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The cohomology of augmented algebras and generalized Massey products for 𝐷𝐺𝐴-Algebras

1966; American Mathematical Society; Volume: 122; Issue: 2 Linguagem: Inglês

10.1090/s0002-9947-1966-0192496-2

1088-6850

J. P. May,

Advanced Topics in Algebra

We will here describe a new procedure for the calculation of the cohomology H*(A) = Ext¿(X, X) of a graded augmented connected (A0 = X) algebra A of finite type over a field X.Our procedure, which is developed in the first half of the paper, involves the application of a spectral sequence defined for any DGAalgebra U to the special case U= C(A), the cobar construction of A. In the second half of the paper we define Massey products in U and relate them to the differentials in the cited spectral sequence.To motivate the construction, we recall that any basis for Hy(A) =Tot¡\,(K, K) is in one-to-one correspondence with a minimal set of generators of A and any basis for H2(A) is in one-to-one correspondence with a minimal defining set of relations for A (see Wall [11,).Now the same result holds for bigraded augmented connected algebras, and in particular for H*(A).Thus knowledge of Hy(H*(A)) and H2(H*(A)) would essentially determine H*(A).We will prove the following theorem.Theorem 1.There exists a spectral sequence {ErC(A)} of differential coalgebras having the properties:(i) EpqC(A) = Hp q(H*(A)) as a graded K-module (where H*(A) is graded with lower indices so that q = 0: H*(A)q = H~q(A));(ii) E2C(A) = H*(H*(A)) as a coalgebra ;(iii) The differentials satisfy 8r: Ep qC(A)-> Ep_r q+r_yC(A);(iv) Ep°iqC(A) = 0ifq^-p;(v) Ep°^pC(A) = (E_p,»A)*, where E°A denotes the associated graded algebra of A with respect to the augmentation ideal filtration F _pA =(IA)P if p>0,F_pA = AifpúO;(vi) EoeC(A) = (E°A)* as a coalgebra.By (i), EpqC(A) = 0 if p < 0 or if p + q > 0. In principle, knowing Hl(A) and the relations in H2(A), we can calculate Ep^pC(A) for all p. Calculation of