Hopfological algebra for infinite dimensional Hopf algebras

FARINATI, MARCO ANDRÉS

Algebras and Representation Theory

SPRINGER

Lugar: Berlin; Año: 2021 vol. 1 p. 1 - 35

1386-923X

We consider "Hopfological" techniques as in [K] but for infinitedimensional Hopf algebras, under the assumption of being co-Frobenius. Inparticular, H=k[Z]#k[x]/x2 is the first example, whosecorepresentations category is d.g. vector spaces. Motivated by this example wedefine the "Homology functor" (we prove it is homological) for any co-Frobeniusalgebra, with coefficients in H-comodules, that recover usual homology of acomplex when H=k[Z]#k[x]/x2. Another easy example of co-FrobeniusHopf algebra gives the category of "mixed complexes" and we see (by computingan example) that this homology theory differs from cyclic homology, althoughthere exists a long exact sequence analogous to the SBI-sequence. Finally,because we have a tensor triangulated category, its K0 is a ring, and weprove a "last part of a localization exact sequence" for K0 that allows usto compute -or describe- K0 of some family of examples, giving light of whatkind of rings can be categorified using this techniques.