Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,493 questions
12
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Is completeness of a field an algebraic property?
Pretty straitforward:
If a field has a metric in which it is complete can it have a metric in which it is not complete?
By metric I mean field norm of course
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1
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371
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How to construct a ring with global dimension m and weak dimension n?
Given two integers $m,n$ such that $n < m$, it is easy to construct a ring with global dimension $m$ or weak dimension $n$. But I wonder whether there exists a ring satisfying both the conditions?
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3
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2
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How to compute the ring of invariants of SO_3(k) acting on a polynomial ring
Let $k$ be a field and let $A$ be the polynomial ring over $k$ in $3n$ variables: $A = k[X_{ij} \vert i=1,2,3 \quad j=1,2,\cdots,n]$.
${\rm SO}_3(k)$ acts on $A$ in the following way: Given $g \in {\...
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3
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Is being torsion a local property of module elements?
Say $R$ is a ring, not necessarily a domain, and $M$ is an $R$-module. All rings are commutative with 1. An element $m\in M$ is called torsion if $r.m=0$ for some regular element (non-zerodivisor) $r\...
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0
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0
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165
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Support sets along a ring homomorphism.
Let $(R,m)$ and $(S,n)$ be commutative noetherian local rings, and $f: R\rightarrow S$ be a local homomorphism (i.e., $f(m) \subseteq n$) with $S$ flat as $R$-module. If $M$ is a finite generated $R$-...
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3
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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 ...
2
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2
answers
866
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Modules over a Gorenstein ring
$A$ a Gorenstein ring, $M\neq 0$ a finite $A$-module with finite injective dimension. According to Bruns, this implies that $M$ has finite projective dimension. How do I see that?
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3
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3
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670
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Algebraic, analytic, formal modules
Consider torsion free modules over the germ of a fixed isolated algebraic hypersurface singularity {$f=0$}$\subset\mathbb{C}^n$.
There are natural functors (using categories of finitely generated ...
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0
votes
1
answer
502
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Finiteness of injective hull of residue field for Artin local ring
$(A,\mathfrak{m})$ an Artin local ring, $E(A/\mathfrak{m})$ the injective hull of $A/\mathfrak{m}$. How do I see that $E(A/\mathfrak{m})$ is a finite $A$-module?
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2
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671
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Do Gorenstein rings necessarily have a finite projective dimension (as a module over itself)? [closed]
Do Gorenstein rings necessarily have finite projective dimensions?
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Projective & injective dimensions
$A$ a Noetherian local ring, $M\neq 0$ a finite $A$-module. I'm not quite sure about the relation between finiteness of projective and injective dimensions of $M$. Does the finiteness (or infiniteness)...
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3
votes
2
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Depth and dimension
$A$ a Noetherian local ring, $M\neq 0$ a finite $A$-module. Then is it true that $\mbox{depth }M\le\mbox{depth }A$ just like $\mbox{dim }M\le\mbox{dim }A$? I don't see any relation between an $M$-...
- 2,857
5
votes
2
answers
731
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What is the completion at a family of ideals?
Let $A$ be a (commutative with unit) noetherian ring. If $I$ is an ideal of $A$, the $I$-adic completion of $A$ is by definition
$$
\widehat{A} := \underset{\leftarrow}\lim A/I^n.
$$
This operation is ...
- 3,704
13
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4
answers
4k
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Size of a Groebner basis
I've spent some time recently looking at some Groebner bases for some specific ideals coming from problems in computer vision. The generators are not sparse, and they all have the same degree (...
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An elementary question about the Krull dimension of modules [closed]
Let $R$ be a commutative ring. If $0\rightarrow M'\rightarrow M\rightarrow M''\rightarrow 0$ is an exact sequence of modules, we have that $\operatorname{Supp}M=\operatorname{Supp}M'\cup \operatorname{...
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2
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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 $...
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14
votes
4
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When is an algebra of commuting matrices (contained in one) generated by a single matrix?
Let C be an nxn matrix, then the polynomials in C (with appropriate coefficients) form an algebra of commuting matrices. I feel that I should know if the converse is true but I do not. So my first ...
- 30.1k
2
votes
1
answer
980
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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 ...
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2
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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 ...
- 63.1k
5
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2
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689
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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 $I$ ...
- 3,093
43
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5
answers
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Explicit elements of $K((x))((y)) \setminus K((x,y))$
In an answer to the popular question on common false beliefs in mathematics
Examples of common false beliefs in mathematics
I mentioned that many people conflate the two different kinds of formal ...
- 65.4k
9
votes
1
answer
792
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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, ...
3
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2
answers
2k
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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 ...
- 2,857
2
votes
2
answers
420
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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. ...
- 2,857
7
votes
2
answers
2k
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Ideals in the ring of single-variable Laurent polynomials with integer coefficients
I'm writing some software to automatically compute things like Alexander modules, Alexander ideals, Milnor signatures and Farber-Levine pairings for 1 and 2-knot complements. So among other things I'...
- 44.4k
14
votes
3
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2k
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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
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2
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2k
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Existence of a minimal generating set of a module
Does a module (over a commutative ring) always possess a minimal generating set? When the module is not finitely generated, the typical Zorn's lemma type argument doesn't seem to work. More precisely, ...
- 2,857
36
votes
4
answers
12k
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Flatness and local freeness
The following statement is well-known:
Let $A$ be a commutative Noetherian ring and $M$ a finitely generated $A$-module. Then $M$ is flat if and only if $M_{\mathfrak{p}}$ is a free $A_{\mathfrak{p}}$-...
- 2,857
4
votes
2
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2k
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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/\...
- 2,857
12
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2
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Is tensor product of local algebras local?
In general, the tensor product of two local rings is not local. For example, $\mathbb{C} \otimes_{\mathbb{R}} \mathbb{C}\ $ is not a local ring.
Let $\mathbb{F}_{p}$ denote the finite field with $p$ ...
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13
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3
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3k
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Zero divisor conjecture and idempotent conjecture
Let $G$ be a torsion-free group and $C$ the ring of complex numbers. The zero divisor (idempotent, resp.) conjecture is that there is no nontrivial zerodivisor (idempotent, resp.) in $CG$.
The wiki ...
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16
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4
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3k
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Cardinal of maximal linearly independent subsets of a free module
Is it true that the cardinality of every maximal linearly independent subset of a finitely generated free module $A^{n}$ is equal to $n$ (not just at most $n$, but in fact $n$)? Here $A$ is a nonzero ...
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2
votes
0
answers
797
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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?
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6
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3
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3k
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Tor and projective dimension
Is it possible that $\mbox{Tor }^{r+1}(M,N)=0 \ \ \forall N$ yet $\mbox{proj. dim }M>r$?
What I do know is that if $(A,\mathfrak{m})$ is Noetherian local and $M$ is finitely generated over $A$ ...
- 2,857
1
vote
4
answers
2k
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Reduced rings, idempotents and their prime spectrum
Let $B$ be a commutative unitary reduced ring and let $A$ be a subring of it. Let $e$ be an idempotent of $B$. Then we have a natural surjective ring homomorphism $A\rightarrow Ae$ defined by $a\...
- 2,275
4
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0
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179
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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=\...
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15
votes
1
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Countable Hom/Ext implies finitely generated
Today I learned this interesting fact from Jerry Kaminker: If $A$ is an abelian group such that $\mathrm{Hom}(A,\mathbb{Z})$ and $\mathrm{Ext}(A,\mathbb{Z})$ are both countably generated, then in fact ...
- 56.6k
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2
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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.
- 2,857
14
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0
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899
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Frobenius upper shriek/flat of a dualizing complex
Let $X$ be a separated connected scheme of characteristic $p > 0$. I am going to assume that $F : X \to X$ (the absolute Frobenius) is a finite map. This condition is called being $F$-finite.
...
48
votes
4
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4k
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Are there more Nullstellensätze?
Over which fields $k$ is there a reasonable analogue of Hilbert's Nullstellensatz?
Here is a more precise formulation: let $k$ be an arbitrary field, $n$ a positive integer, and $R = k[t_1,..,t_n]$. ...
- 65.4k
9
votes
3
answers
1k
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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 ...
4
votes
2
answers
883
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Group & modules of arbitrary cardinality [closed]
How do I see that there is a group of an arbitrary cardinality? Is this also true for abelian groups? Also, given a commutative ring $R\neq 0$ how do I see that there is an $R$-module of arbitrary ...
- 2,857
13
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0
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496
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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 ...
- 30.5k
2
votes
1
answer
918
views
Is a formally smooth morphism a filtered inductive limit of smooth algebras?
Given a unital commutative ring $A$ (not necessarily noetherian) and a formally smooth morphism of rings $f:A \to B$, where $B$ is not necessarily noetherian, is (or when is) $B$ a filtered inductive ...
2
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1
answer
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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 ...
- 427
1
vote
0
answers
534
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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
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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 ...
- 1,207
28
votes
6
answers
5k
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Expressing $-\operatorname{adj}(A)$ as a polynomial in $A$?
Suppose $A\in R^{n\times n}$, where $R$ is a commutative ring. Let $p_i \in R$ be the coefficients of the characteristic polynomial of $A$: $\operatorname{det}(A-xI) = p_0 + p_1x + \dots + p_n x^n$.
I ...
- 427
20
votes
1
answer
3k
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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 ...
- 47.3k
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 $x=yz$?
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