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7 votes
2 answers
637 views

An algebraic proof of Mumford's smoothness criterion for surfaces?

(Disclaimer: I'm a beginner in this area, so welcome corrections.) Let $(X,x)$ be a germ of a complex surface (i.e. locally the zero set of some holomorphic functions) and assume that $x$ an isolated ...
Graham Leuschke's user avatar
13 votes
2 answers
2k views

Galois group of a product of polynomials

How can I compute the Galois group of the polynomial $fg\in K[x]$ assuming that I know the Galois groups of $f\in K[x]$ and $g\in K[x]$? Let's suppose for simplicity that the field $K$ is perfect.
roger123's user avatar
  • 2,782
32 votes
7 answers
5k views

Invariant polynomials under a group action (hidden GIT)

Let's say I start with the polynomial ring in $n$ variables $R = \mathbb{Z}[x_1,...,x_n]$ (in the case at hand I had $\mathbb{C}$ in place of $\mathbb{Z}$). Now the symmetric group $\mathfrak{S}_n$ ...
babubba's user avatar
  • 1,993
9 votes
1 answer
1k views

First-order UFD (factorial ring) condition / pre-Schreier rings

All rings in this post are commutative and with $1$. Everyone knows the definition of a factorial ring, a. k. a. unique factorization domain (UFD). I have been wondering about some variations ...
darij grinberg's user avatar
27 votes
2 answers
1k views

Limit of a series of singularities

The $A_\infty$ and $D_\infty$ plane curve singularities have defining equations $x^2=0$ and $x^2y=0$. These equations are "clearly" natural limiting cases of the equations for $A_n$ singularities $x^...
Graham Leuschke's user avatar
11 votes
3 answers
1k views

Existence of non-commutative desingularizations

Let $R$ be normal, local ring of dimension at least $2$. Let $M$ be a reflexive $R$-module and let $A=Hom_R(M,M)$. Suppose $A$ has finite global dimension. Then one can view $A$ as a weak non-...
Hailong Dao's user avatar
  • 30.5k
52 votes
7 answers
8k views

"Algebraic" topologies like the Zariski topology?

The fact that a commutative ring has a natural topological space associated with it is still a really interesting coincidence. The entire subject of Algebraic geometry is based on this simple fact. ...
5 votes
2 answers
1k views

Injective modules and Pontrjagin duals

Forgive me for this naive question. We consider the following lemma and its proof in Lang's algebra, Third Ed., published 1999, Chap. 20, section 4, page 784. Every module is a submodule of an ...
Anweshi's user avatar
  • 7,442
2 votes
1 answer
1k views

monoid ring and some structure within it - how is it called?

I am amateur - mathematics is my hobby, and I find some strange structure working with toy matrices structure so I try to ask some questions regarding it. Let me allow to introduce some structure ...
kakaz's user avatar
  • 1,626
2 votes
2 answers
395 views

When do primes lift uniquely (provided they lift at all)?

Given a ring $R$, a prime ideal $\mathfrak{p}$ of $R$, and an extension ring $S$ (the algebra map $R\to S$ is injective), are there any nontrivial sufficient conditions for the induced map $Spec(S) \...
0 votes
3 answers
1k views

When is the radical of the extension of a prime ideal prime?

(All rings assumed to be commutative and unital) Given a ring $R$, a prime ideal $\mathfrak{p}$ of $R$, and an extension ring $S$ (the algebra map $R\to S$ is injective), are there any nontrivial ...
78 votes
9 answers
26k views

Irreducibility of polynomials in two variables

Let $k$ be a field. I am interested in sufficient criteria for $f \in k[x,y]$ to be irreducible. An example is Theorem A of this paper (Brindza and Pintér, On the irreducibility of some polynomials in ...
Hailong Dao's user avatar
  • 30.5k
1 vote
2 answers
394 views

Relations in matrix semigroups

Suppose that $A_1, \dots, A_k \in M_n(\mathbb{Q})$ and $S$ is the semigroup generated by them. Two questions: are there always a finite set of relations $\{R_i\}$ among the $A_j$ such that $S$ is ...
Victor Miller's user avatar
3 votes
1 answer
590 views

Adjunction for underlying reduced subschemes

Let $k$ be a perfect field (so reduced = geometrically reduced) and $f:X\rightarrow \mathrm{Spec}(k)$ a Cohen-Macaulay morphism. Denote by $i:X_{red}\rightarrow X$ the underlying reduced subscheme ...
B. Cais's user avatar
  • 1,609
11 votes
6 answers
1k views

Computing the structure of the group completion of an abelian monoid, how hard can it be?

Cherry Kearton, Bayer-Fluckiger and others have results that say the monoid of isotopy classes of smooth oriented embeddings of $S^n$ in $S^{n+2}$ is not a free commutative monoid provided $n \geq 3$. ...
Ryan Budney's user avatar
  • 44.4k
38 votes
2 answers
11k views

A finitely generated, locally free module over a domain which is not projective?

This is a followup to a previous question What is the right definition of the Picard group of a commutative ring? where I was worried about the distinction between invertible modules and rank one ...
Pete L. Clark's user avatar
7 votes
2 answers
370 views

Wants: Polynomial Time Algorithm for Decomposing a Multiset of Rationals into Two Additive Subsets.

First, allow me to say that this problem was posed to me by a professor in the department. It is related to his research in a way that I do not know. However, since I couldn't come up with anything ...
B. Bischof's user avatar
  • 4,842
34 votes
4 answers
5k views

What is the right definition of the Picard group of a commutative ring?

This is a rather technical question with no particular importance in any case of actual interest to me, but I've been writing up some notes on commutative algebra and flailing on this point for some ...
Pete L. Clark's user avatar
11 votes
1 answer
946 views

Is ΩΣ in {simplicial commutative monoids} group completion?

Let C be the model category of simplicial commutative monoids (with underlying weak equivalences and fibrations), or equivalently the (∞,1)-category PΣ(Top), where T is the Lawvere theory ...
Reid Barton's user avatar
  • 25.2k
4 votes
2 answers
551 views

Normality of an affine semigroup

An affine monoid is a finitely generated commutative submonoid of $\mathbb Z^k$ for some positive integer k. Let S be an affine monoid and let G(S) be the group generated by S. We say the monoid S is ...
user3649's user avatar
  • 111
7 votes
3 answers
1k views

Intersection of finitely generated subalgebras also finitely generated?

Let $k$ be a field and $A$ be a finitely generated (commutative) algebra over $k$. If $A_1$ and $A_2$ are finitely generated $k$-subalgebras of $A$, is it true that $A_1 \cap A_2$ is also finitely ...
pinaki's user avatar
  • 5,339
18 votes
2 answers
3k views

What does primary mean geometrically?

Given a primary ideal I in a ring A, we can consider the subscheme V(I) of Spec(A). It is a nilpotentification (?) of the integral subscheme V(rad(I)) given by the radical rad(I) of I. My question is ...
user3575's user avatar
  • 211
13 votes
2 answers
1k views

When does a quasicoherent sheaf vanish?

Let $F$ be a quasi-coherent sheaf on a scheme $X$. To check that $F$ vanishes it suffices to check that all the stalks of $F$ vanish. I would like to know whether it suffices to check that all the ...
David Treumann's user avatar
14 votes
3 answers
3k views

non-Dedekind Domain in which every ideal is generated by at most two elements

Does anyone know of such a domain?
Neal Harris's user avatar
8 votes
2 answers
217 views

Flipping Hilbert series of semigroup rings

I'll first give intuition, and then give a precise statement. For $|z|<1$, we have $\sum_{i \geq 0} z^i = 1/(1-z)$. For $|z|>1$, we have $\sum_{i<0} (-1) z^i=1/(1-z)$. Thus, the two ...
David E Speyer's user avatar
30 votes
6 answers
8k views

Algebraic stacks from scratch [closed]

I have a pretty good understanding of stacks, sheaves, descent, Grothendieck topologies, and I have a decent understanding of commutative algebra (I know enough about smooth, unramified, étale, and ...
7 votes
1 answer
676 views

Dual of $\mathbb Z^I$ for uncountable $I$

Let $I$ be an infinite set. There is a homomorphism of abelian groups $\mathbb{Z}^{(I)} \to \hom(\mathbb{Z}^I,\mathbb{Z})$ which sends the basis element $e_i$ to the projection $p_i$. If $I$ is ...
Martin Brandenburg's user avatar
31 votes
2 answers
2k views

Should Krull dimension be a cardinal?

A totally ordered finite set $\quad \mathcal P_0 \varsubsetneq \mathcal P_1\varsubsetneq \dots \mathcal \varsubsetneq \mathcal P_n \quad$ of prime ideals of a ring $A$ is said to be a chain of ...
Georges Elencwajg's user avatar
4 votes
1 answer
367 views

Do n-th Witt polynomials generate {P | P' is divisible by n} ?

EDIT: Proved it on my own. It easily follows from the Witt integrality theorem. Sorry for posting. Let $P\in\mathbb{Z}\left[\Xi\right]$ be a polynomial (where $\Xi$ is a family of symbols that we use ...
darij grinberg's user avatar
55 votes
5 answers
3k views

Bizarre operation on polynomials

There I was, innocently doing some category theory, when up popped a totally outlandish operation on polynomials. It seems outlandish to me, anyway. I'd like to know if anyone has seen this ...
Tom Leinster's user avatar
  • 27.7k
12 votes
4 answers
3k views

Atiyah-MacDonald, exercise 7.19 - "decomposition using irreducible ideals"

An ideal $\mathfrak{a}$ is called irreducible if $\mathfrak{a} = \mathfrak{b} \cap \mathfrak{c}$ implies $\mathfrak{a} = \mathfrak{b}$ or $\mathfrak{a} = \mathfrak{c}$. Atiyah-MacDonald Lemma 7.11 ...
CJD's user avatar
  • 1,098
8 votes
3 answers
2k views

When does the group of invertible ideal quotients = the free abelian group on the prime ideals?

I haven't learned that much about primary decomposition, but from I understand about Dedekind domains, we have that all fractional ideals are invertible and all (plain old) ideals factor uniquely into ...
Zev Chonoles's user avatar
  • 6,792
42 votes
4 answers
8k views

Serre intersection formula and derived algebraic geometry?

Let $X$ be a regular scheme (all local rings are regular). Let $Y,Z$ be two closed subschemes defined by ideals sheaves $\mathcal I,\mathcal J$. Serre gave a beautiful formula to count the ...
Hailong Dao's user avatar
  • 30.5k
4 votes
1 answer
412 views

F_q-structures on schemes

Let $k|\mathbb{F}_q$ be a field extension. An $\mathbb{F}_q$-structure on a $k$-algebra $A$ is an $\mathbb{F}_q$-subalgebra $A _0$ of $A$ such that $A _0 \otimes _{\mathbb{F}_q} k \cong A$ via the ...
user717's user avatar
  • 5,243
9 votes
1 answer
884 views

Isolated hypersurface singularities, Chow groups and D-branes

Say a ring $R$ is an isolated hypersurface singularity if $R = k[x_1, \ldots, x_n]_{(x_1, \ldots, x_n)}/(W)$, where $k$ is a field and $W \in k[x_1, \ldots, x_n]$ is such that the ideal $(\partial_1 W,...
Jesse Burke's user avatar
1 vote
1 answer
925 views

Torsion-free and torsionless abelian groups

This question is motivated by my most spectacular answer on MO (: Let $A$ be a module over $\mathbb Z$. $A$ is said to be torsion-free if $na=0$ implies $n=0$ or $a=0$ for any $n\in \mathbb Z, a\...
Hailong Dao's user avatar
  • 30.5k
38 votes
1 answer
10k views

Infinite tensor products

Let $A$ be a commutative ring and $M_i, i \in I$ be a infinite family of $A$-modules. Define their tensor product $\bigotimes_{i \in I} M_i$ to be a representing object of the functor of multilinear ...
Martin Brandenburg's user avatar
3 votes
2 answers
611 views

Computing Integral Closures

I'm wondering if there's an algorithm, or a program I can use, to compute integral closures. Specifically, what I have in mind are variants of questions of the sort: what is the integral closure of &#...
Randy Brown's user avatar
  • 1,386
4 votes
1 answer
734 views

Parametric polynomial solution of a single polynomial equation

Let $P$ be a polynomial in $n$ variables with rational coefficients, $P \in {\mathbb Q}[Z_1,Z_2, \ldots ,Z_n]$, and consider the algebraic set $Z=\lbrace (z_1,z_2,z_3, \ldots ,z_n) \in {\mathbb Q}^n |...
Ewan Delanoy's user avatar
  • 3,595
11 votes
5 answers
8k views

An example of two elements without a greatest common divisor

Is there an easy example of an integral domain and two elements on it which do not have a greatest common divisor? It will have to be a non-UFD, obviously. "Easy" means that I can explain it to my ...
Alfonso Gracia-Saz's user avatar
13 votes
3 answers
978 views

Model Structure/Homotopy Pushouts in topological monoids?

Let $\mathsf C$ be the category of topological monoids, that is, the category of monoids in $(\textsf{Top}, \times)$. Can the model category structure on $\textsf{Top}$ (Serre fibrations, ...
Joey Hirsh's user avatar
  • 1,033
6 votes
3 answers
797 views

Tensor product is to flat as Hom is to ?

Sorry if I'm missing something here, but what do we call $M$ if the functor $H_M:N\mapsto Hom(M,N)$ is exact? Is this in fact equivalent to being flat through some adjointness properties?
Zev Chonoles's user avatar
  • 6,792
23 votes
3 answers
6k views

Does homology detect chain homotopy equivalence?

Is the following true: If two chain complexes of free abelian groups have isomorphic homology modules then they are chain homotopy equivalent.
Stephen Bigelow's user avatar
4 votes
2 answers
1k views

elementary classification of artinian rings

this may be too elementary for mathoverflow, but I'll give it a try. rings are commutative here. it is well-known that every $0$-dimensional noetherian ring is artinian. the standard proof uses a ...
Martin Brandenburg's user avatar
27 votes
5 answers
3k views

Class number measuring the failure of unique factorization

The statement that the class number measures the failure of the ring of integers to be a ufd is very common in books. ufd iff class number is 1. This inspires the following question: Is there a ...
Dror Speiser's user avatar
  • 4,593
11 votes
2 answers
470 views

Localizing at the primitive polynomials?

For any UFD $R$, the concept of a primitive polynomial (gcd of the coefficients is 1) makes sense in $R[x]$. The product of two primitive polynomials is primitive (Gauss's Lemma), and certainly 1 is a ...
Zev Chonoles's user avatar
  • 6,792
11 votes
1 answer
1k views

Flat cohomology and Picard groups

Let $(R,m)$ be a local complete intersection of dimension $3$. Let $X=Spec(R)$ and $U=Spec(R) -\{m\}$ be the punctured spectrum of $R$. I am trying to understand the following comment by Gabber (see ...
Hailong Dao's user avatar
  • 30.5k
7 votes
0 answers
770 views

Artin-Schreier Theorem for Rings

This has been in my mind for quite some time. Looking at the Artin-Schreier Theorem for fields: If $L$ is a field and $K$ its algebraic closure and if $1< [K:L] < \infty$ then $K=L[i]$ and $L$ ...
Jose Capco's user avatar
  • 2,275
53 votes
3 answers
6k views

Is it true that, as $\Bbb Z$-modules, the polynomial ring and the power series ring over integers are dual to each other?

Is it true that, in the category of $\mathbb{Z}$-modules, $\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}[x],\mathbb{Z})\cong\mathbb{Z}[[x]]$ and $\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}[[x]],\mathbb{Z}...
Maharana's user avatar
  • 1,742
14 votes
1 answer
2k views

When are epimorphisms of algebraic objects surjective?

Let $C$ be the category of $\tau$-algebras for some type $\tau$. Consider the statements: Every monomorphism is regular. Every epimorphism in $C$ is surjective. It is easy to see that 1. implies 2. ...
Martin Brandenburg's user avatar