All Questions
6,055 questions
2
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Motivation for definitions of donor and receptor in Salamander Lemma?
$\newcommand{\im}{\operatorname{Im}}$Consider the following (subpart of) a double complex, using the same notation as in George Bergman's pre-print or in these lecture notes:
$$\require{AMScd}\begin{...
- 764
7
votes
0
answers
167
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Does a discriminant condition on $f(x,y)$ imply that $f$ is weighted homogeneous?
[This is an updated version of https://math.stackexchange.com/questions/4522399/.]
Let $f = \sum_{i=0}^n f_iy^i \in \mathbb{C}[x,y]$ be a polynomial (where $f_i \in \mathbb{C}[x]$ with $f_0,f_n \ne 0$)...
- 71
4
votes
1
answer
328
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Example of a certain type of Cohen-Macaulay ring
Let $k$ be a perfect field. I am looking for a $k$-algebra $R$ with the following properties.
$R$ is of finite type over $k$ and is a domain;
for all ${\mathfrak p}\in{\rm Spec}(R)$, the local ring $...
- 301
4
votes
0
answers
216
views
Is an orthogonal direct sum decomposition with respect to two quadratic forms necessarily unique up to isomorphism
Consider two quadratic forms $Q$ and $P$ over a finite dimensional vector space $V$ over a quadratically closed (or perhaps Pythagorean) field $F$. If $V$ can be decomposed as $V = V_1 \oplus V_2 \...
- 4,943
2
votes
1
answer
304
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Normal forms of ADE singularities
Given a surface $X:f(x,y,z)=0\subset \mathbb{A}^{3}_{\mathbb{C}}$ with only ADE singularities, how does one determine the correct singularity type of $X$ by computing the normal forms?
Does a similar ...
- 20.5k
0
votes
0
answers
179
views
Product/intersection of two ideals
Let $I_{1}$ and $I_{2}$ be two ideals in polynomial ring $C[u,v,t]$, defined as $I_1 = \langle t-u^3, (v-u^5)^2\rangle$ and $I_2 = \langle u^{11} + v^{11} + t^{11} + 3\rangle$. Is there a method for ...
- 1
2
votes
1
answer
290
views
Flat scheme-theoretic closure
Suppose $R$ is a discrete valuation ring with fraction field $K$. Let $X\subset \mathbf{P}^n_{C_K}$ be a closed subscheme, flat over $C_K$, a smooth projective curve over $K$.
Let $C_R$ be a flat ...
- 20.5k
0
votes
0
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91
views
Comparison of depth of two monomial ideals
Let $I$ and $J$ be two monomial ideals of $R=K[x_1,\dotsc,x_n]$ where $K$ is a field.
Could we say $\mid\operatorname{depth} (R/I)- \operatorname{depth}(R/(I:J))\mid\leq 1$, where $(I:J)$ is colon ...
- 113
3
votes
0
answers
375
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On the analogy between $p$-derivations and derivations
$\DeclareMathOperator\Spec{Spec}$Let $p$ be a prime number, and $A$ a commutative ring. Recall that a $p$-derivation on $A$, or a $\delta$-ring structure on $A$ is a set map $\delta : A \to A$ such ...
- 64k
0
votes
1
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147
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Are maps into a smooth curve equivalent to relative effective Cartier divisors?
Let $X$ be a smooth curve and $S$ an affine scheme, both over an algebraically closed field $k$.
Given a morphism $f : S \to X$, is it true that the graph morphism $\Gamma_f : S \to X \times S$ is a ...
- 39.3k
2
votes
2
answers
261
views
Examples of stretched artinian local ring
In Sally's Paper stretched artinian local ring is defined as :
Let $(R, \mathfrak{m})$ be an Artin local ring of length $\lambda.$ If $\nu$ is the embedding dimension of $R$, that is, $\nu$ is the ...
- 16.2k
4
votes
1
answer
146
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When is semigroup algebra local?
Let $G$ be a finite semigroup (or monoid if that helps) and $K$ a field.
Question: When is the semigroup algebra $KG$ local?
Here local means that there is a unique maximal right (or left) ideal.
...
- 38.6k
7
votes
2
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784
views
Is there a Hopf algebra-style description of chain complexes?
An old chestnut is that filtered objects are the same as sheaves over $\mathbb A^1 / \mathbb G_m$.
Question: Is there a similar description of chain complexes?
More precisely, if $\mathcal C$ is a ...
- 5,760
2
votes
0
answers
104
views
Kouchnirenko's theorem for non-generic polynomials
In Polyèdres de Newton et nombres de Milnor (Theorem 1.18), Kouchnirenko proved that given Laurent polynomials $f_1, \dotsc, f_k$ in $k$ variables, the number of isolated solutions is less than or ...
- 12.9k
3
votes
0
answers
116
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Intersection numbers of moduli spaces and noncrossing partitions
The coefficients of the monomials $u_1^{e_1}u_2^{e_2} \ldots u_n^{e_n}$ of the partition polynomials (ParPs) $[M=M1]$ on pg. 831 of The Handbook of Mathematical Functions by Abramowitz and Stegun are ...
- 10.5k
8
votes
5
answers
1k
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Additive integer-valued functions on the module category
This is inspired by the theorem mentioned in Why is this theorem attributed to Serre?. But I'm not sure if it's research level. If not, please feel free to vote for closing.
Let $R$ be a ring and ...
- 1,446
5
votes
1
answer
289
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Cataland: Facets and partition polynomials of cluster complexes
Figure 25 on pg. 101 of "Cataland: Why the Fuss?" by Christian Stump, Hugh Thomas, and Nathan Williams depicts cluster complexes (CCs) associated with generalized $(m)$-Narayana / ...
- 2,175
6
votes
1
answer
393
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Algebra generated by transformation matrices
Let $T_n$ be the full transformation monoid of an $n$-set $N_n$ with elements 1,...,n consisting of all functions $f: N_n \rightarrow N_n$.
We can associate to each function $f$ a matrix $M_f$ in the ...
- 18.4k
5
votes
1
answer
152
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Cartan matrix of the full transformation monoid ring
Let $T_n$ be the full transformation monoid of an $n$-set and $A_n=KT_n$ its monoid algebra over the complex numbers.
Question 1: Is the Cartan matrix of $A_n$ known? Im especially interested to see ...
- 26.5k
5
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0
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144
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Is there a good notion of higher-rank archimedean norm?
Let $K$ be a field. I think I know what a norm (archimedean or not) $|-| : K \to \mathbb R_{\geq 0}$ is. In the case where the norm is nonarchimedean, it's equivalent to the data of a valuation of ...
- 64k
3
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0
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225
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Intersection of two modules (and sub-modules) under tensors
I have a (hopefully quick) question regarding an intersection of two tensor modules. Let $K$ be a field and $A, B$ finitely-generated modules over the Laurent series $K((X))$. Let $\tilde{A}$ be a (...
- 31
4
votes
1
answer
182
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Effective bound on "Jacobian rank" for (regular) planar algebraic curves
Let an irreducible, square-free complex polynomial $f\in \mathbb C[x,y]$ be given. It is well known that the curve $\mathcal C:=\{f=0\}$ is nonsingular if and only if $\mathbb C[x,y]=<f,\partial_xf,...
- 5,428
12
votes
2
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775
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Hilbert polynomials of graded algebras evaluated at negative numbers
Let $k$ be a field and let $R$ be a commutative (standard) graded $k$-algebra, that is, $R=\bigoplus_{n=0}^\infty R_n$ with $R_0=k$ (and $R=k[R_1]$). The Hilbert function $h_R:\mathbb{N}\rightarrow \...
- 50.8k
2
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0
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195
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Interpretation of completed tensor product of algebras over lower base
Let $\mathbb F$ be a finite field of order $q = p^n$. It is known that
$$\mathbb F[[x_1]] \mathbin{\widehat{\otimes}_{\mathbb F}} \mathbb F[[x_2]] = \mathbb F[[x_1, x_2]].$$ Geometrically, this is the ...
5
votes
1
answer
481
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Alternative description of strict henselization
Let $R$ be a local ring with field of fractions $K$, maximal ideal $\mathfrak{m}$ and residue field $\kappa = R/\mathfrak{m}$. Let $R^\mathrm{sh}$ be a strict henselization of $R$, and let $L$ be the ...
- 148k
4
votes
1
answer
295
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Finite type injective ring map between domains preserves the open point $(0)$
I am looking for a proof of the following statement without using full power of Chevalley's theorem on constructible sets. We say a domain $A$ is $0$-open if $\{(0)\}$ is open in $\operatorname{Spec}(...
- 28.7k
22
votes
6
answers
8k
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A finitely generated $\mathbb{Z}$-algebra that is a field has to be finite
I was trying to understand completely the post of Terrence Tao on Ax-Grothendieck theorem. This is very cute. Using finite fields you prove that every injective polynomial map $\mathbb C^n\to \mathbb ...
- 12.9k
11
votes
2
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1k
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Is there a relation between Gelfand duality and the spectrum of a ring (with its Zariski topology)?
Compare the following two results:
Thm A) Let $A$ be a commutative $C^*$-algebra and let $X$ be its Gelfand spectrum. Gelfand duality says that there's a natural isometric $*$-isomorphism from $A$ to ...
2
votes
1
answer
174
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Understanding the picture of monoidal space
Ogus in his slides https://math.berkeley.edu/~ogus/preprints/colloqhandout.pdf presents the following picture of a monoidal space $\operatorname{Spec}(\mathbb{N} \longrightarrow \mathbb{C}[\mathbb{N}])...
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1
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1
answer
165
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Sufficient conditions to guarantee finite intersection points in Bezout's Theorem
Bezout's Theorem concludes that if $f_1,f_2,\cdots,f_n\in k[x_1,x_2,\cdots,x_n]$ have finite intersection points, then they have at most $d_1d_2\cdots d_n$ intersection points, where $d_i$ is the ...
1
vote
1
answer
149
views
The map from the ring of integers to the residue field of a valuation subring is surjective
Let $L$ be a number field and let ${\mathcal{O}}_L$ be its ring of integers. Let $B$ be a valuation subring of $L$ and let $k_B$ be the residue field of $B$. Then the map from ${\mathcal{O}}_L$ to $...
- 1,969
3
votes
2
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117
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When do there exist $ n $ commuting, derivations which locally generate $ T_{A/k} $ as an $ A $-module?
Let $ \operatorname{Spec}(A) $ be a non-singular, $ n $-dimensional, affine variety over a field $ k $ of arbitrary characteristic. For the definition of "variety" I use Hartshorne's ...
- 912
5
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1
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345
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Sufficient conditions for $\mathrm{Der}_k(A)$ to be f.g. projective
Let $k$ be a field and $A$ a commutative $k$-algebra. What are sufficient conditions for the module of derivations $\mathrm{Der}_k(A)$ to be finitely generated projective?
I'm looking for conditions ...
- 6,406
2
votes
2
answers
749
views
Excellent property of rings
Let $A$ be a commutative ring. If $A$ is an excellent ring, is the reduced ring $A/\sqrt{(0)}$ also an excellent ring?
- 28.7k
5
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2
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287
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Modulo $x^2 + y^2 - 1$, is every homogeneous polynomial that is a square of a polynomial, necessarily of sum of squares of homogeneous polynomials?
I am hoping this question is alright for Math Overflow. I didn't get a definitive solution in Math Stack Exchange.
Let $f(x, y) \in \mathbb{R}[x, y]$ be a homogeneous polynomial with real coefficients ...
- 6,237
1
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0
answers
72
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Extending prime ideals that lie over the same prime ideal via monomials
Let $(R, \mathfrak{m})$ be a local (Noetherian) ring containing the rationals. Then the formal power series ring $\mathbb{Q}[[x_1, \ldots, x_n]]$ naturally forms a subring of $R[[x_1, \ldots, x_n]]$. ...
4
votes
1
answer
370
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Determining when quotient of a polynomial ring is a Gorenstein ring
I would like to be able to look at the ring $R=\mathbb{Z}[x_1,x_2,\ldots,x_n]/\mathcal{I},$ where $\mathcal{I}$ is generated by a finite number of monomials and say whether $R$ is a Gorenstein ring. ...
- 20.5k
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votes
2
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1k
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Commutative von Neumann algebras and localizable measure spaces
This is not my subject so I apologize if my question is too obvious or understood from other pages.
I read some pages such as
Reference for the Gelfand-Neumark theorem for commutative von Neumann ...
- 17
0
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0
answers
61
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A sequence of polynomials that the variety defined by every $n$ of them is small
Let $\mathbb{C}[x_1,x_2,\cdots,x_n]_{= d}$ denote the set of polynomials in $\mathbb{C}[x_1,x_2,\cdots,x_n]$ of total degree $d$.
Is there exists a sequence of polynomials $f_1,f_2,f_3,\cdots$ in $\...
- 25
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0
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154
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Software for computing invariant rings
I have an linearly reductive algebraic group $G$ acting regularly on an affine variety $X$(over an algebraically closed field of characteristic 0). I want to compute the invariant ring $\mathbb{K}[X]^{...
- 839
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1
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527
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Geometric interpretation of a (standard) commutative algebra fact
Which is your geometric interpretation (if any) of the following commutative algebra proposition?
Proposition. Let $M$ be a finitely generated $A$-module, $I\subseteq A$ an ideal, and $\phi\in \...
- 263
1
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0
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119
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Germs of holomorphic functions and invariant functions
Consider a complex vector space $V \cong {\mathbb C}^n$. Consider the ring of germs of holomorphic functions ${\mathcal O}_0 (V)$ at $0\in V$. We know that this ring is Noetherian.
Now consider a ...
- 803
3
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0
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124
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Picard group of a cusp [duplicate]
$\DeclareMathOperator\Pic{Pic}$For a commutative ring with unity $R$, if $R_{\mathrm{red}}$ is semi-normal then we have an equivalent criterion which states that $\Pic(R) \cong \Pic(R[t])$. Here $\Pic(...
- 167
3
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2
answers
257
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Cancelable commutative monoids with finite maximal subgroups
Suppose $\mathcal{M} = (M, +, 0)$ is a cancelable commutative monoid. Let $G$ be the maximal subgroup of $M$, i.e.
$$G = \{ a \in M \colon (\exists b \in M)\, a + b = 0 \}.$$
For $a, b \in M$ say $a \...
- 38.6k
4
votes
0
answers
391
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Intersection of principal ideals is not principal
This may be an easy question for the right people, but I did not find an answer anywhere. I am trying to figure out what one can say about the ideal $a\mathbb{Z}[\lambda] \cap b \mathbb{Z}[\lambda]$ ...
- 135
3
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0
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217
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An attempt to extend polynomial rings
Suppose $\mathbb{K}$ is a field of characteristic $0$. Let $S_n=\mathbb{K}[[x_1,\dots,x_n]]$ be the ring of formal power series in $n$ variables, $L_n$ be its fraction field and $W_n=\mathbb{K}[x_1,\...
- 1,543
3
votes
1
answer
194
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Flatness over regular local rings of dimension 3
Let $R$ be a regular local ring of dimension $n$. Let $i:\text{Flat}\to\text{Fin}$ be the fully faithful inclusion of the category of flat finitely generated type $R$-modules into all finitely ...
- 19.6k
1
vote
1
answer
228
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On image of map $\text{Ext}^1_R(X,F)\to \text{Ext}^1_R(X,G)$ induced by $R$-linear map of free modules $F\to G$ with entries in the maximal ideal
$\DeclareMathOperator\Ext{Ext}$Let $(R,\mathfrak m)$ be a Noetherian local ring. Let $F,G$ be finitely generated free $R$-modules and $f:F\to G$ be an $R$-linear map such that $f(F)\subseteq \mathfrak ...
CommunityBot
- 1
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votes
0
answers
28
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Differentials for a nonsigular subvarity of a nonsigular variety
Let $A$ be a noetherian regular local ring of dimension $n$, and $P\subset A$ a prime ideal, such that $A/P$ still a regular local ring of dimension $m$.
I want to show $P/P^2$ is a $n-m$ rank free ...
- 39
5
votes
2
answers
555
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Another characterization of tensor products of modules
It is known that the tensor product is characterized by its universality in the category of $A$-modules.
Does the following proposition hold?
Proposition There exists only one operation $\otimes$ ...
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