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votes

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54 views

### Codim r strata in a scheme

My question is what is the definition of codimension r strata for a scheme $X$, with NCD $D$. I heard the term in a lecture but I could not find it nowhere.

**0**

votes

**0**answers

43 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 ...

**1**

vote

**1**answer

142 views

### Commutator with a generator of a free group

Let $F$ be a free group $\langle x_1,...x_m\rangle$.
If $a\in F_2$ and $[a,x_1] \in F_n$ then $a\in F_{n-1}$.
Here, $F_n$ is the $n$-th lower central series term with $F_2=[F:F]$.
How can I prove ...

**0**

votes

**1**answer

100 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

58 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

140 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

216 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

80 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

144 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

80 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

77 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

127 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

104 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

223 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?

**5**

votes

**1**answer

175 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 ...

**1**

vote

**1**answer

106 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

144 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

110 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

313 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

78 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

59 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

118 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

405 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**

votes

**1**answer

111 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 ...

**2**

votes

**2**answers

228 views

### Irreducibility after substitution

I would like to show that when $f(x,y)$ is irreducible over $\mathbb{C}[x,y]$ then $f(x^2,y)$ is irreducible over $\mathbb{C}[x,y]$. I know that this is not true in general, for example, $f(x,y) = ...

**5**

votes

**0**answers

130 views

### Link between abelian groups and endomorphisms

When teaching Algebra, I try to share my fascination about two apparently unrelated questions, which turn out to involve the same theory:
classifying the finitely generated abelian groups,
...

**2**

votes

**1**answer

124 views

### Extending descent data from the special fiber of an extension of DVR's

My question is about the proof of Lemma D.3 on p. 147 of the book "Neron models" by Bosch, Lutkebohmert, and Raynaud. Namely, towards the end of that proof there is the sentence "That $\varphi$ ...

**7**

votes

**1**answer

140 views

### Is there a ring which is not Hermite but is coherent?

Call a commutative unital ring $R$
Hermite if for all $m, n\in \mathbb{N}$ with $m<n$, and all $f\in R^{m\times n}$ such that transpose($f$) is left invertible (with a matrix with entries from ...

**4**

votes

**1**answer

124 views

### Compute adjugate matrix over commutative ring

Let $A$ be a $n\times n$ matrix over a commutative ring. I'm looking for a good method to compute its adjugate matrix.
My current approach is to use the Cayley-Hamilton theorem:
$$\text{adj}(A) = ...

**1**

vote

**1**answer

108 views

### Regular rings and formally smooth algebras

Let $A\rightarrow B$ be a commutative $A$-algebra. If $A$ is a field and $B$ Noetherian and formally smooth over $A$, then it is known that $B$ must be a regular ring. Is there a partial converse of ...

**0**

votes

**0**answers

70 views

### Equidimensionality of stalks of $\operatorname{Proj} S$ when $S$ is equidimensional.

I would like to know a reference of the following statement (or counter example).
Let $S$ be a (commutative) Noetherian standard graded ring over a local ring, i.e., $S = S_0[S_1]$, where $S_0$ is ...

**4**

votes

**1**answer

194 views

### Transcendence degree of the surreals over the subfield generated by the ordinals

Consider the Grothendieck ring $K[\Omega]$ of the semiring $\Omega$ of all ordinals under the operations of natural sum and product. Its quotient field $K(\Omega)$ is naturally a subfield of the ...

**1**

vote

**0**answers

71 views

### Computing the bourbaki ideals

By virtue the Griffith's paper and subsequently e.g. Goto's paper several examples of several desired class of Noetherian normal domains with specific finite length local cohomologies are constructed ...

**1**

vote

**0**answers

218 views

### Affine communication lemma and finite limits in the category of rings

Let $X$ be a scheme and $\mathrm{Spec}(B) = V \subseteq X$ be an open affine subset. When using the affine communication lemma (c.f. Theorem 6.3.2, Vakil's notes, Foundations of Algebraic Geometry), ...

**5**

votes

**2**answers

206 views

### Integral domains with totally ordered spectra

In my research I ended up trying to prove some properties of integral domains such that their spectrum is a totally ordered poset. Are there some nice (ubiqitous/natural) examples of such domains, ...

**1**

vote

**1**answer

129 views

### ideal of maximal minors is cohen-macaulay?

Let $k$ be an algebraically closed field.
Let $A$ be an $m \times n$ matrix with linear forms $a_{ij} \in k[x_1, \ldots, x_p]_1$ as entries. Let $I$ be the ideal generated by the maximal minors of ...

**0**

votes

**1**answer

136 views

### When a proper morphism of schemes is a closed imbedding?

Let $X$ and $Y$ be finitely presented schemes over $\mathbb{C}$. Let $f\colon X\to Y$ be a proper morphism. Let us assume that for any finitely presented scheme $S$ the induced map
$$Mor_{Sch}(S,X)\to ...

**4**

votes

**1**answer

214 views

### Intersection of nonzero prime ideals is zero — does it have a name?

The Rabinowitch trick (in Eisenbud's Commutative Algebra with a view toward Algebraic Geometry, page 132) says that $R$ (commutative unital ring) is Jacobson if and only if for every prime ideal $P ...

**0**

votes

**0**answers

71 views

### Criterion for global dimension of subring

All rings are assumed to be associative and unital.
If $B$ is a commutative sub-ring of $A$ (which itself needs not be commutative) then what properties of $B$ are both necessary and sufficient for ...

**0**

votes

**0**answers

74 views

### Residual Intersections of a complete intersection

Let $R$ be a Cohen-Macaulay local ring and $I=(b_1,\dots,b_s)$ be a complete intersection generated by a regular sequence $\underline{b}$. Let $\mathfrak{a}\subseteq I$ such that ...

**1**

vote

**2**answers

185 views

### Computing the nonsingular projective model of a plane curve

Is there an implemented algorithm available in standard software systems (Sage, Magma, Macaulay, etc.) that will compute the nonsingular projective model of a plane curve over $\mathbb Q$?

**1**

vote

**0**answers

117 views

### Questions on prime integral ideal congruences

Suppose that we are given a fixed pair $a_1, a_2$ of non-zero irrational algebraic integers in some number field $K$ which are independent over $\mathbb{Q}$. Suppose that $\mathcal{P}$ is a prime ...

**0**

votes

**1**answer

159 views

### A condition on isolated singularity

Suppose $F: {\mathbb C}^N \to {\mathbb C}$ defines a singularity at the origin (for simplicity one can assume that $F$ is a quasi-homogeneous polynomial). Suppose it is nondegenerate, i.e., $dF(z) = ...

**1**

vote

**0**answers

71 views

### Does this condition imply a polynomial is a product of linear factors

Let $\Lambda$ be a lattice (i.e. $\Lambda \simeq \mathbb{Z}^n$) with a positive subcone $\Lambda^+$. Let $H: \Lambda^+ \rightarrow \mathbb{C}$ be a function such that $\forall\mu \in \Lambda^+$, ...

**4**

votes

**0**answers

141 views

### How to check whether a scheme of finite type over Spec Z is regular or not [duplicate]

Let $f_1, f_2, \ldots f_k$ be a set of polynomials in $n$ variables, with integer coefficients. These define an affine scheme $X$ of finite type over $Spec \mathbb{Z}$. (We could also consider ...

**1**

vote

**0**answers

44 views

### Name for generalization of bivariate weighted-homogeneous polynomials

A polynomial $f = \sum_j c_j X^{\alpha_j}Y^{\beta_j}\in\mathbb K[X,Y]$ is said weighted-homogeneous if there exist $p$, $q$ and $d$ (where $p$ and $q$ are not both $0$) such that ...

**1**

vote

**0**answers

44 views

### Bass' stable range for Bezout rings

As discussed in this MO topic, every principal ideal domain has stable rank at most 2. The proof in the accepted answer uses the fact that PID is a unique factorization domain, but there can be no ...

**3**

votes

**0**answers

71 views

### Is there some algorithm for calculating the least number of generators of an ideal in a polynomial ring?

My question comes from the context of algebraic geometry. By Krull's principal ideal theorem, the number of generators of the defining ideal of a variety gives an upper bound of its codimension.
...

**0**

votes

**0**answers

57 views

### Isomorphy of finite $R$-algebras under special conditions

Let $R=k[x_1, \ldots, x_n]$ be the polynomial ring over a field of characteristic zero. Let $R \subseteq S$ be a finite ring extension i.e. $S$ is finitely generated as $R$-module (free if that helps) ...

**1**

vote

**0**answers

124 views

### Are there good properties of the divided power completion map?

Let $Y \to X$ be a closed immersion of smooth schemes over, say, the ${\rm Spec}(\mathbb{Z}_p)$. The completion map $$X_{/Y}\to X$$ is an ind-closed immersion (sometimes called pseudo-closed ...