دراسة في مبرهنات التماثل لجداء المخططات الحلقية

Abstract EN

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 X xx (X xs Y) « X xs Y where each of X and Y is S -scheme - and the theorem (4 - 2) which shows the isomorphisms (X xs Y) xY Z ͌ Z xY (X xs Y) ͌ X xs Z ͌ Z xs X where each of X and Y is S -scheme and Z is Y -scheme , theorem (4 - 2) was employed in proving the associative property of schemes product (X xs Y) xs Z ͌ X xs (Y xs Z).

Theorem (4 - 4) shows that if O 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 X xs Y ͌ X xT Y where each of X and Y is S -scheme in addition to the assumption that there is an immersion a:S→T.