Math. Z. 192, 473-490 (1986) Mathematische Zeitschrift 9 Springer-Verlag 1986 Spectral Theory and Sheaf Theory. II Mihai Putinar Department of Mathematics, INCREST, Bd. Pficii 220, 79622 Bucharest, Romania Introduction This paper is a continuation of [23] and deals with a sheaf model in the multidimensional spectral theory of linear operators. The infinite dimensional extension of the coherence property for analytic sheaves was brought out to light by Ramis and Ruget [25] and Leiterer [17~. An analytic sheaf o ~ on 112 n is said to be Fr~chet quasi-coherent, in the terminology of [25], if the following assertions: (i) ~- is a Fr~chet @-module, that is for every open subset V of 117 ~, ~-(V) is a Fr6chet space and the multiplication map (9(V)x~(V)~(V) is con- tinuous, denoting by (9 the sheaf of holomorphic functions on II7 ~, and (ii) For every Stein open subset U of (12 n the topological and natural identification T6r~q(r ifif q>0,q=0' hold true. Thus a Fr6chet quasi-coherent O-module ff on 112 n is completely deter- mined by its global sections space X= Y(Cn), and it admits globally a canoni- cal resolution with topological free O-modules, see w1 in [23]. The Fr6chet (9(~")-module X is nothing more than a commutative n-tuple a=(al, a2, ..., a,,) of linear continuous operators on X, namely the multiplication operators with the coordinate functions. The remarkable fact is that the quasi-coherence conditions (i) and (ii) correspond exactly to the property (fi) of the n-tuple a, as was pointed out in [23]. The property (fi) is due to Bishop [4] and it was generalized to several variables by Frunzgt [11]; it insures a good local spectral theory for the n-tuple a (see [8, 11, 23]), and also it is a first test for the investigation of the spectral decomposability behaviour of the system of oper- ators a (see [5, 11] and Chapter IV in [28]). The Fr6chet quasi-coherent @-module Y associated to the commutative n- tuple a by the rule (ii) was named in [23] the sheaf model of a. Two isomor-