Exercise 16, § 16

a) Prove that the homomorphism $r_{K[X]/K}:$ Br(K) $\rightarrow$ Br(K[X]) induces an isomorphism from Br(K) to a direct factor of Br(K[X]).

$*$b) Suppose that the field K is perfect. Prove that $r_{K[X]/K}$ is an isomorphism (deduce from Tsen’s theorem (VIII, p. 360, Exercise 7) that Br(K[X]) is the union of the groups Br(L[X]$/K[X]$), where L runs through the Galois extensions of K of finite degree, and then apply Exercise 13).

c) Let A be a regular $\mathbf{Q}$-algebra that is an integral domain. Prove that the homomorphism $r_{A[X]/A}:$ Br(A) $\rightarrow$ Br(A[X]) is bijective (use b) and Exercise 15 to prove that the homomorphism Br(A[X]) $\rightarrow$ Br(A) deduced from the surjection $P\mapsto P(0)$ is injective).$*$

d) Suppose that K has characteristic $p >0$ and is not perfect; let $\theta \in K$ - $K^p$. View B = K[U] as a K[X]-algebra via the homomorphism $\rho : K[X]\rightarrow B$ determined by $\rho (X) = U^p-U$. Let $\sigma$ be the automorphism of this K[X]-algebra such that $\sigma (U) = U + 1$, and let S be the quotient K[X]-algebra of $B[V]_{\sigma}$ (VIII, p. 11) by the two-sided ideal $B(V^p-\theta )$. Let L be a K[X]-algebra that is a commutative field. Prove that if the equation $y^p-y= X$ has a solution in L, then $S_{(L)}$ is a matrix algebra over L (let $L'= L[t]/(t^p-\theta )$; consider the homomorphism $S\rightarrow$ End$_L(L')$ that sends V to the multiplication $m_t$ by $t$ and U to $m_y-D$, where D is the L-derivation of $L'$ such that $D(t) =t$). In the opposite case, $S_{(L)}$ is a central simple L-algebra; it is a matrix algebra if and only if $\theta$ is the norm of an element of the extension $L[Y]/(Y^p-Y-X)$ of L (apply Exercise 11).

Deduce that S is an Azumaya K[X]-algebra and that its class in Br(K[X]) does not come from Br(K) (take L = K(X)).