Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,792 questions
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Is this rational function a Hilbert series?
This old MO question deserves renewed attention in my opinion. The original question defined
$$F_n(X) := \sum_{k=0}^n (-1)^k \binom{n}{k} X^{k(n-k)}$$
where $n=2m$ is an even integer, and asked for a ...
- reputation score 96730
Score of 5
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189 views
Why does André–Quillen cohomology classify square-zero extensions?
Let $B$ be an $A$-algebra and let $M$ be a $B$-module, and consider the André–Quillen cohomology groups $D^*(B / A, M)$. I am trying to understand the following fact:
Fact. The first André–Quillen ...
- reputation score 809
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93 views
Conjecture on finite, medial (entropic) magmas
Consider a finite magma $S$ and elements $a, b \in S$, denote by $R_S(a,b)$ the set of all pairs $(x,y) \in S \times S$ such that $x \not= b$ and $y \not= a$ and $ax = by$, and let $r_S(a,b)$ be the ...
- reputation score 1106
Score of 3
0 answers
213 views
Conjecture on finite, commutative, self-distributive magmas
Consider a finite magma $S$ and elements $a, b \in S$, denote by $R_S(a,b)$ the set of all pairs $(x,y) \in S \times S$ such that $x \not= b$ and $y \not= a$ and $ax = by$, and let $r_S(a,b)$ be the ...
- reputation score 1106
Score of 15
2 answers
1181 views
Is every known counterexample to the Jacobian Conjecture in $\mathbb C^n$, $n≥4$, a trivial stabilization of a 3‑dimensional one?
The recent explicit counterexample to the Jacobian Conjecture in $\mathbb C^3$
(L. Alpöge, July 2026 – a Keller map with constant Jacobian and a field
extension $L/K$ of degree $3$ with monodromy $S_3$...
- reputation score 161
Score of 1
0 answers
45 views
Intersection of a principal right ideal in an Ore localization with a quantum torus
Title: Intersection of a principal right ideal in an Ore localization with a quantum torus
Let
$$
A=\mathbb{K}\langle x^{\pm1},y^{\pm1},z^{\pm1}\rangle
$$
be the quantum torus over a field $\mathbb{K}$...
- reputation score 933
Score of 7
0 answers
4028 views
Galois structure of the new counterexample to the Jacobian conjecture: an explicit cubic model with $S_3$ monodromy — is this known? [closed]
Background
Yesterday (July 19, 2026) Levent Alpöge tweeted (source) the polynomial map $F=(a,b,c):\mathbb{C}^3\to\mathbb{C}^3$,
\begin{align*}
a&=(1+xy)^3z+y^2(1+xy)(4+3xy),\\
b&=y+3x(1+xy)^2z+...
- reputation score 151
Score of 23
0 answers
728 views
Does there exist a separated non-affine scheme with finitely many points?
A classic example of a non-affine scheme is to glue two copies of a DVR along the generic point. The resulting scheme has just $3$ points, is not affine, and is also not separated. This begs the ...
- reputation score 557
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158 views
Perturbation of ideal with "too many generators"
Let $(A, \mathfrak{m})$ be a complete regular local ring and let $I$ be an ideal that is minimally generated by $r>\dim(A)$ elements such that $\frac{A}{I}$ is not zero dimensional. Is it true that ...
- reputation score 1115
Score of 5
1 answer
233 views
Can elements in a local ring be expanded over generators of the maximal ideal, using only unit coefficients?
Let $R$ be a (commutative) Noetherian local ring with maximal ideal $ \mathfrak m$. Choose any $f_i$ that generate the maximal ideal: $(f_1, \ldots,f_t) = \mathfrak m$.
Is it true that for every ...
- reputation score 407
Score of 1
0 answers
71 views
Congruency of matrix to block diagonal over local ring
Let $(R, \mathfrak{m})$ be a complete local ring such that $2$ is invertible and let $A$ be an $m\times m$ matrix over $R$. Is there a necessary and sufficient condition for $A$ to be congruent to a (...
- reputation score 1115
Score of 6
1 answer
564 views
Cubic analogue of Euler's four squares identity
Are there polynomials with integral coefficients $P, Q, R,S$, such that the identity
$(a^3+b^3+c^3+d^3)(m^3+n^3+p^3+q^3)=P^3+Q^3+R^3+S^3$ holds? If true, in order to deduce that every integer is the ...
- reputation score 2417
Score of 4
0 answers
254 views
Are there another examples of wild automorphisms of $R[x,y]$ for a ring $R$ other than Nagata's automorphism
I learned about the fact that the automorphisms of $k[x,y]$ for a field $k$ are all tame and in fact composition of triangular and linear automorphisms. Then I saw the Nagata's example as a wild ...
- reputation score 41
Score of 5
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243 views
Number of terms of discriminant of general polynomial
Is there a formula for the number of terms of the discriminant of a polynomial $f=a_0X^n+\dots+ a_n$ with $n \in\mathbb{N}$? More precisely: Let $D_f \in \mathbb{Z}[a_0,\dots,a_n]$ be the discriminant ...
- reputation score 1010
Score of 7
2 answers
386 views
Gorenstein in all but finitely many characteristics
Fix a finite set of nonzero integer primes $S$. Is there a concrete construction somewhere in the literature of a homogeneous ideal $I$ in $R = \mathbb{Z}[x_1,..,x_n]$ for some n such that $R/(I,p)$ ...
- reputation score 343
