# Tagged Questions

**2**

votes

**1**answer

211 views

### Field of definition of an algebraic set

I find this definition in Silverman's book, The Arithmetic of Elliptic Curves:
an algebraic set(in $A^n(\bar{K})$) is called defined over $K$ if its ideal can be generated by polynomials in ...

**4**

votes

**1**answer

86 views

### How to write the map $ℂ[G/U]↪ℂ[B]$ explicitly?

Let $G$ be a reductive algebraic group and $B$ a Borel subgroup of $G$. Let $T$ be a maximal torus of $G$ contained in $B$. The $B=UT=TU$ for some unipotent subgroup $U$ of $G$. We have Bruhat ...

**3**

votes

**1**answer

50 views

### relate shellability of a simplicial complex to the links of its faces

Reisner's criterion give a complete characterization of Cohen–Macaulay simplicial complexes, based on $link$s of faces of the simplicial complex. Is there a known fact that relate shellability of a ...

**4**

votes

**0**answers

128 views

+100

### 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 ...

**16**

votes

**1**answer

321 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 ...

**3**

votes

**0**answers

32 views

### Is the restriction of a graded automorphism linearizable in characteristic zero?

This question follows up a previous one which was answered by Todd Leason. I want to impose two new requirements on the setup.
Let $k$ be a characteristic zero field. Let $A=k[x_1,\dots,x_n]$ be the ...

**2**

votes

**2**answers

293 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, ...

**5**

votes

**1**answer

253 views

### Milnor descent for ring spectra

Suppose given a homotopy cartesian square of (commutative) ring spectra (or (c)dgas)
$\begin{matrix}A & \to & A_1 \\
\downarrow & & \downarrow \\
A_2 & \to &A'\end{matrix}.$
...

**4**

votes

**1**answer

264 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 ...

**7**

votes

**0**answers

237 views

### When is $\lim_{n\rightarrow\infty}\:\mathrm{depth}\:R/\mathfrak{a}^n= \mathrm{depth}\:R/\mathfrak{a}$?

Suppose $(R,\mathfrak{m})$ is a noetherian local ring. I am interested in ideals $\mathfrak{a}$ of $R$ for which $$\lim_{n\rightarrow\infty}\:\mathrm{depth}\:R/\mathfrak{a}^n= ...

**1**

vote

**1**answer

112 views

### Canonical module of a Buchsbaum ring

Is the canonical module of a Buchsbaum ring a Buchsbaum module?

**0**

votes

**1**answer

101 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.)

**1**

vote

**0**answers

136 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 ...

**5**

votes

**1**answer

311 views

### A Hom-Tensor identity - $\text{Hom}_{R}(P,B)\otimes _SC \cong \text{Hom}_{R}(P,B \otimes_S C) $

let $R,S$ be associative algebras over $\mathbb{C}$. Let $\mathcal{C} \subseteq$ $R$-Mod be a full abelain subcategory of $R$-Mod which is the category of $R$-modules. Let $B$ and $C$ be, a ...

**0**

votes

**0**answers

61 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 ...

**0**

votes

**1**answer

179 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 ...

**1**

vote

**0**answers

117 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 ...

**2**

votes

**1**answer

62 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 ...

**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

**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

92 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

373 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 ...

**3**

votes

**1**answer

175 views

### Maximal length of filter regular sequence

Let $A$ be a Noetherian ring and $R$ is a standard graded ring over $A.$ Let $M$ be a finitely generated graded $R$-module and $I$ be a graded ideal of $R.$ Then $x_1,\ldots,x_r\in I$ is called ...

**4**

votes

**1**answer

290 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 ...

**4**

votes

**3**answers

382 views

### Generators of the kernel of multiplication of a ring

Is the following true?
Let $R$ be a commutative ring and $S$ be an $R$-algebra. Let $\mu: S\otimes_R S\to S$ be the multiplication.
Let $I$ be the kernel of $\mu$, and $J\subseteq I$ be the smallest ...

**3**

votes

**2**answers

139 views

### Maximal Cohen-Macaulay modules of type one

Does any body know an example of a Noetherian local ring $(R,m)$ which admits a maximal cohen-macaulay module of type one, but the ring $R$ itself is not CM.
If $C$ is the maximal CM module then the ...

**1**

vote

**1**answer

255 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 ...

**15**

votes

**4**answers

6k views

### Atiyah-MacDonald, exercise 2.11

Let $A$ be a commutative ring with $1$ not equal to $0$. (The ring A is not necessarily a domain, and is not necessarily Noetherian.) Assume we have an injective map of free $A$-modules $A^m \to ...

**7**

votes

**2**answers

319 views

### invariants that can be measured by Local Cohomology

What invariants can be measured by Local Cohomology (and what application it has)?
As an example of what I mean:
Local Cohomology can measure invariants like depth and dim. So in some cases ...

**0**

votes

**0**answers

193 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 ...

**7**

votes

**1**answer

651 views

### A Zero-Multiplicity Problem Related to Foulkes' Conjecture

I'm a combinatorialist that is interested in estimating multiplicities of irreps of $1^{S_{kn}}_{S_k \wr S_n}$ (the action of symmetric group on uniform partitions). I'm aware of the difficulty (or ...

**1**

vote

**0**answers

206 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 $\mathbb{C}[GL_3]$ ...

**72**

votes

**2**answers

7k views

### When is the tensor product of two fields a field?

Consider two extension fields $K/k, L/k$ of a field $k$.
A frequent question is whether the tensor product ring $K\otimes_k L$ is a field. The answer is "no" and this answer is often justified by ...

**2**

votes

**1**answer

137 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

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 ...

**4**

votes

**1**answer

339 views

### Index of the Hecke algebra with operators omitted

This is a spin-off to the question Omitting primes from a Hecke algebra by David Loeffler.
Let $N$ be a positive integer. For a finite set of primes $\Sigma$, let $\mathbb T^{\Sigma}$ be the $\mathbb ...

**4**

votes

**4**answers

1k views

### Jacobson ring = a ring whose nilradical and Jacobson radical coincide?

In Wikipedia it is claimed that "A ring is called a Jacobson ring if the nilradical coincides with the Jacobson radical." Here the word "ring" means a commutative ring.
However, I remember that ...

**3**

votes

**0**answers

168 views

### a generalization of the annihilator of cokernel ideal (some new invariants of modules?) [closed]

Let $R$ be a (commutative, associative, unital) ring, consider a homomorphism of some (finitely generated) free $R$-modules $E\stackrel{A}{\rightarrow}F$. Say $rank(F)=m$.
The basic invariants of $A$ ...

**2**

votes

**1**answer

147 views

### the pfaffian-adjugate and its counterparts for matrices odd size

Let $R$ be a commutative ring (zero characteristic). Take a skew-symmetric matrix $A\in Mat^{skew-sym}(n,R)$.
If $n$ is even, then $\det(A)=Pf^2(A)$ and there exists the "Pfaffian adjugate/adjoint" ...

**8**

votes

**4**answers

1k views

### Example of a projective module which is not a direct sum of f.g. submodules?

This semester I am teaching a graduate course in commutative algebra, and I have been taking the occasion to try to look at the proofs of some the results in my basic source material (Matsumura, ...

**-1**

votes

**0**answers

26 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 ...

**32**

votes

**7**answers

4k views

### Is there a preferable convention for defining the wedge product?

There are different conventions for defininig the wedge product $\wedge$.
In Kobayashi-Nomizu, there is $\alpha\wedge\beta:=Alt(\alpha\otimes\beta)$,
in Spivak, we find ...

**1**

vote

**0**answers

57 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 ...

**3**

votes

**1**answer

95 views

### non-Noetherian r-Noetherian ring with Noetherian total quotient ring

A commutative ring is said to be r-Noetherian if every regular ideal is finitely generated, where an ideal is said to be regular if it contains a non-zerodivisor. Does there exist a non-Noetherian ...

**0**

votes

**0**answers

87 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 ...

**2**

votes

**2**answers

357 views

### Castelnuovo-Mumford Regularity of Ideals of Maximal Minors

Hi,
I have a $m \times 2m$ matrix of linear forms over $\mathbb{C}[x,y,z,w]$. It is of the form $M = ( x I - A z -B w | y I - C z - D w$. Here $A,B,C$ and $D$ are $m \times m$ scalar matrices. Let ...

**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 ...

**3**

votes

**2**answers

567 views

### Group & modules of arbitrary cardinality [closed]

How do I see that there is a group of an arbitrary cardinality? Is this also true for abelian groups? Also, given a commutative ring $R\neq 0$ how do I see that there is an $R$-module of arbitrary ...

**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 $Z_1,...,Z_n \in |\mathcal{O}_X(k)|$ ...