2 1. INTRODUCTION (C2) Reversing the direction of the edges from the collection in the graph yields a graph that still contains no directed cycle. Adapting the terminology from discrete Morse theory we call such a collection of edges an acyclic matching or a Morse-matching on C•. Let us explain how the construction above reflects the algebraic approach de- scribed before. Assume that c and c are vertices corresponding to basis elements in homological degrees i and i − 1 respectively such that c and c are connected by an edge. Assume further that the weight on an edge between c and c is invertible in R. Then we apply the basis transformation in the homological degree i − 1 that replaces c by ∂i(c) and is the identity on all other basis elements. This yields an exact subcomplex 0 → Rc → R∂i(c) → 0. For a collection of subcomplexes of this type, their direct sum can be a subcomplex only if condition (C1) is satisfied. But (C1) is not suﬃcient. Now condition (C2) will be shown to imply the existence of a basis transformation that assures condition (A1). The fact that we have chosen edges that have invertible coeﬃcients means that this basis transformation is indeed invertible over R and therefore (A2) is satisfied. The advantage of this approach is that the calculation of the quotient complex is purely combinatorial: The basis elements of the modules in the quotient complex are in one-to- one correspondence with the vertices not lying in any edge e ∈ M. The coeﬃcient of the basis element c in the differential ∂M(c), taken in the quotient complex, can be calculated by summing up the weights of all directed paths from c to c in the graph where all edges from the matching are reversed. Here the weight of a path is the product of the weights of the edges along the path. It should be mentioned that in practice the calculation of the differential of the quotient complex can be very complicated and technical. But often it is enough to know the basis of the quotient complex: For example, consider the problem Calculate the Betti-numbers of an R-module: If we start with a multigraded free resolution then the multidegree of the remaining basis elements can give information about minimality of the resolution. Thus, in this situation for calculating Betti- numbers, regularity, and Poincar´ e-Betti series it is not necessary to calculate the differential. Recall that Forman’s discrete Morse theory (see [17], [18]) allows starting from a regular CW-complex the construction of a homotopy equivalent CW-complex with fewer cells. When applied to the complex calculating cellular homology of a regular CW-complex algebraic discrete Morse theory describes the impact of Forman’s construction. As an advantage over Formans theory algebraic discrete Morse theory can be applied iteratively: Discrete Morse theory can only be applied to a regular CW-complex, which implies that all coeﬃcients of its cellular chain complex have to be ±1. Algebraic discrete Morse theory only requires certain coeﬃcients to be invertible. Consider the following example. Assume we are given a regular CW-complex and want to calculate its homology with coeﬃcients in a field of characteristic 0. If we apply Forman’s theory to this complex we may arrive at a CW-complex for which certain face incidences give rise to coeﬃcients = ±1, 0 in the cellular chain complex. Discrete Morse theory is no longer applicable to those

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