**1**

vote

**0**answers

27 views

### Saturation of a subalgebra over the Tate-algebra inside the power series ring

Let $A$ be a discrete valuation ring and $\pi$ a uniformizer.
Over $A$ we consider the Tate-algebra
$$A\langle t \rangle =\{ f=\sum_{n=0}^\infty a_nt^n \mid a_n\in A, \lim_{n\to \infty} \lvert ...

**2**

votes

**1**answer

168 views

### The standard topology of a module over a noetherian local ring

Let $A$ be a noetherian local ring with maximal ideal $m$. One says that an $A$-module $X$ is discrete if for every $x\in X$, there is a natural number $n$ such that $m^n.x=0$.
My question is: Given ...

**2**

votes

**1**answer

162 views

### Name and references for a “twisted” addition in a ring

This question may seem a little unmotivated, but it actually isn't something that just occurred to me out of nowhere. It stems from a discussion I had the other day with a friend and is directly ...

**5**

votes

**2**answers

349 views

### Formal completion of the normal bundle

Let me for simplicity start with affine case. If $X=\operatorname{Spec}(A)$ is an affine variety $Z \subset X$ is a closed affine subvariety $Z=\operatorname{Spec}(A/I)$. What conditions are ...

**6**

votes

**2**answers

278 views

### Hochschild homology of upper triangular matrix algebra?

Let $K$ be a field and $A$ the associative unital $K$-algebra of all $n\times n$ upper triangular matrices with entries in $K$. What is $\dim_K$ of its hochschild homology $HH_k(A;A)$?
Is there any ...

**2**

votes

**2**answers

292 views

### intersection of finitely many maximal ideals

For what commutative rings with infinitely many maximal ideals we can say that the intersection of any combination of finitely many maximal ideals is not zero?
Obviously it holds for Dedekind domains ...

**1**

vote

**1**answer

92 views

### Does the going-up theorem hold between flat algebras?

Let $R$ be a commutative Noetherian ring with unit and $S$ a flat $R$-algebra. Does the going-up theorem hold between $R$ and $S$?

**2**

votes

**1**answer

280 views

### geometric interpretation and differences of Gorenstein rings, Complete intersections and regular rings

Let $R$ be a local Noetherian ring.
What is the geometric interpretation of
1- Gorenstein rings
2- Complete intersections
3- Regular rings?
and how can I realize differences by geometric ...

**1**

vote

**0**answers

92 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

242 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

160 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

60 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

429 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

201 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

278 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

198 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

132 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

271 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

189 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

87 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 ...

**13**

votes

**3**answers

454 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 ...

**24**

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

136 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

283 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

90 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

30 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

91 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

200 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

254 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

331 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

99 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

261 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

172 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

188 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

123 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

173 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

77 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

171 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

147 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

257 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

254 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

99 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

82 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

249 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

222 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 $d - 1$?

**9**

votes

**1**answer

414 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

160 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

436 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

133 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

158 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 ...