**0**

votes

**0**answers

17 views

### p-divisibility of the connected component of the Picard group

Let $X$ to be a smooth projective variety over a field of positive characteristic $p>0$, then can one claim $Pic^0(X)$ is p-divisible.

**5**

votes

**1**answer

218 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 $f(g(x_1,x_2,\ldots,x_n))$ ...

**2**

votes

**0**answers

105 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) ...

**2**

votes

**0**answers

43 views

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

Let $A$ be and 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 ...

**0**

votes

**0**answers

74 views

### Tensor product of complexes [on hold]

Let $A$ be a ring and let the modules that are involved be left and right A-modules (not necessarily bimodules over A).
I'll denote as $\mathcal{E}^n_R(M, N)$ the category of n-fold extensions of M ...

**0**

votes

**0**answers

72 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**

votes

**0**answers

58 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**

vote

**1**answer

91 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 ...

**-2**

votes

**0**answers

63 views

### Existence of general element in module case [closed]

Let $(R,m)$ be a noetherian local ring. Let $a_1,a_2,a_3\in R\oplus R$ and $S=R[X_1,X_2,X_3].$ Then is it true that if $z=a_1X_1+a_2X_2+a_3X_3\in S\oplus S$ then $\frac{S}{zS}\cong S[Y].$

**4**

votes

**2**answers

198 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 ...

**0**

votes

**0**answers

47 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**

votes

**2**answers

529 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. ...

**6**

votes

**0**answers

153 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**

votes

**3**answers

317 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 $R$:
...

**1**

vote

**1**answer

87 views

### question about a particular Polynomial ring

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 $R =T \cap L$ ?

**5**

votes

**1**answer

332 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 ...

**0**

votes

**0**answers

100 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?

**2**

votes

**1**answer

173 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 ...

**4**

votes

**1**answer

192 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 ...

**1**

vote

**0**answers

132 views

### Infinite intersection of prime ideals [migrated]

Let $A$ be a commutative ring with identity. Let $p_{i}, i\in I$ and $p$ be prime ideals in $A$, where the index set $I$ is infinite. If we have
$$
p\supset \bigcap_{I}p_{i}
$$
Do we still have ...

**4**

votes

**0**answers

118 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**

votes

**1**answer

161 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 $n-k-1$ ...

**1**

vote

**0**answers

77 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**

votes

**1**answer

272 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 ...

**3**

votes

**0**answers

107 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**

votes

**1**answer

218 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

**3**answers

489 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**

votes

**1**answer

185 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**

votes

**1**answer

163 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**

votes

**1**answer

126 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 ...

**0**

votes

**1**answer

108 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 ...

**2**

votes

**0**answers

65 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 ...

**1**

vote

**1**answer

153 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**

votes

**3**answers

236 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**

votes

**0**answers

88 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**

vote

**1**answer

213 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**

votes

**1**answer

97 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**

votes

**1**answer

82 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 ...

**1**

vote

**0**answers

134 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 $I$⊂$P$?

**1**

vote

**0**answers

107 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**

votes

**1**answer

237 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**

votes

**1**answer

188 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**

votes

**1**answer

141 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 ...

**0**

votes

**1**answer

150 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 ...

**1**

vote

**0**answers

122 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**

votes

**0**answers

351 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 ...

**0**

votes

**0**answers

82 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**

votes

**1**answer

68 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**

votes

**1**answer

121 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**

votes

**3**answers

421 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 ...