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We will define the Alexander duality for strongly stable ideals. More precisely, for a strongly stable ideal $I subset Bbbk[x_1, ldots, x_n]$ with ${rm deg}(mathsf{m}) le d$ for all $mathsf{m} in G(I)$, its dual $I^* subset Bbbk[y_1, ldots, y_d]$ is a strongly stable ideal with ${rm deg}(mathsf{m}) le n$ for all $mathsf{m} in G(I^*)$. This duality has been constructed by Fl$o$ystad et al. in a different manner, so we emphasis applications here. For example, we will describe the Hilbert serieses of the local cohomologies $H_mathfrak{m}^i(S/I)$ using the irreducible decomposition of $I$ (through the Betti numbers of $I^*$).
For a pair $(P,Q)$ of finite posets the generators of the ideal $L(P,Q)$ correspond bijectively to the isotone maps from $P$ to $Q$. In this note we determine all pairs $(P,Q)$ for which the Alexander dual of $L(P,Q)$ coincides with $L(Q,P)$, up to a switch of the indices.
We study the extremal Betti numbers of the class of $t$--spread strongly stable ideals. More precisely, we determine the maximal number of admissible extremal Betti numbers for such ideals, and thereby we generalize the known results for $tin {1,2}$.
Let $K$ be a field and let $S=K[x_1,dots,x_n]$ be a standard polynomial ring over a field $K$. We characterize the extremal Betti numbers, values as well positions, of a $t$-spread strongly stable ideal of $S$. Our approach is constructive. Indeed, g
Let $X$ be a set of points whose coordinates are known with limited accuracy; our aim is to give a characterization of the vanishing ideal $I(X)$ independent of the data uncertainty. We present a method to compute a polynomial basis $B$ of $I(X)$ whi
We define the notion of a power stable ideal in a polynomial ring $ R[X]$ over an integral domain $ R $. It is proved that a maximal ideal $chi$ $ M $ in $ R[X]$ is power stable if and only if $ P^t $ is $ P$- primary for all $ tgeq 1 $ for the prime