On computer-assisted research in homological algebra Original Research Article

We give a survey of how computer algebra can be used to help the mathematician to guess results and to prove theorems in homological algebra. Our main point is that the Poincaré-Betti series of a commutative graded algebra contains much deeper information and is harder to calculate than the Hilbert series of the same algebra. However, in many cases (going far beyond the so-called Koszul algebras), the two series are closely related, and this gives an interesting theory. This theory could hardly have been revealed without an intensive use of the programs MACAULAY by Dave Bayer and Michael Stillman and BERGMAN by Jörgen Backelin. We also present new results and conjectures inspired by these studies and indicate how our results are related to problems in algebraic geometry and algebraic topology.