Section retractions obstruct the fibre-product line topoi

Definitions and mathematical statement

An arithmetic scheme in the source means a scheme of finite type over $B=\operatorname{Spec}{\mathbb{Z}}$. Flach asks for a definition of its Weil--étale topos, using the then-current Weil--étale topology, as a fibred product, and for computations of constant-sheaf integral cohomology, in particular for ${\mathbb{A}}^1_{{\mathbb{Z}}}$ and ${\mathbb{P}}^1_{{\mathbb{Z}}}$.

The relevant product is a $2$-fibre product of topoi: given geometric morphisms $\mathcal X\to\mathcal B$ and $\mathcal W\to\mathcal B$, the topos $\mathcal X\times_{\mathcal B}\mathcal W$ has the universal property of pairs of geometric morphisms equipped with an isomorphism of their composites to $\mathcal B$. Schematically, once a base Weil--étale topos $B_W\to B_{\mathrm{\acute et}}$ is fixed, the ordinary requested construction has the form \[ X_W=X_{\mathrm{\acute et}}\times_{B_{\mathrm{\acute et}}}B_W. \] The analogous Artin--Verdier compactified construction requires the corresponding compactified topoi and compatible morphisms. The question itself does not supply these base definitions or archimedean conventions. For a chosen construction, the target groups are $H^i(X_W,{\mathbb{Z}})$, with ${\mathbb{Z}}$ the constant sheaf and $i\ge0$; ordinary and compactified cohomology must be distinguished.

The supplied research assessment attributes ordinary and Artin--Verdier compactified $2$-fibre-product constructions to Flach--Morin (2012). Its candidate obstruction uses the structural morphism $p:X\to B$ and a standard section $s:B\to X$ for $X={\mathbb{A}}^1_{{\mathbb{Z}}}$ or ${\mathbb{P}}^1_{{\mathbb{Z}}}$. It claims that the induced morphisms at the actual fibre-product-topos level satisfy $p_Ws_W=\operatorname{id}$, and hence induce a split pair \[ R\Gamma(B_W,{\mathbb{Z}})\ \xrightarrow{p_W^*}\ R\Gamma(X_W,{\mathbb{Z}}) \ \xrightarrow{s_W^*}\ R\Gamma(B_W,{\mathbb{Z}}),\qquad s_W^*p_W^*=\operatorname{id}. \] The analogous assertion is made for the compactified construction. Thus base cohomology is a derived direct summand. In the compactified theory the report says that the base has a non-finitely-generated degree-four group, forcing the line-topos $H^4$ to be non-finitely generated as well.

Separately, it attributes to the 2018 Flach--Morin derived theory an unconditional projective-bundle decomposition for ${\mathbb{P}}^1_{{\mathbb{Z}}}$ with finite degree-four cohomology. The resulting claimed non-equivalence is specifically between that later derived complex and integral constant-sheaf cohomology of the 2012 compactified fibre-product topos. It is not an identification of their coefficients or a complete computation of either line's cohomology. The exact base group, complete derived complexes and projective-bundle formula are not supplied here. Their definitions and the compactification functoriality needed for the retraction must be checked in the cited constructions before this reported obstruction is accepted.

Short English statement

Define a Weil--étale topos for every arithmetic scheme and compute the constant-sheaf cohomology of the affine and projective lines. The supplied report describes a fibre-product construction and a section-retraction obstruction distinguishing its compactified integral cohomology from a later derived theory. It does not give a full list of the requested groups.

Sources

• Local UnsolvedMath registry, permanent ID $20000432$, source code catalogue entry; AIM algebraic number theory corpus, aim-algebraic-number-theory-notes.json, notes[83], stored source item 1. Attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0.
• AIM workshop The Tate conjecture, attributed to M. Flach: https://aimath.org/WWN/tateconjecture/tateconjecture.pdf. The supplied verification reports page 5, actual Problem 21 rather than OCR "1"; the trailing "5" is a page artifact. The word "current" is historical source wording, not a claim about a present-day definition.
• The research summary identifies Flach--Morin's 2012 topos construction and 2018 derived theory by authors and years only; titles, precise definitions, formulas and full bibliographic citations are absent. Original remarks and literature fields are empty.
• Registry dated August 14, 2026: statement exact, status partially_solved, classification PARTIAL-PROGRESS; attempt 1 labelled partial result, rigor proved, novelty confidence low. The section-retraction and non-equivalence claims are machine-generated research aids, not peer reviewed, and require independent verification before citation.