All Questions
6,057 questions
2
votes
1
answer
281
views
Does fiberwise exactness imply exactness?
Let $R$ be a local Noetherian domain with fraction field $K$ and residue field $\Bbbk$. Let $C^{\bullet}$ be a bounded complex of free, finitely generated $R$-modules. Suppose that $C^{\bullet} \...
- 7,338
4
votes
1
answer
552
views
Factorization of schemes
Let $k$ be a ring (perhaps a field). Let $M$ be the "set" of isomorphism classes of $k$-algebras and regard it as a commutative monoid with multiplication $\otimes_k $ and unit element $k$. There is ...
- 63.1k
13
votes
1
answer
1k
views
Succesful applications of algebra in combinatorics
Hi. This may be a very general question.
Are there any examples of problems in combinatorics which were open, but which found a solution when stated in algebraic terms?
If yes, could somebody ...
3
votes
1
answer
538
views
Comparation of dimensions of rings
Let $k$ be a field, $A$ be an integral domain, $B \subset A$, and $A, B$ are both finitely generated $k$ algebra. Let $p \subset B$ be a prime ideal. Suppose there exists prime ideals $q \subset A$, ...
- 95
1
vote
2
answers
292
views
Symbolic powers in regular local rings
I am having trouble understanding one of the results in the following paper
http://arxiv.org/PS_cache/math/pdf/0104/0104175v1.pdf
In proposition 3.1, the author says
Let $(R,\frak{m})$ be a ...
3
votes
0
answers
603
views
Norms in Galois extensions
Let $k$ be a field of characteristic 0, and $\overline k$ be a fixed algebraic closure of $k$.
Let $k\subset F\subset E$ be a tower of finite Galois extensions in $\overline k$,
where both $\mathrm{...
- 14.2k
2
votes
3
answers
688
views
Metrizability of $\mathfrak{a}$-adic topology
Let $A$ be a ring, $\mathfrak{a}\subset A$ an ideal. Then is the $\mathfrak{a}$-adic topology on $A$ necessarily a metric space? I can see that it is true when $A$ is a DVR, but is it true in general?
- 2,857
4
votes
1
answer
1k
views
checking if F[x]/I is isomorphic to F[x]/J
Let $F$ be a field. Let $p$ and $q$ be monic members $F[x]$. Let $I = \{p\cdot r : r\in F[x]\}$ and $J = \{q\cdot r : r\in F[x]\}$. I know that if $F[x]/I$ is isomorphic to $F[x]/J$ then ($\...
user5810
4
votes
1
answer
711
views
Faltings' category of almost modules
Hi,
Let $V$ be an integral domain with an ideal $m\subset V$ and put $K = S^{-1}V$ where
$S = {1}\cup m$ (a multiplicatively closed subset). Is it true that the category of almost
$(V,m)$-modules is ...
- 2,842
8
votes
1
answer
3k
views
Number of graphs with a given number of nodes, edges and triangles
Hi. Does anyone know if it is possible to enumerate the set of labeled/unlabeled graphs (loopless, undirected, only one edge between pairs of nodes) having a given number of nodes, edges and triangles?...
- 902
18
votes
4
answers
4k
views
Flatness of normalization
Let $X$ be a noetherian integral scheme and let $f \colon X' \to X$ be the normalization morphism. It is known that, if non trivial, $f$ is never flat (see Liu, example 4.3.5).
What happens if we ...
- 3,704
5
votes
1
answer
220
views
When are these rings regular?
Let $R$ be a noetherian regular domain. Suppose that $a, b \in R$, with $b \neq 0$, and consider the ring $S:=R[\frac{a}{b}]=R[X]/(bX-a)$. Is $S$ regular? If this is not the case are there some ...
- 3,704
1
vote
0
answers
138
views
Bases of Ideals With no Monomials
Let $K$ be an algebraically closed field and $K[\underline{x}]$ its ring of polynomials in $n$ variables $x_1,\cdots, x_n$. Let $J\leq K[\underline{x}]$ be an ideal such that there are no monomials in ...
- 345
7
votes
3
answers
2k
views
Products of Ideal Sheaves and Union of irreducible Subvarieties
Assume I have a nonsingular, irreducible, algebraic variety $X$ and irreducible, nonsingular subvarieties $Z_1,\ldots,Z_k\subseteq X$. Let $\mathcal{I}_i$ be the ideal sheaf of $Z_i$ and $\mathcal{I}:=...
- 4,902
10
votes
1
answer
1k
views
Formally smooth morphisms, the cotangent complex, and an extension of the conormal sequence
I'm reading Daniel Quillen's paper "Homology of commutative rings," in which he proves:
A finitely presented morphism of rings $A \to B$ is
Formally etale iff $L_{B/A}$ (this denotes the cotangent ...
- 25.6k
2
votes
0
answers
245
views
Is simplicity preserved under completion of the base ring?
Let $(A,\mathfrak{m})$ be a noetherian local ring and $R$ be an $A$-algebra, which is finitely generated generated as an $A$-module (module finite $A$-algebra). Let $\widehat{A}$ be the $\mathfrak{m}$-...
- 1,391
11
votes
1
answer
4k
views
Does completion commute with localization?
Suppose $A$ is a Noetherian (not necessarily local) ring and $\mathfrak{m}\subset A$ a maximal ideal. Then is it true that $$\hat{A}_{\hat{\mathfrak{m}}}=\widehat{A _{\mathfrak{m}}},$$ where hats ...
- 2,857
4
votes
1
answer
662
views
Modules with flat duals
Let $R$ be a commutative ring, $M$ an $R$-module, $M^*=Hom_R(M,R)$ its dual. What are sufficient (and possibly necessary) conditions on $M$ that ensure that $M^*$ is flat? Is there a name for such ...
- 12.4k
37
votes
3
answers
3k
views
What does it mean geometrically that an element in a domain is irreducible?
Consider a domain $A$ and a non-zero element $f\in A$. That element $f$ is prime if and only if the subscheme $V(f)\subset \operatorname{Spec}(A)$ is integral and this is a completely satisfactory ...
- 47.5k
24
votes
4
answers
4k
views
Is there a Galois correspondence for ring extensions?
Given an ring extension of a (commutative with unit) ring, Is it possible to give a "good" notion of "degree of the extension"?. By "good", I am thinking in a degree which allow us, for instance, to ...
- 1,415
11
votes
1
answer
1k
views
Do I need to know what an infinity-Gerstenhaber algebra is, and if so, what is it?
I am in the following situation. I have two (rather explicit and specific) dg commutative algebras $R,S$ over a field of characteristic $0$. In fact, $S$ is an $R$-algebra, in that I have a map $R \...
- 54.6k
15
votes
2
answers
2k
views
Is every poset the poset of prime ideals of a ring?
The answer to this question, as it is, is trivially false, for one necessary condition is the existence of maximal element(s), i.e., maximal ideals exist and are prime.
My question was inspired from ...
- 899
3
votes
1
answer
2k
views
Multiplicity of a singular point
Let $X$ be a smooth projective complex variety. Assume that $Z$ is a subvariety. Let $T$ be a generic complete intersection of codimension $\dim Z-1$. Assume that $p$ is a point in $Z_T:=T\cap Z$. Is ...
- 2,444
3
votes
1
answer
268
views
Universal catenarity and Laurent algebras
A Noetherian (commutative) ring $A$ is called universally catenary if every $A$-algebra of finite type is catenary. If one wants to know whether $A$ is universally catenary, then this definition ...
- 6,700
11
votes
1
answer
840
views
Which cluster algebras are coordinate rings of double Bruhat cells?
Background
A uselessly vague paragraph follows. A cluster algebra is a commutative algebra $A$ with a distinguished set of generators called cluster variables. These cluster variables are grouped ...
- 13k
6
votes
1
answer
1k
views
reference for p-local and p-complete integers
Can anyone suggest a good thorough reference for $p$-localization and $p$-completion of the integers? I'm an algebraic topologist who's found himself washed up without any intuition.
In particular, ...
- 499
2
votes
0
answers
506
views
Zariski's Main theorem [closed]
Sir,
I am studying Zariski's Main theorem. May i know good source of problems related to Zariski's main theorem to understand it better?
Thanks in advance.
- 161
24
votes
6
answers
5k
views
Pythagorean 5-tuples
What is the solution of the equation $x^2+y^2+z^2+t^2=w^2$ in polynomials over C ("Pythagorean 5-tuples")?
There are simple formulas describing Pythagorean n-tuples for n=3,4,6:
n=3. The formula ...
- 1,488
3
votes
0
answers
474
views
Jacobson-Bourbaki correspondence
The Jacobson-Bourbaki correspondence induces the traditional, finite Galois correspondence by suitable restriction; I've been pondering two things: 1. Are there any (other) interesting applications of ...
3
votes
0
answers
473
views
Infinite Galois correspondence "according to Artin"
Ever since Artin's lectures on Galois Theory one knows how to set up and derive the usual Galois correspondence in the finite(-dimensional) case using just a bit of elementary Linear Algebra, and ...
13
votes
3
answers
8k
views
$fgf = f$, $gfg = g$, $fg$ not necessarily identity, what is this called?
A very simple question, I just totally forgot how it was called, and Google is not helping.
There's a pair of functions $f:X\to Y$, $g:Y\to X$.
$fgf = f$, $gfg = g$, but $fg$ and $gf$ don't need to ...
- 241
13
votes
6
answers
7k
views
Applications of commutative algebra
Hi. I'm preparing a thesis in commutative algebra, and when I say this to my friends they always ask me what are the applications to "real-world", and I don't know what to answer. This let me think ...
9
votes
2
answers
667
views
irreducibility of generic linear combination of polynomials?
I would be shocked if the following were not true, but I can't seem to see a proof.
Claim:
Let $R$ be an integral domain containing an AC (uncountable if you wish) field $k$. Let $a, b \in R$, and ...
- 525
10
votes
1
answer
3k
views
Rings with all modules projective ?
Is there a classification of the commutative rings (with unit) such that each module over the ring is projective ?
- 16.2k
12
votes
2
answers
3k
views
Vector bundles on affine scheme
I have already asked similar questions before, but now I realized that there a nice
general way to ask what I want. Namely let $X$ be a normal affine variety over
a field $k$. Assume first that $k$ is ...
- 7,037
11
votes
3
answers
2k
views
When is a blow-up Cohen-Macaulay?
Let $X$ be a smooth projective variety and $Z$ a closed subscheme. Let $X'$ be the blow-up of $X$ with center $Z$.
Under what conditions on $Z$ is $X'$
Cohen-Macaulay?
In the case $Z$ is non-...
- 11.6k
5
votes
2
answers
491
views
Is independence meaningful for commutative $C^*$-algebras?
I don't know very much about spectral theory so probably the answer to my question has a basic reference which I would appreciate.
Let's say I have two self-adjoint operators on a Hilbert space and ...
- 2,243
1
vote
1
answer
272
views
Criteria for Preservation of a Module Structure under Extension of Scalars.
Let $A\to B$ be a morphism of (commutative) algebras and $M$ a $B$-module. The $A$-bilinear map $B\times M\to M$ given by $(b,m)\mapsto bm$ induces a surjective homomorphism $B\otimes_{A}M\to M$.
...
- 486
18
votes
2
answers
961
views
How to prove that $e^{zw}$ can not be written as a rational expression in functions in $z$ and in $w$?
Let $K$ be the field of fractions of
$\mathbb{C}[[z]]\otimes_{\mathbb{C}}\mathbb{C}[[w]]\subset \mathbb{C}[[z, w]].$ Given
a formal power series in $t, f\in \mathbb{C}[[t]],$ is there any simple ...
- 689
2
votes
2
answers
492
views
Model Theoretic Localization
This is a re-post on a previous question I asked. My first question was too vague to warrant detailed responses. Really, I have two specific questions to ask.
1) Let $\sigma = (A; \{0,1\}; +, \times)...
- 761
2
votes
1
answer
521
views
Kahler differentials of a hypersurface over a non-algebraically closed field
The following was recently on my algebraic geometry homework:
Let $k$ be an algebraically closed field, $f\in B=k[x_1,\ldots,x_n]$, and $A=B/(f)$. Show that $\Omega_{A/k}$ is locally free of rank $...
- 6,792
9
votes
3
answers
1k
views
Quasi-compact maps in Number Theory
Can someone give me an example of a non-quasi-compact morphism of schemes which arises naturally in the field of Algebraic Number Theory?
- 761
2
votes
1
answer
259
views
Nuclearity of certain semigroup crossed product C*-algebras
This question is related to this question link.
Suppose we have an (abelian) semigroup $S$ acting by endomorphisms on a $C^*$-algebra A giving rise to a semigroup crossed product $B = A\rtimes S$. ...
- 2,029
6
votes
1
answer
971
views
Reflexive sheaves on singular surfaces
Let $S$ be a normal surface over an algebraically closed field $k$ and let
$s$ be a point of $S$. Let ${\mathcal F}$ be a reflexive sheaf on $S$ of generic rank $n$ . Consider the (derived) fiber of $...
- 7,037
9
votes
2
answers
1k
views
Non-Standard Prime
Hello,
My question is about the non-standard models of the integers. If we add to the Peano's axioms $P$ of arithmetic the following axioms for a fixed constant $c$:
$c \neq 0$, $c \neq 1$, $c \neq 1+...
- 663
6
votes
2
answers
2k
views
When does the conormal bundle sequence split?
Let $X\subset \mathbb{P}^n$ be a smooth projective variety with ideal sheaf $I_X$. The conormal sequence is given by
$$
0\to I_X/I_X^2\to \Omega_{\mathbb{P}^n}|_X\to \Omega_{X}\to 0.
$$
For which ...
- 11.6k
3
votes
1
answer
392
views
Can cones (toric monoids) be built as colimits of their faces?
Suppose $L$ is a lattice (free abelian group) and $\sigma$ is a (pointed) spanning rational cone in $L\otimes\mathbb Q$. Then $M=L\cap \sigma$ is a monoid with $M^{gp}=L$. A monoid of this form is ...
- 24.1k
7
votes
2
answers
512
views
Is the reduction of a flat, finite, surjective scheme over an integral base still flat?
Is the reduction $X_{red}$ of a flat, finite, surjective scheme $X$ over an integral base $S$ still flat?
I could possibly add that I am already aware we can assume the base $S$ to be local and ...
- 1,347
1
vote
2
answers
1k
views
maximal ideal in local subrings
Let $A,B$ be two local rings and put $\mathfrak{m}_A, \mathfrak{m}_B$ their maximal ideals. Now suppose that we have an injection $0 \to A \to B$ and put $\mathfrak{n} := A \cap \mathfrak{m}_B $. It ...
- 379
6
votes
2
answers
850
views
Decomposition of finite algebras over finite fields
Let $K$ be a number field, $Z_K$ its ring of integers, and $p$ a rational prime number. Then $A_p = Z_K/(p)$ is a finite ${\mathbb F}_p$-algebra. Using ideal arithmetic in $Z_K$ and the Chinese ...
- 32.6k