Download A Mathematician and His Mathematical Work: Selected Papers by Chern S.S., Li P., Cheng S.Y., Tian G. (eds.) PDF

By Chern S.S., Li P., Cheng S.Y., Tian G. (eds.)

Those chosen papers of S.S. Chern speak about subject matters equivalent to essential geometry in Klein areas, a theorem on orientable surfaces in 4-dimensional house, and transgression in linked bundles

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8 The sets Br (K |k) and Br (k) equipped with the above product operation are abelian groups. 32 Central simple algebras and Galois descent Before proving the proposition, we recall a notion from ring theory: the opposite algebra A◦ of a k-algebra A is the k-algebra with the same underlying k-vector space as A, but in which the product of two elements x, y is given by the element yx with respect to the product in A. If A is central simple over k, then so is A◦ . Proof Basic properties of the tensor product imply that the product operation is commutative and associative.

In (2), one chooses x = α and y as in the proposition. 6 Reduced norms and traces 37 √ an arbitrary degree m cyclic Galois extension K |k in the form K = k( m a), as in the corollary above. 13 (1)) shows that a cyclic Galois extension of degree p in characteristic p > 0 is generated by a root of some polynomial x p − x − a. In the previous chapter we have seen that the class of a nonsplit quaternion algebra has order 2 in the Brauer group. More generally, the class of a cyclic division algebra (a, b)ω as above has order m; we leave the verification of this fact as an exercise to the reader.

Now observe that for every finite field extension K of k contained in k, ¯ ¯ the inclusion K ⊂ k induces an injective map A ⊗k K → A ⊗k k and A ⊗k k¯ arises as the union of the A ⊗k K in this way. Hence for a sufficiently large finite extension K |k contained in k¯ the algebra A ⊗k K contains the elements ¯ via e1 , . . , en 2 ∈ A ⊗k k¯ corresponding to the standard basis elements of Mn (k) ∼ ¯ ¯ the isomorphism A ⊗k k = Mn (k), and moreover the elements ai j occurring in the relations ei e j = ai j ei defining the product operation are also contained in K .

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