All Questions
6,055 questions
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292
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regular locus of an affine domain
Let $A$ be an affine domain over a field $k$ (need not be algebraically closed). Let $\mathfrak{p}$ be a prime ideal of $A$, such that $A_{\mathfrak{p}}$ is a regular local ring. Does there always ...
0
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0
answers
109
views
Vanishing of the module of differentials of a extension of perfect fields
Let $L|F$ be a extension of perfect fields of characteristic $p$, $\phi_F:F \to F_{\phi}$, $\phi_L:L \to L_{\phi}$ the Frobenius isomorphisms ($F_{\phi}=F$ but considered as $F$-algebra via $\phi_F$). ...
0
votes
0
answers
79
views
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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votes
0
answers
105
views
$\Gamma_Z(\widetilde M)\cong\widetilde{ \Gamma_Z(M)}$
Let $R$ be a Noetherian ring and let $M$ is an $R$-module. Consider the associated affine scheme $(\text{Spec R},\mathcal{O}_{\text{Spec R}})$ and Suppose $Z\subset X$ is a closed subset of $\text{...
- 21
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votes
0
answers
182
views
Zariski open set of linear forms
Let $I$ be a graded homogeneous ideal over $k[x_1, ... ,x_n]$ and $h$ a linear form, let $H$ be the corresponding hyperplane and $I_H$ the restriction of $I$ to $H$.
I am looking for a Zariski open ...
- 73
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votes
0
answers
230
views
A question about the unbounded derived category of the polynomial ring in infinitely many variables
In this moment I am trying to understand the derived category of the polynomial ring in infinitely many variables over a field $k$, $R=k[x_{1},x_{2},\dots]$ and I am wonder if it is true that $Hom_{D(...
- 11
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votes
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answers
350
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Reference: A nowhere vanishing section of a vector bundle is locally split
Well-known fact:
If $(A, \mathfrak{m})$ is a local Noetherian ring, $E$ is a finitely generated free $A$-module, and $e\in E$ is an element not contained in $\mathfrak{m}E$, then $E/eA$ is also a ...
- 7,338
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0
answers
136
views
Monoid action on an uncountably infinite set
The action of a monoid on a finite set is equivalent to a finite state machine, however I would like a categorical way to think about an uncountably infinite state machine (a state transition system?)....
- 101
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votes
1
answer
119
views
Colon operation after adjoint variables
Let $R$ be a commutative Noetherian ring and $M$ a finitely generated $R$-module. Let $I$ an ideal of $R$. We have
$$0:_MI = \cap_x(0:_Mx),$$
where $x$ runs a set of generators of $I$.
Now set $S = ...
- 2,773
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answers
68
views
Let $(R, m)$ be noetherian local, $\dim(R)=1$. Show $CH^1(R)=\mathbb{Z}/(\gcd([k_i, R/m]))$, where $k_i$ are residue fields of normalization
Let $R$ be a $1$-dimensional noetherian local domain. Then we have that $CH^1(R)=\mathbb{Z}/(\gcd([k_i, k]))$ where the $k_i$ are residue fields of the normalization and $k$ is the residue field of $R$...
- 841
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votes
0
answers
138
views
Profinite Local Ring inside Polynomial Ring
This is a "technical" question that I came across in my research.
Let $A = \textbf{Z}_{p}[\![t_1, \cdots, t_a ]\!]<z_1, \cdots, z_b>$ be the $(p, t_1, \cdots, t_a)$-adic completion of the ...
- 61
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0
answers
355
views
Cubic field and the corresponding cubic binary form
I am currently reading about binary cubic forms and cubic number fields (mainly about using binary cubic forms with integer coefficients to parametrize orders in the cubic field) and I thought it ...
- 21
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0
answers
320
views
Invariants of the Determinant Form
Consider a form of degree $r$ in $n$, that is, a homogeneous polynomial
$$f(x_1, \ldots, x_n)=\sum_{i_1+\ldots i_n=r}\alpha_{i_1 ... i_n}x_1^{i_1} ... x_n^{i_n}
$$
After the linear change of ...
- 801
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0
answers
72
views
Decomposition results for locally commutative semigroups
Every finite abelian group is the direct product of its cyclic groups of prime order, and every commutative monoid divides a product of its cyclic submonoids. Could these results generalized to ...
- 798
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votes
1
answer
368
views
The closure of an effective Cartier divisor in a special situation
I am studying first order deformations and a natural question arises.
Situation: Let $X_1$ be a scheme. $\pi: X_1 \to {\rm Spec}~ k[t]/(t^2)$ is a flat morphism of finite type, where $k$ is an ...
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0
answers
548
views
Fitting ideal sheaves and determinant bundles
I am working on a problem in algebraic geometry which comes down to a fact in commutative algebra that I am hoping is well-known.
Suppose $F$ is a coherent sheaf on a smooth variety $S$, and that the ...
- 5,857
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votes
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answers
52
views
Cohen-Macaulayness of inseparable isogeny k-algebras
Let $R$ and $S$ be 2 associated, commutative, and unita $k$-algebras where $k$ is an algebraically closed field of characteristic $p$. We call these algebras inseparable isogeny or $F$-isomorphism if ...
- 127
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0
answers
181
views
A question on binary polynomials
This is probably a well-known result but I was not able to find a reference on my search. My question concerns general polynomials $f(x,y) \in \mathbb{Z}[x,y]$ such that $f$ cannot be written as a ...
- 26.9k
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0
answers
255
views
Image of critical points
Let $K$ be a field of characteristic $0$, $f:K^n\rightarrow K^n$ be an algebraic function, that is, $n$ polynomial functions in $n$ variables. Let $S$ be the set of critical points of $f$. If $K=\...
- 3,741
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259
views
Ring algebraically closed in its completion.
First I would like to be clear about the definition, which I am having trouble finding.
What does: The local ring $A$ is algebraically closed in $B\supset A$. (e.g. for $B:=\hat{A}$, the completion ...
- 807
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votes
0
answers
93
views
Ring of even characteristic.
Is possible to choose three units $u,v,w$ of a ring $R$ (not containing a field) with even characteristic
such that $u+v+w=0$.
Thanks in advance.
- 335
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votes
0
answers
99
views
Example of a ring whose minimals are annihilators of idempotents?
I'm looking for examples† of rings with the property that for each
$P={\rm Ann}_R(a)\in{\rm Min}(R)$ then $a\in R$ is idempotent (ie $a^2=a$)
† other than domains!
- 189
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votes
1
answer
322
views
Height unmixed ideal
Suppose $R$ is a regular local ring and $I$ is a non-zero ideal such that $I$ is a radical ideal and $I$ is height unmixed. Suppose $J$ is any radical ideal contained in $I$ and with the same height ...
- 3
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votes
0
answers
123
views
Irreducibility of superelliptic curves
Let $k$ be an algebraically closed field of characteristic zero, let $a,d$ be integers, and let $f\in k[x]$ be a separable polynomial of degree $d$.
Question: a) Is the affine plane curve $y^a=f(x)$ ...
- 23
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0
answers
166
views
The intersection complex and the Cohen-Macaulay property
Let $\Delta:Y\rightarrow X$ a closed immersion of $k$-schemes of finite type and equidimensionnal.
We assume that $\Delta^{*}[-d]IC_{X}=IC_{Y}$, if $X$ is Cohen-Macaulay, does it imply that $Y$ is ...
- 3,472
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votes
0
answers
355
views
Can we find a Groebner Basis?
I would like to ask the following. Given only the leading terms of an ideal $I$, namely the set $LT(I)$, is it possible to find a Groebner Basis of $I$? If not always, then when is it possible? We ...
- 1
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1
answer
468
views
Finite extensions of residue fields of Henselian DVRs
Let $K$ be an Henselian discrete valuation field such that its completion is separable over $K$. Let $F$ be its infinite residue field. Is it true that a finite extension of $F$ is a simple extension ...
- 2,032
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votes
0
answers
245
views
Notation Problem, Fixed Rings and Fields
I am trying to make sense of the notation and certain sets in two articles by Annick Valibouze whose results I'm using for my bachelor's thesis, I hope it's relevant enough to merit an answer.
In one ...
- 119
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votes
0
answers
355
views
abstract algebra for component wise operations on "vectors" or what it might be called
I have a quite tough problem to solve and need an algebra that allows to "vectors" following operations:
- multiplication between two vectors are componentwise that means v=(v1, v2, v3,...) multiplied ...
- 254
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votes
0
answers
244
views
Properties of Gorenstein ideal
Fix an integer $k>4$. For any integer $r>0$, denote by $S_{r}:=\mathbb{C}[X_0,X_1,X_2,X_3]_{r}$ the vector space of degree $r$ polynomials in $X_i$ with coefficients in $\mathbb{C}$. Let $W$ be ...
- 1,040
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votes
0
answers
87
views
Algorithm for computing basis of zero dimensional ring?
If given a zero dimensional ring over a field, for example, a polynomial ring $A=k[x_1,\ldots,x_n]/(f_1,\ldots,f_n)$ such that $A$ is 0-dimensional, is there an algorithm to compute a monomial basis ...
- 1,157
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0
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235
views
Power of ideals and exact sequences
Hello, I'm reading about analytic sheaves and I've a problem to understand something that's related with commutative algebra:
Let $\mathfrak{a}\subset R$ an ideal and $M$ an $R$-module. Then,
$\...
- 987
0
votes
0
answers
315
views
Definitions for Oddness
In the thread Even Xor Odd Infinities I defined odd models of Modular Arithmetic (MA) as models satisfying the axioms of MA and two first order statements. Even XOR Odd Infinities?
$O1) \forall x(x=0 ...
- 687
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votes
0
answers
383
views
Pseudo-cauchy sequence and valuation
Let $k$ be a field and $x$ is transcendental over $k$. Can we construct a pseudo-cauchy sequence $(a_{i})$ convergent to $x$ with each $a_{i}$ is algebraic over $k$ and $k(a_{i})\subseteq k(a_{i + 1})$...
- 173
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votes
0
answers
178
views
The transcendence degree of the algebras of invariants
Let $V_n,V_m$ be the vector $\mathbb{C}$-spaces of the binary forms of degrees $n,m$ considered as usual $SL_2$-modules. Let $I_{n,m}=\mathbb{C}[V_n \oplus V_m]^{SL_2}$ and $C_{n,m}=\mathbb{C}[...
- 301
0
votes
1
answer
177
views
Laurent series with analytic coefficients
Let $A=H(D(0,1))$ the ring of holomorphic functions on the open unity disc.
I consider the function $f$:
$$f (t)=\sum f_{i}t^{i} \in A[[t]]$$
I suppose that the $t$-adic valuation of it is less or ...
- 3,472
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votes
0
answers
381
views
Completion of commutative rings.
Assume that $(R,\mathfrak{m})$ is a commutative local ring of equal characteristic zero. So $R$ contains the field of rationals. The well known $\mathfrak{m}$-adic completion of $R$ provides a ...
- 591
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votes
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answers
152
views
Kählerdifferentials and normal crossing divisors
Let $k$ be an algebraically closed field of arbitrary characteristic, $X$ a smooth surface over $k$, and $D_i \subset X$ be an regular, effective Divisor such that $D=\sum D_i$
has normal crossings ...
0
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0
answers
243
views
strict henselian and excellent henselian
Hello, everyone. I want to ask a problem about strict henselian ring.
Let $A$ be a strict henselian DVR.
Dose there exist subrings $A_{i}$ of $A$, such that $A=lim_{i} A_{i}$ and where $A_{i}$ are ...
- 1,921
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votes
0
answers
428
views
flat morphism between regular local rings
Suppose $f: A \rightarrow B$ is a local homomorphism of local rings. Assume that $A$ and $B$ are noetherian, regular and $\mathrm{Spec} B \rightarrow \mathrm{Spec} A$ is quasi-finite. Is is necessary ...
- 153
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votes
0
answers
346
views
Length of $\mathfrak{m}$-torsion module
Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring and $M$ is an $R$-module such that $\mathfrak{m}^tM=0$ for some non-negative integer $t$. Then the length of $M$ is finite.
Is that right?...
- 259
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votes
0
answers
236
views
On vanishing orders of an ideal via the restriction
Let $Y$ be a submanifold of a complex manifold $X$, and $a$ be an ideal on $X$ which does not vanish along the entire $Y$. Consider a point $\xi$ on $Y$, there are the vanishing order $ord_{\xi}a$ ...
- 320
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votes
0
answers
87
views
Standard Notation for Monomial Orderings?
Is there a standard way to denote a particular lexicographic (resp. reverse lexicographic) monomial ordering using subscripts or superscripts? For example, I might want to refer to the lexicographic (...
- 326
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votes
0
answers
212
views
Homomophism from Koszul complex to the original ring
In an article, I encounter an isomorphism relation as follows:
Let S be a comm. ring, x an element in S. K[x,S] be corresponding Koszul complex. The article says "this is a classical isomorphism":
$...
- 61
0
votes
0
answers
774
views
Discrete valuation rings.
Given an algebraically closed field $\mathbb F$ of characteristic $p$, let $\mathbb A$ be a discrete valuation ring of characteristic zero having $\mathbb F$ as its residue field ( it does exist, but ...
- 1
0
votes
1
answer
583
views
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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votes
0
answers
179
views
semigroup actions of groups on regular rooted trees
If $G$ is a group which has a semigroup action on a regular rooted tree via prefix-preserving, continuous transformations (I give the tree the path metric), what kinds of algebraic restrictions can we ...
- 125
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votes
0
answers
166
views
Can the zero-degree part of $M_f \otimes_{S_f} N_f$ be identified with $M_{(f)} \otimes_{S_{(f)}} N_{(f)}$?
The isomorphism ${(M \otimes _ {S} N)} _ {f} = M _ {f} \otimes _ {S _ {f}} N _ {f}$ is well-known. Here, $S$ is a graded ring, and $M,N$ are graded $S$ modules.
Now, let $f$ be any homogeneous ...
- 945
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votes
0
answers
551
views
sub ring of algebra over subfield
Let $k$ be a field and $k[a]$ an algebric extension.
If $A$ is a reduced commutative algebra over $k[a]$ and $B$ is a subring which is an algebra over $k$, then is the following true: if there exist ...
- 1
0
votes
0
answers
237
views
resolution of singular points on plane curves and base change
Let $k$ be a field and $C/k$ be an affine plane curve over $k$, namely $C = \mathrm{Spec}(A)$ for some $A = k[x,y]/(f(x,y))$, here $f(x,y) \in k[x,y]$ is an irreducible polynomial. Let $B$ be the ...
- 1,109