FUNDAMENTA MATHEMATICAE 183 (2004) Homological computations in the universal Steenrod algebra by A. Ciampella and L. A. Lomonaco (Napoli) Abstract. We study the (bigraded) homology of the universal Steenrod algebra Q over the prime field F 2 , and we compute the groups H s,s (Q), s ≥ 0, using some ideas and techniques of Koszul algebras developed by S. Priddy in [5], although we presently do not know whether or not Q is a Koszul algebra. We also provide an explicit formula for the coalgebra structure of the diagonal homology D * (Q)=  s≥0 H s,s (Q) and show that D * (Q) is isomorphic to the coalgebra of invariants Γ introduced by W. Singer in [6]. Introduction. It is a basic problem of homological algebra to compute the (co)homology of various augmented algebras. The purpose of this paper is to compute the diagonal homology D ∗ (Q)=  s≥0 H s,s (Q) of the univer- sal Steenrod algebra Q and provide a description of D ∗ (Q) as a coalgebra in terms of invariant theory. Q is a graded algebra arising from algebraic topol- ogy, for it is the algebra of cohomology operations in the category C (2, ∞) of H ∞ -ring spectra. It is an interesting object as it contains Λ, the lambda algebra introduced in [1], as a subalgebra, and the Steenrod algebra arises as a quotient of Q. Hence it would be nice to understand the cohomology algebra H ∗ (Q) = Ext Q (F 2 , F 2 ) and the homology H ∗ (Q) = Tor Q (F 2 , F 2 ), but these computational problems are presently unsolved. What makes such computation hard is the fact that Q is by no means locally finite and most of the methods developed by Priddy ([5]) do not apply. The second author succeeded in computing the diagonal cohomology of Q in [4]. In Section 1 of the present paper a description of the homology groups H s,s (Q), i.e. the groups Tor Q s,s (F 2 , F 2 ), s ∈ N, is provided, with an explicit formula for the coalgebra structure map of D ∗ (Q). In [5] S. Priddy works under the hypothesis of local finiteness for the algebras involved, but the idea we borrow from his work in the proof of Theorem 1 does not depend 2000 Mathematics Subject Classification : 55S10, 18G15, 55T15. Key words and phrases : Koszul algebras, homology of algebras, universal Steenrod algebra. [245]