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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Minimality implies algebraic independence?
$\DeclareMathOperator\supp{supp}$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 ...
- 341
84
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31
answers
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Applications of the Chinese remainder theorem
As the title suggests I am interested in CRT applications. Wikipedia article on CRT lists some of the well known applications (e.g. used in the RSA algorithm, used to construct an elegant Gödel ...
3
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1
answer
164
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Minimality of the Koszul resolution
Let $R = \mathbb{C}[x,y]$ and $V = \mathbb{C}x\oplus\mathbb{C}y$. Then, the Koszul resolution of $R$ (as an $R$-bimodule) is given by
\begin{align*}
0\to R\otimes_{\mathbb{C}}\wedge^2V\otimes_{\mathbb{...
- 985
11
votes
0
answers
616
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Monotonicity of ratio of symmetric polynomials
The complete homogeneous symmetric polynomials of degree $\ell$ in $n$ variables are defined by
\begin{equation*}
h_{\ell}(x_1,x_2,\ldots,x_n) = \sum_{1 \leq i_1 \leq i_2 \leq \ldots \leq i_{\ell} \...
8
votes
1
answer
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Why is $\operatorname{Spec}(\mathbb Z)$ supposed to lie over $\operatorname{Spec}(\mathbb F_1)$ rather than the other way around?
$\DeclareMathOperator\Spec{Spec}$I understand that one major motivation for the field with one element is supposed to be that there should be a map $\Spec(\mathbb Z) \to \Spec(\mathbb F_1)$, which has ...
- 64k
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votes
0
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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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2
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1
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242
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Sheaves which are locally free on subschemes of dimension zero
Let $\mathscr{F}$ be a coherent sheaf on a scheme $X$ with reasonable assumptions.
Obviously if I restrict to any point $x \in X$, the restriction $\mathscr{F}|_x$ is free over $x$.
I am interested in ...
- 635
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votes
0
answers
115
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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 ...
- 31
1
vote
0
answers
80
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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.
...
- 480
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0
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108
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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 ...
- 507
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0
answers
112
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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&\...
- 141
101
votes
31
answers
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Errata for Atiyah–Macdonald
Is there a good list of errata for Atiyah–Macdonald available? A cursory Google search reveals a laughably short list here, with just a few typos. Is there any source available online which lists ...
2
votes
1
answer
158
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How to decompose a given polynomial by ideal generators
Given a finite set of polynomials $f_1, f_2,..., f_n$ of variables $x_1,...,x_m$, generating the ideal $I$, suppose that we have one more polynomial $g\in I$.
What is the algorythm for decomposing $g$ ...
- 728
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1
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639
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A question on algebraic independence
Let $f_1,f_2,\ldots,f_n, g \in \mathbb{F}_q[x_1,...,x_m]$. Assume that $f_1,\ldots,f_n$ vanish at $0$, so that $\mathbb{F}_q[[f_1,...,f_n]]$ is a subring of $\mathbb{F}_q[[x_1,...,x_n]]$. Suppose that ...
- 341
222
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8
answers
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How to memorise (understand) Nakayama's lemma and its corollaries?
Nakayama's lemma is mentioned in the majority of books on algebraic geometry that treat varieties. So I think Ihave read the formulation of this lemma at least 20 times (and read the proof maybe ...
- 14.3k
3
votes
1
answer
331
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Is there a variety which is not locally set theoretic complete intersection?
A variety $X$ is a locally set theoretic complete intersection in a nonsingular variety $Y \supseteq X$ if each point in $X$ has a neighborhood $U$ in $Y$ such that $X \cap U$ is set theoretically the ...
- 5,339
10
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1
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534
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How is Taylor-Wiles patching "horizontal Iwasawa theory"?
I have recently been reading into the proof of modularity of semistable elliptic curves, in particular (what is now known as) the Taylor-Wiles patching argument used to prove the $R=T$ theorem in the ...
- 28.2k
13
votes
0
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261
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Big list of Hochster dual concepts
Let $X$ be a spectral space. Then there is a canonical space $X^\vee$ with the same points, same constructible topology, and the opposite specialization order. This is known as “Hochster duality”, and ...
10
votes
0
answers
190
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Is every UFD a filtered colimit of Noetherian UFDs?
I'm wondering how one could prove or disprove that any non-Noetherian UFD is a filtered colimit of Noetherian UFDs. This would allow for some absolute Noetherian approximation to be applied for ...
- 804
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1
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546
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In an abelian category with no nontrivial Serre subcategories, does every short exact sequence split?
Question: Let $\mathcal A$ be an abelian category. Suppose that the only Serre subcategories of $\mathcal A$ are the zero category and $\mathcal A$ itself. Does it follow that every short exact ...
- 64k
2
votes
1
answer
206
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Noetherian local ring with non-lci formal fibers
I am looking for an example of a noetherian local ring $A$ such that its formal fibers are not complete intersection rings in the sense of https://stacks.math.columbia.edu/tag/09PY (i.e., there is a ...
- 2,923
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votes
2
answers
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Prove that $\Bbb C[x,y]/(x^3+y^3-1)$ is not a UFD
I am posting this question on MO since I haven't received any answers on MSE.
Below is my (very elementary) attempt. Feel free to post a solution using facts in algebraic geometry and facts about ...
- 171
1
vote
0
answers
67
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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 ...
- 359
2
votes
1
answer
203
views
Artin-Schreier theorem for rings (a little different)
Motivation:
Let me recall the well-known Artin-Schreier theorem (AST) for fields in a non-formal way; if $L$ is an algebraically closed field, and $K \subset L$ a subfield not 'much smaller' than $L$, ...
- 748
0
votes
0
answers
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 ...
- 341
17
votes
1
answer
687
views
Multiply an integer polynomial with another integer polynomial to get a "big" coefficient
I have copied this question from StackExchange, in the hope that some experts here can provide some relevant insight. Thanks to Greg Martin for improving the question.
Given $f(x) = a_0 + a_1 x + a_2 ...
- 823
3
votes
2
answers
401
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$R$-Module objects in cartesian closed categories
I am looking for a reference for the following statement.
Theorem. Let
$C$ be a regular, well-powered, countably complete cartesian closed category,
$R$ be a (commutative) ring object in $C$,
$R\...
user103549
2
votes
1
answer
181
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Hilbert–Samuel multiplicity under hypersurface sections
Let $\newcommand{\frakm}{\mathfrak{m}}(R,\frakm)$ be a reduced Noetherian local ring of dimension $d$ and $f\in\frakm^\alpha\setminus\frakm^{\alpha+1}$ a parameter of $R$, i.e. $\dim R/(f)=d-1$. Let $...
- 135
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votes
0
answers
131
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When is a degree-$n$ homogeneous polynomial in $\mathbb{C}[x_1, x_2, \ldots , x_m ]$ the product of $n$ one-forms?
When is a degree-$n$ homogeneous polynomial in $\mathbb{C}[x_1, x_2, \ldots , x_m ]$ the product of $n$ one-forms?
Is there any simple algorithm or criterion to check it?
I have chosen the complex ...
- 171
6
votes
0
answers
235
views
A standard name for the algebraic structure on a projective line?
Question: Is there any name for the natural algebraic structure of the projective line?
Algebraically, a projective line over a field is a set $L$ endowed with two binary operations $+$ and $\cdot$ ...
- 42k
8
votes
2
answers
1k
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A conceptual proof that bounded index subgroups of a bounded torsion abelian group contain bounded index complemented subgroups
Call an abelian group $G = (G,+)$ $m$-torsion for some natural number $m$ if one has $m \cdot x = 0$ for all $x \in G$. A subgroup $H$ of $G$ is said to be complemented if one can write $G = H \oplus ...
- 114k
0
votes
0
answers
47
views
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 ...
- 71
2
votes
1
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199
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Regular sequence in cohomology of Grassmannians
$\DeclareMathOperator\Gr{Gr}$Consider the polynomial ring $\mathbb{Z}[x_1,\dots,x_m, y_1,\dots,y_n]$, I want to prove that the sequence $$x_1 + y_1, x_2 + x_1y_1 + y_2, \dots, x_my_{n-1} + x_{m-1}y_n, ...
- 185
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votes
1
answer
170
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Isn't every algebraic operad equipped with a trivial weight?
In Loday–Vallette "Algebraic Operads" they state the following result (Theorem 6.6.2, Operadic twisting morphism fundamental theorem):
Let $P$ be a connected weight graded differential ...
- 215
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0
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93
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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}=\...
- 367
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votes
0
answers
226
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Decomposing an endomorphism as a tensor product
$\DeclareMathOperator\End{End}$Let $f$ be an endomorphism of the finite-dimensional vector space $V$, over the field $K$. The question of whether $f$ is decomposable, that is, whether $V$ can be ...
- 2,287
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vote
0
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107
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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)=\...
- 2,837
6
votes
1
answer
367
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Do groups of units change base nicely, assuming the fields are algebraically closed?
Let $K$ be an algebraically closed field. Let $X$ be an irreducible affine algebraic variety over $K$. Let $L/K$ be a field extension, where $L$ is also algebraically closed. Suppose the group of ...
- 1,802
1
vote
0
answers
186
views
Is it true that monomorphisms of local Artinian $\mathbb{R}$-algebras are regular?
A Weil algebra is a finite-dimensional real algebra, in which each element is the uniquely sum of a scalar and a nilpotent (so nilpotents constitute the only maximal ideal of codimension 1). In other ...
- 6,962
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votes
1
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1k
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If some powers of polynomials are linearly independent, does it imply higher powers are also independent?
Let $P_1,\dotsc,P_k$ be polynomials. Assume they are pairwise non-proportional (i.e., any two of them are linearly independent). Suppose $N$ is a power such that $P_1^N,\dotsc,P_k^N$ are linearly ...
- 6,237
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votes
2
answers
769
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Are topological PID's Noetherian?
Romain Giquaud has given a counterexample to the general form of the question. The bounty is for a solution for locally compact, metrizable rings. (I suspect the answer may be positive with this ...
- 42.8k
0
votes
0
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97
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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}]\...
- 341
1
vote
1
answer
463
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Vector bundles on $\mathbb{P}^1$
I am considering an alternative proof of Grothendieck's classification of vector bundles on $\mathbb{P}^1$. Given a vector bundle $E$ on $\mathbb{P}^1$ one can associate a graded module $\Gamma(E)$ ...
- 21
2
votes
0
answers
134
views
A pattern in factorization of certrain symmetric polynomials made of sums of squares and products
Secondary-school pupils may learn things like
$$(a+b-c)^2 = a^2+b^2+c^2 +2ab - 2ac - 2bc$$
and that no choice of plus versus minus can give us
$$
(\pm a\pm b\pm c)^2 = a^2+b^2+c^2 - 2ab - 2ac - 2bc,
$$...
0
votes
0
answers
76
views
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
1
vote
1
answer
198
views
Shrinking the base field of an affine variety
This is a question on algebraic geometry/commutative algebra.
Let $K,L$ be fields of characteristics zero and let $K\subset L$ be a field extension (I am interested in the case when this is ...
- 21
1
vote
1
answer
109
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Gorenstein property from initial ideal
My question is:
If $I$ is a homogenous ideal of $S=K[x_1,\dots,x_n]$ and $in_{<}(I)$ is the initial ideal of $I$, with respect to a term order $<$ on $S$, then $S/I$ is Gorenstein if and only if ...
- 115
1
vote
0
answers
205
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Finding if an ideal is the radical of another one
Let's suppose we have, in the ring $\mathbb{Z} [x,y,z,w,v]$, the following polynomials:
$f=xw-yz$,
$g=x^2z-y^3$,
$h=yw^2-z^3$,
$k=xz^2-y^2w$.
The question is to prove that $I=(f,g,h,k)$ is the radical ...
2
votes
1
answer
185
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Finite étale cover of factorial ring
Let $A$ be a regular factorial ring.
Consider $B=A[X]/(P)$ such that $B$ is finite étale over $A$. When do we have that $B$ is also factorial?
- 3,472
1
vote
0
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142
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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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