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When $S$ is a polynomial ring or more generally a standard graded algebra over a field $K$, with homogeneous maximal ideal $𝔪$, it is known that for an ideal $I$ of $S$, the regularity of powers of $I$ becomes eventually a linear function, i.e., $\mathrm{reg}\left(I^m\right)=dm+e$ for $m\gg 0$ and some integers $d$, $e$. This motivates writing $\mathrm{reg}\left(I^m\right)=dm+e_m$ for every $m\ge 0$. The sequence $e_m$, called the of the ideal $I$, is the subject of much research and its nature is still widely unexplored. We know that $e_m$ is eventually constant. In this article, after proving various...

We study finitely generated bigraded Buchsbaum modules over a standard bigraded polynomial ring with respect to one of the irrelevant bigraded ideals. The regularity and the Hilbert function of graded components of local cohomology at the finiteness dimension level are considered.

Let $R$ be an integral domain with quotient field $K$ and $f\left(x\right)$ a polynomial of positive degree in $K\left[x\right]$. In this paper we develop a method for studying almost principal uppers to zero ideals. More precisely, we prove that uppers to zero divisorial ideals of the form $I=f\left(x\right)K\left[x\right]\cap R\left[x\right]$ are almost principal in the following two cases: – $J$, the ideal generated by the leading coefficients of $I$, satisfies $J^{-1}=R$. – $I^{-1}$ as the $R\left[x\right]$-submodule of $K\left(x\right)$ is of finite type. Furthermore we prove that for $I=f\left(x\right)K\left[x\right]\cap R\left[x\right]$ we have: – $I^{-1}\cap K\left[x\right]=\left(I{:}_{K\left(x\right)}I\right)$. – If there exists $p/q\in I^{-1}-K\left[x\right]$, then $(q,f)\ne 1$...