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We develop some foundations of commutative algebra, with a view towardsalgebraic geometry, in symmetric tensor categories. Most results establishanalogues of classical theorems, in tensor categories which admit a tensorfunctor to some tensor category verifying specific conditions. This is in linewith the current program which aims to describe tensor categories by theirtensor functors to incompressible categories. We place particular emphasis onthe notion of observable subgroups of affine group schemes in tensorcategories, which in particular leads to some further insight intoobservability for classical affine group schemes.

Commutative algebra in tensor categories