2019/04/23 by Farinati, Marco A.
#16E35 #16T05 #18D99 #18G99 #19A49 #81R50 #FOS: Mathematics #K-Theory and Homology (math.KT) #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.1904.10430
We consider "Hopfological" techniques as in \citeKo but for infinite dimensional Hopf algebras, under the assumption of being co-Frobenius. In particular, H=k[\mathbb Z]#k[x]/x2 is the first example, whose corepresentations category is d.g. vector spaces. Motivated by this example we define the "Homology functor" (we prove it is homological) for any co-Frobenius algebra, with coefficients in H-comodules, that recover usual homology of a complex when H=k[\mathbb Z]#k[x]/x2. Another easy example of co-Frobenius Hopf algebra gives the category of "mixed complexes" and we see (by computing an example) that this homology theory differs from cyclic homology, although there exists a long exact sequence analogous to the SBI-sequence. Finally, because we have a tensor triangulated category, its K0 is a ring, and we prove a "last part of a localization exact sequence" for K0 that allows us to compute -or describe- K0 of some family of examples, giving light of what kind of rings can be categorified using this techniques.