Curved finite-cover calculus controlling all log-smooth deformations

We record the elementary chart lifting calculation. A saturated local characteristic map $Q\to P$ is integral. In an integral monoid map, if two monomials become equal after adding base exponents, the defining integrality relation factors that equality through a common monomial with base exponents. Consequently each $P^{\mathrm{g}\mathrm{p}}\mathrm{/}{Q}^{\mathrm{g}\mathrm{p}}$-homogeneous piece of $k[P]$ is a filtered colimit of free rank-one $k[Q]$-modules: the objects are its monomials with an allowed base denominator, and the common-factor relation is exactly the common-refinement condition. Thus $k[P]$ is flat over $k[Q]$. A smooth map to this toric chart is flat after any base change. Its log differentials are locally free: the toric part is the quotient lattice tensored with the ring, plus ordinary smooth coordinates; any finite torsion is invertible. Across a square-zero extension, lift its smooth presentation and an invertible Jacobian minor. For an existing map, lift the chart's finitely many monoid generators and their units first. Relations in the group completions reduce to linear equations in the square-zero unit group $1+I$, and finite torsion is solved by division by its order. Lift the ordinary coordinates next and remove the relation errors by the invertible minor. This proves the required local existence and infinitesimal lifting for log maps directly from F1. Differences of two lifts over a square-zero ideal $I$ are exactly pairs $(D,a)$ with $D\alpha (m)=\alpha (m)a(m),a(m+n)=a(m)+a(n),a{\mathrm{\mid }}_Q=0,$ hence sections of $T_{\log }\otimes I$ by the universal property in F1.

Put ${\mathrm{\Omega }}_n=k[t_0,\dots ,t_n,d{t}_0,\dots ,d{t}_n]\mathrm{/}(\sum t_i-1,\sum d{t}_i)\mathrm{.}$ Polynomial TW sections of a sheaf are the families on intersections tensored with ${\mathrm{\Omega }}_n$ whose restrictions agree on faces. Boundary restriction is surjective in every graded degree. To prove this, the $i$-th face ideal is $J_i=(t_i,d{t}_i)$. For a proper subset of the indices eliminate a coordinate outside it; the remaining ideals are independent coordinate-block ideals, so their intersection equals their product by monomial expansion. For the full set, if $\alpha$ belongs to every $J_i$, then $\alpha ={\sum }_i{t}_i\alpha$ belongs to their product by the proper-subset result. The same identity holds after passing to any face quotient. Extend prescribed faces one at a time: their difference on the new face vanishes on its intersections with previous faces, belongs to the product of the preceding face ideals, and that product lifts to the simplex. This proves graded extendability, including coefficients in an arbitrary module.

These calculations apply to affine opens without assuming a global chart. Choose a finite affine strict-étale chart cover of an affine $U_0$. Its finite products over $U_0$ are affine. The augmented Amitsur complex of any quasi-coherent module is exact: after tensoring by the faithfully flat product covering algebra it has the extra degeneracy given by multiplication, and faithful flatness detects exactness. Local lifts from step 1.1 form a descent problem with automorphism kernel $T_{\log }\otimes I$: on double intersections choices of isomorphisms form torsors under that module, and on triple intersections their composition defect is an additive Čech 2-cocycle. One can refine the same strict-étale cover when choosing the isomorphisms; the Amitsur contraction kills the 2-cocycle and then the 1-cocycle for two choices. The corrected transports give effective descent: underlying affine algebras descend by F3, and the monoid sheaves and their maps descend by taking compatible sections on the covering étale site. Units descend because being a unit is faithfully flat local. Smooth chart presentations descend locally, so the resulting object is log smooth. The same argument for a pair of lifts proves affine rigidity, and for isomorphisms proves surjectivity of automorphism reduction. Thus affine local lifts exist at every order, any two are marked-isomorphic, and no globally chosen chart on $U_0$ has been used.

Saturation persists under a nilpotent deformation: a function is a unit precisely when its reduction is, so the characteristic monoids and their base maps are unchanged. For an integral saturated chart the special toric fibre is geometrically reduced: its defining base-monomial ideal is radical. One checks this by the facet valuations of the rational polyhedral cone; a nonradical surviving monomial would give, after adjoining its fractional base root, an element in an integral pushout whose multiple belongs to the pushout although the element does not, contradicting saturation. The same chart calculation gives the dense open union of smooth strata on which relative log vector fields act faithfully on functions. Indeed a character logarithmic derivation annihilating all surviving monomials has zero pairing on the lattice span of each such stratum; these spans together with the base lattice generate the relative lattice. A vector field vanishing on that dense open vanishes everywhere because $T_{\log }$ is locally free and the fibre is reduced. For a nilpotent flat lift the same conclusion follows successively on $I\otimes T_{\log ,0}$. Thus the action on functions of each log-vector-field coefficient is faithful. Properness also persists: nilpotent closed immersions are universal homeomorphisms, so universal closedness and the closed-diagonal condition reduce to the central ones, while finite presentation comes from the lifted charts and finite cover. These observations apply to every deformation in F2; its definition imposes properness, while saturation is not imposed and is shown here to persist.

Choose $k$-linear splittings of these graded boundary restrictions once. Recursive extension over the finite nerve expresses each graded sheaf ${\mathrm{T}\mathrm{W}}^q(\mathcal{F})$ as a finite sum of sheaves $j_{I\ast }(\mathcal{F}{\mathrm{\mid }}_{U_I}){\otimes }_k{V}_I^q,$ where $V_I^q$ is a vector space of relative-boundary polynomial forms. Each $j_I$ is affine, and its intersection with any affine open is affine. F3 proves that these sheaves and their restrictions to the cover intersections have no positive cohomology. They are flat over an Artin base when $\mathcal{F}$ is, and all their graded constructions commute with arbitrary base change. Moreover $q<N$. The augmentation $\mathcal{F}\to \mathrm{T}\mathrm{W}(\mathcal{F})$ is a local resolution: on $U_i$ insert the vertex $i$, use the simplex homotopy $t\mapsto s{e}_i+(1-s)t$, and integrate contraction with ${\mathrm{\partial }}_s$ from zero to one. Polynomial integration is defined over $k$ and gives $dH+Hd=1-\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{l}{\mathrm{e}\mathrm{v}}_i$, with compatible face restrictions. Integration over each simplex gives the natural comparison to the finite Čech resolution; Stokes follows by integrating monomials, using ${\int }_{{\mathrm{\Delta }}_n}{t}_0^{a_0}\cdots t_n^{a_n}d{t}_1\cdots d{t}_n=\frac{\prod a_i!}{(n+\sum a_i)!}\mathrm{.}$ Both are acyclic resolutions, so this is a quasi-isomorphism. All operations and these homotopies commute with scalar extension.

Fix a finite Zariski affine cover $(U_i{)}_{i=1}^N$ of $X_0$. All intersections are affine because $X_0$ is separated. Put $R_r=\mathrm{\Lambda }\mathrm{/}{\mathfrak{m}}_{\mathrm{\Lambda }}^{r+1}$ for every integer $r\ge 0$. Inductively apply step 2.1 to $R_{r+1}\to R_r$ to choose local log-smooth affine models and overlap comparisons reducing to the preceding choices, with specified compatible reduction identifications. The kernel is killed by the maximal ideal, and automorphism reduction is surjective by step 2.1, so previously chosen overlap comparisons can also be lifted. This chooses one compatible tower, not one terminal quotient. The comparisons need not satisfy a triple cocycle. Relative log forms, polyvectors and their operations commute with these local base changes by their explicit chart modules. A log automorphism equal to the identity modulo the maximal ideal is an exponential of a unique nilpotent log derivation. On functions use the finite series $\log (g)$ and $\exp (D)$; the identity that $\log (g)$ is a derivation follows by applying the finite polynomial family $g^s$ to products and comparing its identity at every integer $s$, over the infinite field $k$. On the log structure use the same logarithm of the unit multiplier and the compatibility equation in step 1.1. Conversely the finite exponential solves that equation and gives the log automorphism. This proves the assertion with the actual log structure, not just its underlying scheme.

Tensor local polyvectors and forms with these simplex forms, with Koszul signs. Explicitly, for homogeneous simplex forms $a,b$ and physical polyvectors $\theta ,\xi$, the product has sign $(-1{)}^{\mathrm{\mid }\theta \mathrm{\mid }\mathrm{\mid }b\mathrm{\mid }}$ and the odd bracket has sign $(-1{)}^{(\mathrm{\mid }\theta \mathrm{\mid }+1)\mathrm{\mid }b\mathrm{\mid }}$ in front of $ab$ times the physical operation. Contraction has sign $(-1{)}^{\mathrm{\mid }\theta \mathrm{\mid }\mathrm{\mid }b\mathrm{\mid }}$, and Lie derivative has sign $(-1{)}^{(\mathrm{\mid }\theta \mathrm{\mid }+1)\mathrm{\mid }b\mathrm{\mid }}$. A physical unary map of degree one has sign $(-1{)}^q$ on simplex degree $q$. These give the total-degree Cartan identities $(-1{)}^{\mathrm{\mid }\theta \mathrm{\mid }}{L}_{\theta }=\mathrm{\partial }{\iota }_{\theta }-(-1{)}^{\mathrm{\mid }\theta \mathrm{\mid }}{\iota }_{\theta }\mathrm{\partial },$ ${\iota }_{\theta }{L}_{\xi }=(-1{)}^{\mathrm{\mid }\xi \mathrm{\mid }+1}{\iota }_{[\theta ,\xi ]}+(-1{)}^{\mathrm{\mid }\theta \mathrm{\mid }(\mathrm{\mid }\xi \mathrm{\mid }+1)}{L}_{\xi }{\iota }_{\theta }\mathrm{.}$ They follow first on a decomposable tensor by moving the simplex factors past the physical factors, and hence on all sections. They fix all signs in what follows.

We now glue the graded calculus, without gluing a scheme. Suppose transports have been chosen through one nilpotent order. At the next small order their triple defect is $\exp (\mathrm{ad}{\theta }_{ijk})$, with ${\theta }_{ijk}\in I\otimes {\mathrm{T}\mathrm{W}}^0(T_{\log ,0})$; associativity says $\delta \theta =0$. The same parameter acts on forms by $\exp (L_{\theta })$. By step 2.3 the coefficient sheaf is acyclic, so its Čech $H^2$ vanishes and a 1-cochain correction kills the defect. Apply this successively to the adic quotients. This constructs a true graded cocycle $g_{ij}$ of the full calculus, reducing to the original central one, each factor being a geometric local transport and finite inner exponentials. Two choices differ by a graded 0-cochain of these exponentials because the same coefficient sheaf has $H^1=0$. These assertions hold on any fixed affine-cover intersection too. No differential is required to be preserved at this stage, so this does not assert a global log deformation exists.

Locally write $d_i$ for the simplex differential. Its transport discrepancy is inner: for an inner change use the finite identity $e^{\mathrm{ad}b}d{e}^{-\mathrm{ad}b}=d-\mathrm{ad}\left(\frac{e^{\mathrm{ad}b}-1}{\mathrm{ad}b}db\right)\mathrm{.}$ Geometric chart transports themselves commute with $d$. At each small order the inner discrepancy on overlaps is therefore a 1-cocycle valued in $I\otimes {\mathrm{T}\mathrm{W}}^1(T_{\log ,0})$. Faithfulness from step 2.2 identifies this coefficient, rather than only its adjoint operator. Its $H^1$ vanishes by step 2.3; choose local $a_i\in \mathfrak{m}{\mathrm{T}\mathrm{W}}^1(T_{\log })$ so that $d_i+\mathrm{ad}{a}_i$ and $d_i+L_{a_i}$ glue on polyvectors and forms respectively. Iterate through the adic orders. Denote the resulting global predifferential by $\bar{\mathrm{\partial }}$ and put ${\mathrm{\ell }}_i=d_i{a}_i+\frac{1}{2}[a_i,a_i]\mathrm{.}$ The square is $\mathrm{ad}{\mathrm{\ell }}_i$ or $L_{{\mathrm{\ell }}_i}$, so faithfulness makes ${\mathrm{\ell }}_i$ global; Jacobi gives $\bar{\mathrm{\partial }}\mathrm{\ell }=0$. Physical $\mathrm{\partial }$ is preserved by the transports and anticommutes with $\bar{\mathrm{\partial }}$. We have the curved calculus, not a square-zero predifferential.

Every graded glued sheaf is acyclic, flat over the Artin base, and has graded base change. Indeed at a small extension its kernel is $I\otimes$ the central TW sheaf; step 2.3 and the cohomology exact sequence prove acyclicity inductively. On an affine refinement each section module is flat by the local direct-sum description. The finite augmented Čech sequence of these sections is exact; descending from its final term shows its initial global-section module flat as well, by the short-exact-sequence flatness criterion. Applying the same exact Čech sequence after any base change proves the asserted canonical global-section base change. Equivalently one may lift a central basis, use flatness to identify its kernel reduction, and kill kernel and cokernel by the nilpotent maximal ideal. Perform steps 5.1 and 6.1 successively on the tower $R_r$, retaining all previous choices on reduction; their small-order corrections lift the existing transports and predifferentials. For $A\in {\mathbf{A}\mathbf{r}\mathbf{t}}_{\mathrm{\Lambda }}$, choose $r$ with ${\mathfrak{m}}_{\mathrm{\Lambda }}^{r+1}A=0$ and set $G_A=G_{R_r}{\otimes }_{R_r}A$ and ${\mathcal{A}}_A={\mathcal{A}}_{R_r}{\otimes }_{R_r}A$, using the global-section base-change isomorphism just proved. If $s\ge r$, the specified compatible tower identifies $G_{R_s}{\otimes }_{R_s}{R}_r$ with $G_{R_r}$, and likewise for forms and every operation; associativity of tensor product therefore identifies the definitions from $s$ and $r$. These identifications are transitive. For any Artin arrow $A\to A^{\mathrm{\prime }}$, take one $r$ annihilating both structure maps and use scalar extension at that level. Transitivity makes the result independent of the level and compatible with identities and composition. The later absolute gauges and volume lifts are chosen by the same successive-order procedure. No finite quotient is asserted to dominate all Artin bases, and no tensor interchange with an infinite formal product is used.

We verify actual geometric control. For a global classical solution $\varphi$, the local element $a_i+{\varphi }_i$ solves the ordinary simplex Maurer–Cartan equation. On an affine $U_i$ every positive cohomology of the central log tangent column vanishes by F3 and step 2.3. At the first nonzero nilpotent order its MC equation is closedness, so it is a boundary; subtract that boundary by a degree-zero exponential gauge. Repeating kills the local element completely. The transformed overlap cocycle $h_{ij}$ now commutes with the simplex differential. It differs from geometric transport by $\exp (\mathrm{ad}c)$ with $c$ of simplex degree zero. Commutation with $d$ says $\mathrm{ad}\left(\frac{e^{\mathrm{ad}c}-1}{\mathrm{ad}c}dc\right)=0.$ Faithfulness and invertibility of the parenthesized operator give $dc=0$. The augmentation resolution in step 2.3 identifies its degree-zero closed sections with actual log vector fields. step 3.1 therefore makes $h_{ij}$ an actual log-scheme isomorphism. Its cocycle glues the affine log models to a log-smooth deformation with the central marking.

Finally make the absolute extension required for total-class lifting. On a toric monomial local model take absolute log forms generated by the toric lattice and ordinary smooth coordinates; a monomial relation $z^a=0$ has differential $z^{a}d\log a=0$ already, so this free chart description survives every monomial quotient. Smooth coordinates and their étale relations lift by the Jacobian calculation. Relative forms are the quotient by the base log forms. Each relative log vector field annihilates the base and acts on absolute forms by its absolute Cartan Lie derivative. Thus every geometric transport and every chosen inner gauge extends to the absolute calculus, preserving the base log forms; the same cocycle and predifferential construction applies there with the SAME coefficients $a_i,\mathrm{\ell }$. In the one-parameter case denote the surviving base generator by $\rho$. Interior multiplication by $a_i$ annihilates $\rho$, and the Cartan formulas of step 4.1 give the finite, exact identity $e^{-{\iota }_{a_i}}(d_i+{\mathrm{\partial }}_{\mathrm{a}\mathrm{b}\mathrm{s}})e^{{\iota }_{a_i}}=d_i+{\mathrm{\partial }}_{\mathrm{a}\mathrm{b}\mathrm{s}}+L_{a_i}+{\iota }_{d_i{a}_i+[a_i,a_i]\mathrm{/}2}\mathrm{.}$ The second commutator is ${\iota }_{[a_i,a_i]}$ and all higher commutators vanish because contractions commute. This identifies the corrected absolute total differential with the untwisted one locally, including its kernel on the central relative quotient; the exponential is the identity centrally. It also commutes with multiplication by $\rho$ and with ${\mathrm{\partial }}_u$. Therefore $d_{\mathrm{a}\mathrm{b}\mathrm{s}}+\rho \wedge {\mathrm{\partial }}_u$ on the polynomial extension has exactly the asserted quotient chain maps. Evaluation to absolute forms alone would not be a chain map, which is why the relative quotient is included. All gauges are nilpotent finite differential-order operators, as their definitions show. This is an absolute construction before global geometric gluing, not the de Rham complex of an assumed global deformation.

The given central volume maps to a global section of ${\mathcal{A}}_k^{d,0}$. Lift it through the graded sheaves at successive orders. The obstruction is in $H^1$ of the central degree-zero TW sheaf, hence vanishes by step 2.3. Write the lift $\omega$. Contraction $\kappa (\theta )={\iota }_{\theta }\omega :G_A^{p,q}⟶{\mathcal{A}}_A^{d+p,q}$ is an isomorphism: centrally this is the exterior-power duality for a free log tangent module; at each small extension its kernel and quotient maps are that same central isomorphism tensored with $I$. Define $\mathrm{\Delta }={\kappa }^{-1}\mathrm{\partial }\kappa$ and $y={\kappa }^{-1}\bar{\mathrm{\partial }}\omega$. The local exterior-algebra calculation gives $(-1{)}^{\mathrm{\mid }\theta \mathrm{\mid }}[\theta ,\xi ]=\mathrm{\Delta }(\theta \xi )-\mathrm{\Delta }\theta \xi -(-1{)}^{\mathrm{\mid }\theta \mathrm{\mid }}\theta \mathrm{\Delta }\xi \mathrm{.}$ To check this locally, expand in a basis of the free log tangent module and a volume coefficient: only terms in which the derivation differentiates a coefficient in the other factor survive; they are exactly the negative Schouten terms with the displayed sign. The calculation is invariant under basis change and the TW Koszul rule. Consequently ${\mathrm{\Delta }}^2=0,{\bar{\mathrm{\partial }}}^2=[\mathrm{\ell },-],\bar{\mathrm{\partial }}\mathrm{\Delta }+\mathrm{\Delta }\bar{\mathrm{\partial }}=[y,-],\bar{\mathrm{\partial }}y+\mathrm{\Delta }\mathrm{\ell }=0.$ The last two follow also by applying $\kappa$ and using $\mathrm{\partial }\bar{\mathrm{\partial }}+\bar{\mathrm{\partial }}\mathrm{\partial }=0$ and ${\bar{\mathrm{\partial }}}^2\omega =L_{\mathrm{\ell }}\omega$. Also $\mathrm{\Delta }y=0$ by the $p$-range. Curvature and $y$ vanish centrally. Thus the corrected differential on forms is $D=\mathrm{\partial }+\bar{\mathrm{\partial }}+{\iota }_{\mathrm{\ell }},D^2=0,$ and $D\kappa =\kappa (\bar{\mathrm{\partial }}+\mathrm{\Delta }+(\mathrm{\ell }+y)\wedge )$, with degree shift $d$. This proves the exact volume/BV interface.

Conversely every deformation in F2 is locally marked-isomorphic to the fixed affine models by step 2.1. Its actual overlap cocycle and $g_{ij}$ are isomorphic as graded TW gluing by the degree-zero $H^1$ argument in step 5.1. Transfer its genuine differential to the fixed graded carrier. The difference from $\bar{\mathrm{\partial }}$ is locally inner of simplex degree one by the finite conjugation formula in step 6.1. Those coefficients agree on overlaps by faithfulness, hence give a global $\varphi$. Square-zero of the transferred differential gives exactly its classical MC equation. Changes in these identifications and actual isomorphisms between deformations correspond to degree-zero gauge: order by order the difference is a log derivation, and the same $H^1$ calculation glues its coefficients; its finite exponential supplies the gauge. Conversely such a gauge identifies the local degree-zero closed log algebras and their log structures. Thus the two constructions give inverse maps on marked isomorphism classes, with precisely the indicated gauge action. The local constructions use the same base-changed sheaves and equations, so the bijections are natural on all Artin arrows, including non-surjective ones.

The cover construction above applies when $X_0$ is nonempty. If $X_0=\mathrm{\varnothing }$, use its empty affine cover and define its TW sections as the limit of the empty intersection diagram in modules, namely the zero module. Set every $G_A^{p,q}$ and ${\mathcal{A}}_A^{i,q}$ to zero and take $N=1$ as a degree bound. These modules are flat, commute with every base change, and compute the zero cohomology of every sheaf on the empty scheme. All products, differentials, curvature, contractions and absolute polynomial extensions are zero, so every displayed calculus identity holds and the contraction map is the unique isomorphism $0\to 0$. Any deformation in F2 has central fibre cut out by ${\mathfrak{m}}_A{\mathcal{O}}_X$; this ideal is nilpotent because $A$ is Artinian. A nilpotent closed immersion does not change the underlying point set: every prime contains a nilpotent ideal. Hence an empty central fibre forces $X$ empty. The empty scheme has a unique log structure, marking and isomorphism, so its deformation set is a singleton. On the zero calculus the MC solution set is $\{0\}$, its gauge group is trivial, and the stated bijection is the unique bijection of singletons, naturally in $A$. This also covers the absolute extension, without selecting a cover vertex or a local chart.