Siberian Mathematical Journal, Vol. 39, No. 1, 1998 ON LOCALLY FINITE PARTIALLY ORDERED SETS OF DIMENSION 3t) A. A. Khusainov UDC 512.66 This articleconcerns the homological dimension theory for partiallyordered sets. We give a positive answer to B. Mitchell's question [I, p. 143] as to whether the equality pd AC = 3 + pd A that he established for the n-crown C is valid for every locallyfinitepartiallyordered set C of dimension 3. 1. An auxiliary spectral sequence. Henceforth N stands for the set of natural numbers n >_ 0. Given an arbitrary category `4, we denote by `4op the dual category, by Mor`4 the class of all morphisms, and by Ob `4 the class of all objects. Let `4(-, =) : `4op x `4 --+ Ens be the functor from the Cartesian product of the categories `4op and `4 into the category Ens of sets which to each pair (A, B) of objects in ,4 assigns the set `4(A, B) of all morphisms A --, B and to each pair of morphisms (f : A' ~ A,9: B ~ B') in `4 assigns the mapping `4(f,g) : `4(A,B) ~ ,4(A', B') that on the elements a E `4(A,B) acts in accord with the formula `4(/,g)(a) = g o a o f. If ,4 is an Abe]jan category then the sets `4(A, B) axe Abe]jan groups and the mappings `4(f, 9) are homomorphisms. In that event we treat the functor `4(-, =) as taking values in the category Ab of Abe]jan groups and homomorphisms. Given an arbitrary category .A and a small category C, we denote by CA the category of functors C --* `4 and their natural transformations. Granted an object A E `4, we define the functor AcA : C --* `4 as taking constant values equal to A at objects and to 1A : A --* A at morphisms of C. Let `4 be an Abe]Jan category. Given arbitrary objects A and B of `4 and n E M, we denote by Ext"(A, B) the Abe]jan group of the classes of n-fold extensions from B to A in the sense of N. Yoneda [2]. For every small category C, the category CAb is Abe]Jan and has enough injectives. Hence, the nth derived functors ]Jm~ : CAb --, Ab of the limit functor are defined for all n E N. We consider every partially ordered set C as a category in which the elements of C serve as objects and the pairs a _< b of elements in C serve as morphisms a --+ b. Given an arbitrary element c E C, we put C/c = {x E C :x < c}. For functors, alongside the notation F :C ~ `4 we shall use the record {F(c)}cec. Proposition 1. Let .,4 be an Abe/Jan category and let D be a partially ordered set. If the sets Did are finite for all d E D then, for every object A E ,4 and every fun.ctor F : D ~ `4, there exists a first quarter spectral sequence of type E~ 'q = ]jm~{Extq(a,F(d))}deD ~ ExtP+q(ADA, F). PROOF. We first consider an arbitrary small category C. Recall that the category offactorizations C ~ of C is defined to be the category whose set of objects Ob C t is Mor C and whose sets C'(a, fl) of morphisms between arbitrary a,/~ E Ob C ~ consist of the pairs of morphisms (f, g) in C satisfying the relation g o a o f = 8- The composition and identical morphisms in C ~ are defined componentwise. Let (s, t) : C ~ --+ C ~ xC be the functor that to each a E Ob C' assigns the pair (sa, ta) of the beginning sa and ending ta of a and that acts as (f,a) (/,g) at morphisms. Now, we turn to proving the proposltion. If D is a partially ordered set such that Did are finite for all d E D, then in accordance with [3, Theorem 2.5], for arbitrary functors G,F : D --* `4 in an Abe]Jan category, there exists a spectral 1) The research was supported by the Grant Center for Research in Mathematics at NovosibirskState University. Komsomollsk-na-Amure. Translatedfrom SibirskKMatemaficheskff Zhurnal, Vol.39, No. I, pp. 201-205, January-February,1998.Original article submittedJune 26, 1995. 176 0037-4466/98/3901-0176 $20.00 (~) 1998 Plenum Publishing Corporation