Abstract:  The fundamental theorem of Oka-Cartan (H^q(X, F)=0, q>0, for coherent sheaves F over a Stein manifold X) is the most
basic fact in modern complex analysis and complex geometry.
K. Oka obtained the notion of coherent sheaves (he himself called them ``id'eaux (or modules) de domaines ind'etermin'es'')
and proved three theorems: 1'st Coherence of O_{C^n}, 2'nd Coherence of geometric ideal sheaves (ideal sheaves of analytic subsets), 3'rd Coherence of normalization of complex spaces. (H. Cartan gave another proof for the 2'nd.)

 To develop such a theory, we here take an approach described in the introduction of Oka  (Iwanami version).
 
We deal with the following items:

1) Definitions of holomorphic functions, holomorphically convex domains,
and sheaves.

2) Oka's 1'st Coherence Theorem.

3) Oka's fundamental lemma, H^q(Polydisk, F)=0, q>0.

4) The fundamental theorem of Oka-Cartan over holomorphically
convex domains and Stein manifolds.

5) Finiteness theorems of Cartan-Serre and Grauert.

6) Oka's theorem [IX] (Levi problem or Hartogs' inverse problem)
for Riemann domains over C^n.

7) Some applications.

N.B. The cohomology theory with coefficients in sheaves will
be explained and used without proof.