Lukas Katthän

Stanley depth and the lcm-lattice☆

Abstract In this paper we show that the Stanley depth, as well as the usual depth, are essentially determined by the lcm-lattice. More precisely, we show that for quotients I / J of monomial ideals J ⊂ I , both invariants behave monotonic with respect to certain maps defined on their lcm-lattice. This allows simple and uniform proofs of many new and known results on the Stanley depth. In particular, we obtain a generalization of our result on polarization presented in [16] . We also obtain a useful description of the class of all monomial ideals with a given lcm-lattice, which is independent from our applications to the Stanley depth.

The behavior of Stanley depth under polarization

Let K be a field, R = K X 1 , ? , X n ] be the polynomial ring and J ? I be two monomial ideals in R. In this paper we show that sdepth I / J - depth I / J = sdepth I p / J p - depth I p / J p , where sdepth I / J denotes the Stanley depth and I p denotes the polarization. This solves a conjecture by Herzog 9] and reduces the famous Stanley conjecture (for modules of the form I / J ) to the squarefree case. As a consequence, the Stanley conjecture for algebras of the form R / I and the well-known combinatorial conjecture that every Cohen-Macaulay simplicial complex is partitionable are equivalent.

Edge rings satisfying Serre’s condition (₁)

A combinatorial criterion for the edge ring of a finite connected graph to satisfy Serres condition R_1 is studied.

The Golod property for Stanley–Reisner rings in varying characteristic

We give examples of simplicial complexes ∆, such that Golod property of the Stanley-Reisner ring K[∆] depends on the characteristic of the field K. More precisely, for every finite set T of prime numbers we construct simplicial complexes ∆ and Γ, such that K[∆] is Golod exactly in the characteristics in T and K[Γ] is Golod exactly in the characteristics not in T . Along the way, we also show that a one-dimensional simplicial complex is Golod if and only if it is chordal.

Ehrhart Theory of Spanning Lattice Polytopes

A lattice polytope is called spanning if its lattice points affinely span the ambient lattice. We show as a corollary to a general result in the Ehrhart theory of lattice polytopes that the

LCM Lattices and Stanley Depth: A First Computational Approach

h^*

NORMAL CYCLIC POLYTOPES AND CYCLIC POLYTOPES THAT ARE NOT VERY AMPLE

-vector of a spanning lattice polytope has no gaps, i. e.,

Betti Posets and the Stanley Depth

h^*_i =0

On homology spheres with few minimal non-faces

implies

The structure of DGA resolutions of monomial ideals

h^*_{i+1}=0