All Questions
6,055 questions
10
votes
3
answers
3k
views
Sum of radical ideals
Let $A$ be a commutative ring and endow the closed subsets of $\operatorname{Spec}(A)$ with the Grothendieck topology of finite covers. One may ask if the presheaf $V \mapsto A/I(V)$ is a sheaf. This ...
- 63.1k
6
votes
0
answers
379
views
ring-valued points of locally ringed spaces
of course, one should expect that the concept of ring-valued points is not well-behaved for locally ringed spaces (LRS). I want to see examples for this.
so consider $LRS \to Set^{Ring}, X \mapsto X(-...
- 63.1k
84
votes
31
answers
70k
views
Applications of the Chinese remainder theorem
As the title suggests I am interested in CRT applications. Wikipedia article on CRT lists some of the well known applications (e.g. used in the RSA algorithm, used to construct an elegant Gödel ...
0
votes
3
answers
2k
views
Equality of elements in localization via universal property
I've been studying universal objects of universal algebra in a quite general setting and try to exhibit the structure of their elements just using the universal property. A very nice example for this ...
- 63.1k
13
votes
2
answers
967
views
Does torsion-freeness of class group localize?
Let $R$ be a local normal domain, and let $P \in Spec (R)$. It is well known that $Cl(R) \to Cl(R_P)$ is surjective. However, I do not know any example where $Cl(R)$ is torsion-free, but $Cl(R_P)$ is ...
- 30.5k
35
votes
3
answers
5k
views
Matrix factorizations and physics
I have heard during some seminar talks that there are applications of the theory of
matrix factorizations in string theory. A quick search shows mostly papers written by physicists. Are there any ...
- 30.5k
10
votes
3
answers
2k
views
Rings of integers of function fields
This might be a somewhat silly and inconsequential question, but it's aroused my curiosity. One has the theorem in commutative algebra that the integral closure of a domain $A$ in its field of ...
- 2,168
12
votes
5
answers
5k
views
reduced ⊗ reduced = reduced; what about connected?
Several questions actually.
All rings and algebras are supposed to be commutative and with $1$ here.
(1) Let $k$ be a field, and let $A$ and $B$ be two $k$-algebras. I need a proof that if $A$ and $...
- 33.8k
12
votes
2
answers
658
views
Maps between K-groups induced by rings homomorphism
Let $f: R\to S$ be a map between two commutative Noetherian rings. Let $G_0(R)=K_0(mod R)$ be the Grothendieck group of finite generated modules over $R$. It means $G_0(R)$ is the quotient of the free ...
- 30.5k
3
votes
1
answer
463
views
Decomposition of modules using computer packages
I am interested in computing direct sum decomposition of modules over some quotients of polynomial rings over a field (do not care much about the field at this point). Does any one know a package ...
- 30.5k
5
votes
1
answer
272
views
Classifying Algebra Extensions over a fixed extension?
There are lots of "Ext groups" in homological algebra which measure extensions of various things. I'm sure there must be a homological algebra machine for computing the following, and I'm hoping that ...
- 27.5k
15
votes
5
answers
3k
views
Two-dimensional quotient singularities are rational: why?
I've read that quotient singularities (that is, spectra of invariant subrings of finite groups acting linearly on polynomial rings) have rational singularities. Is there an elementary proof of this ...
- 5,846
3
votes
2
answers
1k
views
About maximal Cohen-Macaulay modules
I´m trying to solve a problem of cancellation of reflexive finitely generated modules over normal noetherian domains. When $R$ is regular domain with $\dim R \le 2$, for finitely generated modules, ...
5
votes
5
answers
3k
views
Field structure for R^n
Hi!
Is it possible to define a product on R^n for n>2 such that R^n can be made into a field?
R is a field in its own right with the standard operations and R^2 can be made into a field by ...
- 67
0
votes
2
answers
1k
views
What is the localization of Q[x]/(x) at 0
Q is a rational field. Q[x] is polynomial ring over Q 。(x) is maximal ideal of Q[x].
Take Q[x]/(x) as a module over Q[x]. Then what is Q[x]-module Q[x]/(x) localize at 0??
I think the result is
Q[x]/...
- 25
29
votes
2
answers
5k
views
Examples of algebraic closures of finite index
So there are easy examples for algebraic closures that have index two and infinite index: $\mathbb{C}$ over $\mathbb{R}$ and the algebraic numbers over $\mathbb{Q}$. What about the other indices?
...
- 545
6
votes
1
answer
1k
views
The inverse limit of locally free module
A is an I-adic complete Noetherian ring. M is a finitely generated A module. For any n>0, $M/I^nM$ is a finitely generated locally free A/I^n-module. Is M necessarily a locally free A-module?
- 1,091
-6
votes
2
answers
1k
views
Can I define the polynomial ring A[x] with an isomorphism f: A ---> A[x]? [closed]
I'm sorry if this isn't an appropriate question for MO. I've been reading here for a while, but I still haven't got a good grasp of what's a good question.Given a field A and the polynomial ring A[x], ...
12
votes
4
answers
7k
views
Definition of étale for rings
Let $A \to B$ be a ring extension.
What is the definition of $B/A$ étale ?
When $A$ is a field, do we get a nice characterization ?
- 1,320
46
votes
4
answers
8k
views
What does "linearly disjoint" mean for abstract field extensions?
All definitions I've seen for the statement "$E,F$ are linearly disjoint extensions of $k$" are only meaningful when $E,F$ are given as subfields of a larger field, say $K$. I am happy with the ...
- 11.3k
1
vote
2
answers
604
views
Homomorphism between exterior powers of a free module of finite rank
I´m looking for homomorphisms between exterior powers of a free module M of rank m
ΛmR M → Λm-1R M
Exactly, I´m looking for an explicit isomorphism
M → Hom R (ΛmR M , Λm-1R M)
I compare the ranks ...
32
votes
4
answers
2k
views
Do there exist non-PIDs in which every countably generated ideal is principal?
The title pretty much says it all: suppose $R$ is a commutative integral domain such that every countably generated ideal is principal. Must $R$ be a principal ideal domain?
More generally: for ...
- 65.4k
6
votes
4
answers
409
views
Is tensoring with a module representable iff it is locally free of finite rank?
Motivation:
It's nice when you can think of the elements of an $A$-module $M$ as sections some $A$-scheme $Y\to Spec(A)$. That is, maps $Spec(A)\to Y$ such that $Spec(A)\to Y \to Spec(A)$ is the ...
- 11.3k
2
votes
1
answer
651
views
Splitting matrix of rank one
Let R a normal domain, that is an integrally closed noetherian domain, like Dedekind domains, UFD, etc
Let A=(a i j ) a matrix with elements in R and dimension n x m.
Suppose
rank A=1 ↔ all 2 x ...
10
votes
4
answers
2k
views
Examples of finite local rings of length 2 or 3
What is an example of a finite local rings, that has length 2 or 3?
I want something different from $F_{q}[x] / x^{i}$ for $i=2, 3$; I'm looking for something more interesting. If you can give me ...
- 2,216
6
votes
3
answers
990
views
Factorization of elements vs. of ideals, and is being a UFD equivalent to any property which can be stated entirely without reference to ring elements?
Why exactly is the unique factorization of elements into irreducibles a natural thing to look for? Of course, it's true in $\mathbb{Z}$ and we'd like to see where else it is true; also, regardless of ...
- 6,792
23
votes
1
answer
966
views
Do DG-algebras have any sensible notion of integral closure?
Suppose R → S is a map of commutative differential graded algebras over a field of characteristic zero. Under what conditions can we say that there is a factorization R → R' → S ...
- 52.6k
19
votes
4
answers
4k
views
Trace map attached to a finite homomorphism of noetherian rings
Let $f:A\rightarrow B$ be a homomorphism of noetherian rings which
makes $B$ into a finite $A$-module. Under what conditions on $f$, $A$,
$B$ can one associate to this map a canonical "trace map"
$$\...
- 1,609
5
votes
1
answer
500
views
Are any finitely generated reflexive module a 2nd syzygy?
Are any finitely generated reflexive module a second syzygy?
(I´m thinking especially in normal noetherian domains)
More general...
Are any divisorial lattice a second syzygy?
(I´m thinking ...
3
votes
1
answer
568
views
When can one localize Ext?
Let $R\to S$ be a ring map such that $S$ is projective over $R$ (I am willing to assume $S=R[X_1,...,X_n]$). Let $M,N$ be finite $S$-modules. Let $P\in Spec R$ such that $M_P$ is $R_P$-flat. Under ...
- 30.5k
16
votes
2
answers
899
views
When is a commutative ring the limit of its local rings?
Let $A$ be a commutative ring. Then we get local rings $A_p$ by localizing at each prime ideal $p$. Moreover, we get $A_p \rightarrow A_q$ when $p$ contains $q$. So we get a big diagram indexed by the ...
- 5,548
9
votes
1
answer
857
views
Some examples of depth
This is related to the question I asked last time. This sounds a bit too specific, I hope this question is still acceptable on MO.
I am still not quite comfortable with the concept of depth, and ...
user709
21
votes
1
answer
2k
views
Two conjectures by Gabber on Brauer and Picard groups
In a paper I need to make reference to two conjectures by Gabber, from
Ofer Gabber, On purity for the Brauer group, in: Arithmetic Algebraic Geometry, MFO Report No. 37/2004, doi:10.14760/OWR-2004-37
...
- 30.5k
2
votes
2
answers
2k
views
Inversion of Laurent series
For a power series $f(z) = \sum_{i=0}^{\infty} a_i z^i$ with $a_1$ nonzero, Lagrange's inversion formula gives an explicit way to compute the Taylor coefficients of the inverse function.
Is there any ...
- 21k
10
votes
2
answers
1k
views
When is every submodule pure?
Recall that a module is called
semisimple if every submodule is a direct summand
pure semisimple if every pure submodule is a direct summand
There is quite a bit of work on semisimple and pure ...
- 3,685
9
votes
6
answers
4k
views
Differences between reflexives and projectives modules
Let R be a normal noetherian domain.
What is the difference between a finitely generated reflexive module and a finitely generated projective module?
Can anybody recommend any references or make ...
5
votes
3
answers
839
views
Is (relatively) algebraically closed stable under finite field extensions?
Let $F\subset F'$ be a field extension such that $F$ is algebraically closed inside $F'$, i.e. if $x\in F'$ is algebraic over $F$ then $x$ belongs to $F$ itself.
Let now $F\subset L$ be a finite field ...
- 51
15
votes
5
answers
4k
views
Generalizing miracle flatness (Matsumura 23.1) via finite Tor-dimension
Let $(A,m_A)$ and $(B,m_B)$ be noetherian local rings and $f:A\rightarrow B$ a local homomorphism. Let $F = B/m_AB$ be the fiber ring and assume that
$$\mathrm{dim}(B) = \mathrm{dim}(A) + \mathrm{dim}...
- 1,609
18
votes
2
answers
2k
views
What does primary decomposition of (sub) modules mean geometrically?
I want to know how I should visualize modules in algebraic geometry. The way we visualize rings, via their spectra, automatically (or by the beauty of its design) depicts primary decomposition of ...
- 11.3k
44
votes
4
answers
12k
views
Classification of finite commutative rings
Is there a classification of finite commutative rings available?
If not, what are the best structure theorem that are known at present?
All I know is a result that every finite commutative ring is a ...
- 2,216
3
votes
3
answers
447
views
Representations of finite commutative band semigroups
I think it's clear that commutative semigroups S that are also bands, i.e. $e^2 = e$ for all e, correspond to finite posets (consider the elements of the semigroups as sets, where the intersection of ...
- 2,216
11
votes
2
answers
1k
views
Class groups of normal domains over finite fields
Let R be a local, normal domain of dimension 2. Suppose that R contains a finite field. I am interested in knowing when the class group of R is torsion. In characteristic 0, this is known to be ...
- 30.5k
25
votes
7
answers
3k
views
When can we prove constructively that a ring with unity has a maximal ideal?
Many commutative algebra textbooks establish that every ideal of a ring is contained in a maximal ideal by appealing to Zorn's lemma, which I dislike on grounds of non-constructivity. For Noetherian ...
- 118k
75
votes
9
answers
17k
views
Why is an elliptic curve a group?
Consider an elliptic curve $y^2=x^3+ax+b$. It is well known that we can (in the generic case) create an addition on this curve turning it into an abelian group: The group law is characterized by the ...
- 9,366
95
votes
11
answers
6k
views
Can a non-surjective polynomial map from an infinite field to itself miss only finitely many points?
Is there an infinite field $k$ together with a polynomial $f \in k[x]$ such that the associated map $f \colon k \to k$ is not surjective but misses only finitely many elements in $k$ (i.e. only ...
- 2,600
10
votes
2
answers
610
views
When is tensoring with a module representable by a scheme?
Consider the following: Let $A$ be a commutative ring, let $M$ be an $A$-module. When is the functor from $A$-algebras to Sets given by $R \mapsto R \otimes M$ representable by an $A$-scheme?
Unless ...
- 5,548
23
votes
6
answers
4k
views
Is projectiveness a Zariski-local property of modules? (Answered: Yes!)
I know that for a finitely presented $A$-module $M$ ($A$ a commutative ring), TFAE:
$M$ is projective;
$M$ is max-locally free, meaning that $M_{\mathfrak m}$ is free for every maximal ideal $\...
- 11.3k
45
votes
5
answers
4k
views
How to think about CM rings?
There are a few questions about CM rings and depth.
Why would one consider the concept of depth? Is there any geometric meaning associated to that? The consideration of regular sequence is okay to me....
user709
8
votes
1
answer
3k
views
Completion of modules of differentials (A strange exercise in Liu's AG textbook)
A is a Noetherian ring, B is an f.g. algebra over A, I is an ideal of A. let $\hat B$ be B's I-adic completion. Prove that $\Omega^1_{\hat B/A}$'s I-adic completion is isomorphic to $\Omega^1_{B/A}$'s ...
- 1,091
11
votes
3
answers
612
views
Can different modules have the same symmetric algebra? (answered: no)
Algebraic geometry allows one to think of an $A$-module $M$ geometrically as a module of functions on the $A$-scheme $\mathrm{Spec}(\mathrm{Sym}(M))$.
I'm wondering if anything is lost in just ...
- 11.3k