High Quality Content by WIKIPEDIA articles! In mathematics, restriction of scalars (also known as "Weil restriction") is a functor which, for any finite extension of fields L/k and any algebraic variety X over L, produces another variety ResL/kX, defined over k. It is useful for reducing questions about varieties over large fields to questions about more complicated varieties over smaller fields. The variety that represents this functor is called the restriction of scalars, and is unique up to unique isomorphism if it exists. From the standpoint of sheaves of sets, restriction of scalars is just a pushforward along the morphism Spec L to Spec k and is right adjoint to fiber product, so the above definition can be rephrased in much more generality. In particular, one can replace the extension of fields by any morphism of ringed topoi, and the hypotheses on X can be weakened to e.g. stacks. This comes at the cost of having less control over the behavior of the restriction of scalars. For any finite extension of fields, the restriction of scalars takes quasiprojective varieties to quasiprojective varieties.

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