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Fields as algebraic objects. For vector and tensor fields, use eg. [dg.differential-geometry]. For physical fields, use eg. [mp.mathematical-physics] or [quantum-field-theory].
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Residual gerbe and field of moduli
(1): There is a natural injection $K\to L$, indeed we have $\operatorname{Spec} L\to \mathcal{G}\to \operatorname{Spec} K$.
(2) and (3): No, it can happen that there are no objects defined on any str …