Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,497 questions
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Pulling back roots from the Completion
Consider the following diagram of regular local rings
$\begin{matrix}
\hat{A} & \xrightarrow{\quad\hat\varphi\quad} & \hat{B} \\
\ \uparrow\scriptstyle\alpha & \circlearrowleft & \ \...
- 4,902
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0
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79
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Stable analytic manifold under simple action
For an integer $m > 1$, let us define the action
$$
f: X_i \to (1+X_i)^{m} - 1
$$
on $C[[X_1,...,X_N]]$, where $C$ is the complex number field. Consider the analytic manifold $V(I)$ defined by the ...
- 1
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2
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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 ...
- 11
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1
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Example of restriction of a finite morphism which is not finite
Every closed immersion is a finite morphism. Therefore, restriction of a finite morphism to a closed subset is always a finite morphism itself. Can you give an example of quasi-projective varieties $X\...
- 422
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Question about modules, quotient rings, and polynomial rings? [closed]
Consider an integer polynomial ring, $A = \mathbb{Z}[t]$, and a ring of fractions, $B = \mathbb{Z}[t, t^{-1}]$; obviously, $A$ is a subring of $B$.
Now we consider two modules over $A$ and $B$, $M$ ...
- 161
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1
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What is correct name of the following construction?
Consider an ideal $I=\langle f_1,f_2,\ldots,f_s\rangle$ in the polynomial ring $\mathbb{Q}[x_1,x_2,\ldots,x_n].$ Build the following set
$$
\{ g_1 f_1+g_1 f_2+\cdots+g_n f_n \},
$$
where $g_i$ ...
- 301
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Decomposition group vs Galois group of completed extension for height > 1 primes
Assume
Let $R$ be a Noetherian normal excellent domain, $F$ its field of fractions.
Let $S$ be a finite $R$-algebra, $L$ its field of fractions.
$L/F$ a (finite) Galois extension
$S$ normal in $L$
...
- 21
1
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1
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Ext modules of coherent sheaves and associated modules
Let $\mathcal{E}$ and $\mathcal{F}$ be two coherent sheaves on a polarized projective variety $(X,\mathcal{O}_X(1))$.
Denote by $E=\Gamma_*(\mathcal{E})=\oplus_{k\in\mathbb{Z}}\mathcal{E}(k)$,
$F=\...
- 2,444
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votes
1
answer
399
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Quotient field extension for an incomplete DVR
Let $R$ be a DVR with maximal ideal $xR$, and assume that $R$ is not complete in the $xR$-adic topology. Let $\hat{R}$ be the completion of $R$ in the $xR$-adic topology. Set $K=Q(R)$, the fraction ...
2
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1
answer
521
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Kahler differentials of a hypersurface over a non-algebraically closed field
The following was recently on my algebraic geometry homework:
Let $k$ be an algebraically closed field, $f\in B=k[x_1,\ldots,x_n]$, and $A=B/(f)$. Show that $\Omega_{A/k}$ is locally free of rank $...
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Question about witt vector of some ring
Suppose $R=Z_p[t]$ , and $\hat{R}$ its p-adic completion, suppose we have Endormorphism $\Phi$ of $\hat{R}$, whose redution mop p is just the absolute Frobenius of $\hat{R}/p\hat{R}$. And $R_{\infty}=...
- 709
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Commutative ring Notes by M. Artin
In 1966, Professor Michael Artin gave a course for first-year graduate students at MIT on commutative algebra. In that course he covered many classical topics, (the Spectrum of a commutative ring, ...
- 165
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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?
- 2,857
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CM module is height-unmixed?
$A$ a Cohen-Macaulay ring (not necessarily local), $M$ a Cohen-Macaulay $A$-module. Then does it necessarily follow that $\mbox{ann}(M)$ is height-unmixed?
- 2,857
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Lattices as invertible modules.
I have asked this question in Math Stack exchange but got no answer. Maybe it fits Mathoverflow better.
All rings below are assumed to be Noetherian.
Let $E$ be an etale algebra over $\mathbb{Q}$....
- 570
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vote
1
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tensor of powers of an ideal
Let $I$ be an ideal of a Noetherian ring $R$. $M$ is a finitely generated $R$-module. I question is:
Does there exist $n_0$ such that for all $n \geq n_0$, the short exact sequence
$$I^n/I^{n+1} \...
- 2,773
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0
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315
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Non trivial definition of bicontinuous functions and the ring of all bicontinuous functions.
At first let me recall that if There are two topology $\tau_1$and $\tau_2$ on a set $X$, the triple $(X,\tau_1,\tau_2)$ is called a bitopological space.
There are many definitions and properties ...
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211
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Open subset in the flat topology on Spec(R)
Let $R$ be a commutative ring and let $S \subset \mathrm{Spec}(R)$ be a subset. Suppose that for each $P \in S$ there exists a Zariski open neighborhood $U$ of $P_P \in \mathrm{Spec}(R_P)$ such that $^...
- 313
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Nontrivial examples of rings of relative stable rank 1
Given a (commutative) ring $R$, one says that the stable rank $sr(R)\leq n$ if any unimodular row $(a_1,\ldots,a_{n+1})$ of length $n+1$ is stable, i.e. there exist $x_1,\ldots,x_n$ such that $(a_1+...
- 3,186
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When is Out$(SL_n(R))$ a torsion group ?
This question is a follow up question to this question. So my question is:
For which rings $R$ (commutative, with unit) (and which integers $n$) is $Out(SL_n(R))$ a torsion group? A consequence of ...
- 11.1k
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a question about flatness
In the book "étale cohomology" by Milne, proposition 2.5 at p.9, it said :
Let $B$ be a flat $A-$algebra where $A$ and $B$ are noetherian rings, and consider $b \in B$. If the image of $b$ in $B/mB$ ...
- 31
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views
On Artin-Hironaka lemma and Galois theory
Let $A=k[[t]]$ Let $B$ a flat $A$-finite algebra which is etale and Galois at the generic point.
Then by Artin lemma 3.12 (ii) in his IHES paper on approximation, we know that there exists an integer ...
- 3,472
1
vote
1
answer
403
views
Prime ideals in univariate polynomial rings
I'm looking for a textbook reference of the following elementary fact (a reference for an excercise in a textbook is also welcome):
Let $R$ be a commutative ring and let $\mathfrak{p}$ be a prime ...
- 16.2k
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vote
0
answers
342
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Passing from Regular sequence to Prime ideal, for power sum symmetric polynomial
Let $S=\mathbb{C}[x_1,x_2,x_3,x_4]$ be a polynomial ring. Let $p_i=x_1^i+\cdots+x_4^i$ be the power sum symmetric polynomial in $\mathbb{C}[x_1,x_2,x_3,x_4]$.
Let $I=(p_1,p_2)$ be an Ideal of $\mathbb{...
- 446
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votes
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Torsion in some specific module over the Laurent polynomials
Consider the ring $\mathbb{Z}[t,t^{-1}]$ of Laurent-polynomials over $\mathbb{Z}$. The abelian group $M:=\prod_\mathbb{Z}\mathbb{Z}$ becomes a module over this ring via
$t\cdot (x_*):=x_{*+1}$.
Is ...
- 11.1k
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1
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Invertible matrices satisfying $[x,y,y]=x$ (take 2).
This is a simpler version of this question. Let $x=\left(\begin{array}{lll} 2 & 0 & 0\\\
0& 1 & 0\\\
0 & 0 & \frac12\end{array}\right)$. Is there a $3\times 3$-matrix $y$ with ...
user6976
4
votes
0
answers
213
views
height of contracted prime ideals in power series rings
Let $K$ be a field of characteristic zero and let $R_n = K[[X_1,\ldots,X_n]]$ be the power series ring in $n$-variables. Let $P$ be a prime ideal in $R_n$. Let $Q = P \cap R_{n-1}$.
Then is $height(Q) ...
2
votes
2
answers
395
views
When do primes lift uniquely (provided they lift at all)?
Given a ring $R$, a prime ideal $\mathfrak{p}$ of $R$, and an extension ring $S$ (the algebra map $R\to S$ is injective), are there any nontrivial sufficient conditions for the induced map $Spec(S) \...
5
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Injective dimension of $\mathcal{O}_X$-modules
Let $(X, \mathcal{O}_X)$ be a regular noetherian scheme of finite Krull dimension (over a field $k$ if needed).
Is it true that any $\mathcal{O}_X$-module (not necessarily quasi-coherent) has a ...
5
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0
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views
Looking for a reference for a generalization of the Weierstrass preparation theorem
I am looking for a reference for the following generalization of the Weierstrass preparation theorem for formal power series. Suppose that $A$ is a noetherian complete local ring with residue field $k$...
- 27k
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Name for generalization of bivariate weighted-homogeneous polynomials
A polynomial $f = \sum_j c_j X^{\alpha_j}Y^{\beta_j}\in\mathbb K[X,Y]$ is said weighted-homogeneous if there exist $p$, $q$ and $d$ (where $p$ and $q$ are not both $0$) such that $p\alpha_j+q\beta_j=d$...
- 456
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views
What are the enforceable models of local artinian rings?
I was reading Hodges' "Model Theory" Chapter 8 a propos existentially closed models of $\forall_2$ theories in a countable first order language $L$. He extends the proof of the omitting type theorem ...
2
votes
1
answer
447
views
Commutator tensors and submodules
Let $k$ be a commutative ring with $1$, and let $B$ be a submodule of a $k$-module $A$.
For every $n\in\mathbb N$ and every $k$-module $V$, let $K^n\left(V\right)$ denote the kernel of the canonical ...
- 33.8k
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votes
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views
When are two ideals in a regular local ring generated by a regular sequence?
Hello!
Let $R$ be a regular local ring, and let $I,J\subset R$ be ideals. I'd like to understand the "meaning" of the existence of a regular sequence $(x_1,...,x_n)$ in $R$ such that $I$ is generated ...
- 2,756
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0
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Centralizer in a matrix algebra over commutative polynomials
Let $A=M_n(F[x_{ij}\mid 1\leq i,j\leq n])$ be the matrix algebra over commutative
polynomials in $n^2$ variables $x_{11},\dots,x_{nn}$ over a (nice enough) field $F$.
I would like to know what is the ...
- 179
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monomorphisms and epimorphisms of local rings
I want to understand the structure of monomorphisms/epimorphisms in the category of local rings (with local homomorphisms), or dually in the category of local schemes. Let $LR$ denote this category.
...
- 63.1k
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Multiplicative groups in field extensions
If we let $K$ be a number field, thank to the fact that we can extend it integer ring to an UFD where the group of units is finitely generated, we can show that
$K^\ast\cong K^\ast_{tor}\times \...
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0
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views
Generalized elementary symmetric functions
The question below came into my mind when I was thinking about this one: A nice generating set for the symmetric power of an algebra.
Let $A$ be a commutative, associative unital algebra over a ...
- 567
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1
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Are there known quantitative descriptions of the fact that the common zero set of some polynomials is empty besides Nullstellensatz?
Working in a polynomial ring, if some polynomials has no common zeros, Nullstellensatz tells us a qualitative description of the propertity of these polynomials, are there known quantitative ...
- 1,528
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cohomological dimension, dimension of modules and arithmetic rank
Let $R$ be a noetherian ring and $I$ be an ideal of $R$ and let $M$ be a finitely generated $R$- module.
I know two fact. first, dimension of $M$(i.e. krull dimension of $R/{\rm ann}(M)$) is greater ...
3
votes
1
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A particular Isomorphism of graded algebras over a regular local ring
In Hartshorne's "Algebraic Geometry", the following statement is a weaker form of Theorem 8.21A (e), which he quotes from Matsumuura's book on commutative algebra:
Proposition. Let $R$ be a regular ...
- 4,902
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Projective dimension of cohomology over regular rings
Suppose $R$ is a regular ring and $F^{\bullet}: 0\to F^0 \to F^1 \to \dots \to F^d \to 0$ is a complex of finite rank free $R$-modules. Is is true that $\mathrm{projdim}H^i(F^{\bullet}) \leq d$ for ...
- 13.1k
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votes
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Hilbert series and resolution of a surface singularity
I have a question about the following theorem in Stanley's paper "Invariants of Finite Groups and Their Applications to Combinatorics".
Suppose that the Cohen-Macaulay $N$-graded $k$-algebra $B$ is ...
- 105
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answers
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Upper semicontinuity of Betti numbers of submodules
Theorem 8.29 in "Combinatorial commutative algebra" by Miller and Sturmfels states the upper-semicontinuity property for Groebner deformations (say, over an algebraically closed field with ...
- 902
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1
answer
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Special case of testing integer polynomials for irreducibility
How much easier is testing polynomials of the form x^n + ax + b for irreducibility (in Z[x]) than testing polynomials in general? I am especially interested in the case where n is prime, which may be ...
- 11
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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
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0
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Testing isomorphism of finitely generated algebras
Let $A=\mathbf{Q}[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over the rational numbers. Let $B=\mathbf{Q}[f_1,\ldots,f_r]$ and
$C=\mathbf{Q}[g_1,\ldots,g_s]$ be two finitely generated $\...
- 7,609
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1
answer
356
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Observable Unipotent Algebraic Subgroups of the Unipotent Upper Triangular Groups
It is well known that any unipotent algebraic group (over a field) can be embedded as a closed subgroup of $U_n$ for some $n$, where $U_n$ denotes the set of all $n \times n$ upper triangular matrices ...
- 511
1
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0
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90
views
Irreducibility of a certain matrix variety
Let $R=\mathbb{Q}[x_{i,j}\,:\, 1\leq i,j\leq n]$. Let $M$ be the $n\times n$ matrix $(x_{i.j})$. Let $\chi(M)$ be the characteristic polynomial of $M$. Finally, let $I$ be the ideal of $R$ ...
- 18.7k
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votes
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Subgroups-ideals correspondence for abelian varieties over $\mathbf{F}_p$
The question I have arose while reading Waterhouse's Thesis (Abelian varieties over finite fields. Ann. Sci. École Norm. Sup. (4) 2 1969 521–560.), and motivates another question I recently asked.
...
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