Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
561
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The space of valuations of a function field
Hello, I'm looking for someone who can help me to understand Zariski's theory of valuations.
First I outline the theory: we take a field $K$ which is a finitely generated transcendent extension of ...
7
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2
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837
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Dimension of a homogeneous polynomial system
Let $m\geq4$ be an even integer, $V\subset\mathbb{C}^{m-1}$ be the solution set of the following polynomial equations:
\begin{cases}
&\sum\limits_{s=1}^{2t-1}z_sz_{2t-s}+\sum\limits_{s=2t+1}^{m-1}...
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Why are minimal resolutions of polynomial ideals important?
Background: Let $k$ be a field and denote by $P = k[x_1,\ldots,x_n]$ the polynomial ring in $n$ (commuting) variables over $k$. A resolution of an ideal $I \lhd P$ is an exact sequence of $P$-modules $...
7
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2
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Double dual of free $\mathbb{Z}_{(p)}$-modules
For an abelian group $A$, put $DA=\text{Hom}(A,\mathbb{Z})$ and $D_{(p)}A=\text{Hom}(A,\mathbb{Z}_{(p)})$. It is a theorem of Specker that when $A$ is free abelian of countable rank, the natural map $...
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Products of Ideal Sheaves and Union of irreducible Subvarieties
Assume I have a nonsingular, irreducible, algebraic variety $X$ and irreducible, nonsingular subvarieties $Z_1,\ldots,Z_k\subseteq X$. Let $\mathcal{I}_i$ be the ideal sheaf of $Z_i$ and $\mathcal{I}:=...
7
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1
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To compute minors of Jacobian of symmetric polynomials
For any $n$ tuple $f_1,f_2,\dots,f_n$ in the polynomial ring $\mathbb{C}[x_1,x_2,\dots,x_n]$
one has Jacobian, expressed by the $(n \times n)$-determinants:
$$
J(f_1,\dots,f_n):=|\frac{\partial}{\...
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Computing cotangent complex
I would like to know if one can compute all the cohomology sheaves of the cotangent complex of a subvariety of the affine space once a resolution of its ideal sheaf is given?
In my precise situation, ...
7
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2
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Criterion for being reflexive via Ext
In this question it was claimed that if a module $M$ over a noetherian domain $R$ satisfies $\rm{Ext}^i(M,R)=0$ for $i=1,2$, then $M$ is reflexive. Is this true? Does someone know a reference or a ...
7
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Pushouts of noetherian rings
Does the category of noetherian commutative rings have pushouts?
Background: If $X/S$ is an abelian scheme, then the relative Picard functor $\mathrm{Pic}_{X/S}$ is only defined on the category of ...
7
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2
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Zermelo's proof for unique factorisation
In Peter Bundschuh's "Einführung in die Zahlentheorie" I came across a possibly well-known but to me rather peculiar proof of unique factorisation, which is attributed to Ernst Zermelo. The proof ...
7
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What is the topology on the set of field orders
Inspired by this question I was wondering whether there is a natural topology on the set of all orders on a field (that extend a given order on a subfield)?
For example for the function field $\...
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0
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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{...
6
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2
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Properties of ring epimorphisms that are true only over commutative rings
I'm interested in knowing/collecting some properties of epimorphisms of rings (with identity) that are true over commutative rings but are false in the non-commutative case.
Example: I learned from ...
6
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Does a variety contain a cartesian product of two curves?
We are given an affine variety $V\subset \mathbb{A}^n\times\mathbb{A}^n$, and wish to know if it contains a product of the form $C_1\times C_2$, where $C_1$ and $C_2$ are two curves in $\mathbb{A}^n$. ...
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Do you know which is the minimal local ring that is not isomorphic to its opposite?
The most popular examples are non-local rings and minimal has 16 elements. I am interested in knowing examples of local rings not isomorphic to their opposite.
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Dimension of polynomial algebras
Let $R$ be a commutative ring of Krull dimension $d$, let $n\in\mathbb{N}$, and let $R[X_1,\ldots,X_n]$ denote the polynomial algebra in $n$ indeterminates over $R$. One can show that then we have $\...
6
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Is $[Im:(x)][Im:(y,z)]\subseteq Im$ in $k[x,y,z]$?
Let $k$ be a field and $S=k[x,y,z]$. Let $m=(x,y,z)$ and $I\subseteq m$ a proper homogeneous ideal in $S$. Is this true that we always have:
$$[Im:(x)][Im:(y,z)]\subseteq Im \ ?$$
In a paper we ...
6
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1
answer
550
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Projective dimension of graded modules
Short version:
Why is the projective dimension of a graded module the same as the projective dimension of its underlying ungraded module?
Longer version:
Let $G$ be a commutative group, let $R$ ...
6
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Valuation Rings and Ultrafilters
I notice there is a certain similarity between the definition of a valuation ring and the definition of an ultrafilter.
To begin, take a field $K$ and let $\mathcal{A}$ be the set of subrings of $K$. ...
6
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Does there exist a discrete valuation subring $R$ of $K((t))$ ($K$ a number field) of residue characteristic $p$ with $\mathrm{Frac}(R) = K((t))$?
Let $K$ be a number field, and let $K((t))$ be the field of formal Laurent series. Let $p > 0$ be a prime.
I have two questions:
Does there exist a discrete valuation subring $R$ of $K((t))$ of ...
6
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2
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Algebraic varieties and UFD
Given an affine algebraic variety $V$ such that $\Gamma(V,\mathcal{O}_V)$ is a UFD, its sheaf of ring can be determined easily since one can show that:
$$\Gamma(D(f_1) \cup \cdots \cup D(f_n),\...
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which automorphisms of a subring extend to those of a ring
(Probably a silly question, but..)
Consider the ring $R=k[[x_1,\dots,x_n]]/I$, (e.g. char(k)=0), and its subring, $R_1$, generated by some of $x_i$'s. In general, an automorphism of $R_1$ does not ...
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Tensor product of two algebras [closed]
Let $A$ and $B$ be algebras over a field $K$. The ideals of the tensor product $A\bigotimes_K B$ are of the form $I\bigotimes_K J$ where $I$ is an ideal of $A$ and $J$ is an ideal of $B$?
6
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0
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Non-crystallographic cluster algebras
Background
Fomin and Zelevinsky have introduced cluster algebras in an influential article. To define a cluster algebra, Fomin and Zelevinsky have defined a mutation of seeds. Here, a seed $(\mathbf{...
6
votes
1
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536
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Augmentation ideal is finitely generated if and only if $A$ is finitely generated as a $k$-algebra?
Let $A \subset k[x_1, \dots, x_n]$ be a subalgebra, which is also a graded subspace $A = \oplus_{i \ge 0} A_i$. One can write $A = A_0 \oplus A_{> 0}$ where we have $A_0 = k^0[x_1, \dots, x_n] = k$ ...
6
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Intuition for thinking about $R$-module of Kähler differentials, universal receptacles, derivations?
Suppose $k$ is a field of characteristic zero, and $R$ is a $k$-algebra. The $R$-module of Kähler differentials $\Omega_{R/k}$ of $R$ over $k$ with generators $\{dr\}_{r \in R}$ is the module subject ...
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Intersection of finitely generated subalgebras also finitely generated?
Let $k$ be a field and $A$ be a finitely generated (commutative) algebra over $k$. If $A_1$ and $A_2$ are finitely generated $k$-subalgebras of $A$, is it true that $A_1 \cap A_2$ is also finitely ...
6
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254
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Algebraization of Bayesian networks?
The algebraization of classical propositional logic is Boolean algebra.
Bayesian networks are a generalization of classical propositional logic with probability truth-values.
What is the ...
6
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1
answer
939
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Exact sequence for divisor class groups
Let $X$ be a either a projective scheme or a compact complex space. Then one has an exact sequence $$ (1) \quad 0 \to \textrm{Pic}(X) \to \textrm{Cl}(X) \to \bigoplus_{x \in \textrm{Sing}(X)} \textrm{...
6
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2
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The coefficient of a specific monomial in the expansion of the following polynomial
Let $a_{n,k}$ be the coefficient of $$X_1^{\frac{k(n-1)}{2}}X_2^{\frac{k(n-1)}{2}}\cdots X_n^{\frac{k(n-1)}{2}}$$ in the expansion of the real polynomial $$\left(\prod\limits_{1\leq i<j\leq n}(X_j-...
6
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2
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521
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A question about homogenous polynomials of degree $\frac{n(n-1)}{2}$
Let $n$ be a positive integer and $S_n$ be the symmetric group on $\{1,2,\ldots,n\}$.
For any $w\in S_n$ and polynomial $f\in \mathbb{R}[x_1,x_2,\ldots,x_n]$, denote $w(f)=f(x_{w(1)},x_{w(2)},\ldots,...
6
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Who defined and who coined "module"?
The title of my Q. says it all:
QUESTION: Who defined and who coined: module?
Would it be Emmy Noether?
EDIT In view of @anon's and KConrad's answers, and as it could have been ...
6
votes
2
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419
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Integer Gelfand-Kirillov dimension
Let $R$ be a (noncommutative) Noetherian affine $K$-algebra. The Gelfand-Kirillov dimension is known to be an integer for many classes of affine Noetherian algebras. I wonder, if this is true for any ...
6
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1
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Assuming $\operatorname{depth}M\ge \operatorname{depth}N$, what can one say about $\operatorname{depth}M_p$ and $\operatorname{depth}N_p$?
Let $(R,m)$ be a Noetherian local ring, $M$ and $N$ finite $R$-modules, $p$ a prime ideal, and $I$ an ideal such that $IM\neq M$.
Definition: The common length of the maximal $M$-sequences in $I$ is ...
6
votes
2
answers
363
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Connectedness of units in finite-dimensional commutative complex algebras
In the following, an algebra will always mean a finite-dimensional associative commutative unital algebra (over some field $k$).
Let $A$ be a $\mathbb{C}$-algebra. I am trying to understand how its ...
6
votes
1
answer
213
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Coloring summands of given n-partition with given weights of colors
Let $\lambda$ and $\sigma$ be partitions of $n$: $\lambda_1+\lambda_2+\cdots+\lambda_l=n$ and $\sigma_1+\sigma_2+\cdots+\sigma_s=n$
Let $M_{\lambda \sigma}$ be the number of ways to colour the parts ...
6
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2
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561
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Isomorphic rings of functions
Let $X$ and $Y$ be two topological spaces with $C(X) \cong C(Y)$ (where $C(X)$ is the ring of all continuous real valued functions on $X$). I know that we can not conclude that $X$ and $Y$ are ...
5
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0
answers
762
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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 ...
5
votes
1
answer
430
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Injective homomorphism of modules and tensor product
Let $R$ be a commutative ring; let $M$, $N$ be $R$-modules and let $f\colon M\to N$ be an injective homomorphism of $R$-modules. Is $f\otimes {\rm id}\colon M\otimes_RN\to N\otimes_RN$ injective?
5
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1
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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 ...
5
votes
2
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Do there exist general conditions for cyclicity of unit groups of quotient rings (generalizations of the primitive root theorem)?
Let $R$ by a commutative ring with $1$, and $I \subset R$ a non-zero integral ideal in $R$. When $R$ has finite quotients, and $I = P$ is prime in $R$, the group of units $(R/P)^{\times}$ of the ...
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485
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Brauer groups of a local ring and of its residue field
This is a question of DeMeyer (see the last paragraph of [1]):
What's an example of a local ring $A$ with residue field $k$ such that the restriction map on Brauer groups $\varphi : \operatorname{...
5
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0
answers
193
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A non-commutative analog of a result concerning a Jacobian pair
Let $k$ be a field of characteristic zero and let $E=E(x,y) \in k[x,y]$.
Define $t_x(E)$ to be the maximum among $0$ and the $x$-degree of $E(x,0)$.
Similarly, define $t_y(E)$ to be the maximum among $...
5
votes
0
answers
91
views
Generalized Puiseux series for diagonal reflections of the curves $y = \frac{x}{(1-ax)(1-bx)^m}$
Reflection of the curve $y = f_m(x) = \frac{x}{(1-ax)(1-bx)^m}$ through the diagonal line $y=x$ in the $xy$-plane can be regarded as local compositional inversion of the curve $y=f_m(x)$. ($x,y,a,b$ ...
5
votes
2
answers
900
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Is every commutative group structure underlying at least one (unitary, commutative) ring structure
From the theorem of classification of finitely generated abelian groups, we can see that every finitely generated commutative group can be considered as the additive structure underlying (at least) ...
5
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3
answers
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Atiyah-MacDonald: exercise 5.29 - "local ring of a valuation ring"
The exercise is the following:
Let $A$ be a valuation ring, $K$ its field of fractions. Show that every subring of $K$ which contains $A$ is a local ring of $A$.
Does anyone know what is meant by "...
5
votes
2
answers
434
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Name and notation for a binary operation
Is there a standard name or standard symbol for the binary operation that combines $x$ and $y$ to give $xy/(x+y)$, or equivalently $1/(1/x+1/y)$? (At least the expressions are equivalent if we ignore ...
5
votes
1
answer
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Why is this theorem attributed to J.-P. Serre?
Page $117$ of Atiyah, MacDonald's Introduction to Commutative Algebra text has the following theorem. Let $P(M,t)$ denote the Poincare- series of $M$.
$\textbf{Theorem.}$ $\bigl(\mathsf{Hilbert-Serre}...
5
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2
answers
228
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Descent of flatness from algebras to monoids
Consider a morphism of commutative monoids $u\colon M\rightarrow N$. We say that $u$ is flat, if the tensor product functor $\bullet\otimes_MN$ from the category of $M$-modules to the category of $N$-...
5
votes
1
answer
610
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Does perfection of rings commute with products?
The perfection of a ring $A$ of prime characteristic $p$ is the perfect ring $A_\rm{pf}=$ lim{$A\to A\to ...$} where all maps are Frobenius, so being a filtered colimit in Rings, the perfection ...