Finite Artin analytic charts and the Kodaira–Spencer comparison

At a point present the deformation germ as $B=A\{z,w\}\mathrm{/}I$. By [F2], choose central coordinates in which its reduction ideal is generated by the coordinates $w_1,\dots ,w_r$, with the $z$ coordinates free. If there are no normal coordinates, this straightening is the identity. For a zero-dimensional central germ, apply the implicit theorem after adjoining one unused complex parameter and restrict that parameter to zero; its stated positive block dimensions are then respected. Lift the normal generators to $f_j\in I$ reducing to $w_j$. Substitution $w_j\mapsto f_j$, $z_i\mapsto z_i$, is an automorphism of $A\{z,w\}$: it is identity modulo $\mathfrak{m}$, and successive corrections modulo ${\mathfrak{m}}^2,{\mathfrak{m}}^3,\dots$ invert it in finitely many steps. Taylor expansion at every step uses only finitely many derivatives of convergent holomorphic functions, hence stays convergent. After this coordinate change, $B=A\{z\}\mathrm{/}J$ with $J\subset \mathfrak{m}A\{z\}$. Apply the tensor exactness in [F2] to $0\to J\to A\{z\}\to B\to 0$, whose right quotient is $A$-flat, and tensor with $A\mathrm{/}\mathfrak{m}$. It makes $J\mathrm{/}\mathfrak{m}J\to \mathbb{C}\{z\}$ injective. This map has zero image since the central quotient has no equations, so $J=\mathfrak{m}J$. Iterating through the nilpotence exponent gives $J=0$. Thus the relative coordinate charts are genuinely $A\{z\}$; no general smoothness criterion or convergent formal effectivity theorem has been assumed.

Extend their transition maps to smooth functions by the finite Taylor rule in the nilpotent holomorphic coordinate corrections, and use the unperturbed central antiholomorphic transition in the barred coordinates. Composition respects this extension by the finite Taylor chain rule, so it defines a sheaf of smooth $A$-algebras. At each order $r$, coordinate changes identity modulo ${\mathfrak{m}}^r$ modulo those identity modulo ${\mathfrak{m}}^{r+1}$ form the additive sheaf $A^{0,0}(T_X^{1,0})\otimes ({\mathfrak{m}}^r\mathrm{/}{\mathfrak{m}}^{r+1})$. It is a smooth function-module sheaf. Explicitly an overlap cocycle $a_{ij}$ is killed by $a_i={\sum }_j{\chi }_j{a}_{ij}$ for a subordinate partition, since $a_i-a_j=a_{ij}$; compactly supported summands extend by zero. Changing trivializations by $\exp (-a_i)$ removes that order. Finite repetition gives a global smooth trivialization. The same partition construction with endomorphism-valued corrections trivializes any smooth relative bundle reducing to a fixed central bundle.

Conversely let $\xi$ satisfy the stated equation. Starting with central coordinates $z^a$, construct $D_{\xi }$-closed coordinates $w^a=z^a$ modulo $\mathfrak{m}$. If they are constructed modulo ${\mathfrak{m}}^r$, their remaining residual in ${\mathfrak{m}}^r\mathrm{/}{\mathfrak{m}}^{r+1}$ is ordinary $\bar{\mathrm{\partial }}$-closed because $D_{\xi }^2=0$. Apply [F3] coefficientwise to remove it on a smaller polydisc. There are finitely many nilpotent orders, so finitely many shrinkings suffice. The substitution $(z,\bar{z})\mapsto (w,\bar{z})$ is invertible by finite Taylor corrections. In these coordinates $D_{\xi }$ kills $w$ and differentiates the barred coordinates, so its kernel consists exactly of holomorphic convergent power series in $w$ with coefficients in $A$. Indeed the scalar kernel is the ordinary Cauchy–Riemann kernel coefficientwise; nilpotent substitution uses only finitely many derivatives of those convergent coefficients. Two systems of such coordinates have $A$-holomorphic transitions because each coordinate lies in this kernel. They therefore glue an analytic deformation locally modeled on $A\{w\}$, which is flat over $A$. Its underlying compact space is $X$, making its map to the one-point Artin base proper. This constructs an actual analytic Artin deformation from $\xi$, without an infinite coordinate iteration.

In this trivialization the relative holomorphic sheaf is the kernel of $D_{\xi }=\bar{\mathrm{\partial }}+{\xi }^i{\mathrm{\partial }}_i$, with $\xi \in L_X^1\otimes \mathfrak{m}$. The square-zero identity for its local Dolbeault operator, transported from the relative coordinate charts, is exactly $d\xi +[\xi ,\xi ]\mathrm{/}2=0$ by [F1]. Changes of the chosen smooth trivialization fix the central antiholomorphic coordinates and induce the finite gauge action of [F1]. For precision, any such nilpotent algebra automorphism $g$ is an exponential of a derivation: its finite logarithm $a=\log g$ exists, and the identity that $\exp (sa)$ preserves products holds for every integer $s$ since these are powers of $g$. It is polynomial in $s$, hence holds for all complex $s$; differentiating at zero gives the Leibniz rule for $a$. This derivation kills $A$ and the central antiholomorphic functions. To justify its vector-field form without an extra analytic comparison theorem, in a coordinate ball fix a point $x$ and apply the real radial integral formula to a smooth function $f$: locally $f(y)-f(x)={\sum }_j(z^j(y)-z^j(x))f_j(y)+{\sum }_j({\bar{z}}^j(y)-{\bar{z}}^j(x))g_j(y),$ where $f_j(x)={\mathrm{\partial }}_{j}f(x)$ and $g_j(x)={\mathrm{\partial }}_{\bar{j}}f(x)$. Apply the derivation and evaluate its $A$-valued coefficient functions at $x$. The Leibniz rule and its vanishing on the barred coordinates give $a(f)(x)={\sum }_{j}a(z^j)(x){\mathrm{\partial }}_{j}f(x)$. Since this holds at every point, $a={\sum }_{j}a(z^j){\mathrm{\partial }}_j$; each coefficient lies in $\mathfrak{m}$ because $a$ is zero after reduction. Under holomorphic coordinate changes this is precisely the tangent-vector transformation rule. Thus the logarithm is an element of $L_X^0\otimes \mathfrak{m}$, and every such element exponentiates back to an allowed smooth coordinate change. Thus the coordinate-change group is the stated gauge group, including stabilizers, not a quotient that discards automorphisms.

Gauge maps send the kernels of the corresponding $D_{\xi }$ operators isomorphically to each other, hence induce marking-preserving analytic isomorphisms in the charts of step 2.2. Conversely a marking-preserving analytic isomorphism between these charts extends to the smooth sheaves by finite Taylor substitution and the fixed central antiholomorphic coordinate action; after the trivializations of step 2.1 it is an automorphism $g$ as in step 3.1. It therefore belongs to the stated gauge group and intertwines the two operators. The two constructions recover their initial holomorphic sheaves as the same kernels, and recover the Maurer–Cartan elements up to these gauge maps. This proves the equivalence of groupoids, including arrows. Every finite coordinate, partition and exponential formula restricts under Artin base change; different permitted trivializations are related by the same marking-preserving gauges, giving the asserted natural compatibility up to isomorphism.

In each relative chart, the ordinary coordinate Dolbeault complex with $A$-valued coefficients resolves the relative holomorphic functions, forms and tangent sections by [F3] applied coefficientwise. Its smooth terms are modules over the central smooth-function sheaf and hence acyclic by [F4]. They therefore compute the relative sheaf cohomology. The trivializations in step 2.1 identify their smooth terms with the central smooth bundles tensored with $A$, and their differentials reduce to central $\bar{\mathrm{\partial }}$. On tangent-valued forms, the graph quotient action is the commutator with $D_{\xi }$, which has coefficient $\bar{\mathrm{\partial }}+[\xi ,-]$ by [F1]. This gives exactly the stated tangent complex and the square-zero perturbation description for relative $p$-forms.

On a relative holomorphic coordinate polydisc centered at zero, for a holomorphic $k$-form $\omega$ with $k>0$ define $(K\omega {)}_x(v_1,\dots ,v_{k-1})={\int }_0^1{s}^{k-1}{\omega }_{sx}(x,v_1,\dots ,v_{k-1})ds\mathrm{.}$ The first inserted vector is the vector $x$ at the point $sx$, specifying the radial-field convention. Differentiating the pullback under $x\mapsto sx$ and integrating in $s$ gives $dK+Kd=1-{\mathrm{ev}}_0^{\ast }$; evaluation contributes zero in positive degrees. The integral preserves holomorphic coefficients and applies coefficientwise in finite-dimensional $A$. In degree zero, a closed holomorphic function is locally constant. Hence the relative holomorphic de Rham complex resolves the constant sheaf $A$. The same displayed formula on a real star-shaped smooth coordinate ball, first for real forms and then coefficientwise over $\mathbb{C}$ and $A$, gives the identical homotopy identity for smooth forms; in degree zero its kernel is the locally constant functions. This explicitly proves the smooth constant-sheaf resolution as well. Its smooth terms are acyclic by [F4], so the actual acyclic-resolution theorem computes constant-sheaf cohomology by global smooth de Rham cohomology. Only at this point apply the global comparison in [F5] to identify that cohomology with $H^k(X;\mathbb{C})\otimes A$; tensoring with finite-dimensional $A$ is a finite direct sum and is natural under Artin base change. The relative fine Dolbeault double resolution in step 5.1 computes the same constant-sheaf cohomology by [F4] and the bounded comparison in [F5]. Zero-dimensional $X$ is a finite set of points with charts $\mathrm{Specan}A$, and the same conclusions are immediate there. All coordinate corrections remain finite at their Artin order; no reduced-parameter analytic effectivity is inferred.