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Monomial ideals whose powers have a linear resolution. (English) Zbl 1091.13013
A graded module \(M\) is said to have a linear resolution if all entries in the matrices representing the differentials in a graded minimal free resolution of \(M\) are linear forms. If an ideal \(I\) has a linear resolution, then necessarily all generators of \(I\) have the same degree \(t.\) In that case one says that \(I\) has a \(t\)-linear resolution. In general, powers of (monomial) ideals with linear resolutions need not to have a linear resolution. In this paper the authors show that if \(I\subset k[x_1,\dots,x_n]\) is a monomial ideal with \(2\)-linear resolution, then each power of \(I\) has a linear resolution.

13D02 Syzygies, resolutions, complexes and commutative rings
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
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