Proper log Hodge degeneration and pregluing total class lifting

Exact construction interface. Fix a finite affine cover of the separated proper X0, its finite nerve, and compatible local log-smooth lift models over each monomial Artin quotient of Lambda=kQ. The local models and relative log forms commute with the actual fixed base maps. Use the same cover, gluing operators and Thom–Whitney carrier after every base change; do not select a new gluing independently. The corrected differential is d=partial+bar-partial+ell contraction on the relative de Rham carrier. Its square is zero by the curved-calculus identities. Locally there is an absolute extension DA, a closed base one-form rho when the base monoid is N, and a nilpotent finite differential-order gauge conjugating the corrected absolute total differential to the ordinary local absolute de Rham/TW differential. These maps preserve the quotient to relative forms and reduce to the identity centrally. The curved Thom–Whitney control premise below supplies precisely these objects, including boundedness, termwise Artin flatness and literal fixed-data base change. Here is the algebra behind the last requirements: degree q TW polynomial forms on a nerve simplex vanish for q greater than its dimension, and the nerve is finite. Boundary restriction of graded polynomial forms is surjective: the face ideals $(t_i,d{t}_i)$ have their expected intersection/product identities, so faces can be lifted recursively. Splitting these graded boundary maps over k expresses each term by sums/extensions of the fixed affine-intersection modules tensored with vector spaces of polynomial simplex forms. These terms are flat and base change along any Artin algebra map, even a nonflat map. The finite-dimensional central cohomology is that of the usual finite Cech/de Rham complex, via polynomial-simplex integration and its elementary integration homotopy. This is the required construction interface, not a conclusion deduced from a hypothetical global family.

A. Central E1, with properness and all characteristic-zero fields retained. Finiteness first requires no analytic comparison. ${\mathrm{\Omega }}_{\log }^{}$ is finite locally free because f0 is log smooth. The actual published root supplier items/thm-proper-pushforward-coherent.md gives finite-dimensional $H^q$ for proper X0 over k. All $p>d$ vanish and a finite affine cover bounds q, so all spectral sequences here are finite. Spread the finitely presented proper scheme, its finite chart/overlap data, and its fs log charts over a finitely generated integral Z-algebra B in k. The same finite monoids retain saturatedness; the log-smooth chart conditions are open finite-presentation conditions, so denominators may be inverted to retain them. Properness can be spread without assuming projectivity: choose a Chow cover X' -> X with X' projective, as supplied by items/lem-chow-lemma-proper-noetherian.md, and spread its projective equations and map. The spread target is separated after localization; the graph makes the map from the projective source proper. Surjectivity holds after localization since the constructible complement has empty generic fibre. Universally surjective proper descent of universal closedness then makes the target proper. Thus this step does not replace X by a projective variety. One may further arrange B smooth over a localization of Z. Indeed the finitely generated characteristic-zero fraction field has a separating transcendence basis, and is a finite separable extension of the rational-function field. A primitive-element equation, with its derivative and all denominators inverted, gives a finite etale model over a localization of a polynomial Z-algebra. All finite coefficients in the preceding construction can be included by further localization. Closed residue fields are finite and perfect, and one may choose residue characteristic $p>d$ outside any further finite list. For clarity, the base-change argument includes de Rham hypercohomology even though d is not $O_X$-linear. A finite affine cover computes it by a bounded complex of flat B-modules: its columns are Cech modules of finite locally free relative log forms on the flat scheme $X_B$. The differential is B-linear. Proper coherent cohomology makes each column cohomology finite, hence makes total cohomology finite by a finite spectral sequence. Here is the required universal finite replacement, without adding an unimported supplier. The smooth finite-type ring B is regular Noetherian of finite dimension, so every finite B-module has a finite resolution by finite projectives: successive finite free syzygies terminate in a projective by the finite global-dimension bound [F3]. Build a finite-projective model of a bounded complex K with finite cohomology by its finite Postnikov truncations. At the first nonzero cohomology degree, use such a resolution of that cohomology module, shifted to that degree. At the next degree j, the truncation triangle is determined by a map $H^j(K)[-j]$ -> ${\tau }_{<j}$ K[1]. Replace $H^j$ by its finite projective resolution $P_j$ and the preceding truncation by its already built finite-projective model E. The connecting map is represented by an actual chain map $P_j[-j]$ -> E[1]: a bounded complex of projectives lifts maps across a quasi-isomorphism by lifting successively against its acyclic mapping cone; the obstruction at each step is a cycle and exactness makes it a boundary. The shifted cone of this chain map is the next finite-projective model. There are finitely many cohomology degrees, so this constructs a bounded finite-projective E and a quasi-isomorphism E -> K. Its bounded mapping cone consists of flat modules and is acyclic. Its cycles are flat by descending induction from the top: in 0 -> $Z^i$ -> $Con{e}^i$ -> $Z^{i+1}$ -> 0, flatness of the quotient preserves tensor exactness, and the injection criterion then makes $Z^i$ flat. Every one of these rows remains exact after tensoring with ANY B-module. Thus the cone stays acyclic and E tensor M -> K tensor M is a quasi-isomorphism for every M. Apply this construction to the finitely many columns and to the total complex. After localization their finite projective terms are finite free; perform Gaussian elimination over Frac(B), then invert its finitely many pivot denominators. The remaining zero differential blocks show that Hodge dimensions and total de Rham dimensions are constant and agree with every remaining closed fibre. This treats d as a base-linear map, not an $O_X$-linear map. Such a closed fibre lifts globally to $W_2(\kappa )$: lift the transcendence coordinates arbitrarily to Witt vectors and solve the primitive-element equation by its invertible derivative modulo $p^2$. Pull back the spread log-smooth scheme. Local relative Frobenius lifts exist on its log-etale toric charts: send monoids by multiplication by p, coefficients by Witt Frobenius, and solve the etale equations by the same invertible-Jacobian lifting. No global Frobenius lift is asserted. Here are the local Cartier and descent calculations used for this $W_2$ lift. On a saturated log-smooth chart Q -> P, the special log fibre has its essential-monomial decomposition. In a relative logarithmic weight e, the de Rham differential is wedge multiplication by the class of e in $(P^{gp}\mathrm{/}{Q}^{gp})$ tensor kappa. If that class is nonzero modulo p, choose a linear functional taking it to 1; its interior contraction contracts that Koszul summand. The zero-character summands are the p-th-power summands, by saturatedness: on an essential face the induced lattice in the quotient is primitive, and a character divisible by p lifts to a p-divisible essential exponent. Equivalently, the saturated base-change chart P ${<mrow><mi\; mathvariant="normal">p</mi><mi\; mathvariant="normal">l</mi><mi\; mathvariant="normal">u</mi><mi\; mathvariant="normal">s</mi></mrow>}_{Q,p}$ Q identifies these summands exactly with the relative Frobenius twist. This is a chartwise lattice assertion, not an assertion that the absolute underlying ring is smooth. Ordinary smooth factors contribute the familiar surviving monomials $x^{p-1}$ dx. Consequently the intrinsic inverse Cartier formulas a |-> $a^p$, da |-> $a^{p-1}$ da, dlog(m) |-> dlog(m) and their exterior products are isomorphisms onto de Rham cohomology. The formulas commute with etale maps and chart changes. For a local Frobenius lift $F_i$, define ${\varphi }_i^1(da)=(d{F}_i(\tilde{a}))$/p modulo p, and ${\varphi }_i^1(dlogm)$=$(dlog{F}_i(m))\mathrm{/}p$. Division is in the $W_2$ lift before reduction. On an overlap put $h_{ij}(da)=(F_j(\tilde{a})-F_i(\tilde{a}))$/p; on a log generator use the first-order ratio of the two lifts. These are derivations relative to Frobenius, satisfy ${\varphi }_j-{\varphi }_i=d$ $h_{ij}$ and $h_{ij}+h_{jk}=h_{ik}$, and hence give a degree-one map into the total Cech de Rham complex. Antisymmetrized tensor powers divided by j! give degree-j maps for $j<p$. Their cohomology is the exterior-power inverse Cartier map, so their direct sum is a quasi-isomorphism ${⨁}_{0\le j<p}$ ${\mathrm{\Omega }}_{X^{\mathrm{\prime }}\mathrm{/}S}^j[-j]$ -> ${\tau }_{<p}$ $F_{\ast }$ ${\mathrm{\Omega }}_{X\mathrm{/}S}^{\bullet }$. Since $p>d$ this is the full complex. This is the local computation and Cech gluing of FFR §§10–11 specialized to the everywhere log-smooth case; no codimension-two extension or reflexive $j_{\ast }$ argument is needed. Finite relative Frobenius and proper coherent cohomology now give, for every n, dim $H_{dR}^n(X_{\kappa }\mathrm{/}{S}_{\kappa })$ = ${\sum }_{p+q=n}$ dim $H^q(X_{\kappa },{\mathrm{\Omega }}_{\log }^{})$. The Frobenius twist has the same Hodge dimensions over the perfect residue field. The finite universal complexes above transfer the equality back to k. A spectral sequence of finite-dimensional vector spaces can only decrease total dimensions on each diagonal by the ranks of its incoming and outgoing differentials; equality between E1 and the abutment on every diagonal forces every differential to vanish. This proves central E1 for arbitrary characteristic-zero k, proper X0, and arbitrary sharp Q.

Compact closed-set patching. Here are the topological sheaf arguments required for F7. For a sheaf $\mathcal{E}$ and a subspace $K$, write $\mathcal{E}(K)=\mathrm{\Gamma }(K,\mathcal{E}{\mathrm{\mid }}_K)$. On a compact Hausdorff space $T$, every finite open cover has a finite closed shrinking: choose, around each point, an open neighborhood whose closure lies in one cover member by regularity, take a finite subcover, and combine the closures assigned to each member. Sections of any abelian sheaf restricted to a finite closed cover glue if they agree on overlaps. Indeed, at a point discard the closed members not containing it; represent the remaining section germs on open neighborhoods and shrink until the finitely many germs agree. These local representatives give the desired section, uniquely because its germ at every point is prescribed. Also, a section on a compact closed subset $K$ of a locally compact Hausdorff space extends to some open neighborhood of $K$. Choose finitely many local representatives on opens $U_i$ and smaller opens $W_i$ whose compact closures lie in $U_i$ and cover $K$. The disagreement set of each pair, intersected with ${\bar{W}}_i\cap {\bar{W}}_j$, is compact and disjoint from $K$. Remove their finite union from ${\bigcup }_i{W}_i$; the representatives now agree and glue. Equality of two such neighborhood extensions is likewise witnessed on a neighborhood of $K$, by their equal germs.

C. Exact analytic log-u kernel contraction. Temporarily take the standard log base N -> $A_m=C[t]\mathrm{/}(t^{m+1})$. In the absolute complex form DA[u], where u is a polynomial variable of degree zero, with $e(\alpha u^s)$ = $d_{abs}(\alpha )$ $u^s$ + s rho wedge alpha $u^{s-1}$, rho=dlog t. The composite evaluation $u=0$ followed by quotient modulo rho is a chain map to $D{R}_A$, hence to $D{R}_k$. Evaluation to DA alone is NOT a chain map. Let R be the kernel of DA[u] -> $D{R}_k$. These maps are termwise surjective and give a short exact sequence of complexes. It suffices to prove hypercohomology(R)=0. Work first in a toric local model, at its distinguished point; ordinary smooth coordinates may be contracted by the radial holomorphic Poincare homotopy with parameters. Write P for its sharp monoid, rho for the image of 1 in $P^{gp}$, $E_m=P$ minus (P+(m+1)rho), $E=P$ minus (P+rho), and $L=P^{gp}$ tensor C. A stalk section of absolute degree k is a finite polynomial in u whose coefficients are convergent monomial sums $z^e$ ${\mathrm{\ell }}_{e,s}$, ${\mathrm{\ell }}_{e,s}$ in ${\mathrm{\Lambda }}^k$ L, e in $E_m$. In the analytic monoid algebra convergence is characterized by an exponential bound $\mathrm{\mid }\mathrm{\mid }{\mathrm{\ell }}_{e,s}\mathrm{\mid }\mathrm{\mid }$ $\le$ C exp(c h(e)) for a positive integral height h. A finite set of smooth factors obeys the corresponding usual convergent-power-series bound. The relative central quotient retains e in E and maps L to L/C rho. Thus the kernel condition is ${\mathrm{\ell }}_{e,0}$ belongs to rho wedge ${\mathrm{\Lambda }}^{k-1}L$, for every e in E. The actual stalk differential is (e ell)_(e,s) = e wedge ${\mathrm{\ell }}_{e,s}$ + (s+1) rho wedge ${\mathrm{\ell }}_{e,s+1}$. To justify this analytically, first use the formula on finite monomial sums; finite truncations converge on a smaller analytic polydisc. Multiplication of coefficients by the linear weight e preserves convergence there. First-order logarithmic derivations and their finite-order composites are continuous under precisely these smaller-radius estimates. Hence the formula extends uniquely to all stalk germs. Nilpotent exponential gauges have finite order and extend by the same argument; one does not tensor an O-linear differential which does not exist. Let ell be a closed kernel section of degree k+1, with u-degree N. For e nonzero choose a linear functional ${\lambda }_e$ with ${\lambda }_e(e)=1$ and norm $\le 1\mathrm{/}\mathrm{\mid }\mathrm{\mid }e\mathrm{\mid }\mathrm{\mid }$; interior contraction $i_{{\lambda }_e}$ satisfies $(e\wedge )i_{{\lambda }_e}$ + $i_{{\lambda }_e}(e\wedge )$ = 1. Set ${\tau }_{e,N+1}=0$ and descend in s from N to zero by $r_{e,s}={\mathrm{\ell }}_{e,s}-(s+1)\rho$ wedge ${\tau }_{e,s+1}$, ${\tau }_{e,s}=i_{{\lambda }_e}$ $r_{e,s}$. Closedness and the already constructed next coefficient give e wedge $r_{e,s}=0$, so this has e wedge ${\tau }_{e,s}=r_{e,s}$. For $s=0$ and e in E choose ${\lambda }_e$ also to annihilate rho; then $i_{\lambda }$ preserves the rho-wedge ideal and ${\tau }_{e,0}$ satisfies the required kernel condition. Such functionals exist uniformly, with saturatedness used essentially. Write the saturated sharp monoid P as the intersection of the nonnegative half-spaces of its primitive integral facet forms $v_F$. For a saturated map N -> P, every $v_F(\rho )$ is 0 or 1. Here is the direct reason: localize at F, so that modulo its unit lattice $P_F$ becomes N, and the base generator has height $a=v_F(\rho )$. If $a>1$, base change N -> N, 1 -> a, gives modulo the same units a monoid generated by v,t with relation a $v=a$ t. In its groupification the nonzero torsion element t-v has a multiple equal to a unit but is not itself in the monoid: its total generator degree is zero, whereas every nonunit represented by nonnegative generators has positive degree. Thus this base-change monoid is not saturated, contrary to saturatedness of the original map. Now e is essential exactly when e-rho is not in P. The facet inequalities imply that some F has $v_F(e)=0$ and $v_F(\rho )=1$. Consequently e lies in the corresponding facet, whose real span excludes rho. Choose that facet for e. There are finitely many facets; their minimum angle with R rho is positive. Consequently one can take $\mathrm{\mid }\mathrm{\mid }{\lambda }_e\mathrm{\mid }\mathrm{\mid }\le 1\mathrm{/}(\gamma \mathrm{\mid }\mathrm{\mid }e\mathrm{\mid }\mathrm{\mid })$ with fixed $\gamma >0$. For e outside E at $s=0$ no kernel constraint is needed and the unrestricted functional suffices. For $e=0$ set ${\tau }_{0,0}=0$ and ${\tau }_{0,s+1}$ = (1/(s+1)) $i_{{\lambda }_{\rho }}$ ${\mathrm{\ell }}_{0,s}$, ${\lambda }_{\rho }(\rho )=1$. Closedness makes ${\mathrm{\ell }}_{0,s}$ rho-divisible for $s>0$, and the kernel condition does so for $s=0$. Thus (s+1)rho wedge ${\tau }_{0,s+1}={\mathrm{\ell }}_{0,s}$. The factor 1/(s+1) is required. The resulting primitive is polynomial in u of degree at most N+1. The recursion gives a uniform bound $\mathrm{\mid }\mathrm{\mid }{\tau }_{e,s}\mathrm{\mid }\mathrm{\mid }$ $\le$ $C_N$ ${\max }_{s\le j\le N}\mathrm{\mid }\mathrm{\mid }{\mathrm{\ell }}_{e,j}\mathrm{\mid }\mathrm{\mid }$ because nonzero lattice vectors have a positive minimum norm, gamma is uniform, and there are at most N+1 recursive steps. This retains the exponential monomial bound and proves convergence of the primitive. Degree-zero closed kernel sections are zero by the same descending recursion, followed by the zero-weight kernel condition. The complex R is therefore acyclic on analytic stalks. Strict etale charts analytify to local analytic isomorphisms, so this calculation proves local acyclicity on every local log-smooth model; the log chart is centered at the chosen point. It does not require a globally toric or globally smooth underlying central fibre.

Global sections and filtered colimits in degree zero. Let $\mathcal{E}={\mathrm{colim}}_i{\mathcal{E}}_i$ be a small filtered colimit of abelian sheaves on compact Hausdorff $T$. The sheaf colimit is the sheafification of the presheaf colimit; its stalks are the filtered colimits of the stalks. If $s\in {\mathcal{E}}_i(T)$ maps to zero, it becomes zero at later stages on an open neighborhood of each point. A finite subcover and filteredness give one arrow out of $i$ on which all these restrictions vanish, hence $s$ vanishes globally. This proves injectivity, on every compact Hausdorff space. For surjectivity choose finitely many opens $U_a$ on which a given section of $\mathcal{E}$ has representatives in one common stage. Choose a closed shrinking $K_a\subset U_a$ covering $T$. Restriction to a subspace commutes with sheaf colimits: its stalks are unchanged, so the natural comparison is an isomorphism on stalks. Apply the injectivity just proved on each compact $K_a\cap K_b$ to make the representatives agree there at one common later stage, then glue by step 1.3. Thus ${\mathrm{colim}}_i\mathrm{\Gamma }(T,{\mathcal{E}}_i)=\mathrm{\Gamma }(T,\mathcal{E})$ naturally. The same holds for sections of restrictions to any closed $K\subset T$.

Softness and its local lifting property. A sheaf $\mathcal{S}$ on compact Hausdorff $T$ is soft if $\mathcal{S}(T)\to \mathrm{\Gamma }(K,\mathcal{S}{\mathrm{\mid }}_K)$ is onto for every closed $K\subset T$. A flasque sheaf is soft: extend a section near $K$ by step 1.3 and then use its surjective open-set restriction map. If $0\to \mathcal{S}\to \mathcal{G}\to \mathcal{H}\to 0$ is exact with $\mathcal{S}$ soft, then $\mathcal{G}(K)\to \mathcal{H}(K)$ is onto for every closed $K$. To see this, take local lifts of a section of $\mathcal{H}{\mathrm{\mid }}_K$ and a finite closed shrinking $K_a$ of their domains covering $K$. Inductively suppose a lift has been glued on $L=K_1\cup \cdots \cup K_{a-1}$. The next lift minus the existing lift on $L\cap K_a$ is a section of $\mathcal{S}$ there, since restriction is exact on stalks. Extend this difference to $T$ by softness, subtract it from that next lift, and glue on the two closed sets. This proves the induction, including $K=T$. If $\mathcal{G}$ is soft as well, the quotient $\mathcal{H}$ is soft: first lift over $K$, then extend its $\mathcal{G}$-lift to $T$.

Soft acyclicity, with a functorial resolution. Under AC every sheaf embeds in an injective sheaf, and injectives are flasque. Embed a soft $\mathcal{S}$ into such an $\mathcal{I}$. Step 2.2 makes $\mathcal{I}\mathrm{/}\mathcal{S}$ soft and makes $\mathrm{\Gamma }(T,\mathcal{I})\to \mathrm{\Gamma }(T,\mathcal{I}\mathrm{/}\mathcal{S})$ onto. The long exact cohomology sequence gives $H^1(T,\mathcal{S})=0$ and $H^{q+1}(T,\mathcal{S})=H^q(T,\mathcal{I}\mathrm{/}\mathcal{S})$ for $q\ge 1$, since positive cohomology of an injective vanishes by its one-term injective resolution. Induction on $q$, applied to all soft sheaves, proves their acyclicity. Use the functorial Godement resolution $C^{\bullet }(\mathcal{E})$: $C^0(\mathcal{E})(U)={\prod }_{x\in U}{\mathcal{E}}_x$, and iterate this construction on cokernels of the injective germ map. Its restrictions are surjective by extending a family by zero, so every term is flasque and hence soft. The germ map is injective, making the resolution exact. An exact resolution by acyclic sheaves computes derived global sections: split it into its kernel short exact sequences and use the same long exact sequence and degree-zero kernel identity, inductively in each cohomology degree.

Compact cohomology and filtered colimits. If the ${\mathcal{S}}_i$ are soft, their colimit is soft: a section over closed $K$ is represented in some ${\mathcal{S}}_i(K)$ by step 2.1, extends to ${\mathcal{S}}_i(T)$, and hence extends to the colimit. Apply the functorial resolution of step 3.1 to a filtered diagram ${\mathcal{E}}_i$. Taking its colimit term by term preserves exactness, because this can be checked on stalks and filtered colimits of abelian groups are exact: each equation, kernel element and preimage involves finitely many witnesses, which hold together at a common stage. The resulting terms are soft by the preceding argument. It is therefore an acyclic resolution of ${\mathrm{colim}}_i{\mathcal{E}}_i$, and step 2.1 identifies its global-sections complex with ${\mathrm{colim}}_i\mathrm{\Gamma }(T,C^{\bullet }({\mathcal{E}}_i))$. Exactness of filtered group colimits also identifies the cohomology of this complex with the colimit of the cohomologies, by the same finite-witness argument for cycles and boundaries. These natural maps prove $H^q(T,{\mathrm{colim}}_i{\mathcal{E}}_i)={\mathrm{colim}}_i{H}^q(T,{\mathcal{E}}_i)$ for every $q\ge 0$, without commuting colimits with a Godement product.

The full locally compact version of F7. Here ${\mathrm{\Gamma }}_c$ consists of global sections with compact support, and $H_c^q$ denotes its $q$-th right derived functor. If $Y$ is compact, then ${\mathrm{\Gamma }}_c=\mathrm{\Gamma }$ and step 4.1 proves the assertion. For noncompact $Y$, let $j:Y\hookrightarrow T=Y\cup \{\mathrm{\infty }\}$ be its compact Hausdorff one-point compactification. Call $\mathcal{S}$ on $Y$ c-soft when its sections on every compact subset extend globally. A flasque sheaf is c-soft by the compact-neighborhood argument in step 1.3. For c-soft $\mathcal{S}$, $j_{!}\mathcal{S}$ is soft on $T$. Indeed, a section over closed $K\subset T$ has compact support $L\subset K\cap Y$, because its germ at infinity is zero. Choose a compact neighborhood $M\subset Y$ of $L$. On the compact closed subset $(K\cap M)\cup \mathrm{\partial }M$ of $Y$, glue the given section on $K\cap M$ to zero on $\mathrm{\partial }M$; they agree on their intersection because $L\subset \mathrm{int}M$. C-softness extends this to $Y$. It is zero on a neighborhood of $\mathrm{\partial }M$ by its zero boundary germs and compactness, so restricting to $M$ and extending by zero gives a compactly supported global section agreeing with the prescribed section on $K$. Extension by zero is exact on stalks. Thus for an injective resolution ${\mathcal{I}}^{\bullet }$ on $Y$, $j_{!}{\mathcal{I}}^{\bullet }$ is a soft resolution of $j_{!}\mathcal{E}$ on $T$; global sections of its terms are precisely ${\mathrm{\Gamma }}_c(Y,{\mathcal{I}}^{\bullet })$. This proves naturally $H_c^q(Y,\mathcal{E})=H^q(T,j_{!}\mathcal{E})$. Since $j_{!}$ is a left adjoint, it commutes with colimits; step 4.1 proves the locally compact assertion of F7. For compact $Y$ every global section has compact support, so $H_c^q=H^q$.

B. Proper algebraic/analytic cohomology comparison needed below. We first reconstruct projective comparison from the actual local suppliers in F4. Cover $P^N(C)$ by its N+1 standard affine charts. Each nonempty intersection is $C^a$ times (C*)^b and is Hartogs pseudoconvex; for example its polydisc boundary radius has negative logarithm equal to a finite maximum of the pluriharmonic functions $-\log \mathrm{\mid }{z}_j\mathrm{\mid }$. F4 therefore makes each intersection acyclic for O(m), which is trivial there. Every holomorphic section has its unique normally convergent homogeneous Laurent expansion with exponent vector $a=(a_0,\mathrm{.}\mathrm{.}\mathrm{.},a_N)$, sum $a_i=m$. Such a mode occurs in the intersection indexed by I exactly when $Neg(a)=\{i:a_i<0\}$ is contained in I. If Neg(a) is nonempty and proper, insert the least vertex j outside Neg(a) with its ordered-simplex sign. The two face-deletion sums cancel except for the identity, giving the Cech contraction. This contraction preserves convergence: group the modes by their finitely many sign sets and chosen vertices, and on each compact output product choose larger outer Cauchy circles and smaller inner Laurent circles. The coefficient estimates are products of geometric ratios $R\mathrm{/}R-prime<1$ and $\rho -prime\mathrm{/}\rho <1$. The inserted coordinate has nonnegative exponents, so the subseries extends across its zero hyperplane. Thus this is a holomorphic contraction, not just a formal monomial argument. The only surviving modes have Neg(a) empty in degree zero, or all N+1 vertices in degree N. They form finite sets of homogeneous monomials for every fixed m, and are exactly the modes of the algebraic standard-chart Cech calculation. The natural inclusion sends each mode to itself, proving the actual comparison for O(m), all m and all degrees. $N=0$ is the point case. The finite twisting-sheaf resolution in F4 remains exact analytically by local-ring flatness; its finite-resolution spectral sequence therefore proves coherent comparison on $P^N$. A closed immersion is exact on underlying sheaves and its analytification has the same local quotient, so this proves comparison for every coherent sheaf on every closed projective subscheme as well. Now let pi:Y -> U be projective with U affine of finite type over C. Embed U in $A^M$ and Y, via its graph, in $P^N$ x $A^M$. A coherent sheaf on Y becomes a coherent sheaf on this product, with a finite graded module over $C[y_1,\mathrm{.}\mathrm{.}\mathrm{.},y_M,x_0,\mathrm{.}\mathrm{.}\mathrm{.},x_N]$, deg(y)=0, deg(x)=1: choose finitely many generators and relations on its finite standard-chart cover and clear their finitely many homogeneous denominators. Sheafification recovers the given finite presentations. Resolve it by finite graded free modules. This polynomial ring is regular of finite global dimension M+N+1 (items/thm-localisation-and-polynomial-extension-of-regular-rings.md); thus the terminal syzygy is finite graded projective. It is a graded direct summand of a finite graded free module: an ungraded splitting can be replaced by its degree-zero homogeneous component. A globally free terminal syzygy or a Quillen–Suslin assertion is unnecessary. Analytification of this finite resolution is exact by the actual local-flatness supplier. Over a small ambient polydisc V in $C^M$, the standard projective intersections are V x $C^a$ x (C*)^b. Their line-bundle cohomology is computed by the same acyclic-intersection Laurent construction as in the preceding projective calculation: V is a polydisc, these products are pseudoconvex, and scalar Dolbeault vanishing applies. The coefficient homotopy is unchanged, with holomorphic y-coefficients. Cauchy estimates on a slightly larger compact polydisc in y give uniform coefficient bounds; the finite choices of negative exponent set and inserted vertex and the geometric bounds in the x variables preserve normal convergence on every smaller compact product. The surviving homogeneous monomials are precisely the finite algebraic ones, with coefficients in O(V). Thus the derived direct images of every O(a) compare; finite resolutions and their terminal idempotent summand give $(R^q{\pi }_{\ast }F)$^an = $R^q$ ${\pi }_{\ast }^{an}$ $F^{an}$ for every coherent F and every q. The map is the natural cohomology comparison, since it is the inclusion of the same monomial representatives and all resolution maps commute with it. This proves relative projective comparison over possibly singular/nonreduced U by working over the smooth ambient $A^M$, not by assuming a singular analytic base-change theorem. Now let Z be any integral closed subscheme of a proper complex X. Chow gives a projective scheme Z' and a proper birational pi:Z' -> Z, an isomorphism on a dense open. The map pi is projective because its graph is closed in Z' x Z. Choose a pi-ample line bundle L from an embedding of Z'. For n large, apply the affine-base Serre-vanishing Statement in F3 to the restrictions of pi over a finite affine cover of Z, and take the maximum of the finitely many bounds. The higher-direct-image sheaves restrict to those of these projective maps, so this gives $R^q$ ${\pi }_{\ast }{L}^n=0$ for $q>0$; relative projective comparison just proved gives the identical analytic vanishing. $G={\pi }_{\ast }{L}^n$ is coherent and has generic rank one on Z. Leray and the actual all-dimensional projective comparison on Z' prove $H^q(Z,G)=H^q(Z^{an},G^{an})$ in all degrees. The comparison property is two-out-of-three in coherent short exact sequences, by exact analytification and long exact cohomology. The actual items/lem-coherent-devissage-one-generic-generator.md therefore gives comparison for every coherent sheaf on proper X. This proof retains proper nonprojective X. For use with TW modules extend this to all quasi-coherent sheaves $\mathcal{F}$. Algebraically $\mathcal{F}$ is the filtered union of its coherent subsheaves, and the noetherian filtered-colimit cohomology lemma in [F2] applies. Analytification preserves this filtered union because it is a left adjoint. The compact Hausdorff filtered-colimit comparison proved in steps 1.3–4.1 applies to $X^{\mathrm{a}\mathrm{n}}$ and the analytified coherent subsheaves. It is a local proof here, rather than an imported theorem input; step 5.1 also proves the full locally compact assertion recorded in F7. Thus coherent comparison before taking colimits yields proper quasi-coherent comparison in every degree. Compactness is essential; no affine-open analytic comparison is asserted.

D. From local models to the pregluing total complex. Apply the finite TW resolution to the absolute local short exact kernel sequence. The polynomial-simplex contraction is a resolution over each affine model. Analytic local-ring flatness preserves its exactness; use the analytification of the ALGEBRAIC TW sheaves here. Do not identify it with a newly formed analytic TW complex or commute analytification with direct image from an affine open. The supplied nilpotent absolute gauge identifies the corrected local kernel total complex with this untwisted TW kernel total complex and is the identity on the central relative quotient. The latter is analytically acyclic by C and the finite double-complex spectral sequence. The algebraic overlap isomorphisms are finite-order nilpotent differential operators. The coefficient/radius estimates in C extend them continuously and uniquely to analytic stalks, including their compositions. Their cocycle equations agree on dense algebraic finite sums and hence on each finitely generated analytic submodule, whose germ topology is Hausdorff. They therefore glue the analytic kernel complex globally even though there is no globally glued scheme $X_A$. Each term has a finite maximal-ideal filtration. Its associated graded is a genuine quasi-coherent $O_{X0}$-module (the gluing reduces to the central gluing), and its analytification is the corresponding associated graded analytic sheaf. B gives cohomology comparison for each graded piece. Induction through the finite filtration gives comparison $H^q(X0,R^p)$ -> $H^q(X{0}^{an},R_{an}^p)$ for every p,q. Polynomial u is a direct sum in degree zero, not a completion; B, using the compact comparison in step 4.1, includes the filtered-colimit comparison needed here. All differential degrees are bounded by the finite log dimension, base-log degree and finite TW nerve degree. The stupid-filtration spectral sequence therefore gives comparison of kernel hypercohomology. The analytic kernel is acyclic, so its algebraic hypercohomology is zero. The long exact sequence of the kernel sequence now proves $H^n(D{A}_A[u])$ -> $H^n(D{R}_k)$ onto. The chain map factors through $D{R}_A$, so $H^n(D{R}_A)$ -> $H^n(D{R}_k)$ is onto. This proves the one-parameter conclusion for the corrected pregluing complex. It does not apply FFR Theorem 1.4 to a nonexistent global deformation.

E. Arbitrary sharp Q: injective finite monomial reduction. For $Q=0$ there is no monomial parameter; the local model and fixed gluing reduce to the central model after base extension, and the same finite minimal-complex argument below applies with no Q-adic perturbation. For Q nonzero put $m=(z^q:q\mathrm{\ne }0)$. To dominate the requested quotient $\mathrm{\Lambda }\mathrm{/}{m}^r$, choose a positive integral height ell:Q -> N and $T\ge 1$ bounding ell on the finite set Q minus $m^r$. Let $F=\{q:\mathrm{\ell }(q)\le 3T\}$, a finite set. Choose an integral linear functional b on $Q^{gp}$ separating every distinct pair in F of equal ell-height. This excludes finitely many proper lattice hyperplanes. On the rational cone of Q, $\mathrm{\mid }b(q)\mathrm{\mid }\le K$ ell(q) for a finite K, because its ell=1 section is compact. Choose an integer $C\ge 2K$ sufficiently large that $h=C$ ell+b also separates the distinct ell-heights in F. Then $(C\mathrm{/}2)\mathrm{\ell }(q)\le h(q)\le (3C\mathrm{/}2)\mathrm{\ell }(q)$, h is positive and injective on F. Let N=$\max h(q):qoutside{m}^r$. If $h(q)\le N$ then $\mathrm{\ell }(q)\le 3T$, so h is injective on ALL retained exponents. The monoid ideal I={q:h(q)>N} is contained in $m^r$. Consequently A0=k[Q]/k[I] -> $B=k[t]\mathrm{/}(t^{N+1})$, $z^q$ |-> $t^{h(q)}$ is an injective local k-algebra map: distinct retained monomials have distinct exponents and all deleted monomials map to zero. Also A0 -> $\mathrm{\Lambda }\mathrm{/}{m}^r$. For $r=1$ the quotient may simply be k and the argument is vacuous. No flatness of A0 -> B is asserted or needed. Use the SAME fixed local-model/TW/gluing data after base change to B. Since f0 is saturated, fs log base change along Q -> N has ordinary underlying scheme base change and preserves log smoothness; relative log forms commute with it. In particular the central relative complex is still the original $C_k$. The author's fixed-data interface gives literally $C_{A0}$ ${\otimes }_{A0}$ $B=C_B$, not merely an unrelated choice of a new complex over B. Here is a complete finite-complex detection argument, including infinite-rank carriers. Every flat module M over an Artin local ring R is free: lift a k-basis of M/mM to obtain $R^{basis}$ -> M. Its cokernel is m times itself and hence zero by nilpotence, without finite generation. Flatness makes the kernel's reduction inject into the source reduction, where it is zero; the kernel is again m times itself and nilpotence kills it. Thus each term of the bounded flat $C_{A0}$ can be identified with A0 tensor $C_k^i$. Split the central vector-space complex as boundaries, representatives $H^i$, and complementary preimages of boundaries; this gives i,p,s with $ps=si=s^2=0$ and 1-ip=d0s+sd0. The given differential is d0+delta with delta in m. Every series in delta s terminates, because m is nilpotent. The standard Gaussian elimination of the contractible pairs gives a homotopy equivalent complex $(H^{\bullet }(C_k)\otimes A0,D)$, $D=p$ delta(1+s delta)^(-1)i, with the convention that the new inclusion is (1+s delta)^(-1)i; direct multiplication using 1-ip=d0s+sd0 verifies the intertwining identities, $D^2=0$, and the homotopy formula. All matrices D are finite because every $H^i(C_k)$ is finite dimensional; D reduces to zero centrally. These finite formulas commute with every fixed base map, including A0 -> B. By D, every central class lifts over B. In each degree, choose cycles in the finite free minimal complex over B lifting the given basis of $H^i(C_k)$. Their coordinate matrix has invertible reduction, hence invertible determinant in the local ring B. The cycles therefore span the entire term, so D tensor $B=0$. Since A0 -> B is injective, every entry of D is zero. Thus $C_{A0}$ has zero minimal differential. Base change from A0 to $\mathrm{\Lambda }\mathrm{/}{m}^r$ and then to any fixed A with $m^r$ $A=0$ gives all central class lifts over A. There is no need to interchange tensor with a formal infinite product over Lambda. This argument also proves finite free total cohomology and its arbitrary Artin base change for these complexes: the homotopy model over every dominating A0 has zero differential, so $H^i(C_A)$ is free of rank ${\dim }_k$ $H^i(C_k)$, with the natural reduction/base-change maps given by the actual original complex. For any map A->A', choose one monomial quotient through which both structure maps factor and perform the same fixed-model argument there; its homotopy model identifies the canonical base-change map with the identity on a zero-differential finite free model. The choice of bases is not claimed canonical; the resulting cohomology comparison map is the canonical one.

F. Descent from C and exact use in the BTT owner construction. For a fixed Artin A over any characteristic-zero k, descend X0, its finite charts, the fixed finite local gluing operators to the needed nilpotent order, A's finite multiplication table, all differential coefficients and compatibility identities to a finitely generated characteristic-zero subfield k0 of k. Embed k0, not k, into C. A finite affine Cech complex computes the coherent central columns, and the finite TW carrier and its polynomial-form contraction commute with scalar extension; tensor with a field is exact. Therefore the central cohomology and the two reduction complexes commute with both k0 -> C and k0 -> k. Complex surjectivity implies k0-surjectivity because the cokernel has faithfully flat zero scalar extension. Extending back to k proves it for the original fixed A. The degree bounds and all monoid arguments are independent of the field. Properness is preserved by the field extensions, and no projective replacement of X0 enters the conclusion. Use A for central Hodge E1, A's first paragraph for finite-dimensionality, and D–F for pregluing total-cohomology surjectivity/freeness/base change. The curved Thom–Whitney control premise supplies the exact absolute calculus, nilpotent gauges and fixed TW base maps specified above. It then uses its already constructed volume contraction and curved/regular/obtuse/semiclassical MC lifting. For an actual MC element its contraction exponential identifies the corrected total complex with the actual de Rham complex; the separate finite filtered-length argument deduces relative Hodge freeness. This note does not treat existence of a global deformation as an input to D, nor infer it merely from central E1.