Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,497 questions
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Is $\mathbb{Z}$ universally definable in any number fields other than $\mathbb{Q}$?
In 2009, Jochen Koenigsmann showed that $\mathbb{Z}$ is universally definable in the field $\mathbb{Q}$. My question is, are there any other number fields in which $\mathbb{Z}$ is universally ...
- 2,051
2
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2
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899
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Number of generators of an ideal in a polynomial ring over a Noetherian ring
Let $R$ be a Noetherian ring. By the Hilbert Basis Theorem the polynomial ring $R[x_1, \ldots , x_n]$ is also a Noetherian ring. What can we say about the number of generators of an ideal $I$ of $R[...
- 35.5k
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Reverse mathematics of Noetherian rings over $\mathbb{Q}$
Take the Hilbert Basis Theorem over the rational numbers in this form in the language of Second Order Arithmetic: For every $n\in N$ every ideal of the polynomial ring $\mathbb{Q}[x_1,\dots,x_n]$ is ...
- 35.5k
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How many solutions are there to the equation $x^2 + 3y^2 \equiv 1 \pmod{p}$?
Let $p$ be a prime. How many solutions $(x, y)$ are there to the equation $x^2 + 3y^2 \equiv 1 \pmod{p}$? Here $x, y \in \{0, 1, \ldots p-1\}$. This paper (https://arxiv.org/abs/1404.4214) seems like ...
- 4,862
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Infinitely many initial ideals for non-Artinian monomial orders?
Consider the polynomial ring $R=\mathbb Z[x_1,\ldots,x_n]$ and an ideal $I\subset R$. Let $<$ be a monomial order, i.e. a total order on the set of monomials in $R$ such that for any monomials $a$, ...
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If $\mathfrak{p}\subset R$ is a minimal prime divisor of $\mathrm{Ann}_R(M)$, then $\mathrm{Ann}_R(M/\mathfrak{p}M)=\mathfrak{p}$
$\DeclareMathOperator\Ann{Ann}$Let $R=\mathbb{C}[x_1,\dots,x_n]$. I am looking for a reference of the following statement.
$(*)$ Let $M$ be an $R$-module, and let $\mathfrak{p}$ is a minimal prime ...
- 2,788
5
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1
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394
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Kähler differentials on an Artinian local ring
Suppose $R$ is a commutative Artinian local ring over an algebraically closed characteristic 0 field $k$. Suppose $f\in R$ is such that $df=0$ (in the sense that the element $df$ vanishes in the ...
- 63.9k
4
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355
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A noneffective descent datum: isomorphism not satisfying the cocycle condition
Let $S,S'$ be schemes, let $\pi : S' \to S$ be a morphism which is faithfully flat and locally of finite presentation, set $S'' := S' \times_{S} S'$ and $S''' := S' \times_{S} S' \times_{S} S'$ with ...
- 984
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matrix congruence and smith normal form
Fixed $n \geq 2$ and consider $A,B \in GL(n,\mathbb{Z}).$ We know that we have the Smith normal form. One can find $U, V \in SL(n,\mathbb{Z})$ such that $A=UDV.$ So as $B$. The Smith normal form is ...
- 121
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General conditions for normality of blow-up
Let $X$ be an integral, affine, normal complex surface. I am looking for conditions on zero-dimensional closed subschemes $Z$ in $X$ such that the reduced scheme associated to the blow-up of $X$ along ...
- 385
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Algebraic structure of the space of multiaffine maps
Let $V$ be a vector space over a field $\mathbb F$ and $k$ some natural number.
It isn't hard to show that the space of multiaffine maps $V^{[k]}\to\mathbb F$ decomposes as a direct sum of vector ...
- 40.2k
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Zero scheme of global sections of vector bundles on affine varieties
I want to understand better the notion of zero scheme of a section of a vector bundle. For simplicity I will consider the case of affine varieties.
Let $\mathbb{K}$ be an algebraically closed field, $...
CommunityBot
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Defining cluster algebras of finite type $\mathrm{A}$ by generators and relations
Consider a cluster algebra of finite type $\mathrm{A}$. The set of all (distinct) cluster variables is of finite cardinality, denote it by $k$, for such algebra. Is it true that, for an arbitrary ...
- 225
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Minimum number of generators of the product of ideals
Let $k$ be an algebraically closed field, let $I, J \subseteq k[x_1,\dots, x_n, y_1,\dots, y_m]$ be ideals, and let $i,j$ be the minimum number of generators of $I$ and $J$, respectively. I have two ...
- 30.5k
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How to compute cup product of derived limits / presheaf cohomology
I have a finite category $\mathcal{C}$, along with a functor $F \colon \mathcal{C}^{\mathrm{op}} \to \mathsf{GradedCommRings}$. If $F_j$ is $j$-th graded piece of $F$, then I write $H^i(\mathcal{C},...
- 37.8k
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$K_0(\mathsf{Nil}(R))$ when $R$ is a field
$\DeclareMathOperator\Nil{\mathsf Nil}\DeclareMathOperator\ker{ker}$I was reading through The $K$- book by Charles A. Weibel. There I found a very interesting category $\Nil(R)$, which consists of ...
- 63.9k
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Non-fractions in faithfully flat extension were non-fractions
All rings are commutative with unity.
Let $\phi:A \to B$ be a faithfully flat ring homomorphism. Let $f \in A$, $g = \phi(f) \in B$, and $\psi:A_f \to B_g$ the induced homomorphism on the ...
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Commutation of tensor products with inverse limits in a specific case
For $X,Y$ sets, let's denote $Y^X$ the set of all mappings $X\rightarrow Y$. If $Y(=R)$ is a ring, $R^X$ is a $R$-module (well, a bi-module but my question is - at first - concerning commutative rings)...
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Integer factorization given modular square root of 2
Let $N$ be composite. It is well-known that if $x^2 \equiv 1 \pmod N$, and $x \neq \pm 1 \pmod N$, then a factor of $N$ is easily found by computing gcd($N$, $x + 1$). I'm curious if there is a ...
- 1,703
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110
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Decomposition an $A$-module to irreducible ones
Let $V$ be a complex vector space (i.e, over the filed of complex numbers) and $A$ be a complex algebra.
Suppose that $V$ is an $A$-module. Under what proper condition(s) there are irreducible ...
- 4,058
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235
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Brauer group of the Henselization
Let $R$ be a Noetherian local ring and let $R^h$ be its Henselization. What can we say about the kernel and range of the map
$$
\operatorname{Br}(R) \rightarrow \operatorname{Br}(R^h)?
$$
Are there ...
- 12.9k
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2
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406
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When splitting of short exact sequence preserves the kernels
This is a problem that I thought at first was obvious but that became less clear the more I thought about it. Assume we have a finitely generated algebra $A$ over a field $k$, and a short exact ...
- 35.2k
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1
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282
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Number of cluster variables
In the paper Hernandez and Leclerc - Cluster algebras and quantum affine algebras, Section 13.5, it is said that when $\mathfrak{g}$ is of type $A_2$ and $\ell=2$, then the corresponding cluster ...
- 12.9k
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Connections to physics, geometry, geometric probability theory of Euler's beta integral (function)
Euler"s integral for the beta function $B(s,\alpha) = $ (with $x = 1$)
$$ \frac{(s-1)!(\alpha-1)!}{(s+\alpha-1)!} x^{s+\alpha-1} = \int_0^\infty t^{s-1}\; H(x-t) \; (x-t)^{\alpha-1} dt = \int_0^x ...
- 10.5k
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On Serre's "Local fields"
While I was reading J.-P. Serre's book "Local Fields" I found something strange in Chapter V. When Serre discusses properties of norm for unramified extensions, he says it is possible to ...
- 63.9k
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1
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Krull dimension and elimination theory over the integers
Let $K:=\mathbb{C}$, and let $R:=K[x_1,\dots , x_n]$.
Then, a system of polynomial equations $p_1=0, p_2=0, \dots , p_r = 0$, where the $p_i$ are polynomials in the $x_j$, has finitely many solutions $...
- 311
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Continuous differentiations of functional algebras
Let $A$ be some algebra (infinite-dimensional) of analytic functions on $\mathbb{C}^n$, and $D$ be some derivation of $A$, i.e. $D(fg)=Df \cdot g + f \cdot Dg)$ (so A may be considered as a ...
- 4,713
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Reference for integral extensions of $\mathbb{Z}/p^k\mathbb{Z}$
I was looking for a reference which discusses the structure of finite integral extensions of $\mathbb{Z}/p^k\mathbb{Z}$. In particular, I am interested in understanding what the abelian group of its ...
- 10.8k
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65
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Constant function on the generic fiber $f^{-1}(\eta)$ is contained in the function field $K(U)$
Let $U$ and $V$ be irreducible varieties and $f\colon V\rightarrow U$ be a proper surjective morphism.
Assume, $f^{-1}(\eta)$ is irreducible ($\eta$ is the generic point of $U$).
$\require{AMScd}$
\...
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Size of largest square divisor of a random integer
Let $x$ be an integer picked uniformly at random from $1 \ldots N$. Write $x = r^2 t$ where $t$ is square-free. How does the expected value of $r$ scale with $N$? Is anything known about the variance ...
- 151k
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Koszul complex of $xy$, $yz$ and $xz$
Has anyone computed the homology of the sequence $xy$, $yz$ and $xz$ in $\mathbb{C}[x,y,z]$?
CommunityBot
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Irreducibility of $\frac{x^{n+1}-(n+1) x+n}{(x-1)^2}$ [duplicate]
The question is motivated by this question.
Consider the polynomials
$$\dfrac{x^{n+1}-(n+1) x+n}{(x-1)^2} = \displaystyle \sum _{k=0}^n (n-k) x^k, n=1,2,3,\dots,$$
Are they all irreducible (over $\...
- 63.9k
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Can we deduce that two rings $R_1$ and $R_2$ are isomorphic if their polynomial ring are isomorphic?
Given two rings $R_1$ and $R_2$ (with or without identity: it's not specified). If $R_1[x]$ is isomorphic to $R_2[y]$ (No such requirement that the isomorphism sends the constant terms to constant ...
- 63.9k
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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
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Pushout and pullback in the category of rings
Consider the following pushout diagram
$\require{AMScd}$
\begin{CD}
A @>f>> B\\
@V g V V @VV V\\
C @>> > D
\end{CD}
in the category of $\textbf{Rings}$ where $f,g$ both are flat ...
- 4,709
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If a morphism from a commutative absolutely flat ring has integral fibers, does it induce an embedding of spectra?
All rings are commutative and unital.
Let $A$ be an absolutely flat ring and $A \rightarrow B$ a ring monomorphism with integral fibers (i.e. for each $\mathfrak{p} \in \operatorname{Spec}(A), B \...
- 1,388
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Online video of some courses
Who knows online video of Riemannian Geometry and Commutative Algebra? If you know, please recommend them to me. I am really eager to learn these courses.
- 30.3k
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Continous morphisms of a local field with conditions in positive characteristic
Let $P$ be a an irreducible polynomial of $k:=\mathbb F_q(T)$, $\Omega_P$ be the completion of an algebraic closure $\overline{k_P}$ of $k_P$, the completion of $k$ for the topology induced by the $P$-...
- 3,065
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Recognizing algebraic independence among Schur polynomials
Given a set of integer partitions $\{\lambda_1, \lambda_2,\dots \lambda_n\}$. Are there combinatorial criteria for deciding whether the associated Schur polynomials $s_{\lambda_1}, s_{\lambda_2},\dots ...
- 7,875
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265
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Sections of smooth morphisms over henselian rings
Let $(A,\mathfrak m)$ be a henselian local ring. Let $R$ and $S$ be $A$-algebras of finite type and $f\colon R\to S$ be a smooth morphism. Assume that the induced morphism $R/\mathfrak m R\to S/\...
- 1,590
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Elliptic units as Euler systems
I’m trying to understand elliptic units in order to work with the Euler systems of the abelian extensions of quadratic imaginary number fields. I’ve looked at few references about the topic, but they ...
- 63.9k
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Is every field the residue field of a discretely valued field of characteristic 0?
Let $k$ be a field of positive characteristic $p$. Is there necessarily a discrete valuation ring of characteristic $0$ with maximal ideal $(p)$ and residue field isomorphic to $k$?
- 9,263
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607
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Prime ideals and localizations of the ring $\mathbb Z[\{\sqrt p: p \text{ prime}\}]$
I have been trying to study the prime ideals of the ring $R:=\mathbb Z [\{ \sqrt{p_n}\}_{n=1}^\infty]$, where $p_n$ denotes the $n$-th prime. This is how far I got: I could conclude, by means of the ...
CommunityBot
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428
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Analytic functions in arbitrary rings?
We have developed a rich theory of analytic functions over $\mathbb{R}^n$ and $\mathbb{C}^n$. This is pretty reasonable, as analyticity here (local representation by power series) is closely linked to ...
- 2,519
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1
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857
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What is the motivation for excellent rings?
First of all I am not formally educated in mathematics so pardon my ignorance if this is obvious and I am skipping something vital, but I am interested nonetheless in what the original motivation and ...
- 321
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71
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Gluing categorical limit over subgraphs
Let $C$ be a category, and $\Gamma$ a graph in $C$. Under good conditions it makes sense to talk about the limit $\lim \Gamma$ of $\Gamma$ in $C$.
Suppose $\Gamma$ is the union of two subgraphs $\...
- 5,230
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770
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Integrally closed factor rings and projective modules
I have a weird vision that comes from reading a paper by Raphael and Desrochers..
Let $R$ be commutative unitary semiprime ring such that for any integral and essential element $a$ of $R$, $R[a]$ is ...
- 938
2
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1
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168
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Adding first generator to Cohen-Macaulay monomial ideal
Let $I$ be a Cohen-Macaulay monomial ideal of $R=K[x_1,...,x_n]$, where $K$ is a field. Can we say the ideal $(x_1)+I$ is Cohen Macaulay?
- 63.9k
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Cohen-Macaulay monomial ideal
Let $R=K[x_1,...,x_n]$ be the polynomial ring over a field $K$ and $I[x_1,...,x_n]=(u_1,...,u_t)$ be a Cohen-Macaulay monomial ideal of $R$. If $m<n$, could we say that $I[x_1,...,x_m,0,0,...,0]$ ...
- 1,313
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Is the Pierce spectrum useful elsewhere in Mathematics?
In Borceaux and Janelidze's Galois Theories, a construction of the Pierce spectrum is given. It is the poset of ideals in a Boolean ring. It's construction is reminiscent of the Zariski spectrum in ...
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