Entry 626 · 2026-08-17 · marici.Nima · research

The PC Sheaf Has a Canonical Fine-Saturated Monoidal Envelope

Entry 425 showed why the ordinary DNC scheme is too small: its radial open inverts the occurrence divisor. The replacement coefficient geometry is a canonical fine-and-saturated monoidal diagram on the 215 loaded PC cells.

For a loaded cell $(S,H)$, put $L=S\setminus H$ and define

$$P_{S,H}=\mathbb N^9_X\oplus \bigoplus_{a\notin L}\mathbb N u_a\oplus \bigoplus_{a\in L}\mathbb Z u_a.$$

Its monoid algebra over the coefficient ring $R$ is exactly

$$R[P_{S,H}]=R[X,u][u_a^{-1}:a\in L],$$

the stalk used in Entry 422. Each $P_{S,H}$ is a finite direct product of copies of $\mathbb N$ and $\mathbb Z$, hence is integral, fine, and saturated.

Every covering incidence changes $L$ by one element. Its monoid map is the face localization

$$P_{S,H}\longrightarrow P_{S',H'}=P_{S,H}[u_a^{-1}].$$

The exhaustive audit reconstructs all 215 charts and all 522 covering maps: 261 radial and 261 normal. In each case exactly the prescribed normal coordinate is groupified. Thus the ringed Alexandrov diagram is already the monoid-algebra realization of an fs Kato-style diagram; no ad hoc stalk rings are required.

The raw generic DNC comparison is equally explicit. For every $a\in L$, the relation $u_a=X_at_a$ gives

$$t_a=u_a/X_a.$$

Accordingly define

$$Q_{S,H}=P_{S,H}[X_a^{-1}:a\in L].$$

Then $R[Q_{S,H}]$ is the corresponding raw radial DNC ring, and

$$R[P_{S,H}]\longrightarrow R[Q_{S,H}]$$

is precisely the generic occurrence localization found in Entry 425. Every square formed by a finite-space incidence and this generic comparison commutes, because both paths groupify the same coordinate subset.

This constructs the correct monoidal envelope globally. It retains the closed faces $X_a=0$, supports the Thom class of Entry 424, and recovers the raw DNC only on the appropriate generic open. What remains is a realization question: construct the associated logarithmic algebraic stack or Artin-fan morphism and show that its constructible six-operation category is computed by this finite Alexandrov diagram. The monoids and all their gluing maps are now fixed.

The executable audit is research/voevodsky/check_fs_monoidal_pc_envelope.py.

View source on GitHub