Lazarsfeld proved a bound for the excess dimension of an intersection of irreducible and reduced schemes. Flenner and Vogel gave another approach for reduced, non-degenerate schemes which are connected in codimension one, using the intersection algorithm of Stiickrad and Vogel and defining a new multiplicity k. Renschuch and Vogel considered a condition to ensure that there is no degeneration for more than two schemes. We define an integer which enables us to unify these methods. This allows us to generalize the result of Flenner and Vogel to non-reduced schemes by comparing the multiplicities j and k. Using this point of view we give applications to converses of Bezout's theorem; in particular we investigate the Cohen-Macaulay case.1991 Mathematics subject classification: Primary 14C17. Secondary 13H15.H. Flenner and W. Vogel gave a first approach to the converse to Bezout's theorem in [1]. Since then there has been other work on this subject (see [2,9,13, 14]). The converse to Bezout's theorem consists of an answer to the following question: under which conditions does the situation of a proper intersection follow from a Bezout'sequality? In this connection, in [13, 14] a relation between the excess dimension and dimensions of vector spaces of linear forms is considered. We want to investigate this connection rather more deeply.We always consider the intersection of two or more than two schemes. This transition from the case of two schemes is not quite trivial, as the discussion of example 12.3.5 of [3, p. 225] in [9, p. 316, Remark 1] shows. In [3, p. 225], one assumption is missed and this is corrected by an unnecessarily strong condition. This condition is commented on and also weakened in [4]. We give here a new condition.In particular, in Proposition 2.4 and Theorem 2.9., we give generalizations of Theorem 4.2 of [1]. These generalizations are based on Proposition 2.2 and Lemma 2.8. In the latter we compare the intersection multiplicities j (see [3,11,12]) and k (see [1], where the notation J was used). This allows us to pass from the intersection of reduced schemes to that of general pure dimensional schemes. The main result of H. Flenner and W. Vogel in [1] and the result of L. O'Carroll in [8] treat the reduced case. Theorem 2.9 especially has some interesting applications. For example, we use this result in Section 3 to give a new proof of Proposition 3.12 of [2] and to study the CohenMacaulay case. Also, we give in Corollary 3.3 a partial answer to the question following Theorem 7 in [14]. We generalize the Theorem of [13] in the geometrically more 427
