A Simple Note on the Yoneda (CO)Algebra of a Monomial Algebra

If A = TV/hRi is a monomial K-algebra, then it is well known that ${Tor}_p^A$ (K,K) is isomorphic to the space V^(p−1) of (Anick) (p − 1)-chains for p ≥ 1. We show that the next result directly follows from well-established theorems on A_∞-algebras, without computations: There is an A_∞-coalgebra model on ${Tor}_{\bullet}^A$ (K,K) such that, for n ≥ 3 and c ∈ V^(p), ∆_n(c) is a linear combination of c₁⨂. . .⨂c_n, where c_i ∈ V ^(pi), p₁ + . . . + p_n = p − 1 and c₁ . . . c_n = c. The proof essentially follows from noticing that the Merkulov procedure is compatible with an extra grading over a suitable category. By a simple argument based on a result by Keller, we immediately deduce that some of these coefficients are ±1.