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## Über dieses Buch

This volume collects contributions by leading experts in the area of commutative algebra related to the INdAM meeting “Homological and Computational Methods in Commutative Algebra” held in Cortona (Italy) from May 30 to June 3, 2016 . The conference and this volume are dedicated to Winfried Bruns on the occasion of his 70th birthday. In particular, the topics of this book strongly reflect the variety of Winfried Bruns’ research interests and his great impact on commutative algebra as well as its applications to related fields. The authors discuss recent and relevant developments in algebraic geometry, commutative algebra, computational algebra, discrete geometry and homological algebra. The book offers a unique resource, both for young and more experienced researchers seeking comprehensive overviews and extensive bibliographic references.

## Inhaltsverzeichnis

### Betti Sequences over Standard Graded Commutative Algebras with Two Relations

Abstract
It is well known that if Q is a standard graded polynomial ring in d variables over a field, and R is a quotient of Q with $$\mathop{\mathrm{proj\,dim}}\nolimits _{Q}R \leq 1$$, then the Betti sequence of every R-module N is constant after d steps. We prove that if $$\mathop{\mathrm{proj\,dim}}\nolimits _{Q}R = 2$$, then the Betti sequence of N is determined explicitly by some initial segment of known length, which can be read off a presentation of N. There are three cases, depending on the degrees of the defining relations of R and on whether these have a common divisor.
Luchezar L. Avramov, Zheng Yang

### Betti Diagrams with Special Shape

Abstract
We consider classes of monomial ideals whose Betti diagrams have a special shape. Monomial ideals with such a Betti diagram satisfy the subadditivity condition for the maximal shifts in the resolution by obvious reasons, and they appear quite frequently in combinatorial contexts. Examples of ideals with special shape are the edge ideal as well as the vertex cover ideal of chordal graphs, whisker graphs and triangulated d-uniform hypergraphs.
Mina Bigdeli, Jürgen Herzog

### Koszul Algebras Defined by Three Relations

Abstract
This work concerns commutative algebras of the form R = QI, where Q is a standard graded polynomial ring and I is a homogenous ideal in Q. It has been proposed that when R is Koszul the ith Betti number of R over Q is at most $$\binom{g}{i}$$, where g is the number of generators of I; in particular, the projective dimension of R over Q is at most g. The main result of this work settles this question, in the affirmative, when g ≤ 3.

### Some Algebras with the Weak Lefschetz Property

Abstract
Using a connection to lozenge tilings of triangular regions, we establish an easily checkable criterion that guarantees the weak Lefschetz property of a quotient by a monomial ideal in three variables. It is also shown that each such ideal also has a semistable syzygy bundle.
David Cook, Uwe Nagel

### Multigraded Generic Initial Ideals of Determinantal Ideals

Abstract
Let I be either the ideal of maximal minors or the ideal of 2-minors of a row graded or column graded matrix of linear forms L. In previous work we showed that I is a Cartwright-Sturmfels ideal, that is, the multigraded generic initial ideal gin(I) of I is radical (and essentially independent of the term order chosen). In this paper we describe generators and prime decomposition of gin(I) in terms of data related to the linear dependences among the row or columns of the submatrices of L. In the case of 2-minors we also give a closed formula for its multigraded Hilbert series.
Aldo Conca, Emanuela De Negri, Elisa Gorla

### A Stronger Local Monomialization Theorem

Abstract
Steven Dale Cutkosky

### The Cayley Trick for Tropical Hypersurfaces with a View Toward Ricardian Economics

Abstract
The purpose of this survey is to summarize known results about tropical hypersurfaces and the Cayley Trick from polyhedral geometry. This allows for a systematic study of arrangements of tropical hypersurfaces and, in particular, arrangements of tropical hyperplanes. A recent application to the Ricardian theory of trade from mathematical economics is explored.
Michael Joswig

### Ideals Associated to Poset Homomorphisms: A Survey

Abstract
In this survey, we give an overview to the various known algebraic properties and invariants of ideals of poset homomorphisms. A particular attention lies on classical related notions that occur as special cases.

### How to Flatten a Soccer Ball

Abstract
This is an experimental case study in real algebraic geometry, aimed at computing the image of a semialgebraic subset of 3-space under a polynomial map into the plane. For general instances, the boundary of the image is given by two highly singular curves. We determine these curves and show how they demarcate the “flattened soccer ball”. We explore cylindrical algebraic decompositions, by working through concrete examples. Maps onto convex polygons and connections to convex optimization are also discussed.
Kaie Kubjas, Pablo A. Parrilo, Bernd Sturmfels

### The Smallest Normal Edge Polytopes with No Regular Unimodular Triangulations

Abstract
The edge polytope of a finite graph is the convex hull of the column vectors of its vertex-edge incidence matrix. In this paper, we discuss the existence of a regular unimodular triangulation of normal edge polytopes of finite graphs. For normal edge polytopes of finite graphs with d vertices, n edges, and no regular unimodular triangulations, we determine the polytope that has the following: (1) the smallest number of vertices (d = 9), (2) the smallest number of edges (n = 15), and (3) the smallest codimension (nd = 4 andd = 17).
Ginji Hamano, Takuji Hayashi, Takayuki Hibi, Koichi Hirayama, Hidefumi Ohsugi, Kei Sato, Akihiro Shikama, Akiyoshi Tsuchiya

### Homological Conjectures and Lim Cohen-Macaulay Sequences

Abstract
We discuss the new notion of a lim Cohen-Macaulay sequence of modules over a local ring, and also a somewhat weaker notion, as well as the theory of content for local cohomology modules. We relate both to the problem of proving the direct summand conjecture and other homological conjectures without using almost ring theory and perfectoid space theory, and we also indicate some other open problems whose solution would yield a new proof of the direct summand conjecture.
Melvin Hochster

### Algebras with the Weak Lefschetz Property

Abstract
This is a survey on some works in which the Weak Lefschetz Property (WLP) for Artinian standard graded algebras is investigated, see for instance (Ragusa and Zappalà, arXiv:1112.1498. To appear in Rend Circ Mat Palermo; Colloq Math 64:73–83, 2013; Favacchio et al, J Pure Appl Algebra 217:1955–1966, 2013). In particular, it is shown that the Hilbert function of an almost complete intersection Artinian standard graded algebra of codimension 3 is a Weak Lefschetz sequence, i.e. it is the Hilbert function of some Artinian algebra with WLP or equivalently it is unimodal and the positive part of their first differences is a O-sequence. Moreover we give both some numerical condition on the Hilbert function and other conditions on the graded Betti numbers in order to force Artinian Gorenstein standard graded algebras of codimension 3 to enjoy the WLP. For Artinian standard graded algebras with the WLP we study the behavior of their linear quotients both with respect to the Hilbert function and to the graded Betti numbers. From this we produce a new property denominated Betti Weak Lefschetz Property (β-WLP) which permits a good behavior of the grade Betti numbers for the linear quotients of Artinian standard graded algebras with the WLP. We find conditions on the generators’ degrees of a complete intersection Artinian graded algebra with the WLP which force the algebra to have the β-WLP.
Alfio Ragusa

### About Multiplicities and Applications to Bezout Numbers

Abstract
Let $$(A,\mathfrak{m}, \mathbb{k})$$ denote a local Noetherian ring and $$\mathfrak{q}$$ an ideal such that $$\ell_{A}(M/\mathfrak{q}M) <\infty$$ for a finitely generated $$A$$-module $$M$$. Let $$\underline{a} = a_{1},\ldots,a_{d}$$ denote a system of parameters of $$M$$ such that $$a_{i} \in \mathfrak{q}^{c_{i}}\setminus \mathfrak{q}^{c_{i}+1}$$ for $$i = 1,\ldots,d$$. It follows that $$\chi:= e_{0}(\underline{a};M) - c \cdot e_{0}(\mathfrak{q};M) \geq 0$$, where $$c = c_{1} \cdot \ldots \cdot c_{d}$$. The main results of the report are a discussion when $$\chi = 0$$ resp. to describe the value of $$\chi$$ in some particular cases. Applications concern results on the multiplicity $$e_{0}(\underline{a};M)$$ and applications to Bezout numbers.

### A Polynomial Identity via Differential Operators

Abstract
We give a new proof of a polynomial identity involving the minors of a matrix, that originated in the study of integer torsion in a local cohomology module.
Anurag K. Singh

### F-Thresholds, Integral Closure and Convexity

Abstract
The purpose of this note is to revisit the results of the paper of Henriques and Varbaro from a slightly different perspective, outlining how, if the integral closures of a finite set of prime ideals abide the expected convexity patterns, then the existence of a peculiar polynomial f allows one to compute the F-jumping numbers of all the ideals formed by taking sums of products of the original ones. The note concludes with the suggestion of a possible source of examples falling in such a framework.
Matteo Varbaro
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