All Questions
6,055 questions
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 ...
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.
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 $\mathfrak{S}_n$ ...
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 ...
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^...
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-...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $n \geq 3$. ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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?
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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,...
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\...
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 ...
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 ...
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 |...
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 ...
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, ...
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?
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.
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 ...
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 ...
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 ...
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 ...
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 $L$ ...
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}...
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. ...