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Given a graded ring $A$ and a homogeneous ideal $I$, the ideal is said to beof linear type if the Rees algebra of $I$ is isomorphic to the symmetricalgebra of $I$. We examine the relations between different notions of sequencesalready known in literature which form ideals of linear type. We observe howthe vanishing of $y$-regularity of the Rees algebra of an ideal is connected tothe ideal being generated by some specific sequences. As an application ofthis, we discuss Koszulness and Cohen-Macaulayness of the diagonal subalgebrasof Rees algebras of ideals generated by some of these sequences. We alsomention Macaulay2 algorithms to check if an ideal is generated by a$d$-sequence, c-sequence, weak relative-regular sequence and for computingbi-regularities.

A note on sequences and y-regularity of Rees algebra