  We study the relationship between the Tor-regularity and the local-regularity
over a positively graded algebra defined over a field which coincide if the
algebra is a standard graded polynomial ring. In this case both are
characterizations of the so-called Castelnuovo--Mumford regularity. Moreover,
we can characterize a standard graded polynomial ring as an algebra with
extremal properties with respect to the Tor- and the local-regularity. For
modules of finite projective dimension we get a nice formula relating the two
regularity notions. Interesting examples are given to help to understand the
relationship between the Tor- and the local-regularity in general.
