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Let denote a finite field and let be a domain of finite type over . Is

for every ideal ?


Let be an abelian variety in positive characteristic and let be a cohomology class of a vector bundle such that the cohomological dimension of the corresponding torsor equals the dimension of . Is it true that can be annihilated by an iteration of the -multiplication?


Let denote a smooth projective variety of dimension over a finite field . Let be a vector bundle on and be a cohomolgy class with corresponding torsor

Does has cohomological dimension if and only if there exists a finite morphism

(where denotes another (smooth?) projective variety of dimension ) such that ?


Give, for every prime number , an example of an -algebra , an ideal and an element such that in , but in for every maximal ideal .

Is this possible for the ring ?

Problem (Miyaoka)  


denote a relative smooth projective curve over an onedimensional affine integral base of mixed characteristic (of finite type over ). Suppose that is a vector bundle on such that the generic bundle over the generic curve (in characteristic ) is semistable. Is it true that for infinitely many points the bundle over the curve (in positive characteristic) is strongly semistable?


Let denote an onedimensional domain of mixed characteristic (of finite type over ) and let be a domain of finite type of relative dimension . Let be an ideal and . Suppose that belongs to the solid closure of inside the generic ring (in characteristic ). Is it true that holds for infinitely many points (in positive characteristic)?