I ask a question about $\prod k$ in Mathematics about several days.https://math.stackexchange.com/q/2766054/453628. And I have the following question:

1.What is the global dimension about $\prod k$?

2.is there an example of a ring which is absolutely flat and the global dimension is infinite?of course,if this exists,it can't be Noetherian.since finite presented flat module is projective. and the global dimension is the supremum of all projective dimension of finite generated modules.

3.can someone help give an example of a ring which has infinite global dimension and every finite generated module has finite projective dimension.

Thank you in advance!