| Résume | A theorem of Deligne guarantees that under some finiteness assumptions, rigid tensor categories over an algebraically closed field admit a fiber functor and are therefore (super-)Tannakian. This, in turn, guarantees that they are relatively close to categories of modules over commutative rings. Beyond the Tannakian case, there is also a general feeling that rigid tensor categories behave "more" like categories of modules over commutative rings than arbitrary tensor categories.
In this talk, I will discuss a K-theoretic failure of this "feeling". More precisely, I will give examples to show that the K-theory of rigid tensor categories lacks one key structural property of the K-theory of commutative rings, by exhibiting failures of the so-called redshift principle (which holds for the K-theory of commutative rings). In the first half of the talk, I will focus on describing the context and discuss examples based on Deligne's category Rep(GL_t), and in the second half, I will discuss a general result that fully computes the K-theory of certain "algebraically closed" rigid tensor categories. |
