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Linear free resolutions and minimal multiplicity. (English) Zbl 0531.13015
If M is a graded module over a polynomial ring with all generators in the same degree, say 0, then the linear part of the resolution of M consists of the syzygies of degree 1, the syzygies on these of degree 2, etc. Linear parts of resolutions are far easier to study than the resolutions themselves, and there are interesting criteria, related to the classical notions of ”Castelnuovo regularity” in algebraic geometry, for a resolution to be equal to its linear part. The first goal of the paper under review is to treat these matters.
Probably the most familiar example of an ideal with linear resolution is that of the \(p\times p\) minors of a \(p\times q\) matrix with linear entries, in the case where the \(p\times p\) minors have generic depth, \(q- p+1\). In the case \(p=2\) these examples appear in geometry as rings of minimal multiplicity (or projective varieties of minimal degree). In fact the classification theorem of Bertini implies that varieties of minimal degree all are Cohen-Macaulay, and their ideals linear resolutions. The second purpose of this article is to give a direct, ring-theoretic treatment of the connection between minimal degree, and these other, homological notions.

MSC:
13H15 Multiplicity theory and related topics
14M12 Determinantal varieties
13D25 Complexes (MSC2000)
13C10 Projective and free modules and ideals in commutative rings
14B15 Local cohomology and algebraic geometry
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
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