Using discrete Morse theory, we give an algorithm that prunes the excess of information in the Taylor resolution and constructs a new cellular free resolution for an arbitrary monomial ideal. The pruned resolution is not simplicial in general, but we can slightly modify our algorithm in order to obtain a simplicial resolution. We also show that the Lyubeznik resolution fits into our pruning strategy. The pruned resolution is not always minimal but it is a lot closer to the minimal resolution than the Taylor and the Lyubeznik resolutions as we will see in some examples. We finally use our methods to give a different approach to the theory of splitting of monomial ideals. We deduce from this splitting strategy that the pruned resolution is always minimal in the case of edge ideals of paths and cycles

Peer Reviewed