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By N.N. Vorob'ev

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The proofs carry over literally. Two results, however, shall be mentioned explicitly. 7 — i) If E is a sheaf of dimension d and pure as an object of the category Cohd,d (X), then there exists a unique filtration in Cohd,d (X) (the Harder-Narasimhan filtration) 0 = E0 ⊂ E1 ⊂ . . ⊂ E = E such that the factors Ei /Ei−1 are semistable in Cohd,d (X) and their reduced Hilbert polynomials satisfy pd,d (E1 ) > . . > pd,d (E/E −1 ). ii) If E ∈ Ob(Cohd,d (X)) is semistable, then there exists a filtration in Cohd,d (X) (the Jordan-H¨older filtration) 0 = E0 ⊂ E1 ⊂ .

3 — Two semistable sheaves E1 and E2 with the same reduced Hilbert polynomial are called S-equivalent if gr(E1 ) ∼ = gr(E2 ). The importance of this definition will become clear in Section 4. Roughly, the moduli space of semistable sheaves parametrizes only S-equivalence classes of semistable sheaves. We conclude this section by introducing the concepts of polystable sheaves and of the socle and the extended socle of a semistable sheaf. 4 — A semistable sheaf E is called polystable if E is the direct sum of stable sheaves.

And secondly, let Fj = Ej for j = 0, . . , i − 1 and let Fj be the preimage of Ej+1 /Ei for j = i, . . , − 1. The induction hypothesis applied to F gives 24 = Preliminaries and Ej /Ej−1 ∼ = j=1 Since E1 ∼ = Ei /Ei−1 , we are done. Ej /Ej−1 . 3 — Two semistable sheaves E1 and E2 with the same reduced Hilbert polynomial are called S-equivalent if gr(E1 ) ∼ = gr(E2 ). The importance of this definition will become clear in Section 4. Roughly, the moduli space of semistable sheaves parametrizes only S-equivalence classes of semistable sheaves.

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