Matthew Elmore tells how to beat poa trivialis in a post on the Rutgers Turf Blog:
Roughstalk bluegrass is a cool-season perennial grass often considered a weed. Light green in color it is most obvious in the early spring when it grows more rapidly than other turfgrasses. As temperatures rise in the summer its growth will slow and it often goes semi-dormant until cooler temperatures and rainfall return in the autumn.
To control roughstalk bluegrass in cool-season lawns and athletic fields, nonselective control with glyphosate (Roundup and others) followed by reseeding is the best option.
Read more and see photos here