Finite truncation tensor-product conjecture for Macaulay rings

Establish whether, for every Macaulay ring S, every variable x not appearing in S, and every positive integer n strictly larger than the largest degree of any element of S, the tensor product S\otimes_K K[x]/(x^n) is a Macaulay ring.

Background

The paper studies when Macaulay properties are preserved under operations on posets and rings, including Cartesian products of posets and tensor products of rings. Earlier work established preservation when tensoring a lexicographically Macaulay ring with a polynomial ring in one variable, corresponding to taking a Cartesian product with an infinite path poset.

The authors report a counterexample to the broader finite-truncation conjecture asserting preservation for arbitrary positive n. They then formulate the restricted version recorded here: no counterexamples were found when n exceeds the largest degree of an element of S. The paper does not resolve this adjusted conjecture.

References

However, we found no counterexamples to the following adjusted version of \Cref{conjecture}. If $S$ is a Macaulay ring, $x$ is a variable not appearing in $S$ and $n$ is stricly larger than the largest degree of any element of $S$. then $S\otimes_K K[x]/(xn)$ is a Macaulay ring.

— Constructions of Macaulay Posets and Macaulay Rings  (2502.15166 - Beall et al., 21 Feb 2025) in Conjecture 2, Section 6, “Further conjectures, examples and counterexamples”