**-3**

votes

**0**answers

40 views

### Maximal ideals of R [x]

Let $R$ be a commutative ring with identity. Is there any relation between maximal ideas of $R[x]$ and maximal ideas of $R$?

**3**

votes

**1**answer

62 views

### Integral domains equal to intersection of their height one localizations

Which integral domains have the property that $R = \bigcap R_P$, the intersection being taken over all height one prime ideals of $R$?
It is a standard fact that Krull domains, and thus noetherian ...

**-1**

votes

**0**answers

31 views

### Why do there is a unique continuous homomorphism? [migrated]

Is this a right place to ask help for an exercise?
Let $n\geq 2$ be an integer and $D=\mathbb Z[1/n]$. Let $A$ be a complete commutative ring with unit for the $I$-adic topology, where $I$ is an ...

**1**

vote

**1**answer

77 views

### Projective dimension of a quotient ring

Assume $A$ and $B$ are commutative algebras with $1$, $B = A[z] = A[Z]/(h(Z))$,
$Z$ an indeterminate.
The first comment in this question says that, if $A$ is noetherian, then
$pd_{B\otimes_A B}(B) ...

**3**

votes

**0**answers

65 views

### Unibranch partial normalization

In a paper I recently read something about the "unibranch partial normalization" of a curve.
Say, $R$ is a local integral domain with maximal ideal $\mathfrak{m}$ and fraction field $K$. Is it ...

**0**

votes

**2**answers

113 views

### Making idempotent element by a relation [on hold]

Let $R$ be a commutative ring with identity and let $a, b \in R$ such that $a=ab$. How can we make a non zero idempotent element of $R$ by this relation?

**10**

votes

**1**answer

421 views

### Is an irreducible ideal in $R$ also irreducible in $R[x]$?

Let $R$ be a commutative Noetherian ring and $I\subset R$ an ideal that is irreducible in the sense that if $I = J_1 \cap J_2$, then $I=J_1$ or $I=J_2$. Is (the ideal generated by) $I$ irreducible in ...

**3**

votes

**0**answers

112 views

### Annihilators of elements in symmetric algebras

Let $M$ be a module over a commutative ring, and $S(M)$ its symmetric algebra. What elements of $S(M)$ annihilate a given element $m\in M$ ? ($M$ is considered as a submodule of $S(M)$.)

**0**

votes

**1**answer

151 views

### Can you detect homological dimensions from homology?

Suppose you are given a bounded chain complex $M$ over a commutative ring $R$.
Is there a clear relation between homological dimensions of $M$ and homological dimensions of its cohomologies?
For ...

**0**

votes

**0**answers

42 views

### trying to understand particular modules, to contruct some nice free resolutions

Let $R$ be a local or graded ring, fix some morphism $\phi\in Hom_R (R^{\oplus n},R^{\oplus m})=:Mat(m,n;R)$.
In various applications one meets modules like $\frac{Mat(m,n;R)}{Span(U ...

**2**

votes

**2**answers

195 views

### Decomposition and valuation rings

I am reading Algebraic Number Theory by A. Fröhlich, M. J. Taylor, it first introduced the theorem:
$(K,u)$ be a field and its absolute value, $(K_u,\bar u)$ be its completion and absolute value ...

**0**

votes

**1**answer

115 views

### perfect modules over polynomial algebra

This may be obvious. My question is short:
$R$ is the polynomial algebra $\mathbb{k}[X_{1},\dots , X_{n}]$. Is the $R$-module $\mathbb{k}$ perfect in the sense that $\mathbb{k}$ is a compact object ...

**6**

votes

**0**answers

151 views

### When is a polynomial ring free over a graded subalgebra?

Keep the setting of my previous question and let $I := k[x_1, \dots, x_n] \cdot A_{>0}$ be an ideal of the algebra $k[x_1, \dots, x_n]$ generated by the set $A_{>0}$. It is clear that $I$ is a ...

**0**

votes

**0**answers

72 views

### A property of the semi-local ring of the normalization of a singular curve

I have two following questions.
1) Let $R$ be a local ring in an algebraic function field of one variable over an algebraic closed field $k$. Let $\bar{R}$ and $m$ be its integral closure and maximal ...

**2**

votes

**1**answer

107 views

### Examples of (non-normal) unibranched rings?

For a local integral domain $R$ the following are equivalent:
a) The integral closure of $R$ in its fraction field (i.e., the normalization of $R$) is again local.
b) The henselization of $R$ is ...

**0**

votes

**1**answer

118 views

### Is there a complete local analogue of the Artin-Tate lemma?

The Artin-Tate lemma states that if $A \subseteq B \subseteq C$ are commutative rings where $A$ and $C$ are Noetherian, $C$ is finitely generated as an $A$-algebra, and $C$ is finitely generated as a ...

**1**

vote

**0**answers

71 views

### Centers of Noetherian Algebras and K-theory

I'll start off a little vauge: Let $E$ be a noncommutative ring which is finitely generated over its noetherian center $Z$. Denote by $\textbf{mod}\hspace{.1 cm} E$ the category of finitely ...

**0**

votes

**0**answers

80 views

### Elementary characterization of Krull dimension

I was reading the following paper: "A Short Proof for the Krull Dimension of a Polynomial Ring. Thierry Coquand and Henri Lombardi"
and came across this corollary. (This is present with a better ...

**1**

vote

**1**answer

120 views

### Socle of Almost Complete Intersections

Let $(A,m)$ be a complete Artinian local ring over a field $K$.
We focus on almost complete intersection ring $A$ of the form
$A = K[[X_1,...,X_N]]/(f_1,...,f_{N+1})$.
We assume that none of $f_i$ ...

**-2**

votes

**0**answers

56 views

### Jacobson radical of an indecomposable commutative ring [migrated]

Let $R$ be a commutative indecomposable ring with identity which has infinit many maximal ideals. Can we deduce that $Jacobson$ radical, $J(R)$, (the intersection of all maximal ideals)of $R$ is th ...

**6**

votes

**0**answers

90 views

### Augmentation ideal is finitely generated if and only if $A$ is finitely generated as a $k$-algebra?

Let $A \subset k[x_1, \dots, x_n]$ be a subalgebra, which is also a graded subspace $A = \oplus_{i \ge 0} A_i$. One can write $A = A_0 \oplus A_{> 0}$ where we have $A_0 = k^0[x_1, \dots, x_n] = k$ ...

**-1**

votes

**0**answers

29 views

### When $ht_SQ \le ht_Rf^{-1}(Q)$? When $ht_R P = ht_S f(P)$?

Let $R$ and $S$ be commutative rings with identity, and $f:R\to S$ be a homomorphism.
Question 1. By what assumptions we can have $ht_SQ =ht_Rf^{-1}(Q)$, for every prime ideal $Q$ of $S$? What ...

**0**

votes

**0**answers

49 views

### The trace ideal of a non zero $R$-module [migrated]

Let $R$ be a commutative ring with identity and $M$ be a cyclic $R$-module, we may deﬁne the ideal $tr(M)$ associated with $M$, the sum of the ideals $f(M)$, for all $R$-homomorphisms $f \in ...

**0**

votes

**1**answer

125 views

### Is it possible to generalize a result of Wang?

Assume $A$ and $B$ are commutative algebras with $1$.
There is a nice result of Wang, Corollary 8, which says the following: "Let $B = A[z] = A[Z]/(h(Z))$. Then $B$ is a separable algebra over $A$ if ...

**2**

votes

**1**answer

105 views

### Projectivity of torsion-free modules over integral group rings

Let $G$ be a torsion-free group and assume that the integral group ring $\mathbb{Z}G$ is torsion-free as well. Let $M$ be a torsion-free, finitely generated module over $\mathbb{Z}G$.
If we assume ...

**2**

votes

**2**answers

252 views

### When $mB \neq B$? $m$ is a maximal ideal of $A$, $A \subseteq B$

The following is a question I have asked here without receiving any comments, therefore I post it here:
Let $A \subseteq B$ be commutative rings, $m$ a maximal ideal of $A$.
When $mB \neq B$?
This ...

**2**

votes

**0**answers

125 views

### Intuition behind if neither $D$ nor $K-D$ are equivalent to an effective divisor, then $\deg(D) = g-1$?

Is there any intuition behind the following fact?
If neither $D$ nor $K-D$ are equivalent to an effective divisor, then $\deg(D) = g-1$.
Here, $K$ is the canonical divisor. It means the degrees ...

**8**

votes

**0**answers

135 views

### Why would we a priori expect $V(I)$ to satisfy axioms to define the closed sets for a topology on $\text{Proj}(S)$?

Reposted from math.stackexchange here.
The topological space $\text{Proj}(S)$ has the underlying set$$\text{Proj}(S) = \{\mathfrak{p} \text{ a homogeneous prime such that }S_+ \not\subseteq ...

**1**

vote

**0**answers

61 views

### Degrees of polynomials defining a Jacobian of maximal rank on a variety

Let $f_1,\ldots,f_{n-k} \in \mathbb{R}[x_1,\ldots,x_n]$ be polynomials of degree at most $d$ defining an algebraic set $A \subseteq \mathbb{C}^n$ which contains an irreducible component $V \subseteq ...

**1**

vote

**2**answers

53 views

### How to determine whether the following sum is nonzero for a given multivariate polynomial?

My research field is combinatorics. I am not very good at Algebra. So I want to ask for a given real multivariate polynomial $f(x_1,x_2,\cdots,x_n)$, is there any algebraic method to compute whether ...

**0**

votes

**0**answers

75 views

### Complement and fibers

Let $\mathcal M \rightarrow S$ be a projective irreducible scheme over the spectrum of a DVR and $U\subset \mathcal M$ an open subscheme surjective on $S$. Is it true for both points (generic and ...

**7**

votes

**3**answers

378 views

### Intuition behind basic facts about homogeneous ideals?

What is the intuition (hopefully, geometric) behind these basic facts about homogeneous ideals? An ideal $I$ in $S$ is homogeneous if an element $f = \sum_{n \ge 0} f_n$ of $S$ lies in $I$ if and only ...

**2**

votes

**1**answer

347 views

### A particularly “natural” algebraic structure with three commutative, pairwise-distributive operations

EDIT: As mentioned in my answer below, I was mistaken in thinking Dirichlet convolution distributes over ordinary convolution. I'm leaving this question here for reference.
I keep stumbling on the ...

**3**

votes

**2**answers

526 views

### Can the projective line be provided with a ring structure?

A definition of multiplication on the projective $1$-points $(a:b)$ of $P_K^1$ with $a$ and $b$ elements of a field $K$ ( e.g. the real or rational numbers ) can be given by mimicking the ...

**1**

vote

**1**answer

201 views

### Is this kind of scheme integral?

Let $X\rightarrow Spec(R)$ an irreducible projective scheme over a dvr. Suppose the generic fiber $X_{\eta}$ is smooth (over the field $Frac(R)$) and irreducible. Is it true that $X$ is integral (i.e. ...

**4**

votes

**2**answers

128 views

### Reference for (co)limit-preserving functor $X\mapsto R^X$

Fix a commutative ring $R$. There's a contravariant functor from finite sets to finite $R$-algebras sending $X$ to $R^X$. Viewed as a covariant functor $\text{set}^{op}\to R\text{-alg}$, this functor ...

**3**

votes

**0**answers

217 views

### When does composing polynomials reduce the degree?

Let $\mathbb{F}$ be the field of size 2. For a function $f : \mathbb{F}^n \to \mathbb{F}$, let $d(f)$ be the smallest integer such that there exists a degree-$d(f)$, $n$-variate, multilinear ...

**6**

votes

**1**answer

265 views

### From polynomial ideal over $\mathbb{Q}$ to polynomial ideal over $\mathbb{Z}$

Is there an algorithm to compute, given a polynomial ideal $I\subset \mathbb{Q}[x_1,\dotsc,x_n]$, the ideal $I\cap \mathbb{Z}[x_1,\dotsc,x_n]$ in $\mathbb{Z}[x_1,\dotsc,x_n]$?
The input and ...

**0**

votes

**1**answer

97 views

### quasi-projective and separated as topological properties

Let $X$ be a non-reduced noetherian scheme over $\mathbb{Z}$ or $\mathbb{C}$. Assume that $X^{red}$ is quasi-projective and separated, does the same hold for $X$ ?
(By the way, projective implies a ...

**1**

vote

**1**answer

107 views

### Purely inseparable field extensions of degree p

Take a field $k$. If $k'/k$ is a field extension of degree $p$, it is known that there are many possibilities for the isomorphism class of $k'$. See
...

**0**

votes

**0**answers

88 views

### Irreducible component of a scheme over a dvr

Let $\mathcal M$ be a (reduced) quasi-projective scheme over a dvr (of mixed caracteristics), $R$. Suppose that the generic fiber $\mathcal M_{\eta_R}$ is (nonempty) smooth and irreducible of ...

**3**

votes

**0**answers

47 views

### Catenarity of monoid algebras

Let $R$ be a commutative ring, let $M$ be a commutative monoid, and let $R[M]$ denote the corresponding monoid algebra. Suppose further that $R$ is universally catenary. One may ask for conditions on ...

**4**

votes

**0**answers

151 views

### Generation of cohomology of graded algebras

Let $A$ be an unital, associative, graded algebra over a base ring $k$. I'm happy to assume that $k$ is a field if need be, and will insist that $A$ free and of finite rank in each degree (locally ...

**0**

votes

**0**answers

111 views

### Degrees of generators of radical ideals

Let $I \subseteq \mathbb{C}[x_1,\ldots,x_n]$ be an ideal generated by polynomials $f_1,\ldots,f_r$ of degree at most $d$. Is it possible to generate the radical $\sqrt{I}$ of this ideal with ...

**0**

votes

**1**answer

96 views

### section of reduced structure map

Let $R$ be a commutative ring whose characteristic is either prime or $0$, such that $R/N$ is an integral domain, where $N$ is the nilradical, and $p: R \rightarrow R/N$ the canonical map. Is there a ...

**4**

votes

**1**answer

193 views

### Milnor descent for ring spectra

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

**0**

votes

**1**answer

211 views

### Determinants of tensors [closed]

Consider a tensor of dimension $[d]\times[d]\times[d]$ which is symmetric with respect to every permutation of the indices. Are there any $\textbf{explicit}$ formulas for notions like determinant-like ...

**0**

votes

**1**answer

63 views

### Some references for f-ring

A commutative ring $R$ is said to be an $f-ring$ if every pure ideal is generated by idempotents. (Recall that the ideal $I$ is said to be pure if for each $a\in I$ there
is a $b\in I$ such that $ab = ...

**0**

votes

**0**answers

107 views

### Question about Castelnuovo-Mumford regularity

let $R$ be a Noetherian ring an $I$ an ideal of $R$.
If $n,m\in N$ and $reg(G(I))=n$, then what can we say about $reg(G(I^m))$?
Here $G(I)$ is the associated graded ring.

**2**

votes

**1**answer

99 views

### Classification of commutative ring ideal closure operators?

First, some setup:
So: given a commutative ring $R$, let $Ideals(R)$ be set of ideals of $R$ and let $IdealClosure(R)$ be the set of closure operations $cl: \mathcal{P}(R) \rightarrow Ideals(R)$. In ...