**1**

vote

**0**answers

78 views

### Algebraic set given by sequence of polynomials

When working on some problem, I have end up with a following situation. Suppose
$P(z)=z^d+a_{d-1}z^{d-1}+\ldots+a_1z+a_0$ is a complex polynomial $d\geq2$ and that $\gamma$ and $\delta$ are non-zero ...

**3**

votes

**1**answer

175 views

### operations on ideals in a subring of number field

For three ideals $I, J$ and $K$ of a subring $R$ in a number field $L$,
does this equality hold in general?
$(I+J) \cap K = (I \cap K) + (J \cap K)$
I have no counterexample yet but I couldn't prove ...

**2**

votes

**0**answers

128 views

### K-theory and completion

I asked this question also on math.stackexchange. But maybe it's better to ask the Mathoverflow community.
I vaguely remember the existence of a statement that relates the $K$-theory (in the sense of ...

**5**

votes

**0**answers

57 views

### K-Theory and completion [duplicate]

I vaguely remember the existence of a statement that relates the $K$-theory (in the sense of Quillen) of a noetherian local ring $A$ with maximal ideal $\mathfrak{m}$ with the $K$-theory of the ...

**3**

votes

**2**answers

415 views

### An identity in an arbitrary commutative ring

This fact might be either trivial, wrong, or well known.
Let $R$ be a commutative ring. Let $u_1,\dots,u_{s-1},u_s\in R$ and $m,M\in R$. Let us assume that $m,M$ satisfy
$$(m-u_1) \dots ...

**4**

votes

**2**answers

147 views

### stability of linear systems of quadrics

A pencil of quadrics in $\mathbb{P}^n$ is a line in $\mathbb{P}^N$, where $N=\frac{n(n+3)}{2}$. So the space of pencil of quadrics is the Grassmannian $Gr(2,N+1)$. The group $SL_{n+1}(\mathbb{C})$ ...

**3**

votes

**3**answers

243 views

### Dimension of a ring after localization

Let $R$ be a Noetherian domain of dimension $\ge 1$. Let $\mathfrak{p}_i$, $i = 1, 2, ...$ be prime ideals of height one. Let $T = R[[X]]$ with $X$ is a indeterminate. For each $i \ge 1$ we set ...

**3**

votes

**1**answer

176 views

### Tor dimension in polynomial rings over Artin rings

I found this tricky problem in trying to understand some properties of local rings at non-smooth points of embedded curves. But this would be a very long story. So I make it short and I try to go ...

**0**

votes

**0**answers

126 views

### wedge product in integral ring extensions

Let $A \subseteq B$ be a finite extension of noetherian rings and let $I$ be an ideal of $B$. We consider $I$ and $B$ as finitely generated $A$-modules. The inclusion map $I \hookrightarrow B$ ...

**3**

votes

**1**answer

236 views

### When is the completion of an integral domain still integral?

I have a ring $R=k[S]$ which defines an affine toric variety $X_\sigma$, where $S=M\cap \sigma^\vee$ is the semigroup from a rational polyhedral cone $\sigma$. Let $I$ be the ideal for the toric ...

**1**

vote

**1**answer

167 views

### Failure of Noether normalization

I was in a class recently where we were trying to roughly count the dimensions of certain spaces of rational maps from algebraic curves into closed subschemes $Z \subseteq \mathbb{A}^n$. One way to ...

**2**

votes

**1**answer

60 views

### Kernel of the induced map of the wedge product

Let $A$ be a noetherian ring and let $M$ be a finitely generated $A$-module. Let $F$ be a free $A$-module and let $d: F \to M$ be a homomorphism which maps a basis of $F$ to a minimal set of ...

**11**

votes

**3**answers

407 views

### Varieties where every algebra is free

I'd like to know more about varieties (in the sense of universal algebra) where every algebra is free. Another way to state the condition is that the comparison functor from the Kleisli category to ...

**21**

votes

**5**answers

1k views

### (Short) Exact sequences with no commutative diagram between them

This question was asked by a student (in a slightly different form), and I was unable to answer it properly. I think it's quite interesting.
The problem is to produce an example of the following ...

**1**

vote

**0**answers

127 views

### Descending chain condition for radical ideals

For which integral domains $R$ (not filed) the ring $R[x_1, \ldots, x_n]$ satisfies descending chain condition for radical ideals? I am not expert in Ring Theory and I need an answer to construct some ...

**7**

votes

**0**answers

272 views

### name for a degree-like invariant of a power series over a commutative ring

Let $R$ be a commutative ring, and let $f \in R[\![X]\!]$ be a formal power series. Sometimes (and for example, this will always be possible if $R$ is Noetherian), one may write $f$ in the form $$
f ...

**0**

votes

**0**answers

84 views

### Family of curve singularities whose generic mebers are smooth

Let $f: (X,x)\rightarrow (\mathbb C,0)$ be a deformation of a curve singularity $(X_0,x)$, and let $f: X \rightarrow T$ be a sufficiently small representative. Assume that $(X,x)$ is reduced and pure ...

**2**

votes

**1**answer

19 views

### Separations between notions of rank for modules over commutative (semi-)rings with no zero divisors.

Let $M$ be an $m$-by-$n$ matrix, here are three definitions$^5$ that we could use for rank:
$rk(M) = \min k$ such that for matrices $P$, and $Q$ with $P$ of size $m$-by-$k$ and $Q$ of size ...

**1**

vote

**1**answer

86 views

### Decomposition of skew-symmetric maps

Let $A$ be a ring and let $F$ be a finitely generated, free $A$-module. Let $\alpha: F \to \textrm{Hom}_A (F, A)$ be a skew-symmetric homomorphism, i.e. $\alpha(x)(y)=-\alpha(y)(x)$ for all $x,y \in ...

**3**

votes

**2**answers

171 views

### Are seminormal rings regular in codimension 1?

Let $A$ be a seminormal ring. (Assume that $A$ is a finitely generated $k$-algebra, if it helps.) Is it true that $A$ is regular in codimension 1? I know this is true for normal rings. If the ...

**4**

votes

**1**answer

191 views

### Tannaka–Krein duality

First I would like to stress that maybe I don't have a necessary background from the theory of Lie groups. I met the topic of Tannaka–Krein duality while reading the book of Gracia–Bondia, Varilly and ...

**3**

votes

**1**answer

309 views

### What sort of ring-theoretic properties does the representation ring of a compact Lie group possess?

Recall the definition of the representation ring $R(G)$ of a compact Lie group $G$. I'd like a reference that gives me basic ring-theoretic properties that $R(G)$ always has, or enough info that I can ...

**2**

votes

**0**answers

91 views

### syzygy of a generalized cohen-macaulay module

Let $R$ be a local, noetherian ring of dimension $d$ and suppose it is generalized cohen-macaulay. Is it true that For any finitely generated $ R $-module $ M $, which is maximal generalized ...

**3**

votes

**1**answer

240 views

### For what varieties do we have results on the category of singularities?

Let $X$ be a singular variety. Define the (triangulated) category of singularities (as in Orlov's paper)
as the Verdier quotient of the derived category of coherent sheaves on $X$ modulo the full ...

**1**

vote

**1**answer

156 views

### Extend morphism between coherent sheaves in $\mathbb{P}^n$

Let $\mathcal{F}_1, \mathcal{F}_2$ be coherent sheaves over $\mathbb{P}^n_{\mathbb{C}}$ for $n \ge 3$. Now, $\Gamma_*(\mathcal{O}_{\mathbb{P}^n})=\mathbb{C}[X_0,...X_n]$. Denote by $U_0$ the affine ...

**1**

vote

**1**answer

160 views

### Does $\mathbb P^1 \times \mathbb P^1$ admit an Ulrich bundle?

In an answer to a MathOverflow question on the following link
Vector bundles on $\mathbb{P}^1\times\mathbb{P}^1$, it is mentioned that $\mathbb P^1 \times \mathbb P^1$ has an Ulrich sheaf. However, ...

**1**

vote

**0**answers

103 views

### Classification of rings between a PID and its field of fractions?

Let $D$ be a PID and let $\mathrm{Frac}(D)$ be its field of fractions. I want to classify the intermediate rings $D\subseteq R\subseteq \mathrm{Frac}(D)$.
Theorem: Every such ring $R$ is a ...

**5**

votes

**1**answer

153 views

### Kernel of the differential in de Rham complex in positive characteristic

Roughly, I'd like to ask how does the first terms in de Rham complex behaves for singular varieties.
Let $Y$ be a potentially singular integral scheme over a perfect field $k$ of characteristic $p$ ...

**1**

vote

**0**answers

71 views

### The structure of symmetric powers of finite-dimensional local rings

Fix an algebraically closed field $k$ of arbitrary characteristic $p$ and let $R$ be a finite-dimensional local $k$-algebra (so in particular $R$ is Artinian and Noetherian). Let $S_n$ be the ...

**1**

vote

**1**answer

121 views

### Does the normalization morphism induce isomorphism on residue fields?

The question is basically coming from the following situation:
Let $C$ be an integral curve over a field $k$ (EDIT and assume that $k$ is not algebraically closed) and let $\phi\colon C^N\to C$ be the ...

**0**

votes

**1**answer

127 views

### Irreducibility of a resultant of real and imaginary parts of a characteristic polynomial

The following question is motivated by the study of a stability border for a robust linear time-invariant control system.
Let us we have an affine family of $n\times n$ matrices with indeterminate ...

**3**

votes

**2**answers

245 views

### Counterexample to Openness of Flat Locus

Let $A$ be a commutative Noetherian ring and $B$ a finitely generated $A$-algebra. Then the set $$U\colon=\{P\in\operatorname{Spec}B\mid B_P\ \mathrm{is\ flat\ over}\ A\}$$ is open in ...

**3**

votes

**1**answer

150 views

### Is the induced ring homomorphism surjective for a finite injective morphism between affine varieties?

Let $X$ and $Y$ be affine varieties over $\mathbb C$, and consider a
morphism $f:X\to Y$ and the induced homomorhism
$$ \varphi=f^*:B=\mathbb C[Y]\to A=\mathbb C[X]. $$
It is very easy to see that ...

**1**

vote

**1**answer

90 views

### Morphisms between a globally generated sheaf and a coherent sheaf(Edited)

Let $X$ be a quasi-projective irreducible scheme, $\mathcal{F}_1$ a globally generated $\mathcal{O}_X$-module and $\mathcal{F}_2$ a coherent sheaf over $X$. Suppose that $\mathcal{F}_1$ is globally ...

**1**

vote

**0**answers

78 views

### When is a power series of two variables formally a rational function?

If we have a (formal) power series of two variables with positive coefficients.
Is there any necessary and sufficient condition for this to be (formally) a rational function?

**1**

vote

**1**answer

239 views

### Cohomology after completion

I've been scouring google and asking friend about something I was certain must be absolutely the easiest thing to people who do homological algebra, and none seem to know the answer to this, so if ...

**6**

votes

**1**answer

196 views

### A question on non noetherian ring

Let $R$ be a reduced commutative non noetherian ring of dimension $d$ and $a$ a non zero divisor. Can I say that Krull dimension of $R/(a)$ is at most $d - 1$?

**8**

votes

**1**answer

367 views

### Commutative algebras whose bidual is commutative

Let $k$ be a commutative ring and $A$ a commutative $k$-algebra. Call $D(A) := \mathrm{Hom}_k(A,k)$ the dual of $A$ as a $k$-module, and $DD(A) := \mathrm{Hom}_k(D(A),k)$ the dual of the latter. Let ...

**2**

votes

**0**answers

127 views

### Weak assassins and essential morphisms

Let $R$ be a commutative ring and let $M\rightarrow N$ be an essential morphism of $R$-modules. Then, $M$ and $N$ have the same associated primes.
Over non-noetherian rings the notion of associated ...

**5**

votes

**3**answers

350 views

### Finite index free subgroups of $\mathrm{SL}(3,\mathbb{Z})$

Does $\mathrm{SL}(n,\mathbb{Z})$ have a free subgroup of finite index for some $n \geq 3$? I know that $\mathrm{SL}(3,\mathbb{Z})$ has many free subgroups and that in the case of ...

**3**

votes

**1**answer

117 views

### Toric ideal of slice of a polytope?

Given a collection $A:=\{a_1, \ldots ,a_n \}$ of different integer points in $\mathbb{N}^d$, which span an affine hyperplane when viewed in $\mathbb{R}^d$, one can define a toric ideal $I_A$ from a ...

**0**

votes

**0**answers

132 views

### Is the universal enveloping algebra of the free Poisson algebra generated by finite set (left)-noetherian?

Let $P$ be the free Poisson algebra over $k$ (a field) generated by a finite set $x_1,\dots,x_n$. Let's consider the universal enveloping algebra $P^e$ of the free Poisson algebra $P$. A Poisson ...

**2**

votes

**1**answer

221 views

### Localisation of $\mathbb{Z}_p[[X]]$ at ideal $(p)$

Let $R=\mathbb{Z}_p[[X]]$ where $\mathbb{Z}_p$ denotes the $p$-adic integers and $p$ is a prime. Then what is $R_{(p)}$ $(R$ localised at the ideal $pR)$ $?$

**0**

votes

**2**answers

231 views

### Tensor powers of an algebra all isomorphic

Let $k$ be a commutative ring with total quotient ring $K$, and let $A$ be a commutative $k$-algebra such that the multiplication map $A \otimes_k A \longrightarrow A$ is an isomorphism.
EDIT: Assume ...

**2**

votes

**2**answers

179 views

### Condition for a local ring whose maximal ideal is principal to be Noetherian

The statement "a local ring whose maximal ideal is principal is Noetherian" is (I think) false. The ring of germs about $0$ of $C^\infty$ functions on the real line seems to be a counterexample since ...

**4**

votes

**1**answer

388 views

### Is there a relative version of Artin's approximation theorem?

I've been thinking about the following situation. I have schemes $X$ and $Y$, smooth of dimension $n$ over a base scheme $S$, together with sections of the structure maps, which are closed embeddings ...

**4**

votes

**1**answer

219 views

### Torsors and the fpqc topology

Fix a scheme $S$, a group scheme $G/S$ (let us say smooth, maybe even affine with some finiteness conditions if you like), and suppose I have some other $S$-scheme $P$ with a right $G$-action. We want ...

**8**

votes

**1**answer

276 views

### Presenting $\mathbb{Q}[[t]]$ as an explicit colimit of smooth $\mathbb{Q}$-algebras: an explicit example for the Popescu's theorem

By the seminal Popescu's theorem, $R=\mathbb{Q}[[t]]$ is a filtered colimit of smooth $\mathbb{Q}$-algebras. Could you give me a hint: which $\mathbb{Q}$-algebras can yield such a colimit? My problem ...

**3**

votes

**0**answers

196 views

### Is there fppf descent of locally free modules

Being locally free is a property of quasi-coherent modules which
does not descend in the fpqc topology (see Remark Tag 05VF). But what happens for fppf coverings? More precisely we ask:
Suppose $A ...

**4**

votes

**0**answers

114 views

### Is pushforward along a closed immersion in the fppf topology exact?

Let $i : Z \to X$ be a closed immersion of schemes. Is $i_* : Ab((Sch/Z)_{fppf}) \to Ab((Sch/X)_{fppf})$ an exact functor?
The answer is yes in the \'etale or syntomic topology. It seems likely the ...