By Dan Laksov

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We write Grass1 (F ) = P(F ), and we call P(F ) the projective space associated to F . A scheme X over S is projective over S if there is a locally free OS –module F of finite rank such that X is a closed subscheme of P(F ) and the structure morphism of X is induced by the structure morphism of P(F ). 22) Note. We have earlier used the projective r–dimensional space P(E) over Spec A, where E is a free A–module spanned by vectors e0 , . . , er . 21). Indeed, the latter is covered by affine schemes V(Ei∗ ⊗OSpec A G) = Spec(SymOSpec A Gi ), where Ei = OSpec A ei and Gi = OSpec A e0 ⊕ · · · ⊕ OSpec A ei−1 ⊕ OSpec A ei+1 ⊕ · · · ⊕ OSpec A er , in exactly the same way as P(E) is covered by the affine schemes Spec A[ xx0i , .

9) Note. 8) that the subfunctor G of F is locally closed if and only if there, for every scheme X over S and every element ξ ∈ F (X), is a subscheme XG,ξ of X such that a morphism g: T → X factors via XG,ξ if and only if F (g)(ξ) ∈ G(T ). 10) Note. It follows from the Definition of a locally closed subfunctor that the associated scheme XG,ξ is unique. 11) Note. Given a locally closed subfunctor G of F . Let Hξ : hX → F be the morphism associated to an element ξ ∈ F (X), and let i: XG,ξ → X be the corresponding subscheme of X.

Consider the sequence p1 QuotF (T ) → i∈I → QuotF (Ti ) − −→ Quot(Ti ∩ Tj ). p2 i,j∈I Given (fi )i∈I ∈ i∈I QuotF (Ti ), where fi is represented by surjections FTi → Gi on XTi . If p1 (fi ) = p2 (fi ) we have that the restriction of FTi → Gi to fT−1 (Ti ∩ Tj ) is equivalent to the restriction of FTj → Gj to fT−1 (Tj ∩ Ti ), for all i and j. Consequently the kernels of the maps FTi → Gi , for all i, define a submodule K ⊆ FT , such that the restriction of the quotient FT → G to fT−1 (Ti ) is equivalent to FTi → Gi for all i ∈ I.