Descent of Hereditary Orders with Involution

Definitions and mathematical statement

Let $R$ be a discrete valuation ring, let $F=\operatorname{Frac}(R)$, and let $k$ be its residue field. The source requires that $2$ be invertible, but its formulations do not consistently locate this condition: $2\in R^\times$ and $2\in F^\times$ give different hypotheses. The intended choice remains unresolved below.

An $R$-order $A$ is a unital $R$-algebra that is finitely generated and torsion-free as an $R$-module and spans its finite-dimensional generic algebra $A_F=A\otimes_R F$. It is hereditary if every left ideal is projective as a left $A$-module. In the usual $R$-linear interpretation, an involution is an $R$-linear map $\tau:A\to A$ satisfying \[ \tau(ab)=\tau(b)\tau(a),\qquad \tau(1)=1,\qquad \tau^2=\operatorname{id}_A. \] An isomorphism of orders with involution is an $R$-algebra isomorphism $f:A\to A'$ such that $f\circ\tau=\tau'\circ f$. For an $R$-algebra $T$, write $(A_T,\tau_T)=(A\otimes_R T,\tau\otimes\operatorname{id}_T)$.

Let $J(A)$ denote the Jacobson radical. An involution preserves this radical and therefore induces an involution $\overline\tau$ on $\overline A=A/J(A)$. Here anisotropy means \[ \forall x\in\overline A,\qquad \overline\tau(x)x=0\ \Longrightarrow\ x=0. \] Use the analogous notation and definition for $(A',\tau')$.

Principal descent question. For every pair of hereditary $R$-orders with involution $(A,\tau)$ and $(A',\tau')$, suppose that \[ (A_F,\tau_F)\cong(A'_F,\tau'_F) \] and that there exists a finite étale $R$-algebra $B$ for which \[ (A_B,\tau_B)\cong(A'_B,\tau'_B). \] If both $\overline\tau$ and $\overline\tau'$ are anisotropic, must \[ (A,\tau)\cong(A',\tau') \] over $R$? All displayed isomorphisms are isomorphisms of algebras with involution. The source does not explicitly require $B$ to be nonzero or faithfully flat; this qualification must be settled to specify the intended étale descent hypothesis. Over a discrete valuation ring, a nonzero finite étale algebra is faithfully flat.

Further questions, incompletely specified by the source. Does the same descent conclusion hold when the residual-anisotropy hypothesis is replaced by the condition that every closed fiber of the automorphism group scheme is almost anisotropic? The natural automorphism functor of a pair is \[ T\longmapsto\operatorname{Aut}_{T\text{-alg},\tau_T}(A_T), \] but the source does not specify whether the condition concerns this functor for $A$, the corresponding functor for $A'$, or both. Nor does it define "almost anisotropic." For a group scheme over $\operatorname{Spec}R$, there is only one closed base point, with fiber over $k$; whether a geometric closed fiber is intended is also unspecified.

Finally, the source asks whether every central simple $F$-algebra with involution $(C,\sigma)$ admits a hereditary $R$-order with involution whose generic fiber is isomorphic to $(C,\sigma)$ and which is "such an order." It does not specify whether this phrase requires residual anisotropy, almost anisotropy of an automorphism-group fiber, or another preceding property. The permitted action of involutions on the center is likewise unspecified. Consequently this existence question, and the proposed group-scheme generalization, cannot yet be given a unique precise formulation.

Short English statement

The principal question asks whether two hereditary orders with involution that are isomorphic over the fraction field and after a finite étale extension must already be isomorphic over the discrete valuation ring when their radical quotients have anisotropic involutions. The source also asks for an almost-anisotropic group-scheme generalization and an existence statement for orders. Several hypotheses and the targets of these extensions require clarification. This is a statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $30003793$, title "Descent of Hereditary Orders with Involution," exact source problem code catalogue entry; supplied collection: Oberwolfach Reports Open Problems. No individual proposer is named. The supplied proposed year is 2018.
• Supplied report citation: Quadratic Forms and Related Structures over Fields (2019), Oberwolfach Reports, DOI 10.4171/owr/2018/21, pp. 1235--1291. https://doi.org/10.4171/owr/2018/21. The supplied title year is 2019, while the DOI contains 2018; this discrepancy has not been resolved.
• Supplied literature source, labeled "orders with involution": https://arxiv.org/abs/1804.09527.
• Supplied literature source, labeled "valuations on algebras with involution": https://doi.org/10.1007/s00208-010-0580-9.
• Supplied literature source, labeled "workshop report": https://ems.press/content/serial-article-files/46742.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and describes the anisotropic or residually-anisotropic strengthening as a conjecture. It states that related valuation and Grothendieck--Serre results do not prove this exact hereditary-order assertion. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; the status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review. These are supplied review claims and do not establish mathematical correctness or resolve the qualifications identified here.