$K$-Theory of $D$-Crystals

Definitions and mathematical statement

Write $\mathcal C_D(X)$ for the $\infty$-category of $D$-crystals on the scheme $X$ intended by the cited source, and write $K(\mathcal C_D(X))$ for its algebraic $K$-theory spectrum with the source's exact or stable structure. The target, attributed in the supplied assessment to Barwick's Conjecture 17, is \[ \forall X\text{ a scheme},\qquad K(\mathcal C_D(X))\simeq K'(X), \] with an equivalence natural in $X$ for the class of morphisms and variance intended in the source. An equivalence of spectra induces isomorphisms on all homotopy groups in the degrees of the chosen theory; the question asks for the spectrum-level natural equivalence, not merely an equality of Grothendieck groups.

For a noetherian scheme, the usual interpretation of $K'(X)$ is coherent $K$-theory, also called $G$-theory, built from coherent sheaves. The extract's quantifier is every scheme; it supplies neither a general-scheme definition of $K'(X)$ nor the definition, finiteness conditions and exact structure of $D$-crystals. Thus the noetherian interpretation explains common notation but does not replace the stated universal scope by a verified restriction. The symbol $D$ is source-specific and is not identified here with differential operators, ordinary crystals, or arbitrary quasi-coherent complexes.

The supplied literature assessment says that details were not checked beyond the Cohen--Macaulay case. This qualification is retained without asserting that the general equivalence, or even a fully specified theorem for that special case, has been established by the supplied extract.

Short English statement

Ask whether the algebraic $K$-theory spectrum of $D$-crystals is naturally equivalent to coherent $K$-theory for every scheme. The source-specific category, general-scheme conventions and naturality maps are missing, and the reported Cohen--Macaulay limitation remains explicit.

Sources

• Supplied report: Algebraic K-Theory (2007), Oberwolfach Reports, DOI 10.4171/owr/2006/32, pp. 1941--1990; proposed year 2006. https://doi.org/10.4171/owr/2006/32.
• Supplied reference: Barwick, D-crystals, https://www.maths.ed.ac.uk/~cbarwick/papers/D-crys.pdf; the registry's assessment identifies Conjecture 17. The linked PDF has not been accessed in preparing this draft.
• Local UnsolvedMath registry, permanent ID 30000529, catalogue entry; created 25 August 2026, updated 16 September 2026. The corrected_verified note says the equivalence question is explicit. Its 21 August 2026 literature triage labels the general assertion open and reports that details were not checked beyond the Cohen--Macaulay case. This dated assessment is supplied provenance requiring independent verification.