Questions tagged [ac.commutative-algebra]

Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.

9
votes
1answer
200 views

Derivative of an algebraic power series in positive characteristic

Let $K$ be a field. It is easy to see that if the characteristic of $K$ is $0$ and $f(T)=\sum_{n\ge0}a_nT^n$ is a power series algebraic over $K(T)$, then $f'$ belongs to $K(T)(f)$. Indeed let $P(X)=\...
2
votes
0answers
66 views

Continuous extension of the derivation in positive characteristic

Let $\Omega$ be the completion of an algebraic closure of $\mathbb F_q\left(\left(\frac1T\right)\right)$ for the topology induced by the valuation $-\deg$. Does there exist a derivation on $\Omega$ ...
2
votes
0answers
96 views

Converse to Tannaka duality for rings

Let $k$ be a field. It's well known and easy to prove that a $k$-algebra $R$ may be recovered from the functor $F: R-Mod \to Vect_k$ as the endomorphisms of $F$. Now suppose $\mathcal{C}$ is a ...
3
votes
0answers
101 views

Algorithm telling when an affine curve is planar

I am sorry, I asked a misguided question here: Reference request: smooth affine curves are planar, here is my attempt at a better question. Let $\mathfrak{p}$ be a prime ideal of $\mathbb{C}[x, y, z]$...
2
votes
0answers
179 views

Concerning certain Keller maps of $k[x,y]$

Let $k$ be a field of characteristic zero. Let $(x,y) \mapsto (p,q) \in k[x,y]$ be a Keller map, namely, a $k$-algebra endomorphism of $k[x,y]$ with $\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in k-\{0\}...
9
votes
2answers
335 views

On a morphism from the Brauer group to the Picard group

Suppose that $k$ is a commutative ring and that $A$ is an Azumaya $k$-algebra. Then there is a well-known morphism from $Aut_{Alg_k}(A)$, the group of algebra automorphisms, to the Picard group $Pic(k)...
3
votes
3answers
582 views

Does there exist another form of the derivative for polynomials?

Let $F : \mathbb{R}[X] \rightarrow \mathbb R[X]$ be a linear map and let $H \in \mathbb{R}[u,x,y,z]$ be a polynomial. Suppose that $$ F(P \cdot Q) = H(F(P),F(Q),P,Q)$$ for all $P, Q \in \mathbb{R}[X]...
0
votes
0answers
50 views

Homogeneous basis on a polynomial subalgebra

Let $k$ be an algebraically closed field of characteristic $0$ and $A=k[X_1, \ldots, X_n]$ with the grading induced by the total degree. Let $B$ be a graded $k$-subalgebra of $A$, ie, if $(A_k)$ is ...
3
votes
0answers
69 views

Polynomial equations parametrized by binary forms

Consider the equation $$\displaystyle Ax^p + By^q = Cz^r, A,B,C \in \mathbb{Z}, \gcd(x,y,z) = 1, p,q,r \geq 2.$$ When $p^{-1} + q^{-1} + r^{-1} > 1$, the above equation is called spherical and ...
3
votes
0answers
77 views

A bound for $[\mathbb{C}(x,y,z):\mathbb{C}(p,q,r)]$, where $\operatorname{Jac}(p,q,r) \in \mathbb{C}^{\times}$

Y. Zhang (in his PhD thesis) and P. I. Katsylo proved the following nice result; the two proofs are different, see: Zhang's thesis and Katsylo's paper: Let $f: (x,y) \mapsto (p,q)$ be a $k$-algebra ...
-2
votes
1answer
104 views

Module such that every finitely generated submodule is semisimple [closed]

Is there an example of a module $M$ (over a commutative ring) that is not free, and such that each of its finitely generated submodule is semisimple (i.e. such that any submodule of any finitely ...
2
votes
1answer
158 views

Every radical ideal in the ring of algebraic integers a finite intersection of prime ideals

Is every radical ideal in the ring of algebraic integers (i.e. the integral closure of $\mathbb{Z}$ considered as a subring of $\mathbb{C}$ via the unique homomorphism of unital rings $\mathbb{Z}\...
2
votes
0answers
43 views

Homomorphism or derivation conserving irreducibility

Let $R$ be a integral domain and $\phi$ be an automorphism of $R$. For a given element $x \in R$, we consider a sequence $(\phi^n(x))_{n=0}^{\infty}$. I wonder if there is any related theory to ...
3
votes
0answers
60 views

Inverse limit and graded functor commute?

I am trying to understand a proof where there are graded algebras and inverse limit involved. In one of the steps it seems to commute this two elements. Is there any reference where this is stated. $...
3
votes
0answers
146 views

Simple description of a Grothendieck topology on the opposite of f.p. complex algebras

Let ${\cal A}$ be the category of finitely presented $\mathbb{C}$-algebras. Let $J$ be the largest subcanonical Grothendieck topology on ${{\cal A}^{op}}$ such that the local algebras in $\cal A$ are ...
3
votes
0answers
71 views

Differential criterion for regular sequences

Let $R$ be a subalgebra of the polynomial ring $\mathbb{C}[X_1,\ldots,X_n]$. Suppose $\theta=(\theta_1,\ldots,\theta_p)$ is a sequence of of elements in $R$. If I want to test for their algebraic ...
1
vote
2answers
222 views

Regularity of certain schemes

In a book I am reading, "Travaux de Gabber sur l'uniformisation locale et la cohomologie etale des schemas quasi-excellents" by Luc Illusie, Yves Laszlo, Fabrice Orgogozo (https://arxiv.org/abs/1207....
0
votes
0answers
253 views

On the product in the power series ring

Let $A_n \colon= K[[X_1,\ldots,X_n,Y_1,\ldots,Y_n]]$ be a power series ring over a field $K$ in $2n$ variables and ${\frak m}_{A_n}$ be the unique maximal ideal of $A_n$. Suppose we have two ...
0
votes
0answers
110 views

Homomorphisms from $k[x,y]$ to $k[x,x^{-1},y]$

Let $k$ be a field of characteristic zero and let $R_{-1}:=k[x,x^{-1},y]$ be the $k$-algebra of polynomials in $x,y$ containing the inverse of $x$, denoted by $x^{-1}$. Let $f: k[x,y] \to R_{-1}$ be ...
2
votes
0answers
345 views

Why are there elementary equations that are not solvable in closed form?

Elementary equations and closed-form solutions can be important in mathematics and in the natural sciences, e.g. for discovering and describing certain relationships. $\log\colon x\mapsto\log(x)$; $x\...
7
votes
1answer
248 views

Finite dimensional commutative algebras containing infinitely many nilpotents whose $d$-way products are nonzero

I'm interested in the following strange question: for some $d > 1$, what is the minimum dimension of a commutative $\mathbb{C}$-algebra containing infinitely many elements that square to zero, but ...
7
votes
0answers
87 views

Catenarity and epimorphisms of rings

Let $R$ be a commutative ring. The following are well-known: If $R$ is catenary and $\mathfrak{a}\subseteq R$ is an ideal, then $R/\mathfrak{a}$ is catenary. If $R$ is catenary and $S\subseteq R$ is ...
2
votes
0answers
73 views

What is the difference between total integral closure and integral closure?

I was advised here to make this a new question: What is the difference between total integral closure and integral closure (geometrically, in the context of rigid analytic geometry)? I have read in ...
4
votes
1answer
119 views

Significance of integrally closed in an affinoid algebra

A Tate affinoid k-algebra is defined as a pair $(R,R^+)$ where $R$ is a Tate algebra and $R^+$ is an open and integrally closed subring of $R$ contained in the ring of powerbounded elements. See for ...
1
vote
0answers
41 views

Non-separated Nygaard filtration

Let $S$ be a quasiregular semiperfectoid ring, then on its prism we may define the Nygaard filtration (Definition 12.1 in this preprint). What is an explicit example where it is not separated?
2
votes
0answers
123 views

Properties of rings of global functions of open subschemes

It is known (although maybe not so well) that there are nice algebraic varieties whose ring of global functions is not finitely generated over the ground field. One can find examples on the web, but ...
4
votes
1answer
173 views

Generalised CRT - How to compute the cokernel?

Let $R$ be a commutative ring of dimension one with minimal prime ideals $P_1,\ldots,P_n$. We have the canonical injective map $$\phi_n: R/(P_1 \cap \ldots \cap P_n) \to \prod_{i=1}^n R/P_i.$$ My ...
1
vote
1answer
72 views

Converging sequence of polynomials

Let $P$ be an irreducible polynomial of $\mathbb F_q[T]$ of degree $2$. Does there exist two polyomials $\alpha,\beta\in\mathbb F_q[T]$ (not both zeroes) such that the sequence $(\beta T^{q^{2n}}-\...
5
votes
1answer
224 views

Is completion of isolated singularity isolated?

Let $K$ be an algebraically closed field and let $A=K[x_1,\dots,x_n]/I$ be a $K$-algebra of finite type which has only an isolated singularity at the origin. Let $\mathfrak{m}=(x_1,\dots,x_n)$ and ...
0
votes
0answers
84 views

Intersection of an ideal and a subring

Is there a Grobner basis method that can compute the intersection of an ideal $I$ of a polynomial ring $R$ and its subring $R^\prime$? For example, I have an ideal $I=(x+y+z^2,1+xyz+yz+xz)$ of $\...
8
votes
1answer
243 views

Example of a ring where every module of finite projective dimension is free?

I'm interested in seeing an example of a ring which is not self-injective where every module admitting a finite projective resolution is free, or at least projective. Note that self-injectivity says ...
4
votes
1answer
111 views

Mapping cone and derived tensor product

This question is in some sense a continuation to this question: Derived Nakayama for complete modules For the setting: Let $A$ be a ring and let $I$ be some finitely generated ideal in $A$. Let $f\...
1
vote
1answer
102 views

Ideal in ring of power series

Let $K$ be a field of characteristic $p$ and $A_n \colon= K[[X_1,\ldots,X_n]]$ be a $n$-variable formal power series ring over $K$ such that $n, p \geq 3$. Consider the ideal $I$ defined by \begin{...
1
vote
0answers
67 views

Proper ideals are invertible

I am reading through Cox's book Primes of the form $x^2+ny^2$ and I am stuck with some proofs in Chapter 7 (I have the 2nd edition). There, the author presents the following Lemma: Lemma 7.5: Let $...
7
votes
0answers
269 views

Foundational Questions on Adic Spaces

There are some foundational questions on adic spaces that I can't find in the literature. It seems that these questions are pretty natural, so I guess that an answer should be known to the experts in ...
4
votes
1answer
109 views

Etale map has image whose complement is the vanishing locus of a finitely generated ideal

While working through a proof of this paper, at the end of page 46, the author seems to claim along the lines that the following is true: Let $A\rightarrow B$ be an etale map of rings. Then the ...
3
votes
0answers
34 views

Graded commutative PBW bases

A Poincaré–Birkhoff–Witt (PBW) basis is a particularly nice basis of a quadratic algebra that can be used to prove that it is Koszul (see Priddy's 1970 paper "Koszul resolutions", Trans. Amer. Math. ...
2
votes
1answer
104 views

Special submodules over almost Dedekind domains

An integral domain $R$ is an almost Dedekind domain if for each maximal ideal $m$ of $R$, the ring $R_m$ is a Dedekind domain, where $R_m$ is the localization of $R$ at $m$. Question: Let $M$ ...
2
votes
0answers
83 views

Finite dimensionality of fibers of etale ring map

While working through a proof of this paper, at the middle of page 46, the author introduces a dimension notion which seems to claim that the following is true: Let $A\rightarrow B$ be an etale map ...
1
vote
1answer
104 views

Map between localizations induces map on underlying modules for Zariski covering

While working through a proof of this paper,1 at the middle of page 45, the author's claim of a short exact sequence seems to amount to the following problem: Let $A$ be a commutative ring and let $...
1
vote
0answers
75 views

Etale algebra whose local rank is constantly zero is the zero algebra

While working through a proof of this paper, at the middle of page 46, the author seems to claim the following is true: Let $A\rightarrow B$ be an etale map of rings. Suppose that for every prime $...
3
votes
0answers
98 views

Noetherian affine schemes for which localization computes the values of the structure sheaf

Is there a non-tautological characterization of the class of commutative unital Noetherian rings satisfying the following property: for any open subset $U$ of the topological space of $\mathrm{Spec}\,...
2
votes
1answer
74 views

Preimage of a constructible set in spectrum of a subring

While working through a proof of this paper, at the beginning of page 42, the author seems to claim the following is true: Let $R\subset S$ be rings, where $R$ is a finite type algebra over $\...
2
votes
0answers
123 views

Structure of Complete Local Rings

Let $X$ be a proper $n$-dimensional $k$-scheme and $x \in X$ a closed point. Consider the stalk $\mathcal{O}_{X,x}$. We consider now it's completion $O_{X,x}^{\wedge}$ wrt it's maximal ideal $m_x$. ...
0
votes
0answers
79 views

Structures like vector spaces but closed under heterogeneous products

The category of (pseudo-)Euclidean vector spaces (vector spaces with a nondegenerate but not necessarily positive-definite quadratic form) is not closed under products because $R^n$ over $R$ and $Z_2^...
1
vote
0answers
79 views

Is a Maximal Cohen-Macaulay sheaf on a Cohen-Macaulay scheme locally free?

Let $F$ be a Maximal Cohen-Macaulay module on a Cohen-Macaulay scheme $X$. By definition $\mathrm{depth}_{\mathcal{O}_{X,x}}(F_x)=\mathrm{dim}(\mathcal{O}_{X,x})$ and $\mathrm{depth}(\mathcal{O}_{X,x})...
0
votes
2answers
180 views

Power series ring and monomials

Let $A_n \colon= K[[X_1,\ldots,X_n]]$ be a formal power series ring over a field $K$ of characterisc $p > 0$ in $n$ variables. For a given positive number $\epsilon > 0$ we call a monomial $X_{...
3
votes
0answers
90 views

Reduced Noetherian ring is the intersection of its localizations at primes associated to a nonzero-divisor

I shall quote proposition 11.3 of Eisenbud: Commutative algebra If $R$ is a reduced noetherian ring then an element $x\in K(R)$ belongs to $R$ if and only if the image of $x$ in $K(R)_P$ belongs to ...
9
votes
2answers
458 views

Involutions in $\mathbb{F}_p[[x]]$

Question: For a prime $p$, is every involution in $\mathbb{F}_p[[x]]$ with a zero constant term a reduction modulo $p$ of some involution in $\mathbb{Z}[[x]]$? Here involution in $A[[x]]$ means $f\in ...
7
votes
2answers
416 views

Basis for free modules over an affine domain

Let $A=k[x_1,\cdots,x_r]/I$ for some prime ideal $I$ and some field $k$. Consider the free $A$-module $A^n$. Question 1. Given an element $e\in A^n$, is there a method to tell whether $e$ can be ...