Algebraic Stories — Lefschetz properties of graded algebras

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a graded algebra $A=\bigoplus_{i\geq0}A_i$, multiplication by a generic linear form $\ell\in A_1$ has maximal rank if its rank is the minimum of the dimensions of its domain and codomain. The weak Lefschetz property (WLP) requires maximal rank for $\ell:A_i\to A_{i+1}$ for all $i\geq0$. The strong Lefschetz property (SLP) requires maximal rank for $\ell^e:A_i\to A_{i+e}$ for every $i\geq0$ and $e\geq1$.For $n,d,k\geq1$, put \[T_{n,d,k}=\mathbb C[x_1,\ldots,x_n]/(x_1^d,\ldots,x_n^d)^k. \] The exponent $k$ denotes the ordinary ideal power.

Statement recorded in the supplied source.

Characterize all triples $(n,d,k)$ for which $T_{n,d,k}$ has the WLP, and all triples for which it has the SLP. [S2]

Short English statement

Determine exactly when the stated powers of monomial complete-intersection ideals give weak or strong Lefschetz algebras.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1500013 (AMR-014-0013). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. Fröberg, S. Lundqvist, A. Oneto and B. Shapiro, Algebraic Stories from One and from the Other Pockets, arXiv:1801.01692 (2018), §2.3, Problem H. https://arxiv.org/abs/1801.01692.