# zbMATH — the first resource for mathematics

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.

##### MSC:
 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)
##### Keywords:
monomial ideals; $$t$$-linear resolution; powers of ideals
Full Text: