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posicDate: Thu, 4 Jul 2002 19:08:39 +0200 (CEST)
From: Leonid Positselski
Subject: morphisms of graded algebras
Dear Jan-Erik,
Several weeks ago you asked me a question about generalization of results from my paper in K-theory preprint archive to non-Koszul commutative algebras. Now it seems that I understand these things somewhat better.
Namely, if f: R\to S is a map of commutative graded algebras, then the DG-algebra S\otimes_R k (the derived tensor product) is bar-dual to the DG-algebra UKer(Lie(S)\to Lie(R)), where U means the universal enveloping, Lie(*) is the DG Lie algebra dual, and Ker means kernel of the map, made surjective by replacing Lie(S) with a quasi-isomophic DG Lie algebra.
In particular, the theory of Golod maps is related to the following observation about DG-algebras: if DG-algebras U and C are bar-dual to each other, then H(U) is a free algebra if and only if all the Massey operations, including the product, in H(C) is zero. This is indeed true and easy to see.
Furthermore, let f: R\to S be a surjective map of commutative quadratic algebras with f^*: Ext_S(k,k) \to Ext_R(k,k) also surjective. Then f is a sub-Golod map in your terminology [i.e., R_1=S_1 and the Hopf algebra kernel of the diagonal part S^! \to R^! of Yoneda algebras map is free] iff all the Massey products of elements of Tor_{n,n-1}^R(k,S), n = 1,2,... are zero.
Best regards,
Leonid.
From: Leonid Positselski
Subject: morphisms of graded algebras
Dear Jan-Erik,
Several weeks ago you asked me a question about generalization of results from my paper in K-theory preprint archive to non-Koszul commutative algebras. Now it seems that I understand these things somewhat better.
Namely, if f: R\to S is a map of commutative graded algebras, then the DG-algebra S\otimes_R k (the derived tensor product) is bar-dual to the DG-algebra UKer(Lie(S)\to Lie(R)), where U means the universal enveloping, Lie(*) is the DG Lie algebra dual, and Ker means kernel of the map, made surjective by replacing Lie(S) with a quasi-isomophic DG Lie algebra.
In particular, the theory of Golod maps is related to the following observation about DG-algebras: if DG-algebras U and C are bar-dual to each other, then H(U) is a free algebra if and only if all the Massey operations, including the product, in H(C) is zero. This is indeed true and easy to see.
Furthermore, let f: R\to S be a surjective map of commutative quadratic algebras with f^*: Ext_S(k,k) \to Ext_R(k,k) also surjective. Then f is a sub-Golod map in your terminology [i.e., R_1=S_1 and the Hopf algebra kernel of the diagonal part S^! \to R^! of Yoneda algebras map is free] iff all the Massey products of elements of Tor_{n,n-1}^R(k,S), n = 1,2,... are zero.
Best regards,
Leonid.
