Trivial Restrictions of Twists in Nilpotent Group Algebras

Definitions and mathematical statement

Let $G$ be a torsion-free nilpotent group of finite cohomological dimension $n$, and let $\alpha\in H^2(G,{\mathbb{C}}^\times)$. In the usual twisted-group-algebra convention, the coefficient action is trivial. Choose a normalized cocycle $a:G\times G\to{\mathbb{C}}^\times$ representing $\alpha$, so $a(g,h)a(gh,k)=a(h,k)a(g,hk)$ and $a(1,g)=a(g,1)=1$. The algebra ${\mathbb{C}}^\alpha G$ has ${\mathbb{C}}$-basis $\{u_g:g\in G\}$ and multiplication $u_gu_h=a(g,h)u_{gh}$; changing the representative cocycle gives an isomorphic algebra. Write $\operatorname{gl.dim}R$ for global dimension, conventionally the supremum of projective dimensions of left $R$-modules, and interpret cohomological dimension conventionally as $\operatorname{cd}_{{\mathbb{Z}}}G=\operatorname{pd}_{{\mathbb{Z}} G}{\mathbb{Z}}$ unless the report specifies another coefficient ring. Suppose \[1\leq r\leq n,\qquad \operatorname{gl.dim}({\mathbb{C}}^\alpha G)=r.\] Must there exist a subgroup $H\leq G$ such that \[\operatorname{res}^G_H\alpha=0\in H^2(H,{\mathbb{C}}^\times),\qquad\operatorname{gl.dim}({\mathbb{C}} H)=r?\] Trivial restriction is a cohomological assertion: it means that $a|_{H\times H}$ is a coboundary, for example $a(h,k)=b(h)b(k)/b(hk)$ for some $b:H\to{\mathbb{C}}^\times$, not necessarily that the originally chosen cocycle is pointwise one. No finite generation of $G$, normality or finite index of $H$, or equality of $H$ with a maximal such subgroup is required by the extract.

The supplied source fields conflict: the original and clean/current statements give this complete question, and the corrected_verified note says its cohomology/twisted-algebra/global-dimension notation was restored. The 2026-08-21 literature assessment instead describes an extraction containing only the word conjecture with no recoverable claim and says this was retained as open for that reason. Both descriptions are preserved; the assessment is not treated as a verification of the displayed conjecture's current mathematical status.

Short English statement

If a twisted complex group algebra of a torsion-free nilpotent group has global dimension $r$, ask whether the twist becomes cohomologically trivial on some subgroup whose ordinary group algebra has exactly that global dimension. The literature summary conflicts with the supplied complete statement.

Sources

• Local UnsolvedMath registry, permanent ID 30000879, source problem catalogue entry, Trivial Restrictions of Twists in Nilpotent Group Algebras, proposed year 2007; no individual proposer supplied.
• Supplied citation: Mini-Workshop: Arithmetik von Gruppenringen (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/55, pp. 3209--3240: https://doi.org/10.4171/owr/2007/55. Supplied literature label: original Oberwolfach report.