Operations on Sheaves Via the Structure Data

Let 7 be a small category, S a scheme, and V = V(L Sch/51. To study sheaves on objects of 7, it is convenient to utilize the structure data of them, and then utilize the usual sheaf theory on schemes. 

We say that a family of morphisms (tpj)ie/ between structure data 

Proof [sketch). It is not so difficult to show that it suffices to show that 

First we prove this for the case that P — PM. For an open subset U of Xi, this composite map evaluated at U is 

Next we consider the case = Mod. First note that the diagram 

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Operations on Sheaves Via the Structure Data agrees with... 

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