All Questions
6,053 questions
1
vote
2
answers
2k
views
Real solutions to underdetermined system of polynomial equations
I have the system of multi-variable polynomial (quadratic) equations with real coefficients. The number of equations is given scales as $K$ and the number of unknowns goes as $K^2$. So for for large $...
CommunityBot
- 1
9
votes
3
answers
3k
views
Why are divisible abelian groups important?
I just quote wikipedia:
"Divisible groups are important in understanding the structure of abelian groups, especially because they are the injective abelian groups."
I am asking for detail ...
- 57.6k
-1
votes
2
answers
671
views
Do Gorenstein rings necessarily have a finite projective dimension (as a module over itself)? [closed]
Do Gorenstein rings necessarily have finite projective dimensions?
- 2,857
9
votes
1
answer
792
views
Reconstructing a polynomial from resultants
I am trying to compute a monic polynomial $f(x)$ with integer coefficients and known degree $d$. I am given $n$ pairwise coprime polynomials $g_1(x),\ldots,g_n(x)$, also with integer coefficients, ...
CommunityBot
- 1
1
vote
2
answers
2k
views
The structure of the module of Kähler differentials of R[[x]] over R
It seems that $\Omega_{R[[x]]/R}^1$ is rather big. For example, take $R$ to be the rational numbers.
We can see that $\Omega_{R((x))/R}^1$ is a $R((x))$-vector space of infinite rank. As by results ...
CommunityBot
- 1
2
votes
1
answer
980
views
Spectral sequence for Ext
Why is $H^p(X, \mathcal{E}xt^q(F, I))=0$ for $p>0$ and $I$ an injective sheaf and $F$ an arbitrary sheaf? I'm trying to check the hypothesis in the grothendieck spectral sequence applied to the ...
- 12.6k
7
votes
2
answers
542
views
Does every nontrivial sheaf of rings have a maximal ideal?
Let $R$ be a sheaf of rings on a topological space $X$. Assume $R \neq 0$. Does then $R$ have a maximal ideal? So this is a spacified analogon of the theorem, that every nontrivial ring has a maximal ...
CommunityBot
- 1
5
votes
2
answers
689
views
Irreducible components of quotients of Cohen-Macaulay rings of the "correct" dimension
Suppose $R$ is a Cohen-Macaulay ring. It is well known that if $I$ is an ideal of $R$ generated by $n$ elements, and $I$ has codimension $n$, then $R/I$ is also Cohen-Macaulay.
Now suppose that $I$ ...
- 5,137
3
votes
2
answers
2k
views
Extension problem
As I understand, if $0\rightarrow A\rightarrow X\rightarrow B\rightarrow 0$ is a short exact sequence of abelian groups, $\mbox{Ext }_{\mathbb{Z}}^{1}(B,A)$ gives all the isomorphism classes of what ...
- 33.8k
2
votes
2
answers
420
views
Homological dimensions of module
$(A,\mathfrak{m})$ a Noetherian local ring, $M\neq 0$ a finitely generated $A$-module. As I understand, $\mbox{Ext }^{j}(A/\mathfrak{m}, M) = 0$ for $j<\mbox{depth }(M)$ and for $j>\mbox{inj. ...
- 5,846
14
votes
3
answers
2k
views
Intuition for Model Theoretic Proof of the Nullstellensatz
I recently read the model-theoretic proof of the Nullstellensatz using quantifier elimination (see www.msri.org/publications/books/Book39/files/marker.pdf). I'm convinced that the Nullstellensatz is ...
- 15.4k
4
votes
2
answers
2k
views
Non-finite version of Nakayama's lemma?
Let $A$ be a local ring with nilpotent maximal ideal $\mathfrak{m}$ (i.e., some power of $\mathfrak{m}$ vanishes), and $M$ an $A$-module (not necessarily finitely generated). Let $\bar{S}\subset M/\...
- 7,108
2
votes
0
answers
797
views
How can I prove R[x] is integrally closed iff R is integrally closed ? (R: integral domain) [closed]
$R$ is a integral domain. How can I prove $R[x]$ is integrally closed iff $R$ is integrally closed?
- 65.4k
13
votes
1
answer
2k
views
Can an infinite commutative ring have a finite (but nonzero) number of non-nilpotent zero-divisors?
By a theorem of Ganesan, if a commutative ring not a domain has only finitely many zero-divisors, then the ring must be finite. (There are analogous results for non-commutative rings.)
There are ...
- 14.5k
2
votes
0
answers
348
views
How much can we say about the number of nilpotents in a finite local commutative ring?
A commutative ring is local if it has a single maximal ideal. If the ring is finite, this implies that all elements are either units or nilpotents. Further, all finite local rings have prime power ...
- 14.5k
20
votes
1
answer
3k
views
On a theorem of Jacobson
In a comment to an answer to a MO question, in which Bill Dubuque mentioned Jacobson's theorem stating that a ring in which $X^n=X$ is an identity is commutative (theorem which has shown up on MO ...
CommunityBot
- 1
4
votes
0
answers
179
views
Global dimensions of orders over non-Gorenstein centers
This question concerns the following Lemma 4.2 in this paper by Van den Bergh:
Lemma: Let $R$ be local, normal Gorenstein ring of dimension $d$. Suppose $M$ is a reflexive $R$ module such that $A=\...
- 30.5k
6
votes
2
answers
2k
views
Cardinality of maximal linearly independent subset
M a finitely generated module over a commutative ring A. I can't think of an example of two maximal linearly independent subsets of M having different cardinality. I know that they all have the same ...
- 2,857
5
votes
2
answers
2k
views
Is there a non-projective flat module over a local ring?
Is there a non-projective flat module over a local ring?
Here I assume the ring is commutative with unit.
- 12.3k
13
votes
0
answers
496
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 ...
CommunityBot
- 1
9
votes
3
answers
1k
views
Vector spaces with natural bases
Sergeib's question asks about vector spaces without a natural basis.
Actually, I would claim (apparently in accord with many comments and answers to Sergeib's question ) that this is the default ...
CommunityBot
- 1
1
vote
0
answers
534
views
Integral element in the quotient of a polynomial ring
Hello,
I'm writting a "report" (to learn) on algebraic geometry, and was looking to write a proof for the following statement :
Theorem : Let $K$ be an algebraically closed field and $C_1, C_2 \...
- 11
11
votes
1
answer
1k
views
The associated prime ideals of $Ext^i_R(M,N)$
If R is a commutative noetherian ring, M and N are modules with M finite. It is well known in commutative algebra that $AssHom_R(M,N)=Supp(M)\cap Ass(N)$. But I want to know whether there is a ...
- 30.5k
2
votes
1
answer
1k
views
For which rings does there exist an invertible Vandermonde matrix?
Suppose $R$ is a commutative ring, and $S \subset R^{n\times n}$ is an $R$-module. We are given $H_0,\dots,H_n \in R^{n\times n}$, and we know that for all $r \in R$,
$$H_0 + r H_1 + \dots r^n H_n \in ...
- 55.9k
5
votes
4
answers
388
views
Familiar equations in more general settings
What equations, or results about equations, generalize in interesting ways from number theory or geometry to more abstract settings? The motivating example for this question was as follows:
...
- 5,227
4
votes
1
answer
914
views
Elements of trace zero in a field extension
Let $K=F_q$ and $F=F_{q^3}$, define the set A={$x \in F$ : $Tr_{F/K} (x)=0$}.
Is it true that for every $x \in A$ there are $y,z \in A$ such that $x=yz$?
- 55.9k
2
votes
0
answers
165
views
Ideals weak equivalence and "finite" equivalence
Let $R$ be an order in a number field. Two $R$-ideals $I$ and $J$ are weak equivalent if there exist (necessarily invertible) ideals $X$ and $Y$ such that $I X=J$ and $J Y=I$.
This is equivalent to ...
0
votes
1
answer
630
views
Useless question on rank
What is the rank of $A^{n}$ if A is the zero ring? It's clearly not $n$ as many careless authors claim, since it's not even invariant. I don't think it's 0 either because it does have a linearly ...
- 33.8k
7
votes
2
answers
1k
views
Is the category of affine schemes (over a fixed field) Cartesian closed?
This is probably a trivial question, but I don't see the answer, and I haven't found it on Wikipedia, nLab, nor MathOverflow.
Let $\text{ComAlg}$ denote the category whose objects are commutative ...
CommunityBot
- 1
1
vote
1
answer
219
views
Name for a module with only one associated prime
In EGA IV2, Def. 3.2.4, Grothendieck defines a quasicoherent sheaf over a locally Noetherian scheme to be "irredondant" if it has a unique associated point. Presumeably, a module over a Noetherian ...
- 20.8k
16
votes
1
answer
2k
views
Does ZF prove that all PIDs are UFDs?
Main Question:
Does ZF (no axiom of choice) prove that every Principal Ideal Domain is a Unique Factorization Domain?
The proofs I've seen all use dependent choice.
Minor Questions:
Does ZF + ...
- 856
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 ...
- 4,736
4
votes
0
answers
1k
views
Associative binary operations on natural numbers
Which are all the associative binary operations on natural numbers ?
Certain results in this regard can be found in arxiv:math/0508215.
It appears that such associative operations cannot grow too fast....
- 118k
5
votes
3
answers
4k
views
How to prove these two rings are not isomorphic
In fact, it is a simple problem. I just want to know whether there are some interesting proof.
$Z[x_1, x_2, ......, x_{n^2-1}]$ and $Z[y_{11}, ......, y_{1n}, y_{21}, ......, y_{nn}]/(det(y_{ij})-1))...
CommunityBot
- 1
13
votes
3
answers
6k
views
Ranks of free submodules of free modules [duplicate]
Possible Duplicate:
Atiyah-MacDonald, exercise 2.11
The following question came up during tea today.
Let $R$ be a commutative ring with an identity and let $M \subset R^n$ be a submodule. ...
CommunityBot
- 1
5
votes
0
answers
437
views
Primary decomposition for non-affine schemes
I will call a (nonzero) ring primary if every zero divisor is nilpotent. (This implies that the prime spectrum is irreducible, although the converse does not hold.) An irreducible scheme I will call ...
- 7,328
6
votes
1
answer
376
views
Checking locally whether a homomorphism is a localization
All rings below are commutative with $1$.
Suppose $A\subset B$ is a subring and that $A\rightarrow A'$ is a faithfully flat ring homomorphism. [You may assume the rings are actually ${\mathbb C}$-...
- 27k
3
votes
2
answers
2k
views
Finitely-generated algebra over Z
Let A be an artin ring which is also a finitely generated algebra over Z.
Show that $|A|<\infty$.
If A would have been a field then I know how to prove it. I know that A is a product of local ...
- 12.6k
9
votes
1
answer
2k
views
Symmetric polynomials theorem
Hello all, I would appreciate comments on the following question:
A main theorem of symmetric functions might be formulated: Let k be a field of char. 0. Then $k[x_1,...,x_n]^{S_n} = k[s_1,...,s_n]$,...
- 1,304
12
votes
0
answers
529
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
$$
v_{m,...
- 181
6
votes
2
answers
610
views
What is the set of possible values of the degree of the sum of two algebraic numbers with fixed degrees?
This question is related to Degree of sum of algebraic numbers and algebraic numbers of degree 3 and 6, whose sum has degree 12.
In this last question I asked a very special case of the following ...
CommunityBot
- 1
1
vote
1
answer
262
views
$\omega$-monoids
Does the notion of $\omega$-monoid exist, analogous to the notions of $\omega$-groupoid and $\omega$-category? If so, some references would be appreciated.
This is an attempted rephrasing of question:...
CommunityBot
- 1
3
votes
3
answers
1k
views
question about tensor of two fields
Let $k$ be a field, $L$, $H$ extension fields of $k$, and $G=L\otimes_k H$. I wonder why (I want to know the proof but I can't find) the prime ideal of $G$ must be maximal, and its properties:
a) if $...
- 14.5k
3
votes
1
answer
3k
views
Isomorphism between direct sum of modules
Let $M$, $N$ be two modules over ring $A$. If $M\oplus M\cong N\oplus N$, can we conclude $M\cong N$? In the case that $M$, $N$ are completely decomposable (e.g. finite-length module by Krull-Schmidt ...
- 1,198
2
votes
0
answers
234
views
Flatness of module
$A\rightarrow B$ a ring homomorphism, $N$ a $B$-module which is flat over $A$. $\mathfrak{q}\subset B$ a prime ideal, $\mathfrak{p}\subset A$ its contraction in $A$. Then is it true that $N_{\mathfrak{...
- 2,857
2
votes
0
answers
333
views
Localization of module
M an A-module, $S\subset A$ a multiplicative subset. Is it possible for $S^{-1}M$ to have an $S^{-1}A$-module structure satisfying $\frac{a}{1}\cdot\frac{m}{1}=\frac{am}{1}$ other than the "usuall" ...
- 63.1k
3
votes
2
answers
335
views
Gaps in Dimension Polynomials
There are several notions of rank/dimension defined on differential fields. However, we do not have a reasonable way to estimate these typically ordinal valued invariants. Especially, we do now know a ...
- 21
1
vote
0
answers
268
views
Rational map defined over K leads to algebra question
Hello,
Concrete algebraic question : Let $K$ be a perfect field, $\bar{K}$ a fixed algebraic closure and let $f \in \bar{K}[x_1,\ldots,x_n]$. I was wondering when there exists another polynomial (non-...
- 11
13
votes
2
answers
5k
views
Primitive element theorem without building field extensions
Is there are nice way to prove the primitive element theorem without using field extensions?
The primitive element theorem says that if $x$ and $y$ are algebraic over $F$ and $y$ is separable over $F$...
- 156k
0
votes
1
answer
2k
views
Dimension of tensor product of modules
$A\rightarrow B$ a ring homomorphism of Noetherian rings, where $A$ is local. $M$, $N$ finitely generated and nonzero $A$- and $B$- modules, respectively. Then I seem to get $\mbox{dim}_ {B}(M\...
