This chapter provides the basic definitions of the theory. After introducing pure sheaves and their homological aspects we discuss the notion of reduced Hilbert polynomials in terms of which the stability condition is formulated. Harder-Narasimhan and Jordan-Hölder filtrations are defined in Section 1.3 and 1.5, respectively. Their formal aspects are discussed in Section 1.6. In Section 1.7 we recall the notion of bounded families and the Mumford-Castelnuovo regularity. The results of this section will be applied later (cf. 3.3) to show the boundedness of the family of semistable sheaves. This chapter is slightly technical at times. The reader may just skim through the basic definitions at first reading and come back to the more technical parts whenever needed.

Some Homological Algebra

Let X be a Noetherian scheme. By Coh(X) we denote the category of coherent sheaves on X. For E ∈ Ob(Coh(X)), i.e. a coherent sheaf on X, one defines:

Definition 1.1.1 — The support of E is the closed set Supp(E) = {x ∈ X|Ex ≠ 0}. Its dimension is called the dimension of the sheaf E and is denoted by dim(E).

The annihilator ideal sheaf of E, i.e. the kernel of Ox → εnd(E), defines a subscheme structure on Supp(E).

Definition 1.1.2 — E is pure of dimension d if dim(F) = d for all non-trivial coherent subsheaves F ⊂ E.

Equivalently, E is pure if and only if all associated points of E (cf. [172] p. 49) have the same dimension.