# The Situation over ℤ: Questions

# The Situation over ℤ: Questions

This chapter considers that there is another sense in which we might ask about “situations over ℤ,” namely we might try to mimic the setting of a theorem of Pink [Ka-ESDE, 8.18.2] about how “usual” (geometric) monodromy groups vary in a family. For each geometric point s in a normal noetherian connected scheme S, we have the closed subgroup Γ(*s*) ⊂ *GL*(*n*,ℚℓ¯) which is the image of π₁(*X*ₛ; *x*ₛ) in the representation corresponding to Ƒₛ. The assertion is that these groups Γ(*s*) are, up to *GL*(*n*)-conjugacy, constant on a dense open set of *S*, and that they decrease under specialization. This chapter treats the following question: suppose a normal noetherian connected scheme *S* which is of finite type over ℤ [1/𝓁], an object *N* in the derived category Dcb((𝔾ₘ)ₛ;ℚℓ¯), and an integer *n* ≤ 1.

*Keywords:*
number theory, noetherian connected scheme, monodromy groups

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