Local coherent acyclicity on sufficiently small polydiscs

Given: Full AC, $N\ge 0$, a point $x$ in an open set $U\subseteq {\mathbb{C}}^N$, and a coherent analytic module $\mathcal{F}$ on $U$. All cohomology below is ordinary right-derived global-sections cohomology on the indicated open set.

The Axiom of Choice: every family of nonempty sets admits a choice function. This is the ambient axiom used by the sheaf-theoretic supplier theorems.

Coherent module sheaves on complex manifolds: coherence means local finite type together with local finite type of the kernels of all finite-rank free module maps into the sheaf.

Oka coherence and local coherent free resolutions: under full AC, the holomorphic structure sheaf on a finite-dimensional complex manifold is coherent; coherent analytic modules are closed under kernels, cokernels and finite extensions. For every coherent module, point and finite integer $\mathrm{\ell }\ge 0$, there is a chart neighborhood with a partial finite-rank free resolution exact at the module and at every free term except possibly the leftmost. No global free resolution is asserted.

For $N\ge 1$, The local Dolbeault lemma on nested polydiscs gives local exactness in every positive antiholomorphic degree. At degree zero, Bigraded complex forms and the Dolbeault operators says that $\bar{\mathrm{\partial }}f=0$ is exactly the Cauchy–Riemann system for a smooth function. Holomorphic functions are smooth by Holomorphic functions of several variables are smooth and their complex derivatives are holomorphic, and For $C^1$ functions, holomorphy, complex linearity of the real derivative, and the Cauchy–Riemann system agree says that a $C^1$ function is holomorphic exactly when that system holds at every point, with holomorphy defined by Holomorphic functions on an open subset of ${\mathbb{C}}^m$. Thus the full smooth Dolbeault sequence is an exact resolution of ${\mathcal{O}}_D$.

Sheaves of smooth-function modules are cohomologically acyclic: sheaves of modules over the smooth-function sheaf on a smooth manifold are acyclic for global sections.

The acyclic-resolution theorem for right derived functors: a sheaf resolution by acyclic sheaves computes the right-derived functors of global sections by its global-section complex.

Positive-degree Dolbeault cohomology vanishes on polydiscs: on every coordinate polydisc in ${\mathbb{C}}^N$ with $N\ge 1$, every smooth $\bar{\mathrm{\partial }}$-closed $(0,q)$-form is $\bar{\mathrm{\partial }}$-exact for $1\le q\le N$. This is a statement about the global Dolbeault complex; its identification with sheaf cohomology is established in step 1.3.

Enough injective abelian sheaves and Sheaf cohomology as right derived global sections: abelian sheaves admit injective resolutions, and $H^q(D,-)$ denotes the right-derived functors of global sections. Finite direct sums of acyclic sheaves are acyclic, since finite direct sums of injective resolutions remain injective and global sections and cohomology commute with finite direct sums.

A sequence of abelian sheaves is exact exactly when it is exact on every stalk: exactness of a sequence of sheaves is equivalent to exactness on every stalk; restriction to an open subset preserves a short exact sequence.

Long exact sequence of sheaf cohomology: every short exact sequence of sheaves induces the natural long exact sequence in derived sheaf cohomology.

Ordinary sheaf cohomology on Euclidean open sets vanishes above real dimension: if $D$ is a nonempty open subset of ${\mathbb{R}}^{2N}$ and $\mathcal{G}$ is any sheaf of abelian groups on $D$, then $H^r(D,\mathcal{G})=0$ for $r>2N$.