?? from wpa on "weil restriction" ... ??seems interesting for various reasons ... ????...

Restriction of scalars over a finite extension of fields takes group schemes to group schemes. In particular, the torus

\mathbb{S} := \mathrm{Res}_{\mathbb{C}/\mathbb{R}} \mathbb{G}_m

where Gm denotes the multiplicative group, plays a significant role in Hodge theory, since the Tannakian category of real Hodge structures is equivalent to the category of representations of S. The real points have a Lie group structure isomorphic to \mathbb{C}^\times. See Mumford–Tate group.


??? wpa on "hodge atructure" ...

One can define a notion of a morphism of mixed Hodge structures, which has to be compatible with the filtrations F and W and prove the following theorem.

Mixed Hodge structures form an abelian category. The kernels and cokernels in this category coincide with the usual kernels and cokernels in the category of vector spaces, with the induced filtrations.

Moreover, the category of (mixed) Hodge structures admits a good notion of tensor product, corresponding to the product of varieties, as well as related concepts of inner Hom and dual object, making it into a Tannakian category. By Tannaka-Krein philosophy, this category is equivalent to the category of finite-dimensional representations of a certain group, which Deligne has explicitly described.


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The machinery based on the notions of Hodge structure and mixed Hodge structure forms a part of still largely conjectural theory of motives envisaged by Alexander Grothendieck. Arithmetic information for nonsingular algebraic variety X, encoded by eigenvalue of Frobenius elements acting on its l-adic cohomology, has something in common with the Hodge structure arising from X considered as a complex algebraic variety. Sergei Gelfand and Yuri Manin remarked around 1988 in their Methods of homological algebra, that unlike Galois symmetries acting on other cohomology groups, the origin of "Hodge symmetries" is very mysterious, although formally, they are expressed through the action of the fairly uncomplicated group R_{\mathbf {C/R}}{\mathbf C}^* on the de Rham cohomology. Since then, the mystery has deepened with the discovery and mathematical formulation of mirror symmetry.

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??? ... back to "weil restriction" ... ???....

???? confusion here (just me, i mean ....) about ... ???... analogy to "total space of bundle" ... ???? .... ??preservation of cartesian products ??? .... ???? ....

?? some other analogy ... ??? .... ?? that we used to think about ??? ... ???? ....


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.

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