Spaces of Generators for Azumaya Algebras with Unitary Involution

Abstract

Let $A$ be a finite dimensional algebra (possibly with some extra structure) over an infinite field $K$ and let $r\in\mathbb{N}$. The $r$-tuples $(a_1,\dots,a_r)\in A^r$ which fail to generate $A$ are the $K$-points of a closed subvariety $Z_r$ of the affine space underlying $A^r$, the codimension of which may be thought of as quantifying how well a generic $r$-tuple in $A^r$ generates $A$. Taking this intuition one step further, the second author, Reichstein and Williams showed that lower bounds on the codimension of $Z_r$ in $A^r$ (for every $r$) imply upper bounds on the number of generators of \emph{forms} of the $K$-algebra $A$ over finitely generated $K$-rings. That work also demonstrates how finer information on $Z_r$ may be used to construct forms of $A$ which require many elements to generate. The dimension and irreducible components of $Z_r$ are known in a few cases, which in particular lead to upper bounds on the number of generators of Azumaya algebras and Azumaya algebras with involution of the first kind (orthogonal or symplectic). This paper treats the case of Azumaya algebras with a unitary involution by finding the dimension and irreducible components of $Z_r$ when $A$ is the $K$-algebra with involution $(\mathrm{M}_n(K)\times \mathrm{M}_n(K), (a,b)\mapsto (b^{\mathrm{t}},a^{\mathrm{t}}))$. Our analysis implies that every Azumaya algebra with a unitary involution over a finitely generated $K$-ring of Krull dimension $d$ can be generated by $\lfloor \frac{d}{2n-2}+\frac{3}{2} \rfloor$ elements. We also give examples which require at least half that many elements to generate, by building on the work of the second author, Reichstein and Williams. Our method of finding the dimension and irreducible components of $Z_r$ actually applies to all $K$-algebras $A$ satisfying a mild assumption.