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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Commutative monoid gradings via group scheme actions
$\newcommand{\Spec}{\mathrm{Spec}}$Recall the following result, proved in Section 2.9 of Neil Strickland's Formal schemes and formal groups, and in Lemma 1.3.2 of Eric Peterson's Formal Geometry and ...
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Matrix factorizations and physics
I have heard during some seminar talks that there are applications of the theory of
matrix factorizations in string theory. A quick search shows mostly papers written by physicists. Are there any ...
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Decomposition of augmentation ideal in a group ring
Let $R$ be a ring and $G$ be a finite group with invertible order in $R$. Assume that the augmentation ideal $\Delta (G)$ of group ring $RG$ has a decomposition $M_1\oplus M_2\oplus \cdots M_k$ as an $...
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Exactness of functor $ Et(B) \to \operatorname{(Ab)}, \ C \mapsto \mathcal{F}(C) $ (Etale Cohomology and the Weil Conjecture by Freitag, Kiehl )
I have question about a statement from Etale Cohomology and the Weil Conjecture by Freitag, Kiehl
at the top of page 16. It seemingly uses the same notations as introduced at the bottom of page 15
and ...
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Generalized Euler phi function
Let $n$ be an integer, there is a well-known formula for $\varphi(n)$ where $\varphi$ is the Euler phi function. Essentially, $\varphi(n)$ gives the number of invertible elements in $\mathbb{Z}/n\...
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Why the sign in the definition of the discriminant?
Consider the split monic $f=\prod_{i=1}^n(x-x_i)\in \mathbb Z[x_1 ,\dots ,x_n,x]$. Its discriminant is usually defined as $$(-1)^{n(n-1)/2}\prod_{i=1}^nf^\prime(x_i)=\prod_{1\leq i<j\leq n}(x_i-x_j)...
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Intersecting a ideal generated in degree $\leq a$ with one generated in degree $\leq b$ in a polynomial ring
Question 1. Let $R$ be a polynomial ring over a field $k$. Assume that $R$ is graded in the usual way (i.e., each variable has degree $1$). Let $I$ and $J$ be two ideals of $R$ such that $I$ is ...
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Computation of Quillen K-groups for mod R
The recent paper K-Groups for Rings of Finite Cohen-Macaulay Type by H. Holm allows us to compute the Quillen $K$-group $K_1(\text{mod}\hspace{.1 cm}R)$ as a quotient of the abelianization of the ...
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Rings such that torsion-free/flat/projective modules are flat/projective/free
While thinking about this question (and specifically YCor's remarks), I tried to remember what can be said about rings such that every torsion-free module is free, and I could not; and such things, ...
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Efficient computation of "higher order" Jacobi symbols
Let $N> 1$, and suppose $a \in (\mathbb{Z}/N\mathbb{Z})^{\times}$. There are efficient algorithms to compute the Jacobi symbol $\left(\frac{a}{N}\right)$ using quadratic reciprocity. Fix $k \geq 1$....
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intersection of two height 2 primes must contain a non-zero prime?
I saw in some contexts the following statement, which I do not have a reference for this:
"Kaplansky asked if in a Noetherian domain the intersection
of two height 2 primes must contain a non-...
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Resolvent is minimal polynomial for universal splitting algebra
Given a degree $n$ monic $f\in A[x]$ write $\mathrm{Split}_Af$ for its universal splitting algebra, constructed by taking the quotient of $A[x_1,\dots ,x_n]$ by the Vieta formulas. This is the initial ...
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Descent for the "localizations at all primes" ring map
Let $A$ be a ring. Is the sequence \begin{align} \textstyle A \to \prod_{\mathfrak{p}} A_{\mathfrak{p}} \rightrightarrows \prod_{\mathfrak{p}_{1},\mathfrak{p}_{2}} A_{\mathfrak{p}_{1}} \otimes_{A} A_{\...
- 149k
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$(q,t)$-Fibonacci polynomials: area & bounce statistics
This is related to my earlier (unanswered) MO post. Preserve notations from there.
We take advantage of the one-to-one correspondence between the $(s,s+1)$-core partitions and $(s,s+1)$-Dyck paths. ...
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Fibonacci embedded in Catalan?
Given a partition $\lambda$ and its Young diagram $\pmb{Y}_{\lambda}$, we say $\lambda$ is a $(t,s)$-core partition provided that neither $t$ nor $s$ is a hook length in $\pmb{Y}_{\lambda}$. We now ...
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The existence of two maximal ideals with the same set of idempotents
Let $R$ be a commutative ring with identity and $A$ and $B$ be two proper ideals of $R$ such that $A+B=R$ and for each $r^2=r\in R$ we have either $r\not\in A$ or $r-1\not\in B$. How can we prove the ...
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Factoring $N$ given a solution to $x^2 + y^2 \equiv 0 \pmod{N}$
Let $N$ be a composite integer, and suppose we are given a randomly generated solution $(x, y)$ of the equation $x^2 + y^2 \equiv 0 \pmod{N}$. By randomly generated, I mean that $(x, y)$ is selected ...
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Finitely generated modules over Noetherian local ring that become isomorphic after faithfully flat base change
Let $(R,\mathfrak m)$ be a Noetherian local ring. Let $S$ be a Noetherian ring which is a faithfully flat $R$-algebra. If $M,N$ are finitely generated $R$-modules such that $M\otimes_R S \cong N \...
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Lifting of flat lci maps
Suppose $R$ is a Noetherian ring and $I$ a nontrivial ideal of $R$, and $A_0\to B_0$ a finite faithfully flat lci map of smooth $R_0 := R/I$-algebras.
We fix a smooth $R$-algebra $A$ lifting $A_0$ and ...
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Vanishing of Kahler differentials vs. surjective Frobenius?
Let $A$ be an $\mathbf{F}_p$-algebra such that $\Omega_{A/\mathbf{F}_p}=0$. Is the Frobenius map on $A$ surjective?
Some context:
i. The converse is clearly true.
ii. The answer is yes if $A$ is a ...
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Chain complexes split in the derived category over rings of global dimension 1
Let $R$ be a ring of global dimension $1$. Then I have seen the claim (in a paper, and in this MO post When do chain complexes decompose as a direct sum?) that any chain complex over $R$ is equivalent ...
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Subsets of the integers which are closed under multiplication
Let $S$ be a subset of the integers which is closed under multiplication. There are many possible choices of $S$:
$S = \{-1, 1\}$.
$S$ is the set of integers of the form $a^k$, where $a$ is fixed and ...
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Cohen-Macaulay coordinate rings defined by regular sequences
Consider the polynomial ring $R = k[x_1, \ldots, x_n]$ in $n$ indeterminates over an algebraically closed field $k$ (my relevant case is the complex numbers). Furthermore, consider an algebraic ...
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Non-empty closed subsets with empty special fiber
Let $R$ be a dvr and $U\to \text{Spec}(R)$ an affine smooth $R$-scheme with non-empty special fiber $U_0$.
Let $Z\subset U$ be a closed subset. Assume the intersection of $Z$ with $U_0$ is empty.
Is $...
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Extension of hyperderivatives
Let $a$ be algebraic over $K:=\mathbb F_q\left(\!\left(\frac1T\right)\!\right)$. Can one extend continuously the hyperderivatives on $K(a)$?
Recal that the hyperderivative $D_h$ over $K$ is defined by
...
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Has anyone studied this possible generalisation of the Singular Value Decomposition to all commutative $*$-rings?
I don't know any abstract algebraists personally, which is why I'm asking this question here.
Let $(R,*)$ be a commutative $*$-ring where $*:R \to R$ is an involution. Each conjecture below is stated ...
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When is the profinite completion of a Noetherian group ring also Noetherian?
Let $G$ be a group, and let $\mathbb{Z}[G]$ denote its group ring. Its profinite completion is the inverse limit over all ideals of finite index. By Benjamin Steinberg's answer here, this profinite ...
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Structure of the automorphism group of an L-rig
This question is a follow-up to Are there infinitely many L-rigs? which is already pretty convoluted.
Define the $\varphi$-evaluation morphism at a complex number $s$ as $\epsilon_{\varphi,s}:F\mapsto ...
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Why are extensions so heavily emphasized in valuation theory?
Whenever I read anything about valuations or things related to them (such as local fields) extensions always occupy a prominent position and a huge amount of effort is expended to derive results about ...
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Valuation theory on semisimple algebras used in the paper of Cohen-Martinet: reference request
I'm currently reading the paper of Henri Cohen & Jacques Martinet "Etude heuristique des groupes de classes des corps de nombres"
On the 2nd section, they recall some facts on valuations, ...
CommunityBot
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Existence of non-zero pseudo-null submodules
Let $p$ be a rational prime, and let $\Lambda_d$ be the Iwasawa algebra in $d$ variables, i.e. $\Lambda_d = \mathbb{Z}_p[[T_1, \ldots, T_d]]$. Let $A$ be a finitely generated and torsion $\Lambda_d$-...
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Injectivity of Keller maps
Let $M: \mathbb{C}[x,y] \to \mathbb{C}[x,y]$,
$(x,y) \mapsto (p,q)$, with $p,q \in \mathbb{C}[x,y]$
satisfying $\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in \mathbb{C}-\{0\}$.
Such a polynomial map is ...
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Is pseudo-rationality preserved by etale morphisms?
Let $f: Y \to X$ be an etale morphism of schemes.
If $X$ has pseudo-rational singularities then does $Y$ also have pseudo-rational singularities?
For the definition of pseudo-rational see, for ...
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Quotients of Gorenstein rings
Let $R$ be a reduced Noetherian ring. Assume $R$ is quasi-excellent and Cohen-Macaulay.
Is $R$ the quotient of a Gorenstein ring?
If the answer is yes, then $R$ has a dualizing complex. The question ...
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Faithfully flat etale morphism from strictly Henselian ring (from Etale Cohomology and the Weil Conjecture by Freitag/Kiehl)
I have question about a statement found in Etale Cohomology and the Weil Conjecture by Freitag, Kiehl at the end of page 15.
It starts with the Remark 1.18 : Let $A$ be a strictly Henselian ring (i.e. ...
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If the Frobenius endomorphism of a characteristic $p$ ring is epimorphic, is it surjective?
MO question 19282 is about integral epimorphisms of commutative rings, and a counterexample is given to surjectivity. What about the case of the Frobenius endomorphism of a commutative, characteristic ...
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Minimal free resolution over arbitrary varieties
Over projective space, it is well-known that given a graded $S^\bullet$-module $M_\bullet$, where $S^\bullet = k[x_0, \dots, x_N]$, there is a unique minimal free resolution
$$
\cdots \to \...
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Homological algebra for commutative monoids?
Homological algebra for abelian groups is a standard tool in many fields of mathematics. How much carries over to the setting of commutative monoids (with unit)? It seems like there is a notion of ...
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Flatness criterion for $I$-adic ring: $I$-torsion free
Let $R$ be an $I$-adically separated and complete valuation ring, with $I$ finitely generated.
It is used a few times in Bosch, Lectures on Formal and Rigid Geometry e.g. first lines of pg. 164, Cor. ...
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Efficiently computing Gröbner basis to prove no solution to polynomial constraints
In a similar vein to these now quite old questions on advice for calculating a Gröbner basis:
Fast computation of a Groebner basis. What is possible?
What is the state of art in Groebner bases
I am ...
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Maximizing and minimizing the number of positive product $k$-subsets of an $n$-set
The question is simple but require some definitions. I came across resolving a certain inequality. If there is no closed answer is there a related sequence describing the situation?
Let
$$S\ :=\ \{X=...
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Eichler orders in a certain quaternion algebra
Let us consider a totally real number field $K$ such that $[K \colon {\Bbb Q}] = {\mathrm{odd}}$. We shall consider the quaternion algebra $D$ over $K$ such that $D$ splits everywhere at finite places ...
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Generalization of the second Brauer-Thrall conjecture for arbitrary Artin algebras
Let $k$ be a field with infinite cardinality and $A$ a finite dimensional $k$-Algebra. The second Brauer-Thrall conjecture states the following: There are infinitely many natural numbers $n_1<n_2&...
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Construction of Jacobian Ideal
In Qing Liu's Algebraic Geometry and Arithmetic Curve, we have the following proposition(6.3.13):
Let $S$ be a locally Noetherian scheme. Let $X$, $Y$ be smooth schemes over $S$. Then any immersion $f:...
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Textbooks on SINGULAR and Macaulay 2
Recently, I'm tired of those theoretical parts on commuative algebra. So I hope that someone could recommend me some good textbooks on SINGULAR and Macaulay 2. And I'm wondering whether SINGULAR is ...
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Do relatively prime polynomials $f$ and $g$ in $k[x,y]$ generate an ideal of finite codimension?
Let $k$ be a field and $f,g \in k[x,y]$ be two non-constant polynomials in two variables. Is it true that the following conditions are equivalent:
$f$ and $g$ are relatively prime in $k[x,y]$, in the ...
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Symbolic power of an ideal associated to non-singular algebraic set
Let $Z\subset \mathbb P^n$ be a reduced non-singular algebraic set and $I$ denote the saturated homogeneous ideal of $Z$. I have seen the following result without proof:
For all $ n\geq 1$, $I^{(n)}=(...
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How to show an invariant subfield of rational function field $\mathbb{Q}(x)$ under a certain group action is actually a simple extension? [duplicate]
Let $K=\mathbb{Q}(x)$ be the rational functions in one variable $x$ and let the automorphisms $\phi,\psi$ of $K$ be defined as $\phi(x)=-\frac{1}{x+1}$ and $\psi(x)=\frac{1}{x}$.
Let $G$ be the group ...
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Structures of subgroups of a finite abelian p-group
$\newcommand\la{\langle}\newcommand\ra{\rangle}$Let $G=\mathbb{Z}/p^{i_1}\times\cdots\times\mathbb{Z}/p^{i_r}$ with $i_1\leq\ldots\leq i_r$ be a finite abelian $p$-group. Then there can be many ...
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The soccer splitting problem in arbitrary commutative ring
There's a folklore problem:
Let $x_1, \cdots, x_{23} \in \mathbb{Z}$ be the weights of $23$ soccer players. Now Master Yoda want's to form two soccer teams with $11$ players each. Turns out for ...
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