All Questions
6,055 questions
1
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155
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Universally catenary and all its formal fibers over minimal members are Cohen-Macaulay but it has a nonCohen-Macaulay formal fiber
Please help me to find a Noetherian local ring $R$ such that: $R$ is universally catenary and all its formal fibers over minimal members of $Spec(R)$ are Cohen-Macaulay but $R$ has a nonCohen-Macaulay ...
- 89
1
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0
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119
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finiteness of the homology of an augmented Koszul complex
Let $(A,m)$ be a local ring $x_1,\cdots,x_n$ elements of $m$ and $M$ a finite $A$-module. Let $K(x)$ be the Koszul complex associated with $x_1,\cdots,x_n$. We define a new complex by $K(x) \otimes M$....
- 398
1
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0
answers
296
views
What is your expectation of the depth?
Let $S=k[x_1,...,x_9]$ be a polynomial ring over field $k$. Set $q_1=(x_1,x_2,x_5,x_6)$, $q_2=(x_1,x_2,x_6,x_7)$, $q_3=(x_2,x_3,x_7,x_8)$, $q_4=(x_1,x_5,x_6,x_7)$, $q_5=(x_1,x_6,x_7,x_8)$, $q_6=(x_2,...
- 21
1
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0
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157
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Indecomposability of image transformations (pure algebra). Open questions
W-transformations -- definitions
We will consider a class called finite window transformations $\ T:C^\mathbb Z\rightarrow C^\mathbb Z\ $ defined a paragraph below; $\ \mathbb Z\ $ is the ring of ...
- 7,500
1
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0
answers
83
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lift sections on a thickened curve
Let $X$ a curve over an algebraically closed field $k$ and $D$ a divisor on X.
Fix an integer $N$ and a closed point $x$ on $X$, we assume that $\deg(D)$ is big enough such that we have a surjective ...
- 3,472
1
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0
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282
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Computation of Quillen K-groups for mod R
The recent paper K-Groups for Rings of Finite Cohen-Macaulay Type by H. Holm allows us to compute the Quillen $K$-group $K_1(\text{mod}\hspace{.1 cm}R)$ as a quotient of the abelianization of the ...
1
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0
answers
205
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Example for 1-dim, Noeth., local domain which is unibranched but not analytically irreducible
Does somebody know an example for an 1-dim., Noeth., local domain $D$ which is unibranched (that is, its integral closure $D'$ is local) but not analytically irreducible (that is, its $\mathfrak{m}$-...
- 11
1
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0
answers
190
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regular sequence
Let $I \subset J $ be two monomial ideals in $S=k[x_1,..,x_n]$ minimally generated by $(a_1,...,a_s)$ and $(a_1,...,a_s,b_1,...,b_r)$. I want to show that depth$_S S/I \geq$ depth$_S S/J$.
Let $c = ...
- 287
1
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0
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131
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Rings with the property $\dim R-\dim R/p\leq \text{const}$ for all minimal $p$
I'm curious if there exists a class of rings generalizing quasi-unmixed rings. I guess a generalization of quasi-unmixed rings can be done as follows:
For a fixed integer $i$
$$\forall p\in\...
- 189
1
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0
answers
358
views
Proof that a finitely generated projective module over a Von Neumann Regular ring is free
I'm searching for a proof that a finitely generated projective module over a Von Neumann Regular ring such that all the localizations have the same rank is free. I know that this result is true, ...
- 163
1
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1
answer
234
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Relation between $H^i_I(-)$ and $H^i_J(-)$ when $I\subset J$
What is the relation between $H^i_I(-)$ and $H^i_J(-)$ (cohomological functors) when $I\subset J$ are ideals of a (local) noetherian ring?
- 189
1
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0
answers
698
views
Is a complete intersection satisfying Jacobian matrix smooth criterion a smooth variety?
Every scheme here is over complex number.
Let $X \subset (\mathbb{C}^*)^n$ be a complete intersection with $X$ defined by the ideal $I \subset \mathbb{C}[x_{1}^{\pm},\dots,x_{n}^{\pm}]$ generated ...
- 3,472
1
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0
answers
220
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the topology of power series ring
Hi, everyone.
Let $A$ be a complete DVR with uniformizer $t$, $R:=A[[X]]$. What is the natural topology of $R$ ?
- 43
1
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0
answers
71
views
Integral Leray Number?
The Leray number of a finite simplicial complex $K$ relative to a field $\Bbbk$ is defined to be the least $d\geq 0$ such that $\widetilde H^n(C,\Bbbk)=0$ for all $n\geq d$ and all induced ...
- 38.6k
1
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0
answers
136
views
de Rham complex of closed immersion between smooth schemes
Hi,
Let $R$ be a $\mathbb Q$-algebra and let $P$ and $Q$ be (EDIT: smooth) $R$-algebras such that there is a surjective
map of $R$-algebras $Q\to P$. The following proof cannot possibly be correct, ...
- 2,842
1
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0
answers
80
views
smooth algebras and triviality of de Rham complex
Hi,
Let $R$ be a $\mathbb Q$-algebra and let $A$ be a smooth $R$-algebra. If $A$ is a polynomial algebra
$A = R[T_1,\dots,T_n]$, then it is easy to see that the natural map
$R \to \Omega^\bullet_{A/R}...
- 2,842
1
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0
answers
255
views
Fitting ideal/ determinantal variety
Let $R$ be an integral domain, "nice" (regular for instance). Consider a homomorphism
$$
f: R^m \to R^m
$$
of two rank $m$ free $R$ modules. Assume that $\ker f =0$ and that the cokernel is $M$. Now ...
- 131
1
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1
answer
320
views
Hochschild (co)homology and Kahler differentials
Suppose $A$ is an augmented commutative algebra over a field $k$. What is the relation between Hochschild homology $H_n(A,k)$ and Kahler differential $\Omega_{A/k}$? The same question is also asked ...
- 97
1
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0
answers
1k
views
Algebraic Independence of Polynomials in n Variables with Real Coefficients
I am considering the problem of determining the algebraic independence of $n$ polynomials in $m$ variables with real coefficients, where $m \geq n$. The variables will be denoted by $a_{1}, a_{2}, ... ...
- 11
1
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0
answers
411
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a question about Beauville-Laszlo
Hi,
let $V$ be a complete DVR with uniformizer $\pi$. Let $m$ be a NON zero integer, $a\in V[[u,v]]/(uv-\pi)^{\times}$ and $f=\pi^{m}a$. Consider $F$ as the kernel of the diagram
$$
V[[u,v]]/(uv-\pi)...
- 11
1
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0
answers
204
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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
1
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0
answers
118
views
Is there a standard name for functions of the form $x^\alpha p(x)$, where $p(x)$ is a polynomial?
Is there any existing standard terminology for functions of the form $x^\alpha p(x)$, where $p(x)$ is a polynomial and $\alpha$ is e.g. a complex number? I haven't been able to come up with any good ...
- 1,488
1
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0
answers
477
views
Comparing the homogeneous defining ideals of multiple embeddings of a projective scheme
If $X$ is a projective scheme over a field $k$ (which we may assume is algebraically closed), then under an embedding $i: X \hookrightarrow \mathbb{P}^n_k$, we may write $X = Proj(R/I)$ where $R = k[...
- 11
1
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0
answers
212
views
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 ...
1
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0
answers
221
views
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
1
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0
answers
111
views
Factorization of Gegenbauer polynomials
For each natural number $n$ there is a Gegenbauer polynomial of degree n, depending on
a spectral parameter $\lambda$. They fulfill many recurrent relations. The question is
how to recognize their ...
- 11
1
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0
answers
639
views
when is a sum of idempotents idempotent in commutative ring theory?
As this question demonstrates that the sum of idempotents is idempotent iff every pairwise product is zero, for finite matrices with complex entries.
What additional restrictions do we need to put ...
- 2,083
1
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0
answers
133
views
Efficient algorithm for computing the integral closure of a computable domain
what is known? even talking about efficiency relatively to the complexity of the computation of the domain itself?
1
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0
answers
242
views
separability of commutative rings
Before discussing on the main Question I should recall two notions in the area of commutative rings.
By $Max(R)$, we mean the set of all maximal ideals of the commutative unitary ring $R$.
...
- 1,788
1
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0
answers
234
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Set of Curves Passing through a smooth point of a Variety is Zariski-Dense
In a paper by F. Pop he claims the following fact-
Let $K$ be a field. The set (by which I believe he means the union) of all smooth $K$-curves passing through a smooth $K$-rational point of an ...
- 11
1
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0
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245
views
Sums of Strongly z-ideals
In the rings of continuous functions,i.e.$(C(X))$ an ideal $I$ is called strongly $z$-ideal if it is an intersection of some maximal ideals of $C(X)$. i.e. $$I=\cap_{\alpha \in A} \mathcal{M_{\alpha}}$...
- 1,788
1
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0
answers
169
views
Algebraic properties of the semiring of open subsets.
Does anyone know of a useful general topological application of the algebraic properties of the semiring of open subsets of some topological space? Or examples of any such nontrivial properties at all?...
- 3,513
1
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0
answers
315
views
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 ...
- 1,788
1
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0
answers
181
views
unfolding as resolution
Has anyone described 'unfolding' as used in mathematical physics (e.g. on-shell AND off-shell) as analogous to a resolution in algebra - higher derivatives are unfolded in terms of new variables?
- 3,880
1
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0
answers
233
views
projective dimension of finitely generated modules over char 0
In the paper "On modules of finite projective dimension over complete intersection" Dutta proved that a finitely generated module over a local complete intersection ring over char $p>0$ has finite ...
- 2,444
1
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0
answers
196
views
Extending commuting endomorphims of a complete discrete valued field to the algebraic closure?
Is it true that any two commuting endomorphisms of a complete discrete valued field extend to commuting automorphisms of the algebraic closure?
- 341
1
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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
1
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0
answers
98
views
Explicit operations with correspondences
Let $X: y^2=f(x)$ be an hyperelliptic curve over a finite field $k$. Consider two non-fibral correspondences $C,D\subset X\times X$. It is well known that they induce endomorphisms $\phi_C,\phi_D:...
- 293
1
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0
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257
views
level of rings and stable range of rings
The level $s(A)$ of a ring $A$ with unity $1$ is the smallest natural number $s$
such that $-1$ is a sum of $s$ squares in $A$. (If $-1$ is not a sum of squares in $A$, we
say that $s(A) =\infty$.). ...
- 11
1
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0
answers
562
views
Is $gcd(zx,zy)=zgcd(x,y)$ (i.e. does the left hand side of this equality 'exist' if the right hand side does).
This should be a simple question on basic definitions.
In an integral domain $R$ we will say that $gcd(x,y)$ exists and is equal to some $r\in R$ if $r$ divides $x$ and $y$, and any common divisor $...
- 16.9k
1
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0
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107
views
Regularity in terms of jets?
Is there any criterion of regularity for rings in terms of jets?
More precisely: It is known that a local ring $B$ (with some hypothesis) is regular if and only if the module of differentials $\...
- 55
1
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0
answers
666
views
Structure theorem for Finitely Generated modules over PID's using localization
I have been trying to work out a proof of structure theorem for finitely generated modules over PID's using localization. This is what my plan is:
1.Prove that every finitely generated torsion free ...
- 11
1
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0
answers
96
views
Explicit expression for multivariable meromorphic series
Let $K$ be a field. For $m>1$, elements in the ring of power series in $m$ variables over $K$, i.e., $K[[X_1, \cdots, X_m]]$ look like -
$$ \displaystyle\sum_{(i_1, \cdots, i_m)\in (\mathbb N_{0})^...
- 899
1
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0
answers
531
views
Krull's intersection theorem for commutative local not necessarily noetherian rings
Is there a characterisation of those commutative local not necessarily noetherian rings that satisfy Krull's intersection theorem ? How can the intersection theorem be phrased in terms the module ...
1
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0
answers
345
views
Is the ideal of denominators preserved under flat pullback?
Let $\phi \colon A \to B$ be a flat homomorphism of rings (commutative, with unit). Let $R$ be the total ring of fractions of $A$ (obtained by inverting all nonzerodivisors), and let $S$ be the total ...
- 7,338
1
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0
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383
views
Size of an abelian permutation group with generators of order 2 [closed]
Let $g_1, \ldots, g_k$ be distinct permutations on a set $\Omega$. Suppose that $G = \langle g_1, \ldots, g_k \rangle$ is an abelian permutation group with only elements of order at most 2. Is it ...
- 11
1
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0
answers
169
views
Choosing generators of a submodule with divisibility properties
Looking at an open subset $U$ of the plane, containing $0 \in \mathbb{C}^2$, with coordinates $x$ and $y$.
Given a quotient sheaf $O_U^n \rightarrow T$, with $supp(T)=\lbrace0\rbrace$. Let $K$ be the ...
- 1,391
1
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0
answers
263
views
In a Noetherian local ring $(R,m)$ with a prime ideal $P\neq m$, $P^{(n)}=P^n:m^{\infty}$
I had asked this question on math.stackexchange.com, but I have not received a response. Hoping to get some help here.
I am trying to prove this result and I am stuck at one step.
Let $(R,m)$ be a ...
1
vote
0
answers
417
views
Absolute Irreducibility in Characteristic 2
Let $\mathbb F$ be a field and $\mathbb F[x_1,\dotsc,x_n]$ the ring of multivariate polynomials in $n$ variables over $\mathbb F$. A polynomial $P\in\mathbb F[x_1,\dotsc,x_n]$ is said absolutely ...
- 456
1
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0
answers
273
views
Depth of intersection
Let $I$ be an ideal in $S=K[x_1,\dots,x_n]$. Can we compute $\operatorname{depth}(I\cap K[x_1,\dots,x_r])$ with $r \leq n$? Is there any relation between depth $I$ and $\operatorname{depth}(I\cap K[...
- 287