Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,493 questions
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$F=\mathbb{C}(u,v)$ satisfying: For every $a,b \in \mathbb{C}[y],c,d \in \mathbb{C}[x]$: $\mathbb{C}(x,y)=F(ax+b)=F(cy+d)$
Let $u,v \in \mathbb{C}[x,y]$, where $u$ and $v$ are algebraically independent over $\mathbb{C}$ and $F=\mathbb{C}(u,v)$. Of course, $d:=[\mathbb{C}(x,y):F] < \infty$.
Denote the following ...
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Transcendence degree and Krull dimension for homomorphic images of power series rings
Let $k$ be a field of characteristic zero and $I$ be a radical ideal of $k[[x_1,\ldots, x_n]]$. Let $P$ be a minimal prime ideal of the reduced ring $R:=k[[x_1,\ldots, x_n]]/I$. Then, $R_P$ is a field ...
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Example of non injective module over Noetherian local ring with trivial vanishing against residue field?
Is there an example of a module $M$ over a commutative Noetherian local ring $(R,\mathfrak m, k)$ such that $\text{Ext}_R^{>0}(k, M)=0$ but $M$ is not an injective $R$-module?
I know that for such ...
- 480
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Alternating forms on abelian groups
Let $G$ and $H$ be abelian groups. By an alternating form, I mean a bilinear function $A\colon G\times G\to H$ such that $A(x,x)=0$ for all $x\in G$.
Question. If $A\colon G\times G\to H$ is an ...
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Computing the minimal polynomial of roots of polynomials with algebraic coefficients
Let $p(x) = \sum_{i=0}^{n} c_i x^i$ with $c_i \in \mathbb{A}$ with $q_i(c_i) = 0$ and all $q_i \in \mathbb{Q}[x]$ being minimal polynomials of the coefficients.
Let $r$ be a zero of $p(x)$. Is there ...
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If $E \subseteq F=k(x_1,\ldots,x_r)$, satisfies $E(x_1^{i_1},\ldots,x_r^{i_r})=F$, for every $(i_1,\ldots,i_r) \neq (0,\ldots,0)$, then $[F:E] \leq 2$
For $r \geq 2$, let $A_r=\mathbb{C}[x_1,\ldots,x_r]$,
$F_r=\mathbb{C}(x_1,\ldots,x_r)$ the field of fractions of $A_r$, and $E_r \subseteq F_r$ an arbitrary subfield of $F_r$ with $[F_r:E_r] < \...
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Minimal injective resolution and change of rings
Let $R$ be a commutative Noetherian ring. For an $R$-module $M$, let $0\to E^0_R(M)\to E^1_R(M)\to \ldots $ denote the minimal injective resolution of $M$. I have two questions:
(1) If $I$ is an ...
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When $x=\frac{u(f_i,g_j)}{v(f_i,g_j)}$ implies $x=\frac{u(f_i(x,0),g_j(x,0))}{v(f_i(x,0),g_j(x,0))}$ ($x=\frac{xy}{y}$ does not imply $x=\frac{0}{0}$)
Let $f_i=f_i(x,y), g_j=g_j(x,y) \in \mathbb{C}[x,y]$,
$1 \leq i \leq n$, $1 \leq j \leq m$, be such that
$f_i(x,0) \neq 0$ and $g_j(x,0)=0$.
Assume that $\mathbb{C}(f_1,\ldots,f_n,g_1,\ldots,g_m)=\...
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Localization of quasi-excellent rings are quasi-excellent?
If $R$ is a quasi-excellent ring, then is $R_{\mathfrak p}$ also quasi-excellent for every prime ideal $\mathfrak p$ of $R$ ?
I think Matsumura's commutative ring theory book says that localization of ...
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Integral preimages of topologically noetherian, affine schemes
Let $A\to B$ be a ring homomorphism, $d\in \mathbb{Z}_{>0}$ and let $C=\bigotimes^d_A B$ the $d$-fold tensor product of $B$ over $A$. Then $\mathfrak{S}_d$, the symmetric group of $d$ elements, ...
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Finding $\mathbb{C}(u,v)$ such that $\mathbb{C}(u,v,x^p+y^p)=\mathbb{C}(x,y)$, for every prime number $p$
Denote the set of prime numbers by $P$, $P=\{2,3,5,7,\ldots\}$.
Let $F \subseteq \mathbb{C}(x,y)$ be a subfield of $\mathbb{C}(x,y)$, and for $w \in \mathbb{C}[x,y]$ denote by $F(w)$ the subfield of $\...
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Formalization of the independance of products in a (commutative) semigroup
1/ It is well known that associativity implies that the result of the product of an ordered finite set of elements in a semigroup does not depend of the order of composition of the partial products.
...
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135
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Artinian Gorenstein subrings with same socle degree
I am looking for examples of Artinian Gorenstein subalgebras with the same socle degrees.
More precisely, let $A$ be an Artinian Gorenstein $k$-algebra (with $k$ algebraically closed of characteristic ...
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Description of prime ideals in $\operatorname{Spec}(\mathbb{Z}[x_1, \dots, x_n])$
Edit: This seems to be wrong, as pointed out by Will Sawin in the comments.
The prime ideals of $\mathbb{Z}$ and $\mathbb{Z}[x]$ are well-known. It is also not too hard to compute the underlying set ...
- 461
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Conormal module of a commutative Koszul algebra
Does the conormal module $I/I^2$ of a commutative Koszul algebra $A=k[x_1,\dots,x_n]/I$ have a linear minimal free resolution? Is there a formula for the Betti numbers of $I/I^2$, or, even better, a ...
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Localization of totally acyclic complex or projective modules remain totally acyclic?
Let $R$ be a commutative Noetherian ring. An acyclic complex $P$ of projective $R$-modules is called totally acyclic if for every projective $R$-module $Q$, the complex Hom$_R(P, Q)$ is also acyclic.
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Algorithm for finding generating sets of projective modules
Suppose $R$ is a (Dedekind) domain and $M$ is a projective module of constant rank over $R$. We know that $M$ is finitely generated over $R$. I'm wondering is there any algorithms to produce a (...
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Integers as polynomials in infinite variables
This question is more of a request for reference or ideas than else. Forgive (or correct) if there are imprecisions or blatant mistakes.
The main idea is that the unique factorization theorem for $\...
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Examples of semirings where the additive neutral element is not absorbing for multiplication
In the case of a non unital ring, the additive 0 must be absorbing for the multiplication because we have a⋅0 = a⋅(a − a) = a⋅a − a⋅a = 0 and similarly on the other side.
In the case of a unital ...
- 2,655
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Steinitz isomorphism theorem for non-Dedekind domains
(Cross-posted from https://math.stackexchange.com/questions/4931582/steinitz-isomorphism-theorem-for-non-dedekind-domains)
Fix a Dedekind domain $R$ and fractional ideals $I, J$. It's a classical ...
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Question about the sum of odd powers equation
Quite surprisingly the following question appears while studying the billiard dynamics.
Assume we have $2n$ real numbers: $ x_1, x_2,..., x_{2n}$.
Assume also that $S_k=0$ for any odd positive integer ...
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Who and when proved Artin's Theorem on alternative rings?
I am interested in the history of the proof of Artin's Theorem (on the diassociativity of alternative rings).
Question. When has Artin proved this theorem and where was it published for the first ...
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Relationship between equation of integral dependence of an element and its inverse
Let $A$ be a reduced, Noetherian ring. Let $B$ be its integral closure. Let $b\in B$ and let $v\in B$ be its inverse. Let $b^n+\ldots a_0=0$ be an equation of integral dependence for $b$. Is there any ...
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Is there a "weak" fundamental theorem of algebra for matrices?
Let $R$ be the ring of complex $n\times n$ matrices, where $n>1$.
Does every nonconstant polynomial in $R[X]$ have a root in $R$?
Note: The "strong" fundamental theorem of algebra for ...
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Do étale coordinates give rise to a regular sequence of diagonal elements?
Fix an algebraic extension $k\subseteq K$ of fields of characteristic zero and consider a map of commutative rings $\phi\colon K\left[T_{1}^{\pm},\dots,T_{n}^{\pm}\right]\to A$ which is étale. Now ...
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A subfield $R \subseteq \mathbb{C}(x,y)$ with 'many' generators $w$, $R(w)=\mathbb{C}(x,y)$
Let $R \subseteq \mathbb{C}(x,y)$ and assume that $R=\mathbb{C}(u,v)$, where $u,v \in \mathbb{C}[x,y]$ are algebraically independent over $\mathbb{C}$.
Here $\mathbb{N}$ includes $0$.
Assume that $R$ ...
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Can the completion of a local domain which is not a field be a field?
I would like to prove/disprove the following claim:
Let $A$ be an equicharacteristic local domain, and denote by $\widehat{A}$ its completion with respect to its maximal ideal. If $\widehat{A}$ is ...
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A certain subfield of $\mathbb{C}(x,y)$
Let $A=\mathbb{C}(x+y,xy)$, the subfield of symmetric polynomials with respect to the involution $\alpha: (x,y) \mapsto (y,x)$.
Denote $G_A=\{w \in \mathbb{C}(x,y) \ | \ \mathbb{C}(x+y,xy,w)=A(w)=\...
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Explicit description of transfer for $K_1$
Let $R$ be a commutative regular ring and let $s \in R$ be an element such that $R / s$ is also regular. Then we have a long exact localization sequence
$$
\ldots \rightarrow K_i(R/s) \rightarrow K_i(...
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Do there exist these real polynomials?
Do there exist real polynomials $P(x)$ and $Q(x)$ with nonnegative coefficients, and $n>20$ a natural number such that
$$\left(\sum\limits_{k=0}^{n} x^k\right)^2=(x-3)^2P(x)+(x+4)^2 Q(x)?$$
I have ...
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The $K_1$-group of integer valued polynomials
Let $R=$ Int$(\mathbb{Z}) = \{f \in \mathbb{Q}[x]| f(\mathbb{Z}) \subset \mathbb{Z}\}$. I am interested to find $K_1(R)$. I list my trials below:
Let us construct a Milnor square $$\matrix{R&\...
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A commutative ring with unity which does not have relatively pseudo-injective ideals with zero intersection
Let $R$ be a ring with $1$ and $M$ and $N$ be any right $R$-modules. We say that $M$ is pseudo-$N$-injective if every $R$-monomorphism $f:X \to M$ from a submodule $X_R$ of $N_R$ can be extended to $N$...
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English translation of Borel-Serre's "Théorèmes de finitude en cohomologie galoisienne"?
Is there an English translation of this text, or at least some English language paper that proves the same results?
I especially need a proof of the following fact which is in this paper: Say $k$ is a ...
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Explicit computation of Čech-cohomology of coherent sheaves on $\mathbb{P}^n_A$
$\newcommand{\proj}[1]{\operatorname{proj}(#1)}
\newcommand{\PSP}{\mathbb{P}}$These days I noticed the following result of (constructive) commutative algebra, which I think is probably well known ...
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Sum of Betti numbers and certain short exact sequence of modules of finite length over regular local ring
Let $N$ be a module of finite length over a regular local ring $R$ of characteristic $0$. Let $M$ be an $R$-module which fits into a short exact sequence $0\to N^{\oplus a}\to M \to N^{\oplus b}\to 0$ ...
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416
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Tensor product over $\mathbb{Z}$ and p-adic integer ring $\mathbb{Z}_p$
Thanks for your reading. Suppose we have two $\mathbb{Z}_p$-modules $A,B$. Do we always have $A \otimes_{\mathbb{Z}} B \simeq A \otimes_{\mathbb{Z}_p} B$, as abelian groups or $\mathbb{Z}_p$-modules? ...
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Cancellation in polynomial composition
Let $k$ be a field. Suppose $P,Q,R\in k[x]$ satisfy $P\circ Q=P\circ R$. What can we conclude about $Q$ and $R$?
It may not be the case that $Q=R$; for example, if $P=x^2$, any polynomials $Q,R$ with $...
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Books one can read for 2nd course in Commutative Algebra ( Self Study)
I am a student who has completed master's but couldn't take admission to a PhD program due to some unfortunate reasons.
I have done 1 course in Commutative Algebra where I followed the book " ...
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Algebraic independence and substitution for quadratics
Let $f_{1},...,f_{n-1} \in \mathbb{F}[x_1,...,x_n]$ such that $\{ f_1,..., f_{n-1},x_n \}$ is algebraically independent over $\mathbb{F}$. Let $G \in \mathbb{F}[x_1,...,x_n,y_1,...,y_{n-1}]\...
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Picard group of almost module category
I am very new to the world of almost mathematics and I am curious about the following:
Fix an almost mathematics situation $(R,I)$ throughout. Very generally, the almost module category comes with a ...
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Is the derived support $\{x\in X\:|\: \mathsf{L}x^* M\neq 0\}$ closed?
Let $X$ be a scheme and consider an object $M$ of its derived category $\mathsf{D}_\text{qc}(X)$, defined as the full subcategory of $\mathsf{D}(\textsf{Mod}(\mathcal{O}_X))$ consisting of the ...
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When can $RHom^\bullet_{A}(B/I^k\overset{L}{\otimes}_B A, A)$ be computed using formal completions?
Let $\varphi:B\to A$ be a ring homomorphism between Noetherian rings. Let $I\subset B$ be an ideal. Let $B^{\wedge}=\varprojlim_n B/I^n$ be the $I$-adic completion of $B$, and $A^{\wedge}=\...
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Bounding the length of an R-module of matrices
Loosely related to this: Bounding the length in a module of evaluated skew polynomials
Let $C$ be an $\mathbb{F}_q$-vector subspace of $m \times n$ matrices over $\mathbb{F}_q$. Assume WLOG that $m \...
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A very specific quotient of a determinantal variety
I'm interesting in knowing whether a certain variety defined by maximal minors is irreducible. The specific construction is as follows: let $n \geq 2$ and let $R = \mathbb{C}[a_1,b_1,c_1,d_1,e_1,f_1,...
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120
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Normalization and ordinary double points
Let $X$ and $Y$ be two integral projective complex varieties and $f:X\to Y$ be a finite morphism. I assume that
(1) $X$ is smooth,
(2) $f$ is the normalization morphism of $Y$, and
(3) each fiber of $...
- 389
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76
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Largest set of monomials whose span is "co-prime" to a given polynomial
Let $K$ be a number field, and let $F \in K[x_1, \cdots, x_n]$ be a polynomial. For a positive integer $d \geq 3$, define $M(F;d)$ to be the largest positive integer such that there exists a set $S$ ...
- 26.9k
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280
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Freeness of a quotient module over a regular local ring
Let $R$ be a regular local ring with maximal ideal $m$. Let $t\in m\setminus m^2$. Let $N$ be a submodule of a finitely generated free $R$-module $M$ satisfying
$$ tM \subseteq N \subseteq M.$$
...
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113
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Relation between minimality and algebraic independence for binomials?
$\DeclareMathOperator\supp{supp}$Given $f_1,...,f_n \in \mathbb{F}[x_1,...,x_n]$ such that
$f_1 = x_1 + q_1$
$f_2 = x_2 + q_2$
$\cdot \cdot \cdot$
$f_{n-1} = x_{n-1} + q_{n-1}$
$f_{n} = q_n$
such that ...
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2
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199
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Determining the multiplication via addition and some unary operation
It is known that the addition operation in a skew-field $F$ (more generally, in a quasifield) is uniquely determined by the multiplication operation and the unary involutive operation $1_{-}:F\to F$, ...
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148
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Nilpotent polynomial matrices over $F_q$ - polynomial count variety ? ( Nilpotent cone for Hitchin-Gaudin like integrable system)
Context: Number of nilpotent $n\times n $ matrices over $F_q$ is $q^{n(n-1)}$ classical result due to Ph.Hall, M.Gerstenhaber (see very nice exposition by T.Leinster at n-cat-cafe/arxiv) which have ...
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