The Lefschetz properties.

*(English)*Zbl 1284.13001
Lecture Notes in Mathematics 2080. Berlin: Springer (ISBN 978-3-642-38205-5/pbk; 978-3-642-38206-2/ebook). xix, 250 p. (2013).

Let \(A = \bigoplus_{0 \leq i \leq c} A_i\) be a graded Artinian algebra over a polynomial ring \(R\). We say that \(A\) has the Weak Lefschetz Property (WLP) if there exists a linear form \(L\) such that the multiplication map \(\times L : A_i \rightarrow A_{i+1}\) has maximal rank for all \(i\). We say that \(A\) has the Strong Lefschetz Property (SLP) if there exists a linear form \(L\) such that the multiplication map \(\times L^d : A_i \rightarrow A_{i+d}\) has maximal rank for all \(i\) and all \(d\). In both cases, \(L\) is called a (weak or strong) Lefschetz element. These Lefschetz properties have connections to an amazing range of properties and topics in many different areas of mathematics, and this beautiful book does a great job of describing many of these connections. As an interesting side-note, the reviewer and U. Nagel recently wrote an expository paper giving an overview of the Lefschetz properties and related topics [J. Migliore and U. Nagel, J. Commut. Algebra 5, No. 3, 329–358 (2013; Zbl 1285.13002)]. The authors of the book under review state that “it is amazing that there is little overlapping between the topics dealt with in their survey and those in this monograph. The fact is an evidence for the diversity of the Lefschetz properties.”

We give a rough (and by no means complete) description of the contents of the book.

\(\bullet\) The first chapter of the book gives a short treatment of poset theory. The authors discuss the Dilworth number, the Sperner property, and related topics.

\(\bullet\) The second chapter surveys many important results in the theory of local rings. The focus is on the case of Artinian rings. In addition to basic definitions and first results, the authors talk about complete local rings, Matlis duality, Dilworth and Rees numbers, monomial Artinian rings, the Sperner property, the Dilworth lattice, \(\mathfrak m\)-full ideals, Artinian Gorenstein rings, complete intersections, and Hilbert functions.

\(\bullet\) In the third chapter the authors finally get to the definitions of the Weak and Strong Lefschetz Properties. They discuss Lie algebras, the Clebsch-Gordan theorem, \(\mathfrak s l_2\), the Lefschetz theory in low codimensions, Jordan decompositions and tensor products, the case of almost complete intersections, the case of Gorenstein algebras (which is at the center of some important open problems in the study of Lefschetz properties) and the connection to Hessians.

\(\bullet\) The fourth chapter describes what is known about complete intersections and the SLP. Among other topics they mention the tool of finite free extensions of a graded \(k\)-algebra and power sums of consecutive degrees.

\(\bullet\) Chapter 5 looks at a generalization of the notion of a Lefschetz element, namely weak and strong Rees elements, and closely related topics. In particular, the authors show how to obtain Gorenstein algebras that have the WLP but not the SLP.

\(\bullet\) In Chapter 6 the authors describe a generalization of the SLP and WLP that was introduced by M. Boij, A. Iarrobino and the last author in 1995. This is the notion of \(k\)-SLP and \(k\)-WLP. It could also be interpreted as a generalization of the notion of depth, replacing regular element by Lefschetz element. They give a number of results in this setting that generalized results for the WLP and SLP, for instance results of Weibe, results on the Hilbert function of algebras with \(k\)-WLP and \(k\)-SLP, results on revlex ideals and generic initial ideals, and results on graded Betti numbers.

\(\bullet\) Chapter 7 reviews the Hard Lefschetz theorem for the cohomology rings of compact Kähler manifolds and related topics, including projective space bundles, homogeneous spaces, toric varieties, and \(O\)-sequences. It was in this setting that Richard Stanley gave the first proof of one of the main theorems in the theory of the WLP and SLP, namely that an Artinian monomial complete intersection has the SLP.

\(\bullet\) Chapter 8 reviews invariant theory as it relates to the Lefschetz properties. It discusses reflection groups and coinvariant algebras.

\(\bullet\) Chapter 9 gives the connections between the SLP and Schur-Weyl duality, giving many examples.

\(\bullet\) In an appendix, the authors discuss the WLP of ternary monomial complete intersections in positive characteristic. Indeed, some very interesting connections to combinatorics have been discovered relatively recently. The authors cite [J. Li and F. Zanello, Discrete Math. 310, No. 24, 3558–3570 (2010; Zbl 1202.13013)], as well as [C. P. Chen et al., J. Commut. Algebra 3, No. 4, 459–489 (2011; Zbl 1236.05212)], but it is necessary to add D. Cook and U. Nagel to the list (their paper [“Enumerations of lozenge tilings, lattice paths, and perfect matchings and the weak Lefschetz property”, arXiv:1305.1314] appeared after the publication of this book).

We give a rough (and by no means complete) description of the contents of the book.

\(\bullet\) The first chapter of the book gives a short treatment of poset theory. The authors discuss the Dilworth number, the Sperner property, and related topics.

\(\bullet\) The second chapter surveys many important results in the theory of local rings. The focus is on the case of Artinian rings. In addition to basic definitions and first results, the authors talk about complete local rings, Matlis duality, Dilworth and Rees numbers, monomial Artinian rings, the Sperner property, the Dilworth lattice, \(\mathfrak m\)-full ideals, Artinian Gorenstein rings, complete intersections, and Hilbert functions.

\(\bullet\) In the third chapter the authors finally get to the definitions of the Weak and Strong Lefschetz Properties. They discuss Lie algebras, the Clebsch-Gordan theorem, \(\mathfrak s l_2\), the Lefschetz theory in low codimensions, Jordan decompositions and tensor products, the case of almost complete intersections, the case of Gorenstein algebras (which is at the center of some important open problems in the study of Lefschetz properties) and the connection to Hessians.

\(\bullet\) The fourth chapter describes what is known about complete intersections and the SLP. Among other topics they mention the tool of finite free extensions of a graded \(k\)-algebra and power sums of consecutive degrees.

\(\bullet\) Chapter 5 looks at a generalization of the notion of a Lefschetz element, namely weak and strong Rees elements, and closely related topics. In particular, the authors show how to obtain Gorenstein algebras that have the WLP but not the SLP.

\(\bullet\) In Chapter 6 the authors describe a generalization of the SLP and WLP that was introduced by M. Boij, A. Iarrobino and the last author in 1995. This is the notion of \(k\)-SLP and \(k\)-WLP. It could also be interpreted as a generalization of the notion of depth, replacing regular element by Lefschetz element. They give a number of results in this setting that generalized results for the WLP and SLP, for instance results of Weibe, results on the Hilbert function of algebras with \(k\)-WLP and \(k\)-SLP, results on revlex ideals and generic initial ideals, and results on graded Betti numbers.

\(\bullet\) Chapter 7 reviews the Hard Lefschetz theorem for the cohomology rings of compact Kähler manifolds and related topics, including projective space bundles, homogeneous spaces, toric varieties, and \(O\)-sequences. It was in this setting that Richard Stanley gave the first proof of one of the main theorems in the theory of the WLP and SLP, namely that an Artinian monomial complete intersection has the SLP.

\(\bullet\) Chapter 8 reviews invariant theory as it relates to the Lefschetz properties. It discusses reflection groups and coinvariant algebras.

\(\bullet\) Chapter 9 gives the connections between the SLP and Schur-Weyl duality, giving many examples.

\(\bullet\) In an appendix, the authors discuss the WLP of ternary monomial complete intersections in positive characteristic. Indeed, some very interesting connections to combinatorics have been discovered relatively recently. The authors cite [J. Li and F. Zanello, Discrete Math. 310, No. 24, 3558–3570 (2010; Zbl 1202.13013)], as well as [C. P. Chen et al., J. Commut. Algebra 3, No. 4, 459–489 (2011; Zbl 1236.05212)], but it is necessary to add D. Cook and U. Nagel to the list (their paper [“Enumerations of lozenge tilings, lattice paths, and perfect matchings and the weak Lefschetz property”, arXiv:1305.1314] appeared after the publication of this book).

Reviewer: Juan C. Migliore (Notre Dame)

##### MSC:

13-02 | Research exposition (monographs, survey articles) pertaining to commutative algebra |

13A02 | Graded rings |

13A50 | Actions of groups on commutative rings; invariant theory |

06A07 | Combinatorics of partially ordered sets |

06A11 | Algebraic aspects of posets |

14M15 | Grassmannians, Schubert varieties, flag manifolds |

14F99 | (Co)homology theory in algebraic geometry |

14M10 | Complete intersections |

14L30 | Group actions on varieties or schemes (quotients) |

17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |