The following is a list of the most important definitions and results in the course, in the order they appear in the Lecture notes. The points marked with * are especially important.
Chapter 1
Presheaves, Sheaves, sheaf saturation,
*Stalks and germs,
When a map of sheaves is injective/surjective
ker, im and coker sheaves, quotient sheaves
Examples where im/coker fails to be a sheaf
*Left exactness of \(\Gamma\), and failure of its right exactness.
*B-sheaves.
*Sheafification and its universal property.
Pushforward and inverse image of a sheaf. Adjoint properties.
Chapter 2
The Zariski topology on Spec A
V(I) and D(f)
*Maps between rings vs morphisms of Spectra.
Spec of a DVR
*Definition of the structure sheaf on Spec A.
Definition of a scheme and morphisms of schemes
*The category of affine schemes vs the category of commutative rings
Open and closed subschemes
Chapter 3
Gluing of sheaves
Gluing of schemes
Gluing morphisms of schemes
*Maps from schemes into an affine scheme (Theorem 3.6)
Chapter 4
P^n
A non-affine scheme
Affine line with doubled origin
Blow-up
Hyperelliptic curves
Chapter 5
Connectedness, Irreducible, Reduced, Integral, Noetherian, quasi-compact, Finite type, Finite morphisms
The dimension of a scheme.
Chapter 6
*Definition of Proj(R) as a scheme.
*Maps between Proj's
The Veronese embedding is an isomorphism
Chapter 7
*Definition of fiber product for schemes
*Existence of fiber product for affine schemes. Superficial knowledge about the construction of the fiber product in general.
Scheme theoretic fiber
Chapter 8
The various constructions for O_X-modules: sum, tensor product, Hom, ker, ..
Modules over Spec DVR.
Direct and inverse images of O_X-modules
*The \(\sim\) functor and its properties
*Quasi-coherent sheaves
Coherent sheaves
*Quasi-coherent sheaves on affine schemes
Equivalence between categories QCoh_X and Mod_A and the sketch of the proof.
Functorial properties of Quasi-coherence (f^* and f_*, ..)
*Closed immersions vs quasi coherent sheaves of ideals (only a sketch of the proof)
Chapter 9
Locally free sheaves.
*Invertible sheaves and Pic(X).
Chapter 10
The graded $\sim$ functor and its properties
*O(1). Sections of O(m) correspond to elements of R_m.
The associated graded module of a sheaf.
*The relation between graded modules on R and quasi-coherent sheaves on Proj(R).
The correspondence between closed subschemes of Proj(R) and saturated ideals of R.
Chapters 11 and 12
*Cech cohomology
*Cohomology of quasi-coherent sheaves on Spec A for A noetherian
*Cohomology of O(m) on P^n.
*Basic examples of using cohomology to get geometric information (e.g., plane curves, hyperelliptic curves..)
Chapter 14
Weil divisors
Cartier divisors
Linear equivalence, Cl(X), CaCl(X), Pic(X)
Relations between Weil divisors, Cartier divisors and invertible sheaves
Divisors on A^n, P^n