Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,337
questions
4
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235
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Height of a conductor ideal
We say an extension of Noetherian rings $R\subset S$ is elementary subintegral if $S=R[b]$ for some $b\in S$ with $b^2,b^3\in R$. The conductor ideal is defined to be $\operatorname{Ann}_R(S/R)$. What ...
3
votes
0
answers
126
views
Composition of Frobenius $n$-homomorphisms, characteristic-free?
This question is, as so often, a crossbreed of curiosity and laziness. The
former has led me to an interesting, but somewhat unsatisfactory paper by
Khudaverdian and Voronov
(arXiv:2002.02395v2) and, ...
0
votes
0
answers
64
views
Hensel lifting of roots of a biquadratic polynomial
Let $5$ divide $p-1$. Therefore, we have $$1+x+x^2+x^3+x^4=(x-\alpha)(x-{\alpha}^2)(x-\alpha^3)(x-\alpha^4)=f_1f_2f_3f_4$$ over $F_p,$ where $\alpha$ is an element of order $5$ in ${F_p}^\times.$ We ...
3
votes
1
answer
157
views
Kernel of a map of Tate algebras
Let $A$ and $B$ be a pair of noetherian Tate algebras over $\mathbb{Q}_p$, and assume $\text{dim}_{\text{Krull}}(B) > \text{dim}_{\text{Krull}}(A)$. Is it true that any morphism $B \longrightarrow ...
2
votes
1
answer
140
views
The presentations of finite complete local rings
Suppose that $R$ is a commutative ring such that there is a surjection $ \pi:\mathbf{Z}_p[[T_1,\cdots,T_n]]\to R$ of rings where $\mathbf{Z}_p[[T_1,\cdots,T_n]]$ is the ring of formal power series ...
1
vote
0
answers
85
views
Regarding the common zeros of the system of equations
Consider the following two systems of n homogeneous polynomials in n variables of degree $d$ with complex coefficients:
System 1 ($S_1$):
$f_1(x_1,\dots,x_n) = 0$,
$f_2(x_1,\dots,x_n) = 0$,
$\vdots$
$...
3
votes
2
answers
338
views
$R$-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\...
1
vote
0
answers
50
views
Can a surjective morphism between complete intersection rings be given by adding terms to a regular sequence?
Given a surjective morphism
$$\frac{\mathcal{O}[[X_{1},\dots ,X_{n}]]}{I}\twoheadrightarrow \frac{\mathcal{O}[[X_{1},\dots ,X_{n}]]}{J}$$
where $I,J$ are genereated by regular sequences.
Question
Can ...
0
votes
1
answer
223
views
Unpacking the plethystic substitution $h_n[n\mathbf{z}]$ in a paper by Aval, Bergeron and Garsia
I'm not familiar with the formalism of plethysm, so I need some help in unpacking the plethystic substitution $h_n[n\mathbf{z}]$ found in eqns. 5.6 and 5.9 of "Combinatorics of labelled ...
6
votes
3
answers
879
views
A polynomial identity for $\displaystyle \sum_{k=0} ^m (-1)^ka^{m-k}b^k$
I asked this question on MSE here.
I was solving this problem:
Show that if $\gcd(a, b) = 1$ and $p$ is an odd prime, then $\displaystyle \gcd\left(a+b, \frac{a^p+b^p}{a+b}\right)=1$ or $p$.
This ...
4
votes
0
answers
276
views
What is known about the number of elements needed to generate a given ideal in $k[X_1,\dots,X_n]$?
In Algebraic Geometry by J.S. Milne, after he proves Hilbert's Basis Theorem, he makes the following aside:
One may ask how many elements are needed to generate a given a given ideal $\mathfrak a$ in ...
1
vote
0
answers
94
views
Algorithms to decompose a graded module over $R[x]$, where $R$ is a PID
There is a certain class of objects, which can be thought of either as modules over a ring $R[x]$ or as functors. A few equivalent definitions are given below. The question is what computer algorithms ...
10
votes
1
answer
357
views
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 ...
1
vote
1
answer
426
views
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 $\Gamma(E)$ ...
4
votes
0
answers
106
views
Length of dual module
It is well known that, given a commutative ring $R$ and an $R$-module $M$, the dual module $M^\vee = \operatorname{Hom}_R(M, R)$ does not always satisfy $M^\vee \cong M \ (1)$, and not even $M^{\vee \...
0
votes
1
answer
99
views
$S/I$-freeness of $I/I^2$ vs $I/I^{(2)}$, where $I$ is a radical ideal of regular local ring $S$
Let $I$ be a radical ideal of a regular local ring $S$. Put $R:=S/I$. Let $I^{(n)}$ be the $n$-th symbolic power of $I$. It is well-known that $I^n \subseteq I^{(n)}$.
Is it true that $I/I^2$ is $R$-...
3
votes
0
answers
138
views
Taylor-Wiles systems for higher dimensional deformation rings
Let $R$ be a deformation ring and $M$ be a finitely generated $R$-module.
A strategy for proving the theorems $R=T$ is to associate with $(R,M)$ a Taylor-Wiles system denoted $(R_{Q},M_{Q})$. Here I'm ...
1
vote
1
answer
255
views
Is the completed tensor product (over a complete dvr) of two reduced complete Noetherian local rings again reduced?
To be more specific, Let $\mathcal{O}$ be a finite extension of $\mathbb{Z}_{p}$. Let $A=\mathcal{O}[[X_{1},\ldots, X_{n}]]/\left( f_{1},\ldots,f_{r}\right) $ and $B=\mathcal{O}[[Y_{1},\ldots, Y_{m}]]/...
0
votes
0
answers
58
views
Symbolic polyhedron of a monomial ideal
$\DeclareMathOperator\maxAss{maxAss}\DeclareMathOperator\conv{conv}$Let $I$ be a non-zero monomial ideal and $P$ $\subseteq$ $\mathbb R_+ ^ {n+1}$ be its symbolic polyhedron: then
$$
\alpha(P)= \min \{...
1
vote
0
answers
45
views
generating set of polynomial ring
I am considering the polynomials $P=P[x_1,x_2,\ldots,x_n]$ with coefficients in a ring $R$. Consider a subset $S=\{p_1,p_2,\ldots,p_k\}$ of $P$. There is a map $f\colon P[x_1,x_2,\ldots,x_k] \to P$ ...
0
votes
1
answer
135
views
On the solutions of system of homogeneous polynomials of degree $d$ in $n$ variables
Consider the following two system of n homogeneous polynomials in n variables of degree $d$ with complex coefficients:
System 1 ($S_1$):
$f_1(x_1,\dots,x_n) = 0$,
$f_2(x_1,\dots,x_n) = 0$,
$\vdots$
$...
0
votes
0
answers
85
views
Totally isotropic space for bilinear pairing over ring
A duplicate of this:
Consider the following well-known inequality: Let $b$
be a non-degenerate symmetric bilinear pairing over a (finite-dimensional) $\mathbb F$-vector space $V$ and $W$
a totally ...
2
votes
1
answer
111
views
Projective dimension and subrings
$\DeclareMathOperator\pd{pd}$Suppose that $R$ is a commutative ring and $R'$ is a subring of $R$ such that $R$ is a free $R'$-module of finite rank. Assume that both $R$ and $R'$ are regular local ...
3
votes
0
answers
161
views
Amalgamation of commutative subrings
Let $A$ and $B$ be commutative subrings of a non-commutative ring $X$.
Is there always a commutative ring $Y$ containing $A$ and $B$ preserving their intersection?
This is equivalent to ask if in the ...
1
vote
1
answer
83
views
Perturbing pole of Laurent polynomial/series in a single summand
I am working with the ring of Laurent polynomials $\mathbb{F}[X,X^{-1}]$ over $\mathbb{F}$ for some algebraically closed field $\mathbb{F}$ of any characteristic. I encountered a problem emerging from ...
1
vote
0
answers
331
views
Amitsur's theorem for generalized Severi–Brauer varieties
Let $k$ be a field of characteristic zero and assume that $A$ is a central simple algebra of index $2^n > 2$. We denote by $\operatorname{SB}_i(A)$ the $i$-th (generalized) Severi–Brauer variety of ...
8
votes
1
answer
598
views
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 ...
1
vote
1
answer
208
views
Zero divisors in the boolean polynomial ring $\mathbb{F}_2[x_1,x_2,...,x_n]/(x_1^2-x_1,x_2^2-x_2,...x_n^2-x_n)$
Related to this question.
Let $n$ be positive integer and let $K$ be the boolean polynomial ring
$\mathbb{F}_2[x_1,x_2,...,x_n]/(x_1^2-x_1,x_2^2-x_2,...x_n^2-x_n)$.
For all $a$ in $K$ we have $a= -a$ ...
1
vote
0
answers
123
views
Gelfand's representation on matrices: construct maximal ideal in matrix algebra
I would like to see a constructive proof (some algorithm?) of the following statement:
Let $A_1, A_2, \dotsc ,A_k \in M_n(\mathbb C)$ be some commuting matrices, let $B$ be the commutative algebra (...
0
votes
0
answers
104
views
Affine scheme over ring of meromorphic functions with finite poles on unit circle
I am looking into the set $S$ of meromorphic functions with a finite number of poles on the unit circle (i.e., rational functions with poles on the unit circle). I assume that any $h\in S$ has the ...
3
votes
1
answer
135
views
Formal étaleness along Henselian thickenings
Assume that $f:X\to Y$ is an étale map between smooth varieties and $(S,I)$ is a Henselian pair. Let $\alpha\in X(S/I)$. Can we say that the lifts of $\alpha$ to $X(S)$ are in bijection with the lifts ...
1
vote
0
answers
86
views
Computing simplicial resolution of rings
As the title says, I would like to ask how we can give "convient" simplicial resolutions of rings. In the category of modules this is often true: ifI have a ring $R$ and some ideal $I$ ...
1
vote
0
answers
56
views
Factorization of the symmetric function identity $E(t)=1/H(t)$ with the refined Euler characteristic polynomials of associahedra / Lagrange inversion
I've come across two matrix identities, flagged with daggers below, relating the two sets of elementary and complete homogeneous symmetric polynomials/functions via the two sets of refined Lah and ...
8
votes
1
answer
938
views
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 ...
1
vote
0
answers
162
views
Does the symmetric algebra functor preserve inclusions?
Theorem: For any compact abelian group $G$, the homogeneous component $%
H^{2}\left( B_{G};%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
\right) $ of degree $2$ is naturally ...
2
votes
1
answer
109
views
height of sum of prime ideals
Suppose $R$ is a Cohen-Macaulay local ring and $P,Q$ are prime ideals in $R$. Let the height of $P$ and $Q$ be $m$ and $n$ respectively. Then is it true that the height of $P+Q$ is at most $m+n$?
2
votes
1
answer
73
views
Is uniform dimension monotonic in quotients when there is a unique indecomposable injective?
The notion of uniform or Goldie dimension is something I’ve only seen discussed for categories of modules, but I believe the theory works the same way in any Grothendieck category $\mathcal C$. Recall ...
1
vote
0
answers
78
views
Hasse invariant of subalgebra of division algebra over local field
This question didn't receive an answer on MathSE.
Let $K$ be a $p$-adic field, or more generally a local field. Let $D$ be a $d^2$-dimensional division algebra over $K$. Then $D$ is necessarily of the ...
6
votes
1
answer
355
views
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 ...
2
votes
1
answer
86
views
A perfect complex over a local Cohen--Macaulay ring whose canonical dual is concentrated in a single degree
Let $R$ be a complete local Cohen--Macaulay ring with dualizing module $\omega$. Let $M$ be a perfect complex over $R$. If the homology of $\mathbf R\text{Hom}_R(M,\omega)$ is concentrated in a ...
0
votes
0
answers
255
views
Smooth morphisms under base change, Qing Liu's proposition 4.3.38
I have a concern about the first assertion in the proof of proposition 4.3.38 of Qing liu's "Algebraic Geometry and Arithmetic Curves". Referring to smooth morphisms, he says "The ...
4
votes
1
answer
161
views
If $\pi$ is a prime of a UFD $A$, is $\text{Spec }A$ a coproduct of $\text{Spec }A[\pi^{-1}]$ and $\text{Spec }A_{(\pi)}$ over $\text{Spec Frac }A$?
Let $A$ be a UFD (unique factorization domain) with fraction field $K$. Let $\pi\in A$ a prime. Let $A_{(\pi)}$ be the localization at the ideal $\pi$, and let $A[\pi^{-1}]$ be the localization w.r.t. ...
5
votes
1
answer
176
views
Are module finite algebras over semiperfect rings again semiperfect?
Let $S$ be a Noetherian semiperfect ring (https://en.m.wikipedia.org/wiki/Perfect_ring). Let $R$ be a module finite associative $S$-algebra. Then, is $R$ also a semiperfect ring? (Clearly, $R$ is ...
0
votes
0
answers
120
views
Nilpotent elements of $(\mathbb{Z}/n \mathbb{Z})[x_1,...,x_m]/\langle f_1(x_1,...,x_m),f_2(x_1,...,x_m),...f_k(x_1,...,x_m)\rangle$
This is generalization of the univariate case
and also related to open problem.
Let $n,k,m,B>1$ be positive integers and $f_1(x_1,...,x_m),f_2(x_1,...,x_m),...f_k(x_1,...,x_m)$ be polynomials with ...
3
votes
1
answer
150
views
Question on the extension theorem from Humphreys' book on linear algebraic groups
On page 2 of Humphreys' book "Linear algebraic groups" he presents the "extension theorem". I will copy it below:
Extension theorem. Let $R/S$ be an integral extension, $K$ an ...
3
votes
1
answer
130
views
A question about minimal primes in the integral group ring of a finite abelian group
Let $G$ be a finite abelian group of order $n$ and let $R=\mathbb{Z}G$ denote the integral group ring of $G$. Let $R'$ denote the localization of $R$ with respect to the multiplicatively closed set ...
4
votes
0
answers
166
views
Nilpotent elements of $K=(\mathbb{Z}/n \mathbb{Z})[x]/f(x)$
This is related to an open problem.
Let $n$ be integer and $f(x)$ polynomial with integer coefficients and set $K=(\mathbb{Z}/n \mathbb{Z})[x]/f(x)$.
Let $S$ be the set of degree 2 nilpotent elements ...
0
votes
0
answers
114
views
A question on a system of quadratic polynomials
Consider the following system of quadratic polynomials $f_1,...,f_n \in \bar{\mathbb{F}}_2[x_1,....,x_n]$ :
$f_1 (\bar{x}) = x_1 + x_n^2 + q_1$
$f_i(\bar{x}) = x_i + q_i$ for $i \in \{2,...,n-1 \}$
$...
2
votes
0
answers
70
views
From exact triangles in the stable category of maximal Cohen--Macaulay modules to short exact sequences
Let $R$ be a local Gorenstein ring. Let $\underline{\text{CM}}(R)$ be the stable category of maximal Cohen--Macaulay modules, it is known to carry a triangulated structure. My question is: If $M\to N\...
0
votes
0
answers
67
views
"Approximating" ring of semi-invariants
I'm trying to calculate the semi-invariant ring for certain types of quivers. For a very brief introduction to semi-invariant rings of quiver please have a look at this wikipedia article at the ...