Denote Sch the site of schemes over Q endowed with the fppf topology. Let F = Q(x_1, x_2, x_3, …) be the purely transcendental extension of Q generated by countably many elements. Let X = Spec(F). Let G = Z[X] be the free abelian sheaf on (the sheaf represented by) X. This sheaf has the following amusing property: If k is a field then

• G(Spec(k)) = 0 if trdeg(k/Q) finite, and
• G(Spec(k)) is not zero else.

The reason is that Mor_{Sch}(Spec(k), X) = ∅ if the transcendece degree of k is finite.

Here is another amusing abelian sheaf H. For any scheme S in Sch let I_S be the category of arrows f : T —> S where T is a nonempty connected scheme which is locally of finite type over some field of finite transcendence degree over Q. (Yes, this is a bit contrived.) A morphism (f : T —> S) —> (f’ : T’ —> S) in I_S is a morphism a : T —> T’ such that f = f’ o a. Define H(S) to be the set of maps σ: Ob(I_S) —> Z such that σ(f : T —> S) = σ(f’ : T’ —> S) if there is a morphism between f and f’ in I_S. In other words, σ is constant on “connected components” of Ob(I_S). In the case that Ob(I_S) = ∅ we set H(S) = 0. I claim that H is a sheaf (see remark below). Then H has the following property: if k is a field then

• H(Spec(k)) = Z if trdeg(k/Q) is finite, and
• H(Spec(k)) = 0 else.

The reason is that if there exists a morphism T —> Spec(k) with T nonempty and locally of finite type over a field of finite transcendence degree over Q, then k has finite transcendence degree over Q.

Remark: Suppose Sch’ ⊂ Sch is a full subcategory consisting of locally Noetherian schemes such that if T is in Sch’ and T’ —> T is locally of finite type, then T’ is in Sch’. Then Sch’ is also a site (with fppf topology) and the inclusion functor u : Sch’ —> Sch is cocontinuous. This gives rise to a morphism of topoi g : Sh(Sch’) —> Sh(Sch), see the chapter on Sites and Sheaves in the stacks project. Warning: this morphism of topoi is in the “wrong” direction. The sheaf H above is the sheaf g_*Z when we take Sch’ the category of schemes which are locally of finite type over a field of finite transcendence degree over Q. (Note that in our example Sch’ does not have all fibre products, but that doesn’t matter.)

Conclusion: The category of all schemes (over a given base) is too large to expect (fppf) sheaves to exhibit any kind of “coherent” behaviour as the input ranges over spectra of fields.