Definition
Let L/k be a finite extension of fields, and X a variety defined over L. The functor from k-schemesop to sets is defined by
(In particular, the k-rational points of are the L-rational points of X.) 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 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.
Read more about this topic: Weil Restriction
Famous quotes containing the word definition:
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)
“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
—François, Duc De La Rochefoucauld (1613–1680)
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)