All Questions
1,947 questions with no upvoted or accepted answers
7
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329
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Computer Algebra solution for simplicial resolutions for André-Quillen cohomology
Hello,
I would like to experiment with André-Quillen (co)homology. Especially for singular rings.
A key problem is that the construction of a simplicial resolution of a ring seems to require a rather ...
- 71
7
votes
0
answers
658
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Invertible elements in generalized fields
Durov's theory of generalized rings also includes generalized fields (5.7.6), which are defined as generalized rings, which are not subtrivial and whose proper strict quotients are subtrivial. For ...
- 63.1k
7
votes
0
answers
518
views
An elementary question in singularities
The following problem came up in something I am working on. It has a really elementary statement but I couldn't crack it in a couple of hours of thinking about it. It isn't clear to me if I am being ...
- 3,758
7
votes
0
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249
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Does there exist a commutative ring R such that SL_3(R) and SL_2(R) have the same finite subgroups?
This question is inspired, of course, by this question, and I don't know enough commutative algebra to know whether it's answered by silence dogood's answer to this follow-up question. If the answer ...
- 118k
7
votes
0
answers
897
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Does the property (x*y)*x = x*y have a name?
The property $(xy)x = xy$ is one of the equations satisified by a directoid. Various properties have names ($xy = yx$ is commutativity, $xx=x$ is idempotency, etc). The wikipedia page for Magma has ...
- 11.8k
7
votes
0
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769
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Artin-Schreier Theorem for Rings
This has been in my mind for quite some time. Looking at the Artin-Schreier Theorem for fields:
If $L$ is a field and $K$ its algebraic closure and if $1< [K:L] < \infty$ then $K=L[i]$ and $L$ ...
- 2,275
6
votes
0
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151
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Can Harrison cohomology be written using Ext?
Just like Hochschild cohomology for associative algebras and Chevalley-Eilenberg cohomology for Lie algebras, it'll be nice (or disappointing?) if Harrison cohomology can be expressed in terms of Ext'...
- 985
6
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235
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A standard name for the algebraic structure on a projective line?
Question: Is there any name for the natural algebraic structure of the projective line?
Algebraically, a projective line over a field is a set $L$ endowed with two binary operations $+$ and $\cdot$ ...
- 41.8k
6
votes
0
answers
632
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Generating functions in countable commutative monoids
Let $f: \mathbb{N}_0 \rightarrow \mathbb{C}$ be a function. The power series of $f$ can be viewed as the function $\mathscr{P}_f : q \mapsto \sum_{n \in \mathbb{N}_0}^{} f(n)q^n$ where $q \in \mathbb{...
- 405
6
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0
answers
178
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Ext for commutative Gorenstein algebras
Let $A$ be a finite dimensional commutative Gorenstein $K$-algebra over a field $K$.
Question 1: Is there an easy example of $A$-modules $M$ and $N$ such that $\mathrm{Ext}_A^1(M,N)=0$ but $\mathrm{...
- 26.5k
6
votes
0
answers
151
views
Which monomials are "leadable"?
Question: Let $k$ be a field, let $f \in k[t_1,\ldots,t_N]$ be a nonzero polynomial. Which monomials
$m_a = t_1^{a_1} \cdots t_N^{a_n}$ appearing in $f$ are leadable in the sense that they are the ...
- 65.4k
6
votes
0
answers
291
views
What is the algebra structure on the pushforward of the structure sheaf along a finite map to $\mathbb{P}^1$?
$\newcommand{\P}{\mathbb{P}}\newcommand{\O}{\mathcal{O}}$ Let $f : C \to \P^1$ be a ramified finite map of degree $d$ of smooth algebraic curves over an algebraically closed field $k$. How can we ...
- 605
6
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628
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On the Erratum to P. Scholze's "$p$-adic Hodge theory for rigid-analytic varieties"
I am trying to understand section (3) of the Erratum to P. Scholze's "$p$-adic Hodge theory for rigid-analytic varieties" in detail. In particular, there is the following sentence on page ...
- 507
6
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0
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190
views
Computing the automorphism scheme of projective space
$\newcommand{\Spec}{\operatorname{Spec}}$I'm trying to understand why $PGL_{n}$ is the automorphism scheme of $\mathbb{P}^{n-1}_{\mathbb{Z}}$.
In Conrad's Reductive Group Schemes, the following ...
- 605
6
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149
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Rings with epimorphism from a finitely generated ring
For a commutative ring with unit $R$ let's say it has property $(*)$ if there is an epimorphism in the category of rings ${\mathbb Z}[X_1,\dots,X_n]\to R$, where the former is the polynomial ring in $...
user473423
6
votes
0
answers
68
views
Vector algebra in a Tarski space
By a Tarski space I understand a mathematical structure $(X,B,E)$ consisting of a set $X$, a ternary betweenness relation $B\subseteq X^3$ and a 4-ary equidistance relation $E\subseteq X^2\times X^2$ ...
- 41.8k
6
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236
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Functorial criterion for local complete intersection morphisms?
Let me state the question for rings (rather than schemes) for simplicity. Let $R$ be a commutative ring with unit and $A$ an $R$-algebra of finite presentation. Recall that $R\to A$ is called a ...
- 4,482
6
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0
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225
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Iterating exact triangles (particularly in Floer homology)
There are several different Floer-homological invariants of 3-manifolds (and knots). The most prominent of these are Heegaard Floer homology, monopole Floer homology, and instanton Floer homology. It ...
- 101
6
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194
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"Cluster algebra" structure for finite distributive lattices
Let $P$ be an $n$-element poset and $J(P)$ the distributive lattice of its order ideals (i.e., the downwards-closed sets).
For each $I\in J(P)$ let $x_I \in \mathbb{R}^{n}$ be the indicator function ...
- 24.2k
6
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1
answer
2k
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$\mathbb Z_n[x_1^\pm,\dots,x_D^\pm]$-modules extended from $\mathbb Z_n$
Let $n$ be a positive integer and let $\mathbb Z_n=\mathbb Z/n \mathbb Z$. Consider the ring of Laurent polynomials $R=\mathbb Z_n[x_1^\pm,\dots,x_D^\pm]$. $R$-modules of the form $M=M_0 \otimes_{\...
- 344
6
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514
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Quasi-syntomic descent and prismatic F-crystals
I am reading Bhatt and Scholze's paper on F-crystals, and they seem to be using the following result in the proof of Theorem 5.6:
let $X \to Y$ be a quasisyntomic cover of formal schemes over $\...
- 461
6
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0
answers
368
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Geometric meaning of localization at $(1+I)$?
Let $I\vartriangleleft A$ be an ideal of a commutative ring. Consider the submonoid $1+I\subset A$. What is the geometric interpretation of localization at this submonoid? How does it relate to the ...
- 10.5k
6
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0
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398
views
Unbounded derived Nakayama lemma
Let $R$ be a (commutative) local ring, which I don't assume to be noetherian. Let $m$ be its maximal ideal, and $k$ its residue field.
Let $X$ be a complex of $R$-modules with finitely generated ...
- 15.8k
6
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230
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Gelfand ring in Bourbaki's exercises
In Bourbaki's General Topology, Chapitre III §6 Exercise 11, they define a Gelfand Ring as a topological ring $A$ such that
The set $A^*$ ($=A^{-1}$) of invertibles is open.
The uniform structure ...
- 3,738
6
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344
views
Uncountable Mittag-Leffler condition?
Let $(X_\alpha)_{\alpha <\kappa}$ be an inverse system of abelian groups.
If $\kappa = \omega$ (or by extension if $\kappa$ is of countable cofinality), then the Mittag-Leffler condition is a ...
- 63.9k
6
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0
answers
461
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Strict Henselization vs base-change to algebraic closure
Let $x$ be a smooth $k$-point on a variety $X$ over a field $k$ of characteristic $0$.
Is the strict Henselized local ring $\mathcal{O}_{X,x}^{\mathrm{sh}}$ the same as $\mathcal{O}_{X,x}^{\mathrm{h}} ...
- 15.4k
6
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0
answers
119
views
Norm forms, slicing, and ideal classes
Let $K$ be a number field, which we may suppose satisfies $n = [K : \mathbb{Q}] \geq 3$. Let $\mathcal{O}_K$ be the ring of integers of $K$, and let $\{\omega_1, \cdots, \omega_{n}\}$ be a basis of $\...
- 26.9k
6
votes
0
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159
views
Ring with different graded and ungraded global dimensions
Let $A$ be a $\mathbb N$-graded ring. One can consider the two categories $M_A^g$ and $M_A^u$ of graded and ungraded modules over $A$. Both have, say, enough projectives, hence one can compute various ...
- 14.7k
6
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answers
220
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Is $\mathrm{End}-\{0\}=\mathrm{Aut}$ for derivation Lie algebra?
Is it true that every nonzero endomorphism of Lie $\mathbb{C}$-algebra $\mathbb{C}[x_1,\ldots, x_n]\partial_{x_1}\oplus\ldots\oplus\mathbb{C}[x_1,\ldots, x_n]\partial_{x_n}$ is an automorphism?
As I ...
- 291
6
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0
answers
205
views
Automorphisms of $\mathbb{C}[x, y, z]$ over $\mathbb C[x]$
What are the automorphisms of $\mathbb{C}[x, y, z]$ fixing $\mathbb{C}[x]$? I.e. those automorphisms $\phi:\mathbb{C}[x, y, z]\to\mathbb{C}[x, y, z]$ s.t. $\phi(x) = x$. I am interested in a complete ...
- 329
6
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0
answers
114
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Hilbert series of special linear sections of Grassmannian $Gr(2,n)$
Consider the Grassmannian $\operatorname{Gr}(2,n)$. I want to know Hilbert series of $H_1 \cap H_2 \dots \cap H_m \cap \operatorname{Gr}(2,n)$ in the Plücker embedding of $\operatorname{Gr}(2,n)$, ...
- 641
6
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0
answers
382
views
Is there a Dedekind domain which has infinite class group and is free of finite rank over a finite quotient PID?
Is there a Dedekind domain $B$ satisfying the following two conditions:
$B$ is an algebra over a finite quotient PID $A$ such that $B$ is free of finite rank as a module over $A$;
$B$ has infinite ...
- 3,813
6
votes
0
answers
963
views
Constructive contents of “the support of a sheaf is closed” or “the flat locus is open”
Out of curiosity I started to go through an introductory course on schemes and to convince myself that I could prove every result (after an appropriate reformulation if necessary) constructively, i.e. ...
- 1,153
6
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0
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265
views
Classification of one dimensional (non-commutative) formal group laws over $k[\epsilon]/(\epsilon^n)$
It's well-known that any one dimensional formal group law over a $\mathbb Q$-algebra or a reduced ring is commutative, but there are one dimensioal non-commutative formal group laws over rings like $k[...
- 6,612
6
votes
0
answers
407
views
Homogeneous regular sequence
Consider a $\mathbb{Z}$-graded polynomial ring $R = k[x_1,\cdots,x_n]$ over a field $k$, where the elements $x_i$ are homogeneous (they may have negative degree). Let $I$ be a homogeneous ideal that ...
- 71
6
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answers
111
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Relation between $\mathrm{Proj}(A[tI])$ and $\mathrm{Proj}(A[tJ])$ for ideals $I$ and $J$ of $A$ with $I^2 \subset J \subset I$
Let $k$ be a field and $A$ a noetherian local $k$-algebra.
Let $I$ and $J$ be two ideals of $A$ with $I^2 \subset J \subset I$.
Let $A[tI] = \bigoplus\limits_{i\geq 0} t^i I^i$ and $A[tJ] = \...
- 523
6
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0
answers
114
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A question about the span of a sequence of polynomials satisfying a linear recurrence
Let F be a finite field and A(n) in F[t], n in N, be defined by a linear recurrence with coefficients in F[t], together with initial conditions. Is there a decision procedure for determining whether ...
- 5,422
6
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0
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285
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On the prime spectrum of $R[[X]]$ when the prime spectrum of $R$ is Noetherian
All rings below are commutative with unity.
If $R$ has a.c.c. on radical ideals i.e. if $Spec R$ is Noetherian under Zariski topology, then so is $R[X]$, this is Theorem 2.5 in the following paper ...
- 412
6
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0
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867
views
How to extend Ritt's theorem on elementary invertible bijective elementary functions?
The elementary functions according to Liouville and Ritt are the functions of a complex variable built up by applying exponentiation, logarithms and/or algebraic operations finitely often. That means, ...
- 1,053
6
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0
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233
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Morphism that is surjective on PID points is surjective on every Dedekind domain?
Let $f: X=\Bbb A^n \rightarrow Y=\Bbb A^m $ be a morphism between affine spaces over an algebraically closed field $k$. Assume $f: X(R) \rightarrow Y(R)$ is surjective for any PID $R$ over $k$, under ...
- 6,612
6
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0
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357
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Ideals of orders in number fields
Let $K$ be a number field with ring of integers $\mathcal{O}_K$. Let $\mathcal{O}$ be an order of $K$ and $A$ a proper ideal of $\mathcal{O}$, by which I mean that $\mathcal{O} = \{\alpha \in K : \...
6
votes
0
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204
views
Counting exceptional divisors
Suppose that I blow up an ideal sheaf $J$ in $\mathbb A^2$ via a map $\pi : X \to \mathbb A^2$. I'd like to compute, from the ideal, how many exceptional divisors there are for $\pi$, and be able to ...
- 83
6
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0
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133
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What is the monoid of skew-symmetric trilinear forms on finite abelian groups?
I am interested in triple cup product operations on the cohomology ring $H^*(Y;\Bbb Z/p^r)$ of 3-manifolds. Trying to extract the algebra, I am led to the following question.
Let's fix a prime power $...
- 9,580
6
votes
0
answers
145
views
An ideal that there exists a unique ideal maximal with respect to not containing it
Let $I$ be a non-zero ideal of a commutative ring with identity. Is there any equivalent condition to the property that there exists a unique ideal maximal with respect to not containing $I$?
For ...
6
votes
0
answers
120
views
How does $\mathcal{O}_{C_K} / p \mathcal{O}_{C_K}$ looks like?
I am reading stuff about Fontaine's periods rings. Let $K$ be a $p$-adic field and $C_K = \widehat{\overline{K}}$. Then $\mathcal{O}_{C_K}/p\mathcal{O}_{C_K}$ isn't perfect, otherwise $R = \...
- 133
6
votes
0
answers
866
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Picard group of non reduced scheme
Let X be a non reduced scheme. $X_{red}$ be the reduced scheme. Is it true that Picard group of $X_{red}$=Picard group of X?
Is this map surjective $Pic (X)\rightarrow Pic(X_{red})$?
Is there a ...
user111251
6
votes
0
answers
190
views
Constructive approach to complete intersections
Let $K$ be a field (probably of positive characteristic) and consider the ring $R=K[\![x_1,\dotsc,x_n]\!]$. Suppose we have an ideal $I=(f_1,\dotsc,f_n)$ (with the same $n$ as before). Suppose we ...
- 56.9k
6
votes
0
answers
266
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Universal property of $A_{\mathrm{cris}}/p^n$
It is well known that the ring $A_{\mathrm{cris}}$ of Fontaine is the universal $p$-adically complete divided power thickening of $\mathcal{O}_{\mathbb{C}_p}$ over $\mathbb{Z}_p$; in fact, this is one ...
- 825
6
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0
answers
171
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Name for property of mixed characteristic DVR: admits regular local homomorphism from DVR with finite residue field
Does anybody happen to know if there is already a name in the literature for the following property of a mixed characteristic DVR: that there exists a local homomorphism that is regular into the ...
- 4,111
6
votes
0
answers
223
views
Book or survey on Dedekind-finite rings
I'm seeking a book or a survey providing an overview, as rich as possible, of the literature on Dedekind-finite (or von Neumann-finite) rings (let me recall that a unital ring $R$ is Dedekind-finite ...
- 10.5k