# Lefschetz Properties and Artinian Algebras (16w5114)

Arriving in Banff, Alberta Sunday, March 13 and departing Friday March 18, 2016

## Organizers

Sara Faridi (Dalhousie University)

(Northeastern University)

Rosa-Maria Miro-Roig (University of Barcelona)

(University of Göttingen)

Junzo Watanabe (Professor, Tokai University, Japan)

## Objectives

The theory of Lefschetz properties for Artinian algebras was motivated by the Lefschetz theory for projective manifolds, begun by S. Lefschetz, and well established by the late 1950's. It is possible to look at many graded Artinian rings as cohomology rings of an algebraic variety or manifold. In the last decade several authors have attempted to reconstruct the theory of Artinian rings from this viewpoint using an algebraization of the Hard Lefschetz theorem. This topic has the potential to draw the attention of commutative algebraists, algebraic topologists, algebraic geometers, combinatorialists and representation theorists. It is in this direction that a team of six wrote the Springer LNM 2080 "The Lefschetz Properties,'' that appeared in 2013 [5].

It is timely to organize a small workshop at BIRS with focus the Lefschetz properties of Artinian rings. We believe new researchers will be interested in this topic, and, since the study of Lefschetz properties of rings has a short history, it will be accessible to young people. This subject matter has connections to many branches of mathematics, and in particular, concerns as a central object of study in it are Gorenstein rings, which are of strong interest in commutative algebra, topology, and combinatorics. Finally, this area has many open problems that we believe may become accessible: for example we have conjectured that, all graded complete intersections in characteristic zero have the strong Lefschetz property. This is not even known for rings of coinvariants of complex reflection groups. We plan to focus on three problems which we explain next.

(1) For an Artinian algebra with a specified Hilbert function what families of Jordan type (block) partitions are determined by multiplication by the nilpotent elements? This concerns compatible Jordan types of multiplication by two commuting nilpotent matrices, and it generalizes the question of whether an Artinian ring has the weak or strong Lefschetz property. A broad variety of methods have gone into the partial results already known on these topics [1, 6, 7], [Chapter 5,6][5]. In particular do all graded complete intersections have the weak Lefschetz property? Do all codimension 3 graded Artinian Gorenstein rings have the weak Lefschetz property?

(2) Can the theory of Gordan and Noether on algebraic forms be generalized to the case with more than five variables? The theory of Gordan and Noether on the forms whose Hessian vanishes identically (see [4]) has been studied by J. Watanabe, and the theory of Laplace equations has recently been explored by Miró-Roig et al in [11]. Both of them deal with differential equations. Indeed, the latter establishes a close relationship between two a priori unrelated problems: the existence of artinian ideals $I \subset k[x_0, \cdots , x_n]$ which fail the Weak Lefschetz Property; and the existence of (smooth) projective varieties $X \subset \PP^N$ satisfying at least one Laplace equation of order $s \ge 2$. This work is extended to the Strong Lefschetz property by DiGennaro et al in [3]. Left open is an interesting conjecture about monomial ideals generated by cubics and failing the Weak Lefschetz property [11]. These and related questions are our second topic.

(3) There is a geometric interpretation of flat extensions of Gorenstein algebras as cohomology rings of manifolds in fibrations that are totally homologous to zero. The following is a typical example to which the theorem on flat extensions (Theorem 4.10 in [5]) applies to prove inductively that $S(n)$ has the strong Lefschetz property. $0 \rightarrow K[z]/(z^n) \rightarrow S(n) \rightarrow S(n-1) \rightarrow 0$ Here $S(n)$ is the algebra of coinvariants $\mathbb{F}[x_1, x_2, \ldots, x_n]/(e_1, e_2, \ldots, e_n)$ of the symmetric group $S_n$ and $e_1, e_2, \ldots, e_n$ denote the elementary symmetric polynomials. This sequence corresponds to the cohomology rings of the fibration of homogeneous spaces: $\CC \rightarrow U(n-1)/T^{n-1} \hookrightarrow U(n)/T^n \twoheadrightarrow \CC \PP ^{n-1} \rightarrow \CC.$ Our third problem is to expand this kind of correspondence. For example, what happens if $S(n)$ is replaced by the algebra of coinvariants of other Weyl groups? reflection groups? This has been studied geometrically in [10] for Weyl groups.

On the algebraic side there is a need to generalize the Flat Extension Theorem to the situation $C \hookrightarrow A \twoheadrightarrow B,$ where A is not flat as a C-module: one assumes that any two of the algebras have the Lefschetz property and asks if the third also has it. Also we ask if the results of [10] extend to the complex case.

Why about 20 participants are appropriate for this workshop:

The theory of the Lefschetz properties of Artinian rings has only recently, in the last ten years, begun drawing increasing attention of researchers, so it does not have a long history. We have to say there are not many researchers and this has not yet been recognized as a main branch of algebra, although it is growing. Thus we think a workshop with a limited number of participants is most appropriate. In 2012 Prof. Watanabe organized, with input from the other organizers, a very small workshop with 14 participants in Hawaii at Tokai University college campus. It was a success and resulted in new collaborations and several papers ([12], [15]).
Several of the organizers, led by Junzo Watanabe and Larry Smith, are planning a second small workshop at Gottingen in March, 2015. The proposed BIRS workshop will continue and broaden the participants of this series and facilitate a deepening of this work.

Dedication, and some further history (by Prof. J. Watanabe)

It was David Rees who in the early 80's asked for which ideals $I$, say in a regular local ring, is it true that $\mu(J) \leq \mu(I)$ for all ideals $J$ containing $I$? Here $\mu$ denotes the number of generators of an ideal. Perhaps, with this problem in mind, Rees defined the concept of m-full ideals.'' Soon after that it was recognized that this was a combinatorial problem and that the Sperner theory in combinatorics can be directly applied to answer some cases of Rees' problem. It is possible to consider that the weak Lefschetz property of graded Artinian rings is a generalization of the Sperner property of posets with a rank function. (This was a motivation for [5]). The number of generators of ideals is a long--studied topic in the theory of commutative rings (see e.g., [13]), and its natural counterpart in algebraic combinatorics is the number of elements in an antichain. We regard it as very fortunate that the Sperner theory in combinatorics was available for us.

Prof. Junzo Watanabe met David Rees in 1983 and was strongly influenced by David Rees in his study of m-fullness of ideals and Lefschetz properties of commutative algebras. Without the idea of Rees, the monograph [5] would not have existed. We will dedicate this workshop in honor of David Rees.

#### Bibliography

1. H. Brenner, A. Kaid, A note on the weak Lefschetz property of monomial complete intersections in positive characteristic, Collect. Math. 62 (2011), no.1, 85--93.

2. D. Cook, U. Nagel, Hyperplane sections and the subtlety of the Lefschetz properties. J. Pure Appl. Algebra 216 (2012), no. 1, 108--114.

3. R. Di Gennaro; G. Ilardi, J. Vallès, Singular hypersurfaces characterizing the Lefschetz properties. J. Lond. Math. Soc. (2) 89 (2014), no. 1, 194--212.

4. P. Gordan, M. Noether, Ueber die algebraischen Formen, deren Hesse'sche Determinante identisch verschwindet, Math. Ann. 10 (1876), 547--568.

5. T. Harima, H. Morita, Y. Numata, T. Maeno, A. Wachi, J. Watanabe: The Lefschetz properties, Springer LNM 2080 (2013).

6. T. Harima and A. Wachi, Generic initial ideals, graded Betti numbers, and k-Lefschetz properties, Comm. Algebra 37 (2009), no. 11, 4012--4025.

7. A. Iarrobino, L. Khatami, B. Van Steirteghem, R. Zhao, Nilpotent matrices having a givenJordan type as maximum commuting nilpotent orbit, (2014), ArXiv 1409.2192.

8. A. Kustin and A. Vraciu, The Weak Lefschetz Property for monomial complete intersection in positive characteristic, Trans. Amer. Math. Soc. 366 (2014), no. 9, 4571--4601.

9. J. Li, F. Zanello, Monomial complete intersections, the weak Lefschetz property and plane partitions. Discrete Math. 310 (2010), no.24, 3558--3570.

10. C. McDaniel, The strong Lefschetz property for coinvariant rings of finite real reflection groups, J. of Algebra, 331 (2011), 68--95.

11. E. Mezzetti, R.M. Miró-Roig, G. Ottaviani, Laplace equations and the Weak Lefschetz Property, Canadian Journal of Mathematics, 65 (2013), 634--654.

12. J. Migliore, R--M. Miró-Roig, S. Murai, U. Nagel, and J. Watanabe, On ideals with the Rees property, Arch. Math. (Basel) 101 (2013), no. 5, 445--454. [arXiv:1305.2551]

13. J. Sally, Numbers of generators of ideals in local rings, Lecture Notes in Pure and Applied Mathematics, 35, New York, Marcel Dekker (1978).

14. L. Smith, The strong Lefschetz property and cyclic decompositions of linear maps, Preprint, AG-Invariantentheorie, 2013.

15. J. Watanabe, On a paper of Gordan--Noether on algebraic forms with zero Hessian Proceedings of the School of Science of Tokai University, vol. 49 (2014), 1--21.

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