**3**

votes

**0**answers

72 views

### Sign of bivariate polynomial evaluated over two algebraic numbers

I would like to compute the sign of a bivariate polynomial $f$ evaluated over two algebraic numbers $a$, $b$. The numbers are in "isolating interval representation" meaning that each one is defined by ...

**5**

votes

**2**answers

343 views

### A question about homogenous polynomials of degree $\frac{n(n-1)}{2}$

Let $n$ be a positive integer and $S_n$ be the symmetric group on $\{1,2,\ldots,n\}$.
For any $w\in S_n$ and polynomial $f\in \mathbb{R}[x_1,x_2,\ldots,x_n]$, denote ...

**3**

votes

**1**answer

216 views

### etale localization reference request

I'm looking for a reference for the following statement:
Let $P$ be a property of morphisms of schemes local on the target in the etale topology. Let $f : X\rightarrow Y$ be a morphism of schemes ...

**0**

votes

**1**answer

130 views

### The injectivity of Noetherian ring

Let $R$ be a ring with 1, $M$ be a left $R$-module. Then $M$ is fp-injective if every $R$-homomorphism from a finitely presented left ideal to $M$ extends to a homomorphism of $R$ to $M$ i.e. if ...

**6**

votes

**1**answer

135 views

### What is the cokernel of $O_S \to F_\infty/O_\infty$?

Let $k$ be a field of characteristic $\neq 2$ and consider the quadratic extension $F$ of $k(T)$ generated by $\sqrt{T^3 + 1}$. Let $X$ be the set of all places of $F$. Let $S = \{\infty\} \subset X$ ...

**3**

votes

**2**answers

110 views

### The injective hull of cyclic modules and self injective ring

It is well-known that if $R$ is a Noetherian ring, and the injective hull of every finitely generated $R$-module is projective, then $R$ is a self-injective.
My question is that could one replace ...

**0**

votes

**0**answers

65 views

### A question about divided differences

I want to ask a question about divided differences.
Let $n\equiv0,1 \pmod 4$ is a positive integer. We know that for any polynomial $f\in \mathbb{Z}[x_1,x_2,\cdots,x_n]$,
...

**2**

votes

**3**answers

179 views

### Isolating roots of polynomial system

I would like to isolate the regions which contain the roots of a system of two bivariate cubic polynomials.
I thought I would project the solutions onto $x$ and $y$ axis by means of resultant ...

**0**

votes

**0**answers

132 views

### Structure theorem for non-Noetherian local rings

Is there a structure theorem (like Cohen 's structure theorem) for non-Noetherian local rings?
I am adding what I am looking for as someone asked in the comment.
If $R$ is a local domain (not ...

**1**

vote

**1**answer

122 views

### when there is an injection $0 \to R \to K_R$?

Let $(R,m)$ be a Cohen-Macaulay local ring which possesses the canonical module $K_R$. Then $R$ is said to be an almost Gorenstein local ring, if there is an exact sequence $0 \to R \to K_R \to C \to ...

**7**

votes

**2**answers

315 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 ...

**6**

votes

**0**answers

145 views

### When is a commutative ring the limit of its factor rings?

Let $R$ be a commutative ring. Consider the limit of rings $L = lim_{I \in Spec(R)}(R/I)$. Then there is a canonical map $R \to L$. The question is when this map is an isomorphism.
For example, this ...

**2**

votes

**0**answers

302 views

### About relative normalization in Deligne's definition of a “tangential morphism”

I'm reading Deligne's paper "Le Groupe Fondamental de la Droite Projective Moins Trois Points", specifically in the section "Theorie profinie" (sections 15.13 - 15.27)
I'm specifically interested in ...

**5**

votes

**0**answers

158 views

### Solving a Laurent polynomial functional equation

I'm considering a set of functional equations:
For a given $\phi(x)\in\mathbb {Z}[x,\frac {1}{x}] $ with $\phi(x)=\phi(\frac{1}{x}) $,
$f(x)f(\frac {1}{x})+\phi (x)g(x)g(\frac {1}{x})=1, $ where ...

**5**

votes

**2**answers

158 views

### The coefficient of a specific monomial in the expansion of the following polynomial

Let $a_{n,k}$ be the coefficient of $$X_1^{\frac{k(n-1)}{2}}X_2^{\frac{k(n-1)}{2}}\cdots X_n^{\frac{k(n-1)}{2}}$$ in the expansion of the real polynomial $$\left(\prod\limits_{1\leq i<j\leq ...

**1**

vote

**0**answers

47 views

### if $\Delta$ is pure, then what happens to betti-numbers of $I_{\Delta}$ or $I_{\Delta^v}$

Assume that $\Delta$ is a simplicial complex and $\Delta ^v$ is its Alexander dual.
Let in addition $\Delta$ be pure, then what happens to betti-numbers of $I_{\Delta}$ or $I_{\Delta^v}$?
Is there a ...

**1**

vote

**1**answer

167 views

### A condition like primeness for zero ideal

Let $D$ be an integral domain (zero ideal is prime). Then for
every nonzero element $a,b \in D$, we have $\langle a\rangle\cap \langle b\rangle\neq 0$.
Now in a general case, let $R$ be a commutative ...

**1**

vote

**1**answer

70 views

### A linear subspace of $\mathbb{R}[X_1,\cdots,X_n]$ and its generated set

Let $n$ be a positive integer and $W_n$ be the linear subspace of the real vector space $\mathbb{R}[X_1,\cdots,X_n]$ generated by the following set
$$S_n=\{X_1^{i_1}\cdots X_n^{i_n}:i_1+\cdots+i_n=n\ ...

**2**

votes

**0**answers

77 views

### Exterior power of a torsion-free sheaf on a DVR

Let $R$ be a discrete valuation ring and $X$ be a regular, integral. projective $R$-scheme, flat over $R$. Let $F$ be a torsion-free coherent sheaf on $X$ of rank $n$, flat over $\mathrm{Spec}(R)$. Is ...

**1**

vote

**1**answer

155 views

### Is flatness preserved under exterior power

Let $\phi:A \to B$ be a flat ring homomorphism, $M$ be a $B$-module which is flat when considered as an $A$-module. Is the tensor product $M \otimes_B M \otimes_B ... \otimes_B M$ flat over $A$? If ...

**3**

votes

**1**answer

171 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 ...

**1**

vote

**1**answer

143 views

### Flat family: limit of intersection vs intersection of limits

Consider a $\textbf{flat}$ surjective map $f: X \rightarrow \mathbb{A}^1$. The general fibers $F_{\epsilon}$ are canonically isomorphic, and the special fiber $F_0$ above $0 \in \mathbb{A}^1$ is not ...

**4**

votes

**1**answer

213 views

### Equi-dimensionality of special fibers in a flat family

Given a flat map $f: X \rightarrow Y$ such that $X$ is a projective variety and $Y$ is a smooth curve. Each generic fiber is isomorphic to an irreducible projective variety $A$ of dimension $d$.
The ...

**3**

votes

**0**answers

258 views

### Are prime ideals of finite height in the powers series ring in infinitely variables finitely generated?

Let $A:= {\mathbb F}_p[[X_1,...,X_∞]]$ be the infinitely many variables formal power series ring over ${\mathbb F}_p$, which is UFD.
Consider an arbitrary prime ideal $P$ of $A$ such that the height ...

**4**

votes

**1**answer

76 views

### Projective resolutions for commutative monoids

What is the right notion of a projective resolution of a commutative monoid?
The category Mon of commutative monoids has plenty of projective (and even free) objects. Indeed, for every set $X$ ...

**5**

votes

**1**answer

263 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 ...

**4**

votes

**2**answers

185 views

### Tensor product of monomorphisms is a monomorphism?

Given a commutative ring $k$ and for $i = 1,2$ a homomorphism of $k$-modules $X_i \overset {f_i} \longrightarrow Y_i$ with $X_i$ flat over $k$.
Is the following conclusion true for general $k$? If ...

**7**

votes

**0**answers

116 views

### Associated graded of double Koszul dual

Let $k$ be a field, and let $A$ be a graded, connected, augmented, locally finite $k$-algebra. If $\Omega^* A$ denotes the cobar complex of $A$ (i.e., the dual $Hom_k(B_*(A), k)$ of the bar complex ...

**3**

votes

**0**answers

45 views

### Antichains defining facets of a certain cone

Let $(P,<)$ be a finite poset. Let $V$ be the free $\mathbb{R}$-vector space on $P \times \{0,1\}$; I'll write elements as sums of pairs of the form $(p,0)$ and $(0,q)$, so a general element is $$v ...

**35**

votes

**13**answers

3k views

### Applications of the Cayley-Hamilton theorem

The Cayley-Hamilton theorem is usually presented in standard undergraduate courses in linear algebra as an important result. Recall that it says that any square matrix is a "root" of its own ...

**2**

votes

**0**answers

51 views

### Determining Inconsistency of (first-order) Non-linear System of Equations [closed]

Is there a way I can figure out what values of the coefficients of some system of non-linear equations makes the system inconsistent?
Take the following system of equations as an example. The ...

**2**

votes

**0**answers

104 views

### Flat Quotients of Power Series Rings

I apologize if the question is too elementary. I did not get any response on math stackexchange.
I read the following statement in some algebraic topology notes and I want to know if it is true and, ...

**1**

vote

**0**answers

152 views

### Number of minimal primes for UFD

Let $R$ be a UFD which is NOT noetherian. It is well-known that $R$
is a Krull ring. Let $I$ be an ideal of $R$ such that the height of $I$
is $d$ which is finite.
Question. Is the number of minimal ...

**2**

votes

**1**answer

70 views

### Division of multivariable polynomials by an ideal

Let $K$ be a commutative field and consider an ideal $I$ of $K[X_1,\dots,X_n]$.
Is there a well behaved "reduction modulo I", in the following sense :
Given a well-ordering $\leq$ on the set of ...

**2**

votes

**1**answer

151 views

### Number of generators of ideal if quotient field has certain transcendence degree

I am trying to prove a result which is used in A. Macintyre and A. J. Wilkie (1995), 'On the decidability of the real exponential field', in Odifreddi, P.G., Kreisel 70th Birthday Volume, CLSI, p. ...

**2**

votes

**0**answers

165 views

### Hilbert series of the weight 0 sub-algebra of the algebra of functions on GL(N)

Let $A$ be the algebra of polynomials in $N^2$ variables $x^i_j$, $i,j=1,\dots,N$. It is $\mathbb{Z}^N$ graded, with $\text{weight}(x^i_j)=e_i-e_j$. Here $(e_1,\dots,e_n)$ is the standard basis of ...

**0**

votes

**0**answers

44 views

### when: $depth\ R/I\ge depth\ R/J$ then $depth\ (R/I)_p\ge depth\ (R/J)_p$

Let $(R,m)$ be a Noetherian local ring, $M$ and $N$ finite R-modules, $p$ a prime ideal, and $I$ and $J$ ideals of $R$. Here, Count Dracula proves that in general assuming $depth\ R/I\ge depth\ ...

**0**

votes

**1**answer

95 views

### Functions of several variables over finite fields [closed]

For a finite field $F$ any function $f\colon F\to F$ is given by a polynomial. My question is what happens when we are given a function of two or more variables? Is this necessarily a polynomial ...

**3**

votes

**1**answer

88 views

### Characterization of Lagrangian planes in symplectic vector spaces over finite fields [closed]

EDIT: As L Spice pointed out, there is an error in the observation. The question is void therefore
Let $p$ be a prime and $q=p^r$. Let $V$ be a $\mathbb F_q$-vector space of dimension four, with a ...

**4**

votes

**0**answers

78 views

### Example request: seriously deficient homogeneous spaces

In a previous post, I cite a dimension condition commonly satisfied by homogeneous spaces and claim that a counterexample must have deficiency at least $3$. For convenience, I reproduce the definition ...

**4**

votes

**0**answers

63 views

### A dimension condition on the cohomology of a homogeneous space

The rational cohomology of a homogeneous space $G/K$ admits a homomorphism from $H^*(BK)$ induced from the classifying map $G/K \to BK$ of the principal $K$-bundle $G \to G/K$. Assume the Lie group is ...

**2**

votes

**1**answer

181 views

### Rings such that all quotients by prime ideals are PIDs?

Let $R$ be a commutative ring such that for every prime ideal $P$ of $R$, the ring $R/P$ is a PID. Do you know how these rings are called or another characterization of them?
I know there are a lot ...

**2**

votes

**1**answer

89 views

### Elementary divisors for chains of submodules

Given free modules $N \le M$ of finite rank over a PID $R$, it's well-known that there is a basis $\{x_1,\ldots,x_n\}$ of $M$ and there are $e_1,\ldots,e_n \in R$ such that $\{e_ix_i\mid e_i \neq ...

**6**

votes

**2**answers

680 views

### Tensor product of fields over integers

Inspired by this question we ask;
Is there a name for each of the following properties about fields? what are some examples other than $\mathbb{Q}$?:
1.A field $K$ with the property that ...

**0**

votes

**0**answers

125 views

### if 0→A→A⊕B→B→0 is an exact sequence of finitely generated modules over a commutative Noetherian ring, then the exact sequence does split [duplicate]

Here, Martin Brandenburg says it is not true that "Every short exact sequence of the form $0 \to A \to A \oplus B \to B \to 0$ splits." Then Mohan says in comments that "As a positive result,
If ...

**3**

votes

**2**answers

499 views

### Primary structures in $\mathbb Q$

I'll formulate a topic restricted here to the positive rational
numbers $\ \mathbb Q_{_{>0}},\ $, then will pose a question (Q2) plus some related, to which I don't know the answers nor reference. ...

**5**

votes

**0**answers

151 views

### Flat + locally of finite presentation + monomorphism = open immersion

It is known that the following are equivalent for an epimorphism $A \to B$ in $\mathbf{CRing}$:
Let $S$ be the set of elements $a \in A$ such that $A [a^{-1}] \to B [a^{-1}]$ is an isomorphism. Then ...

**4**

votes

**1**answer

101 views

### Completeness of Localizations of Completions of Commutative Rings

Let $R$ be an integral domain. Let $x,y\in R\setminus\{0\}$ be distinct. Let $\hat R$ be the $x$-adic completion of $R$ (the ring of all sequences $(r_n+Rx^n)_{n\ge0}$ where for $n\ge0$, $r_n\in R$ ...

**1**

vote

**0**answers

91 views

### Invertible elements in a group algebra

Let $H$ be a torsion-free abelian group and let $\mathbb{K}$ be a field with two elements.
I would like to ask the following question:
Is the group of units of the group algebra $\mathbb{K}[H]$ ...

**3**

votes

**1**answer

208 views

### A technical question about affine grassmanian

For a commutative ring $R$, consider $R[[t]]$-modules $$t^k R[[t]]^n \subset M \subset t^{-k} R[[t]]^n \subset R((t))^n.$$ It is known that if $t^{-k} R[[t]]^n / M$ is finitely generated projective ...