \dim R$. I wonder if the converse is true: if $ \lambda (I)= \dim R$, can one ...
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1answer
111 views

Connected curve

Assume we have a normal,connected quasi projective scheme $Y:=X\backslash D$ where $X$ is a quasi projective scheme over field $k$, not necessarily char zero and also $D$ is a simple divisor, not ...
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66 views

variants of ramification groups - need terminology and sources

I've asked this question in several more elementary forums, and haven't get any answer. So I presume this is not so an elementary question. Let $L/K$ be a Galois extension, and $w$ be a valuation of ...
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1answer
174 views

Bounded dg algebra vs unbounded dg algebras

1)Let $Cd_{\geq 0}ga$ be the category of non negatively commutative cochain dg algebra over a field $\Bbbk$ of charachteristic zero. Let $w\: : \: Cd_{\geq 0}ga\to dg_{\geq 0}Mod$ be the forgethfull ...
3
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3answers
252 views

Injective map between two schemes

Assuem we have a finite surjective map between two irreducible, separated schemes, $f:X \rightarrow Y$, and for a dense open $U \subset Y$ and for any $y \in U$, $|X_y| =1$, then can we say $f$ is ...
0
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1answer
175 views

Extension of homomorphism to place in quotient field, or of local ring to valuation ring in quotient field

Let be given a domain $R$ (that can be supposed to be integrally closed if this can help), and $\varphi$ an homomorphism of $R$ into a field $F$. $\varphi$ extends uniquely to a homomorphism ...
1
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1answer
234 views

intuitive interpretation of analytic spread

I am studying analytic spreads from Bruns-Herzog's book. The definition is clear but calculation of the analytic spread of an ideal is hard for me in practice. I wonder if it is hard for you too. ...
2
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1answer
101 views

Any two bivariate algebraically dependent polynomials are always in the same ring generated by some bivariate polynomial?

If $f(x,y)$ and $g(x,y)$ are two algebraically dependent polynomials over some field $k$, is it true that there exists a bivariate polynomial $p(x,y)$ such that both $f(x,y)$ and $g(x,y)$ are in the ...
2
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1answer
89 views

On transforming pair of bivariate polynomials to pair of univariate polynomials by applying polynomial map

We know that a polynomial map $f(x,y), g(x,y)$ is polynomial automorphism if there exists polynomials $p(x,y)$ and $q(x,y)$ such that $f(p,q)$=x and $g(p,q)=y$. Jacobian conjecture tries to ...
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0answers
141 views

On Prüfer domains

Is there any Prüfer domain $R$ that has a prime ideal $P$ that is not finitely generated but $xP$ is subset of a finitely generated ideal $I$,for some $x$ in $R-P$ and $I$⊂$P$?
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0answers
117 views

Criterion for normality of a schematic image

Consider a projective flat morphism $$ f\colon X\to Y $$ between normal varieties. Let's say over the complex numbers. The geometric fibers of $f$ are all irreducible. I would like a criterion to ...
2
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1answer
273 views

A perfect domain that is not integrally closed?

Does there exist an integral domain $R$ of characteristic $p > 0$ that is perfect (i.e., $x \mapsto x^p$ is bijective on $R$) but not integrally closed in its field of fractions?
6
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1answer
204 views

Purely noncommutative algebra-Morita equivalence

Morita equivalence of algebras certainly don't preserve commutativity: even if $A$ is commutative there are plenty of noncommutative algebras which are Morita equivalent with $A$---for example all ...
2
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1answer
148 views

what are the possible approximations for ideals

(Fix some local ring $(R,\mathfrak{m})$ over a field of zero characteristic.) Suppose an ideal $J$ is defined by some complicated formula/procedure. And there is no hope of computing it/or writing ...
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1answer
153 views

Codimension in zero and positive characteristic

Let $F_0,\ldots,F_m\in\mathbb{Z}[x_0,\ldots,x_n]$ be polynomials with integer coefficients and let $p$ be a prime integer. Consider the two ideals: $$I_0:=(F_0,\ldots,F_m)\subset ...
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0answers
147 views

Geometric (or intuitive) interpretation of Almost Gorenstein and Cohen-Macaulay rings

This question is related to This one: Darius Math in his good answer added that Cohen-Macaulay ring's singularities is nice. So I'd like to complete that question and ask: Let R be a local ...
7
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0answers
388 views

What is the etale fundamental group of Spec Z((x))?

I know the etale fundamental group of $\mathbb{Z}$ is trivial. For algebraically closed fields $K$, the etale fundamental group of $K((x))$ is $\hat{\mathbb{Z}}$, since all covers in this case are ...
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102 views

The Euler characteristic of Hilbert series

The Hilbert series of a graded vector space $V=\bigoplus_{n\mathbb Z}V_n$ is the (ordinary) generating function of the dimensions of its homogeneous components, $h_V(t)=\sum_{n\in\mathbb Z}t^n\dim ...
0
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1answer
73 views

Homologue of the Inertia group and of the Frobenius theorem for the group of values of a valuation

As I said previously, I have some problems in the theory of valuations and places. Let L/K be a finite (say) Galois extension, F a place of L, and v a valuation of L. I denote by l and k the residue ...
0
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1answer
143 views

Valuations and places - decomposition and inertia group

I feel very uncomfortable with some aspects of the theory of valuations, places, and valuation rings. Here is one of my problems : Assume that L/K is a finite Galois extension of fields, and that F is ...
5
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3answers
441 views

How to prove that two univariate polynomials are always algebraically dependent?

How to prove that two univariate polynomials(over any field) are always algebraically dependent? Also, how to prove the generalization of this question i.e if number of polynomials are more than ...
4
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1answer
127 views

Minimal length of quotient by parameter ideals

Consider a commutative noetherian local ring $R$ of dimension $d$ and define $$c_R\colon=\min_{(x_1,\ldots,x_d)} \{\mathrm{length}\ R/(x_1,\ldots,x_d)R\mid (x_1,\ldots,x_d)\ \mathrm{is\ a\ system\ of ...
9
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109 views

How useful is knowing every torsionfree $\mathcal O(D)$ module is flat?

One of the corollaries of Weiertrass' factorization theorem plus the theorem of Mittag Leffler is that $\mathcal O(\Bbb C)$, more generally $\mathcal O(D)$ for some region $D$ is such that every ...