Powers of Linearly Presented Primary Ideals

Definitions and mathematical statement

Let $K$ be a field, $n\geq1$, and $S=K[x_1,\ldots,x_n]$ with its standard grading and homogeneous maximal ideal $\mathfrak m=(x_1,\ldots,x_n)$. Let $I\subsetneq S$ be a homogeneous $\mathfrak m$-primary ideal, meaning $\sqrt I=\mathfrak m$, generated in one degree $d\geq1$, so $I=S I_d$. Say that $I$ is linearly presented if it has a graded presentation \[ S(-d-1)^b\longrightarrow S(-d)^a\longrightarrow I\longrightarrow0 \] whose first map has linear-form entries; equivalently, the first syzygies on minimal degree-$d$ generators are generated in degree $d+1$. This imposes no claim that every higher syzygy is linear.

The conjecture is \[ I^{\,n-1}=\mathfrak m^{(n-1)d} \] for every ideal satisfying these assumptions. Powers are ordinary ideal powers. Equivalently, since $I^{n-1}$ is generated in degree $(n-1)d$, all forms of that degree must lie in it: \[ (I^{n-1})_{(n-1)d}=S_{(n-1)d}. \] For $n=1$ the assertion is the convention $I^0=S=\mathfrak m^0$; the substantive exponent is positive for $n\geq2$. No characteristic restriction on $K$ is supplied.

The supplied literature assessment checked 21 August 2026 reports the precise $(n-1)$-power equality proved for $n=3$ and for monomial ideals, while describing the general result only as eventual equality: for a given ideal, $I^t=\mathfrak m^{td}$ for all sufficiently large $t$. Such an ideal-dependent eventual threshold is weaker than the uniform threshold $t=n-1$. The assessment calls the all-dimensional conjecture open. These are dated supplied claims; full theorem hypotheses, including any field qualifications in the references, require review.

Short English statement

Must the $(n-1)$st ordinary power of a degree-$d$, linearly presented primary ideal in $n$ variables contain every form of degree $(n-1)d$? The supplied assessment reports the bound in dimension three and for monomial ideals, but only eventual equality in general.

Sources

• Local UnsolvedMath registry, permanent ID 30000080, exact source code catalogue entry; Powers of Linearly Presented Primary Ideals. Proposed year 2004, status open; corrected_verified extraction retains the displayed conjecture.
• Supplied report citation: Combinatorial Commutative Algebra (2005), Oberwolfach Reports, DOI 10.4171/owr/2004/32, pp. 1703--1746, https://doi.org/10.4171/owr/2004/32. The title year differs from DOI/proposed year.
• Supplied literature assessment checked 2026-08-21: Eisenbud--Huneke--Ulrich, The Regularity of Tor, https://arxiv.org/abs/math/0405373; "Published version", https://doi.org/10.1353/ajm.2006.0022; "Original Oberwolfach report", https://doi.org/10.4171/owr/2004/32. This dated OWR-derived triage requests independent verification of time-sensitive status and cited claims.