The Segre Embedding

Throughout this note all rings are commutative, and A is a fixed ring. If S, T are graded A-algebras then the tensor product S ⊗A T becomes a graded A-algebra in a canonical way with the grading given by (TES,Lemma 13). That is, S ⊗A T is the coproduct of the morphisms of A-modules Sd ⊗A Te −→ S ⊗A T for d, e ≥ 0. The canonical morphisms p1 : S −→ S ⊗A T, p2 : T −→ S ⊗A T are then morphisms of graded A-algebras. Definition 1. Let S, T be graded A-algebras. We define their cartesian product, denoted S×A T , to be the following graded A-algebra: as an A-module it is the sum of the images of the A-module morphisms Sd ⊗A Td −→ S ⊗A T for all d ≥ 0. This is an A-subalgebra of S ⊗A T which is a graded A-algebra with grading (S ×A T )d ∼= Sd ⊗A Td for d ≥ 0. The scheme Proj(S×A T ) is covered by open subsets D+(f ⊗ g) for f ∈ S, g ∈ T homogenous of the same degree d > 0. It is not hard to check that the following are well-defined morphisms of A-algebras φf,g : S(f) −→ (S ×A T )(f⊗g) s/f 7→ (s⊗ g)/(f ⊗ g) ψf,g : T(g) −→ (S ×A T )(f⊗g) t/g 7→ (f ⊗ t)/(f ⊗ g) If h ∈ S, k ∈ T are homogenous of the same degree e > 0 then it is readily checked that the following diagram commutes (the vertical morphisms are the canonical ring morphisms)