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I did not understand number theory or characteristic p-algebraic geometry at all. I just know a little about frobenius homomorphism between two schemes. On the other hand, when I learned something on triangulated category. I found there was also a definition of "frobenius morphism" the definition is as follows:

There are two categories C and D. f_:D--->C, f^ :C--->D is left adjoint to f_, we call f_ is a Frobenious morphism if there exists an auto-equivalence G of C such that composition f^* G is right adjoint to f_*.

First question is:is there any relationship between this two frobenius morphism?

Second question is:does frodenius category play roles in algebraic geometry?

All the comments related to this are welcomed.

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That notion of Frobenius morphism between categories is a generalization of Frobenius algebras (those which have a non-degenerate mulplicative bilinear form) to triangulated categories.

This is quite unrelated to the Frobenius morphism on a scheme. There are lot of things named ater Frobenius!

On the other hand, Frobenius categories show up all the time in geometrical contexts. They provide a nice way to construct triangulated caegories (and the triangulated categories so constrcted are particularly nice: they are ´algebraic´) For example, they are used to construct one of the categories equivalent to the derived category of coherent sheaves on projective space in the canonical example of why derived categories are relevant to geometry! This is explained in the book by Gelfan'd and Manin on homological algebra, if I recall correcty.

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Thank you. Yes, I know the stable category of frobenius category is triangulated category and I know why derived categories are related to algebraic geometry. Can you pointed out the reference which shows what you said about " they are used to construct one of the categories equivalent to the derived category of coherent sheaves on projective space" –  Shizhuo Zhang Jan 1 '10 at 16:56
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