**28**

votes

**0**answers

2k views

### A short proof for $\dim(R[T])=\dim(R)+1$

For a commutative ring $R$ we clearly have $\dim(R[T]) \geq \dim(R)+1$. If $R$ is noetherian, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and non-trivial ...

**28**

votes

**0**answers

622 views

### Enriched Categories: Ideals/Submodules and algebraic geometry

While working through Atiyah/MacDonald for my final exams I realized the following:
The category(poset) of ideals $I(A)$ of a commutative ring A is a closed symmetric monoidal category if endowed ...

**16**

votes

**0**answers

389 views

### Algebraic closure as a fibrant replacement?

Emil Artin's construction of the algebraic closure of a field $K$ is as follows. Let $K_{0} = K$, and inductively let $\{x_f\}$ be a set of indeterminates indexed by the irreducible $f$ in one ...

**16**

votes

**0**answers

950 views

### Two conjectures by Gabber on Brauer and Picard groups

In a paper I need to make reference to 2 conjectures by Gabber
(see Conjectures 2 and 3, page 1975)
http://www.mfo.de/programme/schedule/2004/32/OWR_2004_37.pdf
1) Let $R$ be a strictly henselian ...

**15**

votes

**0**answers

274 views

### Are dualizable modules finitely generated?

Let $A$ be a commutative noetherian ring, and assume that $A$ has a dualizing complex $R$. Let $D(-) := \operatorname{RHom}_A(-,R)$ be the associated dualizing functor, and let $M$ be an $A$-module.
...

**14**

votes

**0**answers

363 views

### Refinement of concept of support of a module

My rings are commutative and noetherian.
The support of a module is usually defined to be the set of prime ideals of the ring such that localization at that prime does not make the module zero. This ...

**13**

votes

**0**answers

425 views

### How many ways can one cover $\mathbb Q_p$ with the images of polynomials?

Define a finite set of polynomials over a field $K$ to cover $K$ if the images of the polynomials, viewed as functions from $K$ to itself, have union the whole set.
Define a minimal cover to be a ...

**13**

votes

**0**answers

440 views

### Bloch-Kato conjecture and Wiles' numerical criterion

I already asked this question some days ago on http://math.stackexchange.com/questions/158747/bloch-kato-conjecture-and-wiles-numerical-criterion but didn't receive any response.
In the introduction ...

**13**

votes

**0**answers

576 views

### Countable Hom/Ext implies finitely generated

Today I learned this interesting fact from Jerry Kaminker: If $A$ is an abelian group such that $\mathrm{Hom}(A,\mathbb{Z})$ and $\mathrm{Ext}(A,\mathbb{Z})$ are both countably generated, then in fact ...

**13**

votes

**0**answers

482 views

### When is every “solid” perfect complex faithful?

Let $R$ be a noetherian commutative ring. Consider $D^{perf}(R)=K^b(R-proj)$ the category of bounded complexes of finitely generated projective $R$-modules, with maps of complexes up to homotopy.
...

**12**

votes

**0**answers

675 views

### Frobenius upper shriek/flat of a dualizing complex

Let $X$ be a separated connected scheme of characteristic $p > 0$. I am going to assume that $F : X \to X$ (the absolute Frobenius) is a finite map. This condition is called being $F$-finite.
...

**12**

votes

**0**answers

433 views

### A commutative monoid associated with a finite abelian group

Let $M$ be a finite abelian group, and denote by $e_m$, for $m \in M$, the canonical basis of $\mathbb{Z}^M$. For $m, n \in M$ define elements $v_{m,n} \in \mathbb{Z}^M/\langle e_0\rangle$ as
$$
...

**11**

votes

**0**answers

1k views

### Reverse mathematics strength of identically zero polynomials are the zero polynomial

According to wikipedia, the statement "every polynomial over a countable field that is not the zero polynomial has only finitely many roots" is equivalent to RCA0 over RCA0* (which is called ERCA-0 in ...

**10**

votes

**0**answers

298 views

### (When) is isomorphism on differentials enough to guarantee that a map is étale?

I'm sorry if this is too easy for MO.
Let $S$ be a locally noetherian scheme, flat over $\mathrm{Spec}\,\mathbb{Z}$, $X$ and $Y$ be flat $S$-schemes locally of finite presentation, and let $f:X\to Y$ ...

**10**

votes

**0**answers

313 views

### What is the state of art in Groebner bases

How big polynomial systems can we deal with? How do you know when you don't even have to try?
Motivation:
Recently I tried to solve a problem posed in another MO question and ultimately I got stuck ...

**10**

votes

**0**answers

242 views

### Subrings of invariants in divided power algebras

I am wondering to what extent the functors "ring of invariants under a group action $G$"
and "divided power envelope with respect to a $G$-stable ideal" commute.
To be precise, let $R$ be a ...

**10**

votes

**0**answers

637 views

### monomorphisms and epimorphisms of local rings

I want to understand the structure of monomorphisms/epimorphisms in the category of local rings (with local homomorphisms), or dually in the category of local schemes. Let $LR$ denote this category.
...

**9**

votes

**0**answers

180 views

### Is the generation of rings by their units a question in K-theory?

Susan's question When can number rings be spanned (as $\mathbb{Z}$-modules) by units? smells like an algebraic K-theory question in disguise. I'll reformulate the question first:
Given an integral ...

**9**

votes

**0**answers

179 views

### Smallest class of rings closed under familiar operations

Suppose I start out with the ring $\mathbb{Z}$, and call $\mathcal{C}$ the smallest collection of (commutative, unital) rings closed under the following list of operations (which I am aware has some ...

**9**

votes

**0**answers

376 views

### Are the supports of $Ext^i(M,N)$ eventually periodic?

Let $R$ be a Noetherian, commutative ring and $M,N$ be finitely generated $R$-modules.
Question: Do the sets of minimal primes of $\text{Ext}^i_R(M,N)$ (for a fixed pair of $M,N$) become periodic ...

**9**

votes

**0**answers

225 views

### Krull rings and determinantal invariants

During another attempt to come to grips with Hillman's excellent book Algebraic Invariants of Links, I am having difficulty figuring out why Krull rings are the setting for Chapter 3- the natural ...

**8**

votes

**0**answers

86 views

### How useful is knowing every torsionfree $\mathcal O(D)$ module is flat?

One of the corollaries of Weiertrass' factorization theorem plus the theorem of Mittag Leffler is that $\mathcal O(\Bbb C)$, more generally $\mathcal O(D)$ for some region $D$ is such that every ...

**8**

votes

**0**answers

118 views

### Diameter of simplicial complex mirrored in property of Stanley-Reisner ring?

Consider a pure finite abstract simplicial complex $\Delta$. Define its diameter as the maximal distance between any two facets, i.e., between any two faces of maximal dimension $d-1$. The distance ...

**8**

votes

**0**answers

428 views

### Conceptual proofs for the computation of the structure sheaf

The following lemma in commutative algebra is important for the foundations of algebraic geometry:
If $A$ is a commutative ring, $U \subseteq A$ is a finite subset generating the unit ideal, then ...

**8**

votes

**0**answers

429 views

### Proving that a variety is not (isomorphic to) a toric variety

Is there an algorithmic (or other) way to prove that a (projective)
variety is not isomorphic to a toric variety?
I'd be happy with an algebraic answer (for affine or projective varieties),
using the ...

**8**

votes

**0**answers

299 views

### Computing Ext for toric divisors

Ok, I have an affine (normal) toric variety $X = \text{Spec} k[\sigma]$. Suppose that $F, G$ are two torus invariant Weil divisors on $X$. Is there any relatively straightforward way to compute
$$
...

**8**

votes

**0**answers

418 views

### E(n) Deformations of the infinity category Qcoh(X) with it's E(n)-tensor product

Let $X$ be a smooth scheme, then an infinity enchancement of $QCoh(X)$ has an $E_\infty$ structure and in particular an $E_n$ structure for any $n$. In this paper, http://arxiv.org/abs/0805.0157 ...

**7**

votes

**0**answers

351 views

### What is the etale fundamental group of Spec Z((x))?

I know the etale fundamental group of $\mathbb{Z}$ is trivial. For algebraically closed fields $K$, the etale fundamental group of $K((x))$ is $\hat{\mathbb{Z}}$, since all covers in this case are ...

**7**

votes

**0**answers

287 views

### Singularities arising from the Minimal Model Program (an algebraic point of view)

I will start the story by the end:
Is there some characterization of (some of) the singularities arising from the Minimal Model Program (canonical, terminal, log-...) in terms of commutative algebra ...

**7**

votes

**0**answers

243 views

### When is $I \otimes_A \hat{A} \cong I\hat{A}$?

Let $A$ be a commutative ring and $I$ a (finitely generated) ideal in $A$. We denote by $\hat{A}$ the $I$-adic completion of $A$, i.e. $\hat{A} = \varprojlim(A/I \leftarrow A/I^2 \leftarrow \ldots)$.
...

**7**

votes

**0**answers

289 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 ...

**7**

votes

**0**answers

118 views

### Constructively correct notion of unique factorization domain

Recall the well-known proof that a unique factorization domain is a GCD domain:
Let $x, y \in R \setminus \{ 0 \}$. Factor $x$ and $y$ into pairwise non-associated irreducible elements: ...

**7**

votes

**0**answers

163 views

### Square of primary ideals

Is there any example of a $P$-primary ideal $I$ in a noetherian domain $R$ such that $I^2=PI \not=P^2$?

**7**

votes

**0**answers

83 views

### Irreducibility testing and factoring

It is a result of van Hoeij and Novicin (Algorithmica, 2012) that factoring polynomials of degree $d$ over the integers can be done in $O(d^6 + d^4 \log^2 A)$ time, where $A$ is the coefficient bound. ...

**7**

votes

**0**answers

456 views

### Getting a bound via polynomial equations

When studying the existence problem of power residue deference sets, I came across the following system of polynomial equations over $\mathbb{C}$,
\begin{cases}
...

**7**

votes

**0**answers

387 views

### Grothendieck trace on local cohomology?

Let R be an augmented regular local ring over a field $k$ with maximal ideal m. There is the Grothendieck residue symbol:
$$Res: H^n_m(\Omega^n) \to k$$
If $k=\mathbb{C}$ and $R$ is affine space, ...

**7**

votes

**0**answers

215 views

### History of the characterization of commutative Artin rings

When it comes to the world of "classical" (pre-homological) Noetherian commutative algebra, I tend to think of most of the results (Krull's intersection theorem, the principal ideal theorem, etc.) as ...

**7**

votes

**0**answers

236 views

### Uncountable Lüroth problem

Question. Let $F(X)$ be the field obtained by adding an uncountable collection of indeterminates (mutually transcendental elements) to a prime field $F$. Is there an example of a subfield $E$ ...

**7**

votes

**0**answers

229 views

### Computer Algebra solution for simplicial resolutions for André-Quillen cohomology

Hello,
I would like to experiment with André-Quillen (co)homology. Especially for singular rings.
A key problem is that the construction of a simplicial resolution of a ring seems to require a rather ...

**7**

votes

**0**answers

1k views

### Finding ideals of $\mathbb{Z}[x]$ generated by $n$ elements and no fewer

As the title says, I'm trying to find ideals of $\mathbb{Z}[x]$ generated by $n$ elements and no fewer. I suspect $(2^k, 2^{k-1} x, 2^{k-2} x^2, ..., x^k)$ is generated by no fewer than $n=k+1$ ...

**7**

votes

**0**answers

814 views

### Does the property (x*y)*x = x*y have a name?

The property $(xy)x = xy$ is one of the equations satisified by a directoid. Various properties have names ($xy = yx$ is commutativity, $xx=x$ is idempotency, etc). The wikipedia page for Magma has ...

**6**

votes

**0**answers

156 views

### What is a good introduction to cluster algebras from surfaces?

What is a good reference for cluster algebras from surfaces, with a view to their connection to Teichmuller theory?
In my view, that means it should start off with unpunctured surfaces (and in ...

**6**

votes

**0**answers

155 views

### scheme of commuting matrices

Let $k$ be any field. Let $r$ and $n$ be two positive integers.
Consider the functor $F$ from the category of $k$-schemes to the category of sets which sends a $k$-scheme $T$ to the set of matrices ...

**6**

votes

**0**answers

92 views

### Finiteness for separated residually finite modules

Suppose that $A$ is a commutative noetherian Jacobson ring and $M$ is an $A$-module. Suppose in addition that $M$ is $\mathfrak{m}$-adically separated for every maximal ideal $\mathfrak{m}$, and that ...

**6**

votes

**0**answers

199 views

### Localisation of injectives

When working with injective modules, one bad thing is that they do not necessarily behave well with respect to localisation. Consider a commutative ring $R$ and have a look at the following ...

**6**

votes

**0**answers

204 views

### constructive Serre classes

A Serre class (of abelian groups) is a class of abelian groups closed under subgroups, quotients, and extensions. For instance, finitely generated groups and finite groups are both Serre classes.
...

**6**

votes

**0**answers

168 views

### I. Kaplansky, Going up in polynomial rings, unpublished manuscript, 1972

Anyone got a copy of this article?

**6**

votes

**0**answers

586 views

### When UFD implies PID

The following result is too elementary, both to state and to prove, not to be known. Can someone give a reference? Is there any hope if you don't suppose UFD (i.e. move that from the hypothesis to ...

**6**

votes

**0**answers

297 views

### Are all henselian fields algebraic over complete fields?

Motivations and Terminology
The term "henselian field" is ambiguous. What I mean when I say that $K$ is a henselian field is that there exists a henselian DVR $R$, such that $K=Frac(R)$. What I mean ...

**6**

votes

**0**answers

530 views

### Invertible elements in generalized fields

Durov's theory of generalized rings also includes generalized fields (5.7.6), which are defined as generalized rings, which are not subtrivial and whose proper strict quotients are subtrivial. For ...