All Questions
669 questions
5
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Identity relating iterated determinant line bundles
Suppose that $R$ is a (commutative, unital) ring and that $A$ is a (commutative, unital) $R$-algebra that is projective of constant rank $n$ as an $R$-module. Then $A$ has a "determinant line ...
- 2,356
5
votes
0
answers
437
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Primary decomposition for non-affine schemes
I will call a (nonzero) ring primary if every zero divisor is nilpotent. (This implies that the prime spectrum is irreducible, although the converse does not hold.) An irreducible scheme I will call ...
- 7,338
5
votes
2
answers
1k
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Surjectivity of the natural map of injective module to its localization
Lemma 3.3 page 214 in Hartshorne's Algebraic Geometry book states: "If $I$ is an injective module over a Noetherian ring $A$. Then for any $f\in A$, the natural map of $I$ to its localization $...
- 427
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votes
2
answers
2k
views
Is a valuation domain PID when its maximal ideal is principal?
It is known that a valuation domain is a principal ideal ring if and only if its prime ideals are principal. Is it a principal ideal ring when its unique maximal ideal is principal?
- 1,207
4
votes
0
answers
157
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On skew monoid rings and skew ordered series rings
To my knowledge (see, e.g., H.H. Brungs and G. Törner's Skew Power Series Rings and Derivations [J. Algebra 87 (1984), 368-379]), skew polynomial rings were first introduced by Ø. Ore in 1933: Given ...
- 10.5k
4
votes
4
answers
3k
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Subtle examples of morphisms that are finite but not flat
Let $R$ be a ring (commutative noetherian with unit), and let $K(R)$ be its total ring of fractions (obtained by inverting all nonzerodivisors). Thus, $R \hookrightarrow K(R)$. Let $a \in K(R)$ be ...
- 7,338
4
votes
1
answer
327
views
Detecting closed immersions on fibers
Let $R$ be a dvr and $f : X\to S$ a universally closed morphism of $R$-schemes.
Assume $X$ and $S$ are $R$-flat and universally closed.
If the special fiber of $X\to S$ is a closed immersion, is $X\...
user315884
4
votes
1
answer
604
views
chain of prime ideals in polynomial ring $S=\Bbb{R}[x_1,x_2,...,x_n,...]$
Consider the polynomial ring of countable variables with coefficients in the real numbers, i.e, $S=\Bbb{R}[x_1,x_2,x_3,...,x_n,...]$. Is the following question true?
Question: Is there any maximal ...
- 45
4
votes
1
answer
446
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What is a "cusp" ("кусок") in relation to Guba's embedding theorem?
I'm confused by the definition of a "cusp" as found in
V.S. Guba, Conditions for the embeddability of semigroups into groups, Math. Notes 56 (1994), Nos. 1-2, 763-769 (link).
In the words of Mark ...
- 10.5k
4
votes
1
answer
240
views
Duality for rank one modules over a number ring
Let $K$ be a number field, and $R$ an order of $K$. Consider the category $\mathcal{M}$ of all finitely generated $R$-submodules of $K$. If $X$ is an object of $\mathcal{M}$ such that $R=\textrm{End}...
- 2,406
4
votes
2
answers
586
views
Brauer group of $\mathbb{Z}_{(p)}$
This may be a well known result but I could not find it in the standard references. What is the Brauer group of the local ring $\mathbb{Z}_{(p)}$ (the ring of integers localized at $p$)?
- 81
4
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0
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216
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Characterizing atomicity in a commutative domain
In Proposition 1.1 of [Math. Proc. Cambridge Phil. Soc. 64 (1968), No. 2, 251-264], P.M. Cohn famously claimed (without proof) that a commutative domain is atomic if and only if it satisfies the ...
- 10.5k
4
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0
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158
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Is the group ring of an amenable group, viewed as multiplicative monoid, amenable?
Motivated by this question, it seems natural to ask the following:
Question 1: Is there a [finitely generated discrete] torsion-free virtually Abelian (but not Abelian) group $G$ so that the ...
- 4,432
4
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0
answers
732
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Is there fppf descent of locally free modules
Being locally free is a property of quasi-coherent modules which
does not descend in the fpqc topology (see Remark Tag 05VF). But what happens for fppf coverings? More precisely we ask:
Suppose $A \...
user45795
4
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0
answers
362
views
Comparing different Euclidean algorithms on a Euclidean domain
I have posted this question at stackexchange (502413), without responses until now.
In the papers by T. Motzkin: The Euclidean Algorithm, Bull. AMS 55, 1949, pp. 1142--1146 and P. Samuel: About ...
- 2,454
4
votes
1
answer
435
views
Why we can analytically define $ε$ in dual numbers so to distinguish $ε$ from $-ε$ but cannot do so in complex and split-complex numbers?
Why we can analytically augment the algebraic definition of $ε$ in dual numbers so to distinguish $ε$ from $-ε$ but cannot do so in complex and split-complex numbers? Is there a theorem on this?
...
- 10.1k
4
votes
0
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169
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Integral domains with finitely many units
Question: Is there a classification of (noetherian if needed) integral domains with finitely many units ? (of course we can exclude fields as trivial examples)
Probably there are many such domains ...
- 26.5k
4
votes
2
answers
818
views
Double orthogonal complement of a finite module
Crossposted from math.stackexchange since I'm not getting any answer.
Let $W$ be the finite $\mathbb{Z}$-module obtained from $\mathbb{Z}_q^n$ with addition componentwise where $\mathbb{Z}_q$ is the ...
- 141
4
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0
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162
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Prime/irreducible elements in certain (integral) ring extensions
The answer to this question says the following:
Let $R$ be a finitely generated $k$-algebra, where $k$ is a field.
If $p \in R$ is a prime element, then $p$ is a prime element in $\tilde{R}$, the ...
- 2,837
4
votes
2
answers
660
views
Emptyness of a projective variety
Let $S$ be some (fixed) subset of $\mathbb{Z} [X_1, \dots , X_n]$ which contains only homogeneous polynomials, and if $F$ is a field, let $X(F)$ be the set of $ x \in P^{n-1}(F)$ such that $f (x) = 0$ ...
- 7,249
4
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1
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343
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Why do Nakajima and Watanabe claim the induced action of a finite linear group on the invariant subring of the reflection subgroup is linearizable?
I just picked up the paper "The classification of quotient singularities which are complete intersections" by Haruhisa Nakajima and Kei-Ichi Watanabe, which is in the book
Greco, Silvio, ...
- 2,851
4
votes
1
answer
362
views
Two bivariate polynomials (or rational functions) that generate $\mathbb{C}(x,y)$
Let $f=f(x,y),g=g(x,y) \in \mathbb{C}[x,y]$, each of degree $\geq 1$, and $f,g$ are algebraically independent over $\mathbb{C}$ (= their Jacobian $\in \mathbb{C}[x,y]-\{0\}$).
(1) Is there a ...
- 2,837
4
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1
answer
278
views
If $\mathbb{C}(u(x,y),v(x,y),f(x))=\mathbb{C}(x,y)$, for every $f(x) \in \mathbb{C}[x]-\mathbb{C}$, then already $\mathbb{C}(u,v)=\mathbb{C}(x,y)$?
The following question is a direct continuation of this elaborate question; it is mentioned there at the end:
Let $u,v \in \mathbb{C}(x,y)$ or $u,v \in \mathbb{C}[x,y]$, if it is easier to answer in ...
- 2,837
4
votes
1
answer
2k
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Arithmetically Cohen-Macaulay varieties
What do we mean by a variety being arithmetically Cohen-Macaulay? Is every such variety also Gorenstein?
- 61
4
votes
1
answer
194
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When $R $ is a cusp then $K_0(R) \ncong K_0(R[s])$
Quillen's classical result shows that if $R$ is a regular ring then $K_0(R) \cong K_0(R[t_1,...,t_m])$ for all $m \in \mathbb{N}$. So I wanted to construct some elementary examples where $K_0(R)$ ...
- 167
4
votes
2
answers
883
views
Group & modules of arbitrary cardinality [closed]
How do I see that there is a group of an arbitrary cardinality? Is this also true for abelian groups? Also, given a commutative ring $R\neq 0$ how do I see that there is an $R$-module of arbitrary ...
- 2,857
4
votes
1
answer
434
views
Cancellation problem for Laurent polynomial rings and power series rings
Throughout, let $k$ be an algebraically closed field. For two $k$-algebras $A,B$ let us write $A \cong_k B$ to mean that $A,B$ are isomorphic as $k$-algebras.
It is known that if $A$ is an integral ...
user111524
4
votes
1
answer
554
views
Generators vs minimal degree polynomials of ideals
Given an ideal $I$ of $\mathbb{R}[X_1,X_2,X_3,X_4,X_5]$ generated by two unknown polynomials. I know two homogenous polynomials $p_1 \in I$ and $p_2 \in I$ such that
$p_1$ is of degree 2 and up to a ...
- 1,256
4
votes
2
answers
2k
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When a smooth algebra is regular?
Let $A \subseteq B$ be noetherian integral domains, $A$ regular (=every localization at maximal ideal is a regular local ring) and $B$ is a smooth $A$-algebra. For the definition of a smooth algebra, ...
- 2,837
4
votes
1
answer
428
views
Cancellable elements of a power semigroup
For a semigroup $S,$ its power semigroup $P(S)$ is the semigroup of all non-empty subsets of $S$ with the operation given by $AB=\{ab\,|\,a\in A,b\in B\}.$ I would like to know about the cancellable ...
- 1,445
4
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0
answers
76
views
Minimal rank of a permutation resolution of a $G$-lattice
Let $G$ be a finite group.
By a $G$-lattice I mean a finitely generated free abelian group $L$ with an action of $G$.
One says that $L$ is a permutation lattice if $L$ has a $\mathbb{Z}$-basis ...
- 14.2k
4
votes
0
answers
908
views
Methods to check if an ideal of a polynomial ring is prime
Fix $\ell \geq 3$, $r \geq 2$ and $1 \leq k \leq \ell - 1$ and $z_1, \ldots, z_\ell \in \mathbb{C}$ with $z_i \neq 0$ for all $i$ and $z_i \neq z_j$ for all $i \neq j$. Now consider the (irreducible, ...
- 61
4
votes
1
answer
343
views
An analogue of rational functions for Hahn series
For any field $k$, we have both the field $k(t)$ of rational functions (formal quotients of polynomials, i.e. the field of fractions of $k[t]$) and the field $k((t))$ of formal Laurent series (which ...
- 66.8k
4
votes
2
answers
2k
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What are non-trivial examples of non-singular blow-ups of a non-singular variety?
This question arose from the responses to this question. The references to the comments of Karl Schwede and VA are to comments made there.
The blow-up of the variety $X=\mathbb{A}^2$ along the ...
- 3,284
4
votes
2
answers
377
views
Witt coordinates vs Joyal coordinates on the ring of Witt vectors
I am learning about Witt vectors following [K, Ch. 3], but I am having trouble with the presentation of his material (see (Q1) below).
Kedlaya's definition for ring $W(A)$ of the Witt vectors on a ...
4
votes
0
answers
185
views
"standard limit arguments" involved in showing that roughly every DM stack is locally a quotient stack
I am trying to understand proposition 3.6 of this paper (perhaps I am in over my head):
https://arxiv.org/pdf/math/0703310.pdf
If we denote the stack $\mathcal{M}$ and its coarse moduli space as $M$ ...
- 131
4
votes
1
answer
385
views
Which monoids can be realized as the monoid of ideals of a commutative monoid?
Let $H$ be a commutative monoid (written multiplicatively). We say that a set $I \subseteq H$ is an ideal of $H$ if $IH = I$. The set $\mathcal I(H)$ of all ideals of $H$ is made into a (commutative) ...
- 10.5k
4
votes
1
answer
1k
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How to compute the tangent space of a quotient by a finite group
Let $I\subseteq R:=\mathbb C[x_0,\ldots,x_n]$ be a homogeneous ideal defining a subscheme $X\subseteq\Bbb P^n$. As in my previous question, the permutation group $\mathfrak S_{n+1}$ acts on $R$ by ...
- 4,902
4
votes
0
answers
94
views
A dimension condition on the cohomology of a homogeneous space
The rational cohomology of a homogeneous space $G/K$ admits a homomorphism from $H^*(BK)$ induced from the classifying map $G/K \to BK$ of the principal $K$-bundle $G \to G/K$. Assume the Lie group is ...
- 2,995
4
votes
2
answers
625
views
A DVR algebra with weird automorphisms
Denote by $k$ an algebraically closed field. Can one produce a DVR $A$ over $k$ such that
the fraction field of $A$ has an automomorphism not preserving $A$
no non-trivial field extension of $k$ maps,...
user142965
4
votes
2
answers
710
views
commutative algebra, diagonal morphism
can anyone help me with the following statement (it is part of a bigger proof where it is not explained).
Let $B$ be a finite type commutative $A$-algebra (where $A$ is a commutative ring), and ...
- 303
4
votes
1
answer
225
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Significance of integrally closed in an affinoid algebra
A Tate affinoid k-algebra is defined as a pair $(R,R^+)$ where $R$ is a Tate algebra and $R^+$ is an open and integrally closed subring of $R$ contained in the ring of powerbounded elements.
See for ...
- 7,746
4
votes
1
answer
1k
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checking if F[x]/I is isomorphic to F[x]/J
Let $F$ be a field. Let $p$ and $q$ be monic members $F[x]$. Let $I = \{p\cdot r : r\in F[x]\}$ and $J = \{q\cdot r : r\in F[x]\}$. I know that if $F[x]/I$ is isomorphic to $F[x]/J$ then ($\...
user5810
4
votes
0
answers
210
views
A presentation for a subalgebra
Let $K$ be a field, and let $I=(g_1,\ldots, g_r)$ be an ideal in $A:=K[X_1,\ldots ,X_n]$.
Let $\{f_1,\ldots f_m\}$ be a subset of $A$, and let $B$ be the $K$-subalgebra of $A$ generated by $f_1,\...
- 5,039
4
votes
0
answers
691
views
Why are there elementary equations that are not solvable in closed form?
Elementary equations and closed-form solutions can be important in mathematics and in the natural sciences, e.g. for discovering and describing certain relationships.
$\log\colon x\mapsto\log(x)$; $x\...
- 1,053
3
votes
0
answers
144
views
Noncrossing partitions in Hopf algebras/monoids via compositional inversion
Partition polynomials constructed from the face structures of the associahedra (OEIS A133437) and permutahedra (A133314) comprise the antipodes/compositional inverses in a Faa-di-Bruno-type Hopf ...
- 10.5k
3
votes
1
answer
404
views
How many monoids with $n$ arrows exist?
How many monoids with strictly $n$ arrows exist? Is this known? I ask this only out of curiosity. Looking at $n=1,2,3,4$, this number doesn't appear to be very large relative to $n$.
- 41
3
votes
1
answer
196
views
Number of free summands of finite local extensions
Suppose that $(R, m) \subseteq (S,n)$ is a finite extension of normal local domains that is:
etale on the punctured spectrum
not flat / etale at the origin
and such that the residue fields $R/m = S/n$...
- 20.5k
3
votes
1
answer
285
views
Cancellative semigroup on a distributive lattice
Let $(S,\le)$ be a distributive lattice. Is there a semigroup structure on $S$ such that $S$ is cancellative and always $(x\wedge y)(x\vee y)=xy$?
- 1,145
3
votes
1
answer
3k
views
About the Definition of Flat Morphism (Flat Sheaf)
I have a confusion about the definition of flat sheaf of module.
Let $f: X \rightarrow Y$ be a morphism of schemes and $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$ module. Then $\mathcal{F}$ is flat ...
- 165