In this article we introduce the notion of \emph{multi-Koszul algebra} for the case of a nonnegatively graded connected algebra with a finite number of generators of degree 1 and with a finite number of relations, as a generalization of the notion of (generalized) Koszul algebras defined by R. Berger for homogeneous algebras, which were in turn an extension of Koszul algebras introduced by S. Priddy. Our definition is in some sense as closest as possible to the one given in the homogeneous case. Indeed, we give an equivalent description of the new definition in terms of the \textrm{Tor} (or \textrm{Ext}) groups, similar to the existing one for homogeneous algebras, and also a complete characterization of the multi-Koszul property, which derives from the study of some associated homogeneous algebras, providing a very strong link between the new definition and the generalized Koszul property for the associated homogeneous algebras mentioned before. We further obtain an explicit description of the Yoneda algebra of a multi-Koszul algebra. As a consequence, we get that the Yoneda algebra of a multi-Koszul algebra is generated in degrees 1 and 2, so a $\mathcal{K}_{2}$ algebra in the sense of T. Cassidy and B. Shelton. We also exhibit several examples and we provide a minimal graded projective resolution of the algebra $A$ considered as an $A$-bimodule, which may be used to compute the Hochschild (co)homology groups. Finally, we find necessary and sufficient conditions on some (fixed) sequences of vector subspaces of the tensor powers of the base space $V$ to obtain in this case the multi-Koszul property in the case we have relations in only two degrees.
证明了图不同于A $\_1$和A $\_2$的预投射代数的Koszul演算在任何（余）同调度$p>2$中为零.此外，它的（高）上同调演算同构作为一个双模的（高）同调演算，通过交换度$p$和$2-p$，我们证明了一个广义版本的2-Calabi-Yau属性。对于ADE Dynkin图，预投射代数不是Koszul，也不是Ginzburg定义意义下的Calabi-Yau，但它们满足我们推广的Calabi-Yau性质，我们称它们是维数为$2$的Koszul复形Calabi-Yau（Kc-Calabi-Yau）.对于任意维数的Kc-Calabi-Yau（二次）代数，我们证明了一个Poincaré Van den Bergh对偶定理。我们显式地计算了ADE Dynkin图的预投射代数的Koszul演算。
