We study the coordinate ring of an $L$-convex polyomino, determine its regularity in terms maximal number rooks that can be placed polyomino. also characterize Gorenstein polyominoes and those which are on punctured spectrum, compute Cohen–Macaulay type any polyomino rectangles covering it. Though main results algebraic nature, all proofs combinatorial.

For a partition λ of n ∈ N , let I Sp be the ideal R = K [ x 1 … ] generated by all Specht polynomials shape . In previous paper, second author showed that if / is Cohen-Macaulay, then either ( − d ) or and converse true char 0 this we compute Hilbert series for Hence, get Castelnuovo-Mumford regularity when it Cohen-Macaulay. particular, has + 2 -linear resolution in Cohen–Macaulay case.

In this article, we delve into the properties possessed by algebras, which we have termed seeds, that map to big Cohen-Macaulay algebras. We will show that over a complete local domain of positive characteristic any two big Cohen-Macaulay algebras map to a common big Cohen-Macaulay algebra. We will also strengthen Hochster and Huneke’s “weakly functorial” existence result for big Cohen-Macaulay...