**7**

votes

**0**answers

122 views

### Swan K-theory of Z/4

Given a finite group $G$ and a commutative ring $R$, define the Swan $K$-theory $K_0(G, R)$ to be the Grothendieck group of the category finitely generated projective $R$-modules with $G$-action (with ...

**1**

vote

**0**answers

132 views

### What is the ring of integers in $\mathbb Q^c\otimes_K K_\mathfrak p$? [on hold]

Let $K$ be a number field with ring of integers $\mathcal O_K$ and $\mathfrak p$ a prime of $K$. Let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q$ in $\mathbb C$.
If $L$ is a number field ...

**0**

votes

**0**answers

59 views

### Milnor numbers and mixed multiplicities

section 6 of the link
Teissier showed that Milnor numbers of a hypersurface $(X,0)$ with isolated singulraity at 0 is same as mixed multiplicities of the Hilbert polynomial of the filtration ...

**3**

votes

**0**answers

93 views

### Auslander-Reiten-Quivers of representation-finite algebras having different 3-dimensional forms

I am looking for references, where I can find (pictures of) connected Auslander-Reiten-Quivers of representation-finite $k$-algebras ($k$ is a (preferably, but not necessarily finite) field) with one ...

**1**

vote

**0**answers

114 views

### Base change and geometrically generic reduced fiber

Let $k$ be an algebraically closed field of characteristic $p>0$ and $f:X \to Y$ be a quasi-projective morphism between noetherian $k$-schemes. Assume that $Y$ is regular and the geometric generic ...

**0**

votes

**1**answer

178 views

### Field extension and nilpotent element

Let $k$ be an algebraically closed field of characteristic $p>0$, $A$ a regular local noetherian $k$-algebra, $B$ another local noetherian $k$-algebra, $f:A \to B$ an injective ring homomorphism of ...

**4**

votes

**1**answer

144 views

### Schwartz-Zippel lemma for an algebraic variety

Let $X $ be a smooth affine subvariety of $(\overline{\mathbb{F}_q})^n$ defined by a prime ideal $I$. Let $f$ $\in \mathbb{F}_q[x_1,\ldots,x_n]$ be a polynomial such that $f \notin I$.
Let $r_1, ...

**2**

votes

**1**answer

53 views

### Is the restriction of a graded automorphism of a polynomial ring to a polynomial subring linearizeable?

Let $k$ be a field and let $A=k[x_1,\dots,x_n]$ be a polynomial algebra over $k$, and let $B\subset A$ be a graded subalgebra that is itself a polynomial ring, i.e. $B=k[f_1,\dots,f_m]$ for some ...

**-2**

votes

**0**answers

82 views

### proof that ${\rm SL}_n (R)=E_n(R)$ in a local ring? [closed]

I have to prove that ${\rm SL}_n (R)=E_n(R)$ and I need some help.
$R =R_1\cdot R_2\cdots R_n$ , and every $R_i$ is a local ring . $E_n(R)$ is the elementary group and ${\rm SL}_n(R)$ is the special ...

**0**

votes

**0**answers

90 views

### Does there exist an Affinization or Projectivization process for Varieties?

Let us consider the classical isomorphism of real manifolds between $S^2$ and ${\mathbb CP}^1$. First strange thing we have here is that both are varieties, but $S^2$ is an affine and ${\mathbb CP}^1$ ...

**1**

vote

**1**answer

253 views

### Automorphisms of rings fixing all prime ideals

Let $f,g:A \to B$ be two ring homomorphisms of noetherian rings satisfying that for any prime ideal $\mathfrak{q} \subset B$, $f^{-1}(\mathfrak{q})=g^{-1}(\mathfrak{q})=:\mathfrak{p}$ and the induced ...

**1**

vote

**0**answers

62 views

### Need a reference/proof for computing the regularity of ideal of points in $\mathbb P^d$?

In a lecture notes on 'Cohomology modules' i read the following remark:
Given a set $X$ of points in $\mathbb P^d$,using the Local Cohomology modules one can easily compute the reg$(S_X)$ where ...

**0**

votes

**0**answers

190 views

### On the coherence of formal power series ring

Let $A = {\Bbb F}_p[[X_1,X_2,...]]$
be the ring of formal power series with infinitely many variables over the finite field ${\Bbb F}_p.$
$A$ consists of such formal sum elements as $\sum ...

**2**

votes

**2**answers

260 views

### Irreducible algebraic sets via irreducible polynomials

There are many results about irreducible polynomials over finite fields:
we know a cardinality of all irreducible polynomials with given degree, we know explicit examples of irreducible polynomials, ...

**4**

votes

**1**answer

288 views

### an algebraic variety for a boolean circuit

There is a polynomial reduction from a $3-CNF$ $SAT$ problem to some system of polynomial equations over $\mathbb{F}_2$.
I mean there is polynomial reduction $F$ such that for every boolean ...

**0**

votes

**1**answer

95 views

### Reduction of ideal in noetherian local ring

Let $R$ be a noetherian local ring and $I$ an ideal with $\operatorname{ht}I=\mu(I)$. Prove that $I$ is basic. (Recall that an ideal $I$ is basic when it has no proper reduction.)

**0**

votes

**0**answers

157 views

### Exact sequence of vector bundles

Consider the short exact sequences below;
\begin{equation}
0\longrightarrow H^0(\mathbb{P}^4,\mathcal{O}_{\mathbb{P}^4}(d-1)^{\oplus 4})\longrightarrow ...

**2**

votes

**1**answer

136 views

### The center of a(n endomorphism) ring is a PID

Let $A$ be a torsion-free Abelian group, and let $\mbox{End}(A)$ be its endomorphism ring. Denote by $R\doteq\mathbf{Z}(\mbox{End}(A))$ the center of $\mbox{End}(A)$. What would be necessary and/or ...

**0**

votes

**0**answers

25 views

### Is there a criterion for Ideals to be contracted in the completion morphism?

I have a local noetherian ring $R$ with maximal ideal $\mathfrak{m}$ and two ideals $I,J\unlhd R$. I know they are the same modulo $\mathfrak{m}^k$ for all $k\in \mathbb{N}$. There for they generate ...

**1**

vote

**0**answers

56 views

### Computer algebra system that test zero divisors in a quotient algebra

I have an algebra $A$ over a Noetherian ring and an ideal $I=(x,y)$, where $x,y \in A$. I need to examine whether a polynomial $h \in A$ is a zero divisor in $A/I$ or not.
Is there a computer algebra ...

**1**

vote

**0**answers

205 views

### How to check that an ideal of $\mathbb{C}[GL_n]$ is a coideal or not?

Let $I$ be an ideal of $\mathbb{C}[GL_n]$. Are there effective methods or software to check whether $I$ is a coideal or not? Thank you very much.
For example, let I be the ideal of $\mathbb{C}[GL_3]$ ...

**0**

votes

**0**answers

86 views

### Noetherian almost Dedekind domain

A Dedekind domain is an integral domain in which every nonzero proper ideal factors into a product of prime ideals, and an integral domain $R$ is called almost Dedekind whenever $R_m$ is Dedekind ...

**4**

votes

**1**answer

198 views

### How to compute the tangent space of a quotient by a finite group

Let $I\subseteq R:=\mathbb C[x_0,\ldots,x_n]$ be a homogeneous ideal defining a subscheme $X\subseteq\Bbb P^n$. As in my previous question, the permutation group $\mathfrak S_{n+1}$ acts on $R$ by ...

**1**

vote

**0**answers

144 views

### Structure theorem for infinitely generated modules over a PID

This is a refined version of a question I asked days ago and have no answers yet. I am completely illietarte in algebra so, please, don't kill but explain.
The question is all in the title: is there ...

**1**

vote

**0**answers

104 views

### A strong form of Bezout theorem

Let $X$ be a smooth projective variety of dimension $n$, $U \subset X$, non-empty open set. For any integer $k>0$, does there exist $n$-hypersurface sections $Z_1,...,Z_n \in |\mathcal{O}_X(k)|$ ...

**0**

votes

**1**answer

78 views

### Canonical module of a Buchsbaum ring [closed]

Whether the canonical module of a Buchsbaum ring is a Buchsbaum module?

**4**

votes

**1**answer

94 views

### The volume around a singular isolated root when equalities are loosened

Suppose I have a system of polynomial equations in $n$ real variables $f_i(x_1,\ldots,x_n)=0$, $i=1,\ldots,m$, such that $0$ is an isolated solution. Now I replace each of the equations with a ...

**3**

votes

**0**answers

97 views

### Number of free summands of finite local extensions

Suppose that $(R, m) \subseteq (S,n)$ is a finite extension of normal local domains that is:
etale on the punctured spectrum
not flat / etale at the origin
and such that the residue fields $R/m = ...

**0**

votes

**0**answers

50 views

### Decomposition of PID modules

This is (probably) the culmination of a series of questions I posted recently that have lead me to this (probably) final question. As usual, I aplogize for my illiteracy in algebra.
Recall that ...

**6**

votes

**0**answers

246 views

### Geometric interpretation of minimal number of generators of a module

Let $X \subset \mathbb{C}^n$ be an irreducible affine algebraic curve with coordinate ring $$\mathbb{C}[X] = \mathbb{C}[z_1, \ldots, z_n] / (f_1, \ldots, f_m ) $$ with each $f_i \in \mathbb{Z}[z_1, ...

**5**

votes

**0**answers

151 views

### A question on symmetric functions

Let $0 \leq m \leq n$ be integers. The group $S_n$ of permutations acts on the ring $\mathbb{Z}[X_1,\dots,X_n]$ by permuting the coordinates, with fixed subring $\mathbb{Z}[\sigma_1,\dots,\sigma_n]$, ...

**2**

votes

**0**answers

124 views

### When is the torsion submodule a direct factor?

Let $\mathbb{F}$ be a field (of characteristic 0, if needed) and $\mathbf{V}$ an $\mathbb{F}$-vector space. Let $T\in\mbox{End}_\mathbb{F}(\mathbf{V})$ be an endomorphism, and let (following Bourbaki) ...

**1**

vote

**0**answers

27 views

### Is an weakly finite R-module Serre subcategory of the category of R-modules?

A definition for weakly finite $R$-modules is as follow:
Definition: Let ($R$,$m$) a local ring. Let $S$ be the largest class of $R$-modules satisfying the following four properties:
(1) If $M \in ...

**6**

votes

**0**answers

77 views

### Center Picard group non-commutative algebra

I am wondering if there is a way to describe the center of the Picard group of a non-commutative algebra.
Namely, let $A$ be a finitely generated algebra over a field $k$. Denote by ...

**1**

vote

**0**answers

46 views

### Modification of nonfree locus

Let $ R $ be a commutative noetherian ring with identity. Let $ M $ be an $ R $-module. By definition the nonfree locus $ NF(M) $ of $ M $ is defined as the set of prime ideals $ {\mathfrak p} $ of $ ...

**1**

vote

**0**answers

84 views

### Transitivity for algebraic extensions of integral domains?

I'm trying to prove the following result. Let $R_1\subseteq R_2\subseteq R_3$ be integral domains such that $R_1\subseteq R_2$ and $R_2\subseteq R_3$ are algebraic extensions (note: I do not want to ...

**2**

votes

**0**answers

65 views

### Is the positive part of an algebraic bilateral p-adic convergent power series algebraic?

Let $\mathbb{Z}_p \{ X\}$ and $\mathbb{Z}_p \{ X , X^{-1}\}$ be the henselizations of $\mathbb{Z}_p [X]$ and $\mathbb{Z}_p [ X , X^{-1}]$ with respect to the ideals $p\mathbb{Z}_p [X]$ and ...

**0**

votes

**0**answers

130 views

### Rings with a property similar to integral domains

For an integral domain $R$, the intersection of two non-zero ideals is also non-zero, because the product of any two non-zero elements is non-zero.
Is the converse true, i.e. if $R$ has the ...

**1**

vote

**0**answers

68 views

### Cyclic decomposition of an infinitely generated module

My knowledge of algebra is undergraduate linear algebra, so I apologize for my complete ignorance.
Thinking about Jordan normal forms, I unintensionally came to an idea that turned out to be called a ...

**6**

votes

**0**answers

89 views

### Relative variants of the Jacobson radical

Let $B$ be a commutative ring (with 1). The Jacobson radical can be defined as
$$ J(B) = \{b \in B \mid \forall a \in B \colon \quad 1 + a\cdot b \text{ is a unit in } B \} $$ or $$ J(B) =\{ b \in B ...

**1**

vote

**1**answer

228 views

### When are Segre- and Veronese embeddings Gorenstein?

Given a Segre product $\mathbb P^m \times \mathbb P^n$, or more generally $\mathbb P^{m_1}\times\cdots\times\mathbb P^{m_n}$, is there a characterization in terms of $m$ and $n$, or the $m_i$, for the ...

**0**

votes

**1**answer

109 views

### When Hom(M,E) is injective? [closed]

Let $R$ be a commutative Noetherian ring with non-zero identity, $M$ be an $R$-module
and $E$ be an injective $R$-module. When $Hom(M,E)$ is injective?
Thanks.

**3**

votes

**0**answers

95 views

### An operator derived from the divided difference operator $\partial_{w_0}$

Some main definitions and basic facts of divided differences:
In the symmetric group $S_n$, let $s_i$ denote the adjacent transposition $(i,i+1),i\in \{1,2,\cdots,n-1\}.$ Since $S_n$ is generated by ...

**0**

votes

**0**answers

143 views

### Chow classes of non-reduced sub-schemes

I am trying to understand the geometric picture of primary ideals, in particular if and how one can define a notion of multiplicity for the underlying geometric 'sets'. My understanding of the subject ...

**3**

votes

**1**answer

117 views

### is the normalization of a smooth curve in a tamely ramified finite separable extension of the function field also smooth?

Let $X$ be a smooth proper curve over a field $k$, with function field $K$. Let $L$ be a finite separable tamely ramified extension of $K$, and let $Y$ be the normalization of $X$ in $L$. Is ...

**1**

vote

**1**answer

91 views

### Does local cohomology commute with taking the degree-zero component?

Let $S = \oplus_{d \geq 0} S_d$ be a graded (Noetherian) ring, let $I \subset S$ be a homogeneous ideal, and let $f \in S$ be a homogeneous element. Denote by $S_{(f)}$ the subring of degree-$0$ ...

**3**

votes

**1**answer

63 views

### Prime ideal ramified in extension if and only if certain polynomial divides another one?

Let $k$ be a field of characteristic $ \neq 2$, and let $f \in k[T]$ be a polynomial of degree $\ge 1$ which is square free. Let $K$ be the quadratic extension $k(T)(\sqrt{f})$ of $k(T)$. I know that ...

**3**

votes

**1**answer

127 views

### Non-zero coefficients of primitive polynomials

Let $R$ be the finite field with $q$ elements, and let $m,n\in \mathbb{N}$
be positive integers $\geq 2$. I want to prove that there exists a primitive
polynomial $$F(x) = ...

**7**

votes

**1**answer

260 views

### What does the notation $[b_1,b_2]$ in M. Hochster's “Prime Ideal Structure in Commutative Rings” mean?

I'm reading the article
M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142 (1969), 43--60. Freely available here on the journal's website.
But, I can not find the ...

**7**

votes

**1**answer

278 views

### Intersection between integral closures is algebraically closed field

Consider an algebraically closed field $k$, a finite field extension $K$ of $k(T)$, the integral closure $A$ of $k[T]$ in $K$, and the integral closure $A'$ of $k[T^{-1}]$ in $K$. Does it follow that ...