Specialness of Local Koszul Homomorphisms

Definitions and mathematical statement

The intended question concerns local homomorphisms $\varphi\colon Q\to R$ and the source-specific properties called Koszul and special Koszul. Its universal target is \[ \forall\,\varphi\colon Q\to R,\qquad \left(\varphi\text{ is a local Koszul homomorphism}\right) \Longrightarrow \left(\varphi\text{ is special Koszul}\right). \] The quantifier ranges over the local homomorphisms intended by the cited source. The supplied record does not specify the ambient class of rings or provide complete definitions of either property. Consequently, this is a faithful statement of the intended implication, but a self-contained mathematical formulation requires those missing definitions and hypotheses.

The extracted passage supplies the following partial structural description: if $\varphi$ is Koszul, an $A_\infty$-structure on an object $A$ can be chosen together with a free, nonnegatively graded module $V$ and a summand $W\subseteq V\otimes V$, with a quadratic presentation printed in the extraction as \[ \widetilde A=T^a(V)/(W), \qquad m_2\otimes k=\mu\otimes k, \qquad m_n\otimes k=0\quad(n\ne2). \] Here the passage calls the $m_n$ the higher multiplications and $\mu$ the tensor algebra multiplication. The notation $\widetilde A$, the object $A$, the scalar ring of the tensor products, the field $k$, and the precise meaning of $\mu$ in these comparisons are not defined in the supplied excerpt. These displayed conditions must therefore not be treated as a complete definition of either Koszulness or special Koszulness.

For context, the excerpt states that any local homomorphism \[ Q\longrightarrow R=Q[x]/(x^2-ax-b),\qquad a,b\in\mathfrak m, \] is Koszul. It describes $A=R$ as a free resolution, with $V=Qx$ and $W=V\otimes V$, and contrasts the induced multiplication with the multiplication associated with the quadratic presentation: \[ m_2(x\otimes x)=ax+b\ne0=\mu(x\otimes x) \quad\text{when }a\ne0\text{ or }b\ne0. \] The ideal $\mathfrak m$ and the interpretation of the quadratic presentation are not specified in the excerpt.

The excerpt also states that surjective complete intersection maps are Koszul. It describes the resolution \[ A=\operatorname{Kos}_Q(f_1,\ldots,f_c), \] where $f_1,\ldots,f_c$ minimally generate the kernel, and a quadratic presentation with $V=A_1$ and relations of the forms $x\otimes x$ and $x\otimes y+y\otimes x$ for distinct generators. Its printed target $R=Q$ and the indexing of these generators require clarification. Further examples asserted by the excerpt are Golod maps and surjective Gorenstein maps of projective dimension $3$. These contextual assertions are supplied source claims; they do not resolve the missing definition of special Koszulness.

Short English statement

Must every local Koszul homomorphism be special Koszul? The supplied record identifies this universal question, but omits the complete definitions and ambient ring hypotheses needed for a self-contained formulation. It labels the problem open and reports that known examples are special Koszul; those are supplied curation claims requiring review.

Sources

• Local UnsolvedMath registry, permanent ID $30005417$, title Specialness of Local Koszul Homomorphisms; exact source problem code catalogue entry. The supplied record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year $2023$; it names no proposer.
• Resolutions in Local Algebra and Singularity Theory (2023), Oberwolfach Reports, pp. 327--395, DOI 10.4171/owr/2023/6. Supplied report reference: https://doi.org/10.4171/owr/2023/6.
• Supplied literature source: Briggs--Cameron--Letz--Pollitz, https://doi.org/10.1017/FMS.2025.21. No title or precise supporting passage is supplied.
• Supplied literature source: OWR-2023-6, Question 4, https://oa.tib.eu/renate/server/api/core/bitstreams/11773ea5-b1e5-4e9e-b650-808e9d9dc944/content.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the question open, says the known examples of local Koszul homomorphisms are special Koszul, and reports that the cited primary discussion leaves the universal question unanswered. This is supplied dated curation, not an independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that the official report was checked directly. This verification claim is part of the supplied record; no source visit or independent verification is claimed here. The excerpt's internal references [1] and [2] are not bibliographically identified in the supplied material.