The Cyclic Vector Lemma

Let $F$ be a differential field of characteristic zero with algebraically closed constant field $C$. Let $E$ be a Picard--Vessiot closure of $F$, $R \subset E$ its Picard--Vessiot ring and $\Pi$ the differential Galois group of $E$ over $F$. Let $V$ be a differential $F$ module, finite dimensional as an $F$ vector space. Then $V$ is singly generated as a differential $F$ module if and only if there is a $\Pi$ module injection $\text{Hom}_F^\text{diff}(V,R) \to R$. If $C \neq F$ such an injection always exists.Comment: 3 page

Differential Brauer Monoids

The differential Brauer monoid of a differential commutative ring R s defined. Its elements are the isomorphism classes of differential Azumaya R algebras with operation from tensor product subject to the relation that two such algebras are equivalent if matrix algebras over them are differentially isomorphic. The Bauer monoid, which is a group, is the same thing without the differential requirement. The differential Brauer monoid is then determined from the Brauer monoids of R and its ring of constants and the submonoid whose underlying Azumaya algebras are matrix rings

Grothendieck Topology

Non UBCUnreviewedAuthor affiliation: University of OklahomaFacult

The separable Galois theory of commutative rings

The Separable Galois Theory of Commutative Rings, Second Edition provides a complete and self-contained account of the Galois theory of commutative rings from the viewpoint of categorical classification theorems and using solely the techniques of commutative algebra. Along with updating nearly every result and explanation, this edition contains a new chapter on the theory of separable algebras.The book develops the notion of commutative separable algebra over a given commutative ring and explains how to construct an equivalent category of profinite spaces on which a profinite groupoid acts. I