A simple note on the Yoneda (co)algebra of a monomial algebra

Abstract

UDC 512.7

If $A = TV/\langle R\rangle$ is a monomial $K$-algebra, it is well-known that $\operatorname{Tor}_{p}^{A}(K,K)$ is isomorphic to the space $V^{(p-1)}$ of (Anick) $(p-1)$-chains for $p \geq 1$.
The goal of this short note is to show that the next result follows directly from well-established theorems on $A_{\infty}$-algebras, without computations: there is an $A_{\infty}$-coalgebra model on $\operatorname{Tor}_{\bullet}^{A}(K,K)$ satisfying that, for $n \geq 3$ and $c \in V^{(p)}$, $\Delta_{n}(c)$ is a linear combination of $c_{1} \otimes \ldots \otimes c_{n}$, where $c_{i} \in V^{(p_{i})}$, $p_{1} + \ldots +p_{n} = p - 1$ and $c_{1} \ldots c_{n} = c$.
The proof follows essentially 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 $\pm 1$.