Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,497 questions
2
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Hilbert Regularity in relation to degree of generators
Suppose we look at the $\mathbb {C}$ module $\mathbb{C}[x_1,\dots,x_n]=\oplus{S_i}$ where $S_i$ are the polynomials of degree $i$. Then we look at a subring $R$ (also a $\mathbb {C} $ module) ...
- 928
3
votes
1
answer
277
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Base-change of schemes over number rings
Let $S$ be a finite set of maximal ideals in $ O_K$, where $O_K$ is the ring of integers of some number field $K$. Define $A= O_K[S^{-1}]$.
Let $X$ be an arbitrary $A$-scheme. Consider the scheme $...
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structure theorem for modules
Can structure theorem for modules be extended to modules over UFDS , to modules over Neotherian rings ? if yes then can one get the statement and reference?
Since operations on matrices with ...
- 201
2
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1
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434
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Explicit basis for the space of global sections of a twisted arithmetic ideal sheaf
Assume $x\in X=\mathbb{P}^1_{\mathbb{Z}}$ is a closed point with $f(x)=p\in Y$ where $f:X\rightarrow Y$, here $Y=Spec(\mathbb{Z})$. Assume $k(x)=\mathbb{F}_p$ and denote by $I_x$ the ideal sheaf of $x$...
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2
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What is the transcendence degree of Q_p and C over Q?
Is the tr.deg of Q_p over Q 1? and what about C over Q?
- 1,503
3
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1
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401
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Compatibility of connecting homomorphisms for Tor/Ext
This is a simple question about Tor and Ext functors.
Let $R$ be a commutative ring, and let $0 \to M' \to M \to M'' \to 0$ and $0 \to N' \to N \to N'' \to 0$ be short exact sequences of $R$-modules. ...
- 523
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1
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660
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Minimal prime divisors (MinAss R)
Hello All,is This conclusion true?
If $(R,m)$ is a local ring and $ Min Ass R=Ass R$ then can we conclude that $Min Ass \hat{R}=Ass \hat{R}$? ($\hat{R}$ is $m$-adic completion of $R$)
$MinAss$ ...
- 418
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2
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Limit of a series of singularities
The $A_\infty$ and $D_\infty$ plane curve singularities have defining equations $x^2=0$ and $x^2y=0$. These equations are "clearly" natural limiting cases of the equations for $A_n$ singularities $x^...
- 5,846
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1
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Can ⨁_I A be isomorphic to ∏_I A for infinite I?
Suppose $A$ is a non-zero ring (say commutative unital) and $I$ is an infinite set. Can it happen that there is an isomorphism of $A$-modules $\bigoplus_{i\in I}A\cong \prod_{i\in I}A$?
The obvious ...
- 24.1k
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Simple Question on Injective Hulls
Let $R$ be a noetherian local ring with maximal ideal $\mathcal m$ and denote by $E$ the injective hull of the residue field $k$.
Then, as an $R-$module, what is the support of $E$?
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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{...
- 1,207
3
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1
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Depth zero, high dimension
$\textbf{Question: }$We know that the depth of a noetherian local ring is at most the dimension. Do there exist noetherian local rings with high dimension but zero depth? If not, what's the smallest ...
- 3,555
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0
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47
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estimate on degree of generators in cohomology of differential graded module
Let $R=\mathbb{C}[X_1,\ldots,X_r]$ be a polynomial ring and consider a finitely generated and free differential graded $R$-module $M$ with differential $d$. Lets say that the degree of the variables $...
- 11
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polynomial relations between modular functions
$\newcommand{\Qbar}{\overline{\mathbb{Q}}}$
We define a modular function to be a meromorphic modular form of weight 0 for some subgroup (not necessarily congruence) $\Gamma\le\text{SL}_2(\mathbb{Z})$ ...
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question about valuation ring
$k$ algebraically field, $A$ $k$ algebra and valuation ring of $K$ ($K$ field fraction of $A$) and we have the transcendence degree of $K$ over $k$ is one.
i want to ask if $A$ is noetherian ring?
- 1,066
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156
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A family of maximal ideals
Let $m_i $, $i \in I,$ be an infinite family of maximal ideals in a commutative ring with identity (it is not supposed to be Noetherian). When does there exist $j \in I$ such that $\cap_{i\not= j} m_i\...
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2
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1k
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Graded or stacky Serre duality
I am considering the following situation. $A$ is a finitely generated ring over a field $K$ with non-negative grading and $A_0=K$ of Krull dimension n+1, but I don't necessarily assume A is generated ...
- 121
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2
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527
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Are hensel valuation rings N2?
This seems like the kind of thing an expert should be able to answer off the top of their head:
Recall that a valuation ring is an integral domain $A$ such that for every $a \in Frac(A)$ we have ...
- 1,347
6
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126
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Relative variants of the Jacobson radical
Let $B$ be a commutative ring (with 1). The Jacobson radical can be defined as
$$ J(B) = \{b \in B \mid \forall a \in B \colon \quad 1 + a\cdot b \text{ is a unit in } B \} $$ or $$ J(B) =\{ b \in B ...
- 401
2
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1
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460
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Local blowup versus global blowup
Let $R=k[x_1,..,x_n]/I$ and let $X=Spec(R)$ be it's associated affine scheme. Suppose that $X$ has only one isolated singularity, say at the origin $\mathfrak{m}=\langle x_1,...,x_n\rangle$. Now, ...
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115
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Need a reference/proof for computing the regularity of ideal of points in $\mathbb P^d$?
In a lecture notes on 'Cohomology modules' i read the following remark:
Given a set $X$ of points in $\mathbb P^d$,using the Local Cohomology modules one can easily compute the reg$(S_X)$ where $...
- 121
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reduction of an ideal
Reduction of an ideal is an important method on commutative algebra. Let $R$ be a Noetherian ring, $J \subseteq I, J \neq I$ two ideals of $R$. Then $J$ is a reduction of $I$ is there exists $k$ such ...
- 2,773
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1
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152
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Polynomial degree comparison of Nullstellensatz and Positivstellensatz over real algebraic sets
Suppose we have a (finite) system of polynomials $P = \{ p_i \} \subseteq \mathbb{R}[x_1, \ldots, x_n]$. Then it is well known by the Nullstellensatz that either $P$ has a simultaneous zero over $\...
- 539
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Example of inclusion which is not a finite morphism [closed]
Every closed immersion is a finite morphism. Can you give an example of quasi-projective varieties $X\subset Y$ such that inclusion $X\hookrightarrow Y$ is not finite? Same with Y projective?
Thanks!
...
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5
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1
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953
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power series over a polynomial ring
Let $k$ be a field. Consider the ring $A=k[x][[t]]$ of formal power series in a variable $t$ over the polynomial ring $k[x]$. This ring contains the ring $B=k[[t]][x]$ of polynomials in the variable §...
- 620
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0
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221
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How do I check if a sequence of R-modules is exact?
Let R be a ring. For example, take $R=k[x_1,\ldots,x_n]$ or, if possible, $R = \Bbb{Z}[x_1,\ldots,x_n]$.
Consider a sequence of free R-modules
$$R^a \stackrel{f}\to R^b \stackrel{g}\to R^c$$
where $f$...
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1
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Isomorphism between direct sum of modules
Let $M$, $N$ be two modules over ring $A$. If $M\oplus M\cong N\oplus N$, can we conclude $M\cong N$? In the case that $M$, $N$ are completely decomposable (e.g. finite-length module by Krull-Schmidt ...
- 33
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0
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215
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Generation of cohomology of graded algebras
Let $A$ be an unital, associative, graded algebra over a base ring $k$. I'm happy to assume that $k$ is a field if need be, and will insist that $A$ free and of finite rank in each degree (locally ...
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2
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356
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Is the ideal of a closure of a Bruhat cell generated by generalized minors?
Let $G$ be your favorite complex semi-simple algebraic group, and let $B\supset T$ be your favorite Borus. For any $w\in W$, we have the Bruhat cell $BwB$, and its closure $\overline{BwB}$.
Now, ...
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566
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Commutative algebras and Gamma-modules
A commutative algebra (with unity) over a field gives rise to the covariant functor F: Set_f->Vect from finite sets to vector spaces: F(E) := A^{otimes E}.
Is it true that, over complex numbers, a ...
- 809
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1
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286
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A condition on isolated singularity
Suppose $F: {\mathbb C}^N \to {\mathbb C}$ defines a singularity at the origin (for simplicity one can assume that $F$ is a quasi-homogeneous polynomial). Suppose it is nondegenerate, i.e., $dF(z) = 0$...
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712
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Spectral sequences and Koszul complexes in Deformation Theory
I'm reading this paper of A. Vistoli and I have some questions about the discussion in page 5. This is the context (If you don't want to download the paper):
Let $A'$ be a noetherian local ring with ...
- 987
12
votes
1
answer
967
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Lengths over a local ring
Let $A$ be a noetherian domain, $\mathfrak{m}$ а maximal ideal, $s$ a non-zero element of $\mathfrak{m}$, $d= \dim A_\mathfrak{m}$.
Is the following claim true?
Claim:
For any $\epsilon>0$, there ...
- 635
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0
answers
958
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What is the state of art in Groebner bases
How big polynomial systems can we deal with? How do you know when you don't even have to try?
Motivation:
Recently I tried to solve a problem posed in another MO question and ultimately I got stuck ...
- 8,597
9
votes
1
answer
2k
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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]$,...
- 5,562
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0
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25
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computing quotient ideals efficiently over a polynomial ring: (I:J) when J has many generators?
Can someone guide me to a reference where an algorithm for computing I:J where
J has many generators is discussed? I know the method of using one generator at a time and then taking intersections. I ...
- 11
1
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0
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234
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Separability of a simple ring extension
Assume $A=K[x,y]\subset K[x,y][w]=B$, $K$ is a field of characteristic zero, $w$ is integral over $A$ (so $B$ is a f.g. $A$-module), but $w$ is not in the field of fractions of $A$, and $B$ is an ...
- 2,837
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77
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Random variables with values in binary operations or in topologies of a certain set $X$
I wonder if the following situations have already been considered by mathematicians :
Random variables with values in a set of binary operations endowed
with a certain topology (or just with a $\...
- 129
6
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1
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1k
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reference for p-local and p-complete integers
Can anyone suggest a good thorough reference for $p$-localization and $p$-completion of the integers? I'm an algebraic topologist who's found himself washed up without any intuition.
In particular, ...
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2
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1
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451
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Resolution of singularity of polynomials
Let $f$ be polynomial on a vector space $V$. Let $Z$ be the zero set of $f$ in $V$. Let $Z_{sing}$ be the singular part of $Z$.
By Hironaka's desingularization theorem, there exists a birational map ...
- 1,457
10
votes
1
answer
607
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Are cluster variables prime elements?
Cluster algebras introduction
A cluster algebra is a subalgebra $A$ of $k[x_1^{\pm1},...,x_n^{\pm1}]$ generated by a set of cluster variables, which are elements which can be generated from the set $\{...
- 13k
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0
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314
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Can you always find a regular sequence consisting of monomials?
Let $\mathbb{k}$ be a field, and let $S=\mathbb{k}[x_1,x_2,\ldots,x_n]$. Let $M$ be an $S$-module. A sequence $$f_1,f_2,\ldots,f_r$$ of polynomials in the maximal ideal $\langle x_1,\ldots,x_n\rangle$ ...
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636
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Does exterior product commute functor Hom?
Let $M$ be an module over the commutative ring $R$. I'd like to ask do we have the following isomorphism?
$$Hom_R(\wedge^n_RM,R)\simeq \wedge^n_R Hom_R(M,R)$$
We can obviously see it's true for the ...
- 71
0
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1
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152
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Is it possible to generalize a result of Wang?
Assume $A$ and $B$ are commutative algebras with $1$.
There is a nice result of Wang, Corollary 8, which says the following: "Let $B = A[z] = A[Z]/(h(Z))$. Then $B$ is a separable algebra over $A$ if ...
- 2,837
1
vote
0
answers
91
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Cut ideal of two graphs?
Consider a connected graph $G$ and a connected graph $H$. Their graph ideals are their path ideals. The Alexander duality of the graph ideals give the cut ideals. $G$ and $H$ are not connected to each ...
- 143
2
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0
answers
97
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Can we write an element in a super Grassmannian as a pair of matrices?
Super Grassmannians are introduced by Manin, see for example.
Elements in a grassmannian can be written as matrices, see for example.
Can we write an element in a super Grassmannian as a pair of ...
- 6,211
9
votes
1
answer
689
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When and where did the term "module" enter commutative algebra?
Bruns/Herzog "Cohen-Macaulay-Rings" has a note in the notes for Chapter 1, saying roughly that after the influx of homological algebra into commutative ring theory, modules became popular objects (...
- 1,961
2
votes
1
answer
675
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Is there an irreducible integral polynomial in two variables which is reducible for every value of one of the variables?
Is there a polynomial $f(x,y)$ in two variables, with integer coefficients, such that $f$ is irreducible over the complex numbers (i.e., in $\mathbb{C}[x,y]$), but for every integer $n$, the ...
- 5,309
3
votes
1
answer
342
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Splitting as $\mathbb{F}_p[[X]]$-modules
Let $A$ be a finitely generated torsion $\mathbb{Z}_p[[X]]$-module, $B$ = { $x \in A$ such that $px=0$ } and $C=A/B$ where $\mathbb{Z}_p$ denotes the $p$-adic integers. Given $ 0 \rightarrow B/pB \...
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3
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1
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383
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What is the geometric meaning of content or intersection flatness?
The polynomial extension $R \rightarrow R[X]$ ($X$ an indeterminate) has many nice properties beyond faithful flatness. The one I'm most interested in at the moment is the following. Say that a flat ...
- 1,812