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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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:
...
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Neusis constructions
Is there some simple description of which complex numbers are "constructible" with straightedge and compass and neusis?
See http://en.wikipedia.org/wiki/Constructible_number and http://en.wikipedia....
user5810
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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 ...
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Modern algebraic geometry vs. classical algebraic geometry
Can anyone offer advice on roughly how much commutative algebra, homological algebra etc. one needs to know to do research in (or to learn) modern algebraic geometry. Would you need to be familiar ...
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An advanced exposition of Galois theory
My knowledge of Galois theory is woefully inadequate. Thus, I'd be interested in an exposition that assumes little knowledge of Galois theory, but is advanced in other respects. For instance, it would ...
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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 ...
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How would you solve this tantalizing Halmos problem?
$1-ab$ invertible $\implies$ $1-ba$ invertible has a slick power series "proof" as below, where Halmos asks for an explanation of why this tantalizing derivation succeeds. Do you know one?
Geometric ...
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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 ...
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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 + ...
user5810
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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 ...
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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 ...
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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 ...
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Does Smith normal form imply PID?
Let $R$ be a nonzero commutative ring with $1$, such that all finite matrices over $R$ have a Smith normal form. Does it follow that $R$ is a principal ideal domain?
If this fails, suppose we ...
user5810
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What are the units in the ring of Laurent polynomials?
What are the units in $R[X,X^{-1}]$, where $R$ is a commutative ring with $1$? I know that the question for polynomial rings is a standard textbook exercise. However, I couldn't find a reference for ...
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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))...
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Sums of two squares in (certain) integral domains
While giving the first of eight lectures on introductory model theory and its applications yesterday, I stated Hilbert's 17th problem (or rather, Artin's Theorem): if $f \in \mathbb{R}[t_1,\ldots,t_n]$...
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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. ...
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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 ...
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A direct proof that minimal primes are associated
It is a well-known theorem that, for a Noetherian ring $A$, the minimal primes of $A$ are among the associated primes of $A$; i.e., for every minimal prime $\mathfrak{p}$ of $A$, there is an element $...
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Krull Dimension
For all $n$, I need to find examples of rings $A\subset B$ such that:
i) $\dim A-\dim B\gt n$
ii) $\dim B-\dim A\gt n$
(where $\dim$ is the Krull dimension)
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A remark on Cohen's theorem
It is well known as Cohen's theorem that a commutative ring is Noetherian if all its
prime ideals are finitely generated. Is this statement true or false when prime ideals are replaced by maximal ...
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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 ${\mathbb C}$-...
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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 ...
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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 $k[x_1,...,x_n]^{S_n} = k[s_1,...,s_n]$,...
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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,...
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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 ...
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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 ...
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Rank of a module
What's wrong with defining the rank of a finitely generated module over any (commutative) ring to be just the smallest number of generators? All books I know define rank only locally this way. But why ...
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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{...
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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 $...
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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-...
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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" ...
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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 ...
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Pushouts of noetherian rings
Does the category of noetherian commutative rings have pushouts?
Background: If $X/S$ is an abelian scheme, then the relative Picard functor $\mathrm{Pic}_{X/S}$ is only defined on the category of ...
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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 $F$...
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A condition that implies commutativity
Let $R$ be a ring. A notable theorem of N. Jacobson states that if the identity $x^{n}=x$ holds for every $x \in R$ and a fixed $n \geq 2$ then $R$ is a commutative ring.
The proof of the result for ...
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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\...
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Why is the fibered coproduct of affine schemes not affine?
I am confused about the following issue:
Let $X=SpecS$, $U_1=SpecR_1$, $U_2=SpecR_2$. and suppose we have maps $S \rightarrow R_1$, $S \rightarrow R_2$. Let $U_3=Spec (R_1 \otimes_S R_2)$. We have ...
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Formal power series ring & completion
I encountered the following passage in Matsumura's Commutative Ring Theory :
A a Noetherian ring, $B=A[[x]]$ a formal power series ring. $M\subset B$ a maximal ideal, $\mathfrak{m}=M\cap A$. Then $(...
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0 dimensional Dedekind domain?
It seems that the ratio of those authors allowing a field to be a Dedekind domain to those who do not is almost 50 - 50. Why such a bewildering lack of consensus for such an elementary notion?
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Jordan Form Over a Polynomial Ring
Let $X$ be the set of $k\times k$ matrix with entries in $\mathbb{C}$, and let $M\in X$. The group $GL(k,\mathbb{C})$ acts on $X$ by conjugation, and according to the Jordan decomposition theorem (...
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Extra principal Cartier divisors on non-Noetherian rings? (answered: no!)
On the way to defining Cartier divisors on a scheme $X$, one sheafifies a presheaf base-presheaf of rings $\mathcal{K}'(U)=Frac(\mathcal{O}(U))$ on open affines $U$ to get a sheaf $\mathcal{K}$ of "...
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Subrings of rational functions invariant under change of sign
Let $R$ be a ring generated by $k$ rational functions in the
variables $x_1,...,x_n$ over the real numbers.
Is there an algorithm that computes a set of rational functions
$f_1,...,f_l \in R$ which ...
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Does the free resolution of the cokernel of a generic matrix remain exact on a Zariski open set?
"Random" modules of the same size over a polynomial ring seem to always have the same Betti table. By a "random" module I mean the cokernel of a matrix whose entries are random forms of a fixed degree....
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Hilbert Syzygy Theorem - Induction step
Does someone know in which books, lecture notes or ... I can find the induction step of the proof of Hilbert Syzygy Theorem? I'd only found the proof for R[x] (e.g. Weibel) and I haven't really an ...
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Seeking examples or proof: injectivity of Cartan homomorphism for commutative rings?
This question is motivated by some issue raised by David Speyer in this question.
Let $R$ be a ring. Let $K_0(R)$ and $G_0(R)$ be the Grothendieck groups of f.g. projective modules and f.g. modules ...
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$K_{0}(R) =\mathbb{Z}$ but some f.g. projective not stably free?
This question is motived by this recent question.
$K_{0}(R)=\mathbb{Z}$ is often used as a euphemism for saying that every finitely generated projective module is stably free; however, there are some ...
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Number of A Subset of Monomials
I need to count the number of monomials of degree $n$ in $k$ variables, $x_1,\ldots ,x_k$, that contain at least one variable with a power of 1. The monomials need not include all the variables. ...
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A ring such that all projectives are stably free but not all projectives are free?
This question is motivated by this recent question. Suppose $R$ is commutative, Noetherian ring and $M$ a finitely generated $R$-module. Let $FD(M)$ and $PD(M)$ be the shortest length of free and ...
- 30.5k
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Free resolution dimension?
Is there a notion of a dimension associated to free resolutions like projective and injective dimensions associated to projective and injective resolutions? My guess is that it coincides with ...
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