An example of an Azumaya algebra that isn't free over its centre

Question

Azumaya originally defined an Azumaya algebra (which he called a proper maximally central algebra) to be an algebra A which is a free module of finite rank over its centre Z such that the natural map $$A\otimes_Z A^{\mathrm{op}}\to \mathrm{End}_Z(A)$$ is an isomorphism. More modern definitions (e.g., Knus, Quadratic and Hermitian Forms over Rings) replace "free" with "faithfully projective". Is there an example of an algebra which meets this broader definition but does not meet the original one? I.e., an Azumaya algebra that is faithfully projective but not free as a module over its centre. If possible, I'd also like the centre to be a noetherian $k$-algebra, for some algebraically closed field $k$.

Even better would be some method for producing lots of examples of Azumaya algebras "to order", but that's probably asking too much!

Answer 1

It suffices to find a vector bundle $E$ on an affine variety $V$ such that the vector bundle $\mathcal{E}nd(E)$ is nontrivial: then this vector bundle gives a non-free, projective Azumaya algebra over $\mathcal{O}(V)$. This will be the case if some Chern class of $\mathcal{E}nd(E)$, say $c_2$, is nontrivial in the Chow group $CH^2(V)$.

Take $V:=\mathbb{P}^3\smallsetminus S$, where $S$ is a smooth surface of degree $d\geq 4$, and take for $E$ the restriction of $\mathcal{O}_{\mathbb{P}}\oplus \mathcal{O}_{\mathbb{P}}(1)$. An easy computation gives $c_2(\mathcal{E}nd(E))=-h^2$, where $h$ is a hyperplane in $\mathbb{P}^3$. On the other hand we have an exact sequence $$CH^1(S) \xrightarrow{i_*} CH^2(\mathbb{P}^3)\rightarrow CH^2(V)\rightarrow 0$$ where $i$ is the embedding of $S$ in $\mathbb{P}^3$. If $S$ is general $CH^1(S)=\mathrm{Pic}(S)$ is generated by $i^*h$; we have $i_*i^*h=dh^2$, hence $c_2(\mathcal{E}nd(E))$ is nonzero in $CH^2(V)\cong \mathbb{Z}/d$.

Edit : This is actually easier in higher dimension, where you can use Chern classes in cohomology rather than in the Chow group. For instance if $V=\mathbb{P}^n\smallsetminus H$, where $n\geq 4$ and $H$ is a smooth hypersurface of degree $d>1$, then $c_2(\mathcal{E}nd(E))\neq 0$ in $H^4(V,\mathbb{Z})=\mathbb{Z}/d\ $ for the same $E$.