On the Yoneda algebras of piecewise-Koszul algebras

Let $A = \bigoplus_{i \geq 0} A_i$ be a piecewise-Koszul algebra with cohomology degree function $\delta^d_p$ such that $d \gt p \geqslant 2$ and $E(A) = \bigoplus_{i \geq 0} \mathrm{Ext \,}^i_A (A_0, A_0)$ its Yoneda algebra. We introduce a new grading on $E(A)$:\begin{align}\widehat{E}(A) & = \bigoplus_{i\geq 0} \widehat{E}^{\;i}(A) \; \mathrm{with} \; \widehat{E}^{\;i}(A) \\& = \begin{cases}\mathrm{Ext \,}^0_A(A_0 ,A_0), & \hbox{$i=0$;} \\(\mathrm{Ext \,}^1_A(A_0,A_0) \oplus \mathrm{Ext \,}^p_A(A_0, A_0))^i, & \hbox{$i\geq1$.}\end{cases}\end{align}We use “$\widehat{E}(A)$” to replace “$E(A)$” to suggest the new grading. In the paper, we mainly prove that $\widehat{E}(A)$ is a quadratic algebra and $\widehat{E}(M)$ is a quadratic module over $\widehat{E}(A)$, where $M$ is a piecewise-Koszul $A$-module with the same function $\delta^d_p$. Moreover, we provide a concrete example to show that $\widehat{E}(A)$ is not a Koszul algebra in general, which is different from the Koszul and d-Koszul cases.\[E=mc^2\]

Keywords

Koszul algebra, piecewise-Koszul algebra, Yoneda algebra

2010 Mathematics Subject Classification

Primary 16S37, 16W50. Secondary 16E30, 16E40.

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Published 13 April 2015