Algebra/Topology Seminar

Speaker: Toni Annala

Title: Towards genuine algebraic cobordism

Abstract: Algebraic cobordism is a (bivariant) cohomology theory playing the role of complex cobordism in algebraic geometry. We sketch its history, and then introduce a new approach by Annala and Annala--Yokura based on derived algebraic geometry. This is a non-A¹-invariant theory, and it is hypothetically a slice of a larger extraordinary ``motivic cohomology'' theory (``higher algebraic cobordism''). The theory satisfies the projective bundle formula, and therefore admits a well-behaved theory of Chern classes. An analogue of the Conner--Floyd isomorphism then follows: the zeroth algebraic K-theory group can be recovered from the algebraic cobordism ring. Time permitting, we will also discuss more recent results concerning the corresponding homology theory (algebraic bordism) for schemes over fields of positive characteristic (and nice enough DVRs) after inverting the (residual) characteristic exponent e in the coefficients. Interestingly, the algebraic bordism groups are generated by classical cycles rather than derived cycles. This result has several consequences: it allows us to compute the algebraic cobordism ring of a field in degrees [-2, ∞), to prove an extension theorem for bordism classes, and to deduce the A¹-invariance of algebraic bordism theory. The contents of this talk form a part of the PhD-thesis project of the speaker.