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Wm is an integral basis for H (X). Write Jwj Λ wk = c^w, where w 2n is a normalized volume element of X. Let Ν be a subspace of A (X) such 2 that kw® Ν = Λ " ^ ) . fce JV. Let A be the D G subalgebra of A(X) spanned by 1, w, w,, W y , where 1,7 = 1, m. Then the m a p A-+H(A) given by wfi—• w f , νν^ι—^Ο, wi-> w, and dw^i—•O, is a well-defined chain map, which is a homology isomorphism. T h u s E2 = E^. Let 1, Xx, X m, X be the dual basis for H(A)*. Then Η+(ΩΧ) is isomorphic to the DG-free algebra k[Xl9 X2, ], whose differential d is given by dXt = 0, dX = YdcijXiXj- REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] J.

P. M. C o h n , Free ideal rings. J. Algebra 1 (1964), 47-69. R. Diaconescu, Thesis, Dalhousie University, 1973. J. Isbell and B. Mitchell, Exact colimits. Bull. Amer. Math. Soc. 79 (1973), 994^996. O . Laudal, N o t e on the projective limit on small categories. Proc. Amer. Math. Soc. 33 (1972), 307-309. D. Lazard, A u t o u r de la platitude. Bull. Soc. Math. France 97 (1969), 81-128. B. Mitchell, Rings with several objects. Advances in Math. 8 (1972), 1-161. U. Oberst a n d H. Rohrl, Flat a n d coherent functors.

I= 1 Thus a linear combination of morphisms in ( Y, M ) with coefficients summing to one filters α a n d α'. But these morphisms d o not necessarily have a c o m m o n codomain, a n d so their linear combination does not comprise a morphism of aff(Y, M). However this is taken care of by the following lemma. 3. The components of aff C are filtered if a n d only if every diagram in C of the form A can be completed to a (not necessarily commutative) square in C, and for each diagram of aff C of the form A there is an element β of © such that β<χ = β'α'.

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