Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,496 questions
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Coordinate ring of a flag variety
Edited:
[If G here is a simply connected semismple complex algebraic group.
A partial flag variety $G/P$ can be naturally embedded as a closed subset of $\prod_j \mathbb{P} (L(\omega_j)^*)$.
The ...
- 201
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44
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Polynomial representation with shared root
Let $R$ be a polynomial quotient ring of the form $F_3[x_1,x_2,...,x_n]/\langle \{x_i^3-x_i\}_{1\le i\le n} \rangle$ and $\{f_i\}_{1\le i \le m}$ be elements of $R$, where $m > n$. We know that the ...
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Is there a preferable convention for defining the wedge product?
There are different conventions for defininig the wedge product $\wedge$.
In Kobayashi-Nomizu, there is $\alpha\wedge\beta:=Alt(\alpha\otimes\beta)$,
in Spivak, we find $\alpha\wedge\beta:=\frac{(k+l)...
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Ideal generated by a regular sequence
In Boocher and Grifo - Lower bounds on Betti numbers, in example 2.2 they say that if $R=k[x_1,\dotsc,x_n]$ is a polynomial ring and $M=R/(f_1,\dotsc,f_c)$ where $f_i$ form a regular sequence, then ...
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zero divisors of group ring when the group is abelian
Let G be an abelian group with torsion and C[G] be the group ring over complex numbers C. Is there a clear description or classification of zero divisors of C[G]?
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Equivalence between smoothly regular and analytically regular
I think the following statement is true.
Let $M$ be a real analytic manifold. Let $S \subset M$ be an analytic or semianalytic subset. A point $p \in S$ is called smoothly regular resp. analytically ...
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Reference request: Étale base change of differential-graded algebras
I've asked this question on Math.StackExchange, but didn't receive an answer, so I'd like to try my luck here.
I'm looking for a reference for the following fact, which I've recently stumbled upon:
...
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284
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Generic behavior of "polynomialish" models of $\mathsf{Q}$
(This question was originally asked and bountied at MSE - with different notation, some more explicit arguments, and topology in place of forcing.)
Suppose $\mathcal{R}=(R_i)_{i\in\mathbb{N}}$ is a ...
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Dirichlet unit theorem for finite rings
Let us fix a square free positive integer $n\in\mathbb{N}$ and consider the number field $\mathbb{Q}(\sqrt n)$ with ring of integers $K=\mathbb{Z}[\sqrt n]$. Let us denote the Galois norm of elements ...
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Quillen–Suslin theorem in a more general context
Let $A$ be a finite dimensional local Frobenius algebra that is Koszul.
Question: Is it true for the Koszul dual of $A$ that every finitely generated projective module is free? If not, is there a ...
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What is the "intuition" behind "brave new algebra"?
Y.I. Manin mentions in a recent interview
the need for a “codification of efficient new intuitive tools, such as … the “brave new algebra” of homotopy theorists”. This makes me puzzle, because I ...
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Graded analogues of theorems in commutative algebra
Many theorems in commutative algebra hold true in a ($\mathbb{Z}$-)graded context. More precisely, we can take any theorem in commutative algebra and replace every occurrence of the word
commutative ...
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Automorphism group in finite dimensional case
Let $K$ be a field, $G_a := (K, +)$ be the additive group of $K$, and $X$ an affine variety.
I found the following claim: if $X$ admits a non-trivial $G_a$-action and $\dim(X) \ge 2$, then the group $\...
- 63.9k
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On the map $\Phi_M: M\otimes_RM^*\xrightarrow{x\otimes y\mapsto \left\{f\mapsto f(x)y\right\}}\text{Hom}_R(M^*,M^*) $
$\DeclareMathOperator\Hom{Hom}$Let $M$ be a finitely generated module over a Noetherian local ring $(R,\mathfrak m)$. Denote $(\_)^*:=\Hom_R(\_,R)$. There is a natural map
\begin{align} \Phi_M: M \...
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Expressing elements in Verlinde ideal in terms of generators
It is known that the level $k$ Verlinde ring of $SU(n)$ is $R(SU(n))/I_k$, where $I_k$ is the Verlinde ideal. A set of generators of $I_k$ is given by $\{V_{(k+i)L_1}:=\text{Sym}^{k+i}V_{\text{std}}| ...
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Sets of $\mathbb{F}_p$-points of closed subvarieties of $\mathbb{A}^n$
Let $p$ be a prime and let $n\geq 2$ be an integer.
The set $\mathbb{A}^n(\mathbb{F}_p)$ has $p^n$ elements so it has $2^{p^n}$ subsets. How many of those subsets are of the form $V(\mathbb{F}_p)$ ...
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K-projectivity for rings of finite homological dimension
Let $R$ be a Noetherian commutative ring. A complex of $R$-modules $P^{\bullet}$ is K-projective if for any acyclic complex $A^{\bullet}$, the complex of abelian groups $ Hom(P^{\bullet}, A^{\bullet})$...
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Terminology for commutative ring whose Jacobson radical $J$ is nilpotent with semisimple quotient $R/J$
Is there a name for the following property of a commutative ring $R$:
its Jacobson radical $J$ is nilpotent, and $R/J$ is semi-simple?
(It is easily equivalent to: $R$ is a finite product of ...
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Division algorithm for multivariable power series
Let $\mathbb{Z}_p$ be the ring of $p$-adic integers. Consider the ring $R=\mathbb{Z}_p[[T]]$. Let $f,g \in R$ and assume that $f=a_0+a_1T+...$ with $a_i \in p\mathbb{Z}_p$ for $0\le i \le n-1$, but $...
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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 ...
- 1,286
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Complexity: Groebner bases method vs homotopy continuation method
Today, I came to know about homotopy continuation method to solve system of multivariate polynomials. This method finds its roots from the field of Numerical algebraic geometry.
I already know that ...
- 63.9k
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215
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Using equational Jacobson condition to prove element lies in radical of ideal
Recall the Jacobson radical of a ring consists of elements $f\in A$ such that $1-gf\in A^\times$ for every $g\in A$. Say an ideal $I\vartriangleleft A$ is Jacobson if in the quotient $A/I$ the ...
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Proof of Artin–Rees / Krull intersection motivated by universal property of blowup
I was very confused by the proof of Artin–Rees / Krull intersection theorem when I was younger.
Now that I learnt about blow up— I saw the Rees algebra again and I want to now gain a better ...
- 12.9k
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400
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Bound for multiplicities of closed points on scheme
Let $K$ be a perfect field, and let $f_1, \ldots, f_m \in K[X_1,\ldots,X_n]$ be polynomials. Consider the affine scheme
$$X = \mathrm{Spec}(K[X_1,\ldots, X_n]/(f_1,\ldots,f_m))$$
and let $N = \dim(X)$....
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What are the properties of this set of infinite matrices and operations on them?
Consider infinite matrices of the form
$$\left(
\begin{array}{ccccc}
a_0 & a_1 & a_2 & a_3 & . \\
0 & a_0 & a_1 & a_2 & . \\
0 & 0 & a_0 & a_1 & . \\
...
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Multiplicative identity of determinant of multiplicative action of a polynomial on a quotient ring (companion matrices)
Let $A$ be a commutative ring with $f,g\in A[x]$ monics. Consider the $A$-linear endomorphism $\mu_g^{(f)}\in \mathrm{End}_A\tfrac{A[x]}{\langle f\rangle}$ given by multiplication by $g$.
For monics $...
- 6,262
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Uncountable integral domain such that every countable subset is contained in a finitely generated $\mathbb{Z}$-algebra
Is there an uncountable integral domain such its every countable subset is contained in a finitely generated $\mathbb{Z}$-algebra?
- 63.9k
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Can the embedding dimension of a finite local algebra change after restricting to a finite subfield?
The embedding dimension of a commutative $k$-algebra is the minimum $n$ such that it is a quotient of the polynomial algebra in $n$-variables.
The embedding dimension of $\mathbb{F}_2\times \mathbb{F}...
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Is a proper quotient of $\mathbb{F}_{q^n}[x]$ considered as an $\mathbb{F}_q$-algebra always a quotient of $\mathbb{F}_q[x]$?
Let $\mathbb{F}_{q^n}/\mathbb{F}_q$ be an extension of finite fields.
Is a proper quotient of $\mathbb{F}_{q^n}[x]$ considered as an $\mathbb{F}_q$-algebra always a quotient of $\mathbb{F}_q[x]$ (i.e. ...
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Algebraic machinery for algebraic geometry
Hello everybody,
I'm a math student who has just got his first degree, and I am studying algebraic geometry since a few months. Something I have noticed is the (to my eyes) huge amount of commutative ...
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Structure of $\bigwedge^{2}_{\mathbb{Z}}(A)$ with $A$ a local integral domain
I am trying to see the structure of $\bigwedge^{2}_{\mathbb{Z}}(A)$ where $A$ is a local integral domain with small residue field.
Let $A$ be a local integral domain with maximal ideal $M$, residue ...
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Finding an injective envelope containing another injective envelope
Let $R$ be a local principal ideal domain (PID) with only two prime ideals $0$ and $P$, and let $M$ be an $R$-module. Let for $r\in R$ and $m\in M$, $rm\not=0$. Now if $E(rm)$ is a fixed injective ...
- 12.9k
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Ideals of $F_2[x_1, x_2, \cdots, x_n]/(x_1^2, x_2^2, \cdots x_n^2)$
I am interested in the poset of all ideals of the local ring
$$R_n = \mathbb{F}_2[x_1, x_2, \cdots, x_n]/(x_1^2, x_2^2, \cdots x_n^2).$$
$n=1$ is trivial. $n=2$ takes little work and it is shown below....
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Invariants of matrices (by simultaneous $\mathrm{GL}_n$ conjugation) over arbitrary rings
$\DeclareMathOperator\GL{GL}$Let $R$ be a commutative ring, let $R[n] := R[M_d^{\oplus n}]$ be the polynomial ring on $nd^2$ variables corresponding to the coordinates of $n$-many $d\times d$ matrices....
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Why is $M$ torsion-free?
I am studying the following article
https://www.math.nagoya-u.ac.jp/~takahashi/tc9.pdf
The main theorem is the Theorem 3.3. Howewer, I have the following questions about the proof:
How does it help ...
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Mayer-Vietoris sequence from a bicartesian square of commutative rings
An article that I am reading quotes the following theorem (5.3 p.481, reformulated to focus on the commutative case) from Algebraic K-Theory by Hyman Bass:
Let $\require{AMScd}$
\begin{CD}
A @>p_2&...
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On some loci of rings
Let $(R, \mathfrak m)$ be a Noetherian local ring. Let P be a property of $R$. Set
$$ P(R) =\{\mathfrak p \in Spec(R)\,\,\, |\,\,\, R_{\mathfrak p}\, \, \mbox{is } P\},$$
$$ nP(R) =\{\mathfrak p \in ...
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When is the following a formula for local cohomology?
Suppose $R$ is a Noetherian local ring, and $\kappa$ its residue field. For $R$ module $M$, we can consider the module
$$N:=\kappa \otimes_S RHom(\kappa,M)$$ where $S$ is the derived ring of ...
- 2,356
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A variation on $k(x^2,x^3)=k(x)$
Let $k$ be a field of characteristic zero, for example $k=\mathbb{R}$ or $k=\mathbb{C}$.
Of course, $k(x^2,x^3)=k(x)$, since $x=\frac{x^3}{x^2}$.
Let $f_1,\ldots,f_n,g_1,\ldots,g_m \in k[x]$, $n,m \...
- 149k
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There is a ring with multiplication. Can we find a formula for division based on formula for multiplication?
Studying divergent integrals, I found a good formula for their multiplication:
$\int_0^\infty f(x)dx\cdot\int_0^\infty g(x)dx=\int_0^\infty D^2 \Delta^{-1} \left(\Delta D^{-2}f(x)\cdot\Delta D^{-2}g(x)...
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Polynomials which are functionally equivalent over finite fields
Recall that two polynomials over a finite field are not necessarily considered equal, even if they evaluate to the same value at every point. For example, suppose $f(x) = x^2 + x + 1$ and $g(x) = 1$. ...
- 149k
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de Rham cohomology of a specific ring
I'm running into a certain algebraic de Rham cohomology computation I could use some help with. Specifically, what is the algebraic de Rham cohomology of:
$$
\mathbb{C}[x_1,\dots,x_n,y_1,\dots,y_n,(r^...
- 63.9k
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Does the strict henselization satisfy Going-Up?
$\DeclareMathOperator\sh{sh}$This question is cross-posted from Math.SE where it has gone unanswered for a week -- perhaps it is harder than I guessed. My question is this:
Let $A$ be a local ...
- 2,851
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$k[X_1,\ldots,X_n]/Q$ is UFD for non-singular quadratic form $Q$ and $n\ge 5$
I am looking for a reference for the following result. Thanks in advance.
Let $k$ be a field of any characteristic other than $2$.
Klein and Nagata showed that the ring $R:=k[X_1,\ldots,X_n]/Q$ is a ...
- 26.5k
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1
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230
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Properties of the generic matrix - struggles with constructive proofs
Write $A=(x_{ij})$ for the generic matrix (comprised of indeterminates) defined over $\mathbb Z[x_{11},\dots,x_{nn}]$. In their constructive commutative algebra book, Lombardi and Quitte write that ...
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A Nakayama type of claim for countably generated modules on complex affine varieties
Let $U\subset \mathbb{A}^n_{\mathbb{C}}$ be any Zariski open affine subvarity. Let $M$ be an $\mathcal{O}(U)$-module. Suppose $M$ satisfies $M\overset{L}{\otimes}\mathbb{C}_{\mathfrak{M}}\cong 0$ for ...
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502
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Infinite linearly independent set in finitely generated module
Let $R$ be a (commutative, otherwise the answer is easy, see the comment below) ring and let $M$ be a finitely generated $R$-module. Is it possible that $M$ admits an infinite linearly independent set?...
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Examples of solid abelian groups
I am reading through Clausen's and Scholze's Lectures on condensed mathematics. I am struggling to understand the concept of solid abelian groups so I am looking for some examples.
Is the underlying ...
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236
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Is this property of polynomials generic?
Let $n \geq 2$, and consider a polynomial $f$ in $n$ variables, say over a field $K$ of characteristic 0. Recall that $f$ is geometrically irreducible if $f$ is irreducible over the algebraic closure ...
- 26.9k
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Choosing generators of a submodule with divisibility properties
Looking at an open subset $U$ of the plane, containing $0 \in \mathbb{C}^2$, with coordinates $x$ and $y$.
Given a quotient sheaf $O_U^n \rightarrow T$, with $supp(T)=\lbrace0\rbrace$. Let $K$ be the ...
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