1. Scheme theory：
(a) Sheaves：the concept provides a systematic way of keeping track of local algebraic data on a topological space
(b) Schemes：they enlarge the category of varieties
(c) Important properties：integral、noetherian、finiteness、separatedness、properness
(d) Coherent sheaves：controllable sheaves on schemes

2. Intrinsic geometry：
(a) Linear system：an important tool for studying embedding problem
(b) Picard groups：important intrinsic invariants
(c) Sheaves of differential : algebraic version of differential forms
(d) Forma schemes : they carry information about all the infinitesimal neighborhoods

3. Cohomology theory：
We intend to define it by using derived functors and to compute it by introducing Cech cohomology. We also show some important vanishing theorems and combine them with Serre duality to give some applications in flat family、smooth family, including Zariski’s main theorem and base change theorem of cohomology.