For a monomial ideal I, we consider the ith homological shift ideal of I, denoted by \({\text {HS}}_i(I)\), that is, the ideal generated by the ith multigraded shifts of I. Some algebraic properties of this ideal are studied. It is shown that for any monomial ideal I and any monomial prime ideal P, \({\text {HS}}_i(I(P))\subseteq {\text {HS}}_i(I)(P)\) for all i, where I(P) is the monomial localization of I. In particular, we consider the homological shift ideal of some families of monomial ideals with linear quotients. For any \({{\mathbf {c}}}\)-bounded principal Borel ideal I and for the edge ideal of complement of any path graph, it is proved that \({\text {HS}}_i(I)\) has linear quotients for all i. As an example of \({{\mathbf {c}}}\)-bounded principal Borel ideals, Veronese type ideals are considered and it is shown that the homological shift ideal of these ideals are polymatroidal. This implies that for any polymatroidal ideal which satisfies the strong exchange property, \({\text {HS}}_j(I)\) is again a polymatroidal ideal for all j. Moreover, for any edge ideal with linear resolution, the ideal \({\text {HS}}_j(I)\) is characterized and it is shown that \({\text {HS}}_1(I)\) has linear quotients.

对于一个单项式理想我，我们考虑我的第同源移理想我，记\（{\文本{HS}} _ I（I）\） ，即，所产生的理想通过我的第multigraded移位余。研究了该理想的一些代数性质。结果表明，对于任何单项式理想I和任何单项式素理想P，\（{\ text {HS}} _ i（I（P））\ subseteq {\ text {HS}} _ i（I）（P）\）对于所有i，其中I（P）是I的单项式局部化。特别地，我们考虑一些带有线性商的单项式理想族的同系转移理想。对于任何\（{{\ mathbf {c}}} \）有界的Borel理想I和任何路径图的补码的边缘理想，都证明\（{\ text {HS}} _ i（I） \）对所有i都有线性商。作为\（{{\ mathbf {c}}} \）界上的主要Borel理想的示例，考虑了Veronese型理想，并且证明了这些理想的同质移位理想是多拟脉动的。这意味着对于任何满足强交换特性的多拟理想，\（{\ text {HS}} _ j（I）\）仍然是所有j的拟理想。此外，对于具有线性分辨率任何边缘理想，理想\（{\文本{HS}} _Ĵ（I）\）的特征在于，它表明\（{\文本{HS}} _ 1（I）\）具有线性商。