Improved Kunneth Tricks

The Kunneth trick is a formula for the top cohomology of the derived tensor product of two complexes of modules over a ring. In this note we present two improvements of this formula. The first improved Kunneth trick is a formula for the top cohomology of the plain tensor product of two DG modules over a nonpositive DG ring. The second trick handles the derived tensor product of two DG modules over a nonpositive DG ring. The proofs are elementary.