MONICA Conference

Click on the titles to read the abstracts.

• Josep Álvarez Montaner, Universitat Politècnica de Catalunya (Spain): Local cohomology supported on monomial ideals.

  Let R = k[x1, ..., xn] be the polynomial ring in n independent variables, where k is a field. A lot of progress in the study of the local cohomology modules supported on monomial ideals has been made based on the fact that they have a structure as multigraded modules. Another line of research uses their structure as regular holonomic modules over the ring of k-linear differential operators, in particular the fact that they are finitely generated.

  In our lectures we will recall first the basic notions on local cohomology and D-modules that we will need throughout this course. Our aim is to introduce some tools that allow us to study the local cohomology modules for any ideal I.

  For the case of monomial ideals, we will highlight the main results obtained on the description of the structure of local cohomology modules as multigraded module. We will compare these results with the ones obtained using the D-module approach.

  Finally we will turn our attention to the study of Bass numbers of the local cohomology modules. We will introduce some explicit methods to compute them and we will give bounds for the injective dimension of these modules. Similar questions are raised for the case of dual Bass numbers.

• Jürgen Herzog, Universität Essen (Germany): Combinatorial aspects of monomial and binomial ideals.

  This course will consist on the following three lectures:

  • A survey on Stanley decompositions

    In his fundamental paper ``Linear Diophantine equations and local cohomology" (Invent. Math. 68, 1982) Stanley introduced for Z-graded modules a combinatorial invariant which nowadays is called the Stanley depth and he conjectured that the Stanley depth is always greater than or equal to the depth of a module. This conjecture is still widely open, though in the last years the conjecture has been proved in several special cases and methods have been developed to actually compute the Stanley depth in some important cases. In this lecture we review the known cases for which the conjecture is known to be true, give natural lower and upper bounds for the Stanley depth, and consider the Stanley depth of syzygy modules.

  • Generalized Hibi ideals and Hibi rings.

    In his paper ``Distributive lattices, affine semigroup rings and algebras with straightening laws" (in ``Commutative Algebra and Combinatorics" (M. Nagata and H. Matsumura, eds.) Adv. Stud. Pure Math. 11, North-Holland, Amsterdam, 93--109 (1987)) Hibi introduced a class of algebras which in the literature are called Hibi rings. They are toric rings attached to finite posets, and may be viewed as natural generalizations of polynomial rings. Indeed, a polynomial ring in n variables over a field K is just the Hibi ring of the poset [n]={1,2,...,n}. Hibi rings appear naturally in various combinatorial and algebraic contexts, for example in invariant theory. Hodge algebras may be viewed as flat deformations of Hibi rings. In this sense the coordinate ring of the flag variety for GL_n is the deformation of the Hibi ring for the so-called Gelfand-Tsetlin poset. In this lecture we consider a natural extension of Hibi rings and ideals, study their properties and discuss a few open questions.

  • Binomial edge ideals and ideals generated by 2-minors

    In this survey lecture we present results concerning ideals generated by general sets of 2-minors of an (m x n)-matrix X of indeterminates. We discuss the problem when such an ideal is a prime or radical ideal, consider the minimal prime ideals and compute the Gröbner basis. These studies are partly motivated by applications to algebraic statistic. Diaconis, Eisenbud and Sturmfels showed in their paper ``Lattice walks and primary decomposition" (Birkh\"auser, Boston, 1998, pp. 173--193) how primary decompositions of binomial ideals can be used to analyze connectedness of contingency tables via a given set of moves. This bridge between commutative algebra and statistics motivated several authors to study lattice basis ideals and ideals generated by 2-adjacent minors. We will report on recent results related to these topics.

• Adam Van Tuyl, Lakehead University (Canada):An introduction to edge ideals.

  This course will consist on the following three lectures:

  • The dictionary:The first lecture will focus on the basics of edge ideals. Our goal is to introduce the “dictionary”, i.e., how to read the graph theory invariants from the algebraic invariants of the monomial ideal or ring, and vice versa
  • Minimal free resolutions of edge ideals. The minimal free graded resolution of any homogeneous ideal contains numerous invariants, e.g., Betti numbers, regularity, and projective dimension. In this lecture we give a overview of how to interpret some of the homological invariants in terms of the graph.
  • Powers of the cover ideal. The cover ideal of a finite simple graph G is an alternative way to associate to G a monomial ideal.In this lecture, we look at the information encoded into the associated primes of the powers of the cover ideal. This information is related to colourings of G. We will also describe how one can determine if a graph is perfect using the associated primes.