Spectral Sequences for the Classification of Extensions of Hopf Algebras

We construct spectral sequences which provide a way to compute the cohomology theory that classifies extensions of graded connected Hopf algebras over a commutative ring as described by William M. Singer. Specifically, for (A,B) an abelian matched pair of graded connected R-Hopf algebras, we construct a pair of spectral sequences relating H∗(B,A) to Ext∗,∗ B (R,Cotor ∗,∗ A (R,R)). To illustrate these spectral sequences, we examine the special case of B a monogenic graded connected Hopf algebra and also analyze an extension of Hopf algebras given by James P. Lin. Introduction. The category of graded connected R-Hopf algebras has some striking similarities to the category of groups. For instance, there is a version of the Hochschild-Serre spectral sequence for computing the cohomology, Ext∗,∗ H (R,R), of an extension of Hopf algebras A → H → B in terms of the cohomology of A and B [1, XVI.6]. In the case R = Fp, there is also a Steenrod algebra action, of sorts, on the cohomology of such Hopf algebras which behaves well with respect to this spectral sequence [9, §11]. There is even a classification theory for extensions of Hopf algebras, but unlike the group case, the classification is not in terms of the usual cohomology. In 1962 V.K.A.M. Gugenheim described a theory of central extensions of graded connected R-Hopf algebras in analogy with the usual extension theory of groups [4], and this was generalized to arbitrary extensions of graded connected R-Hopf algebras by William M. Singer in 1972 [16]. An analogous theory of extensions of ungraded Hopf algebras was given by M. Feth in 1982 [3] (in German) or [6] (containing an English summary), and a theory of extensions of Hopf monoids in a symmetric monoidal category, which generalizes both Singer and Feth’s work, was given by Pachuashvili in 1985 [14] (in Russian) and [15] (in English). These extension theories all enjoy a very strong self duality, which makes them beautiful, but difficult to calculate. In fact, the author’s work in 1995 [5] is the only computation of which we are aware, and illustrates the difficulties of working in even the simplest cases. In this paper we describe spectral sequences which provide a means of computing the cohomology groups H∗(B,A) described by Singer [16] in terms of the ordinary cohomology of B (as an algebra) and the homology of A (as a coalgebra). This extends the work in [5] and can be found in §1. The main results are theorem 1.1, which constructs a spectral sequence converging to H∗(B,A), and theorem 1.2, which provides a means of computing the E1 term of this spectral sequence in terms of more familiar objects : Theorems 1.1 and 1.2. If (A,B) is an abelian matched pair of graded connected R-Hopf algebras and r is an integer, then there is a first quadrant spectral sequence with rĒ s,t 2 ∼= { Ext B (R,Cotor t,r A (R,R)) if r > 0 0 otherwise which converges to the E1 term of a spectral sequence which converges to H ∗(B,A) : rĒ s,t 2 ⇒E r,s+t−r 1 ⇒H(B,A). Here Cotor A (R,R) is considered an R-module concentrated in degree 0, and so has trivial B-action. Section 1 also gives a description of the differentials in the spectral sequence E∗ in terms of the input to the spectral sequences rĒ∗, and a description of a special case for which the spectral sequences rĒ∗ collapse : 1991 Mathematics Subject Classification. Primary 16W30 57T05 ; Secondary 18G60.