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

  • E. Herscovich Inst. Fourier, Univ. Grenoble, Alpes., France
Keywords: homological algebra, dg (co)algebras, $A__{∞}$-(co)algebras

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 $\pm 1$.

References

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M. J. Bardzell, The alternating syzygy behavior of monomial algebras, J. Algebra, 188, № 1, 69 – 89 (1997), https://doi.org/10.1006/jabr.1996.6813

E. Herscovich, Applications of one-point extensions to compute the $A_{infty}$ -(co)module structure of several Ext (resp., Tor) groups (2016), 20 p., available at https://www-fourier.ujf-grenoble.fr/eherscov/Articles/Applications-of-one-pointextensions.pdf .

E. Herscovich, On the Merkulov construction of $Ainfty$ -(co)algebras (2017), 8 p., available at https://www-fourier.ujfgrenoble.fr/ eherscov/Articles/Applications-of-one-point-extensions.pdf.

E. Herscovich, Using torsion theory to compute the algebraic structure of Hochschild (co)homology, Homology, Homotopy and Appl., 20, № 1, 117 – 139 (2018), https://doi.org/10.4310/HHA.2018.v20.n1.a8

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P. Tamaroff, Minimal models for monomial algebras (2018), 28 p., available at https://arxiv.org/abs/1804.01435

Published
22.02.2021
How to Cite
Herscovich E. “A Simple Note on the Yoneda (co)algebra of a Monomial Algebra”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 2, Feb. 2021, pp. 275 -77, doi:10.37863/umzh.v73i2.6040.
Section
Short communications