Algebraic structures and descent by symmetric monoidal categories and Deligne’s Theory

Let W be a finite dimensional algebraic structure over a field K of characteristic zero (for example an algebra or a graded algebra). In this talk I will explain how to construct a symmetric monoidal category CW which is (up to some categorical data) a complete invariant of W. This category will be a form of RepK-G, where G is the algebraic group of automorphisms of W, over some subfield K0 of K. The field K0 can be thought of as the field of invariants of W, in a way which I will make precise.By using the theory of Deligne on symmetric monoidal categories I will show how one can use this category to construct a generic form of W, and to study scalar invariants of W. Moreover, I will show that forms of the structure W are in one to one correspondence with fiber functors from this category.I will give some examples of this category when W is a central simple algebra or a module over a given central simple algebra. I will also explain how one can use this category to study embeddings of projective varieties in projective spaces and study questions about field of definition.