Study in Isomorphism Theorems of Schemes Product

Abstract

This research studies  the Cartesian diagrams of products, and we clarify by this research the  importance of  the Cartesian diagrams in simplifying proofs of some isomorphism theorems of schemes product. The research starts by employing the Yoneda Lemma to be able to prove some of isomorphism theorems of  schemes product by using product of sets, and that is because the definition of  the product of sets is more simpler in comparison with the definition of schemes product. The corollary (3-5) shows that there is an equivalence between the Cartesian diagrams of sets and the Cartesian diagrams of schemes, theorem (3 – 9) shows how to use the Cartesian diagrams in proving isomorphism between tow schemes. Theorem (4 – 9) was employed in proving  the theorem (4 – 1) - which shows the isomorphism   where each of  and   is -scheme - and the theorem (4 – 2) which shows the isomorphisms  where each of  and is -scheme and  is -scheme  , theorem (4 – 2) was employed in proving the associative property of schemes product   . Theorem (4 – 4) shows that  if  is an immersion then it can be canceled of the left in composing morphisms between schemes, which was employed in proving  the theorem (4 – 5) which shows  the isomorphism   where each of  and   is -scheme in addition to the assumption that there is an immersion .