Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,493 questions
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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 ...
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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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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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]/...
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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?
...
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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?
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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], ...
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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 ?
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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 ...
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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 ...
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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 ...
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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
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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 ...
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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 ...
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3
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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 ...
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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 ...
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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"
$$\...
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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
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1
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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 ...
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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
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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
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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
...
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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 ...
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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 ...
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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 ...
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3
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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 ...
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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}...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\...
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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
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1
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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 ...
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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
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When is a localization of a commutative ring finitely generated as an algebra?
Prove or counterexample: If A is a commutative ring and $A_p$ is a finitely generated algebra over A for all prime ideal p of A, then A is a product of local rings.
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Dense section of sheaves of modules
Here is something that isn't yet very clear to me. Say, I've got a commutative ring A. I consider the affine scheme from A, so it's a sheaf of rings over Spec A.
EDIT: And additionally let's say ...
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Intuition for Nagata's altitude formula?
This is theorem 14.C on p.84 of Matsumura's commutative algebra.
Let $A$ be a noetherian domain, and let $B$ be a finitely generated overdomain of $A$. Let $P \in Spec(B)$ and $p = P \cap A$. Then ...
user709
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1
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Spectra of rings that are projective module over a subring
This question is motivated from my last question here. I wonder if one has a ring A and an over-ring of this ring say B, and if we know that B is a projective A-module can we have a particular idea of ...
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3
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Atiyah-MacDonald: exercise 5.29 - "local ring of a valuation ring"
The exercise is the following:
Let $A$ be a valuation ring, $K$ its field of fractions. Show that every subring of $K$ which contains $A$ is a local ring of $A$.
Does anyone know what is meant by "...
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4
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Are quotients of polynomial rings almost UFDs?
If $K$ is a field then the polynomial ring $K[x_1,\ldots, x_n]$ is a UFD. On the other hand, quotients of such a polynomial ring usually don't enjoy unique factorization: consider, for instance, $\...
- 1,412
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3
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Dirichlet series with integer coefficients as a UFD
I recall the following question from Ulam's book "Unsolved math problems": show that the ring of Dirichlet series with integer coefficients is a factorial ring. I believe that soon after Ulam wrote ...
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5
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Algebraic description of compact smooth manifolds?
Given a compact smooth manifold $M$, it's relatively well known that $C^\infty(M)$ determines $M$ up to diffeomorphism. That is, if $M$ and $N$ are two smooth manifolds and there is an $\mathbb{R}$-...
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When are dual modules free?
Let $A$ be a commutative integral domain, with fraction field $K$. Let $T$ be a torsion-free finitely generated $A$ module, so $T \otimes_A K$ is a finite dimensional vector space $V$. Let $T^*$ be ...
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