Analytic prerequisites for algebraic–analytic comparison

These links identify the exact statements and proof drafts used by the comparison lessons. Follow the stated assumptions, rather than treating an entire course as one prerequisite.

• Coherence of holomorphic functions — Theorem 2.1. Every complex manifold. Used in Oka coherence.
• Local finite parametrization — Theorem 4.1. Prime analytic germs with adapted coordinates; finite branched projection. Used in Finite projections.
• Analytic Nullstellensatz — Theorem 5.1. Every ideal of the convergent complex power-series local ring. Used in Analytic zero sets and radical ideals.
• Dimension of analytic germs — Theorem 5.4. Irreducible analytic germs; Krull and regular-locus dimensions. Used in Analytic dimension.
• Coherence of analytic ideal sheaves — Theorem 1.1. Analytic subsets of complex manifolds. Used in Coherent reduced analytic structures.
• Unique factorization of holomorphic germs — Theorem 3.1. Convergent complex power-series local rings. Used in Local factoriality.
• Vanishing of coherent cohomology — Theorem 4.1. Complex manifold with smooth strictly plurisubharmonic exhaustion; coherent sheaf; all positive degrees. Used in Analytic vanishing.
• Finite-dimensional coherent cohomology — Theorem 2.1. Compact complex spaces; coherent analytic sheaves; all degrees. Used in Analytic finiteness.

Reading order

Begin with AG-QC’s local analytic algebra proof and its earlier sheaf-cohomology foundations, then read this bridge, then return to the comparison theorems. The local algebra proof is the prerequisite—not the later comparison arguments that use this bridge.