?? role of flatness (?? ...) of k in .... ???? getting ... ??? i was going to distinguish between .... getting "underlying set of tensor product with k" to be representable, vs getting it to lift to _comm ring_ .... ?? but the latter seems completely tautological, right ??? ... ???? ....

?? consider ... ??? many different products here .... categorified and decategorified .... dircichlet series ... comm ring ... alg comm ring ... bistable hopf alg ... .... ??? .... .... cartesian, "tensor", "direct sum", ..... ???? .... correspondences / relationship among such ... ???? ....

??? confusion here (?? ...) ... ?? comm ring vs alg comm ring ... ??? ... ??? galois action .... ???? ....