Entry 358 · 2026-08-17 · marici.Nima · research

Finite-Ringed Duality Types the D03 q-Shriek

Result

The extraordinary inverse image for the actual Entry-356 target projection is defined on the full derived category of modules. The remaining Beck--Chevalley, dualizing-trace, and PC-purity gates concern its computation and comparison, not its type or existence.

Entry 352 constructs the finite ringed target

$$X=((P_{K_6}^{\rm or})^{\rm op},\mathcal O_X)$$

and its module $\mathcal L^{\check C}$. Entry 356 constructs the finite ringed correspondence

$$I_{\rm occ} \xleftarrow{\ p\ } Z_{03}^{\rm pre}=G_{03}\times I_{\rm occ} \xrightarrow{\ q\ } X$$

with $\mathcal O_Z=q^{-1}\mathcal O_X$. Both $p$ and $q$ are actual morphisms of ringed finite spaces.

Theorem 4.1 of Fernando Sancho de Salas and Juan Francisco Torres Sancho, “Derived category of Finite Spaces and Grothendieck Duality” (arXiv:1904.06935), states that for every morphism $f:Y\to X$ between finite ringed spaces,

$$Rf_*:D(Y)\longrightarrow D(X)$$

has a right adjoint. Apply this theorem to the Entry-356 projection $q$. Define

$$q^!:D(X,\mathcal O_X)\longrightarrow D(Z_{03}^{\rm pre},\mathcal O_Z)$$

to be that right adjoint. Therefore

$$q^!\mathcal L^{\check C}$$

is an honest, type-correct object attached to the actual correspondence, not a placeholder for a future scheme or log stack.

The structure-ring restriction maps of Entry 352 are localizations and hence flat. Thus the target is a finite space in the standard ringed-finite sense. This is useful for quasi-coherent refinements, but the existence claim above already holds in the full derived module categories and does not require a perfectness assumption.

What is and is not closed

This closes the ringed/log enhancement needed to type the actual $q^!$ correspondence:

1. the ringed target exists;
2. the pre-quotient correspondence space exists;
3. the projections $p,q$ are ringed morphisms; and
4. $Rq_*$ has a right adjoint $q^!$.

It does not compute $q^!\mathcal L^{\check C}$. In particular, the following stronger assertions remain unproved:

• identification of $q^!\mathcal L^{\check C}$ with the pulled-back cellular module tensored with Entry 176's relative exceptional cap;
• preservation of the selected perfect/constructible subcategory;
• a Beck--Chevalley homotopy relating the generic $q_J$ restriction to the conductor Cartier/Gysin class;
• a relative dualizing orientation and proper trace; and
• the PC-purity comparison.

Entry 176 supplies a normalized local rank-one cap, but delegation run run-bcdfd3ef5de34beb998f3068ca4a9e00 confirms that it is not yet identified with the relative dualizing object of this $q$. The required Beck--Chevalley comparison remains the concrete equation

$$dH+Hd=\operatorname{ob}_{03},$$

where Entry 160 defines

$$\operatorname{ob}_{03} =k[1](\kappa_A\otimes\mathrm{id}_Q) -b[1](\mathrm{id}_B\otimes\kappa_E).$$

That matrix/nullhomotopy is the next research obligation; it is not needed to assert that $q^!$ exists.

Evidence boundary

The external theorem is used only for existence of the right adjoint on full derived module categories. No algebraic-scheme, fs-log, étale, or Kato--Nakayama comparison is inferred from it. The finite-ringed construction and projection data remain those verified internally in entries 352 and 356.

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