|Ph.D Student||Itai Cohen|
|Subject||Some Theorems on the Arithmetic Plane|
|Department||Department of Mathematics||Supervisor||Professor Haran Shai|
|Full Thesis text|
We consider an analogue of the category of commutative rings, where the initial object is "the field with one element" F. We show how to relate the categories and by adjusting the commutativity and constructing an idempotent endofunctor in . The pushout generalized ring
is defined under in different ways, which gives different (but isomorphic) ways to go back to commutative rings. It has
as its symmetry group, by permuting additions .
We explain how injects into
, as an analogue of the injection.
The research stems from Weil's approach of the Riemann's Hypothesis for projective smooth curves $\Gamma$ over finite fields , whose 1946 proof [Weil2] uses intersection theory on for the Frobenius graph intersected with the diagonal The proof uses the Castelnuovo- Severi inequality for any divisor on a product surface , for projective smooth curves, with injections
The theory of schemes over constructs as a pro- object in the geometric category of affine schemes parallel to the generalized rings category together with an open dense embedding , and a non-reduced plane:
The research shows the suggested plane is an integral domain and cancelable as a multiplicative monoid.