Minimal free resolutions that are not supported by a CW-complex

In [1] it is shown that every monomial ideal admits a simplicial resolution (Taylor’s resolution) and that some minimal free resolutions are supported in simplicial complexes (Scarf ideals, monomial regular sequences). This idea is generalized in [2] where cellular resolutions are introduced. The authors show that every monomial ideal admits a resolution supported in a regular cell complex (the hull resolution) which is minimal in many cases (for example ideals in three variables) but not in general. More recently, in [3] and [6] discrete Morse theory is used to construct CWcomplexes which support the minimal free resolutions of virtually all classes of monomial ideals for which these resolutions are known. As the class of allowed topological spaces grows (simplicial complexes, regular cell complexes, CW complexes) more minimal free resolutions can be endowed with one of these ”geometries”. As more and more positive examples are found, the following fundamental question (see e.g. [6]) became very interesting: Can every minimal monomial free resolution be supported by a CW-complex? The purpose of the present paper is to explore this question and to answer it in the negative.