Research

• A.Engström, M.Orlich. The regularity of almost all edge ideals. Advances in Mathematics 435 (2023), Paper No. 109355, 34 pp. [arXiv]

Abstract: A fruitful contemporary paradigm in graph theory is that almost all graphs that do not contain a certain subgraph have common structural characteristics. The "almost" is crucial, without it there is no structure. In this paper we transfer this paradigm to commutative algebra and make use of deep graph theoretic results. A key tool are the critical graphs introduced by Balogh and Butterfield. We consider edge ideals of graphs and their Betti numbers. The numbers of the form β_{i,2i+2} constitute the "main diagonal" of the Betti table. It is well known that, for edge ideals, any Betti number below (or equivalently, to the left of) this diagonal is always zero. We identify a certain "parabola" inside the Betti table and call parabolic Betti numbers the entries of the Betti table bounded on the left by the main diagonal and on the right by this parabola. Let β_{i,j} be a parabolic Betti number on the r-th row of the Betti table, for r≥3. Our main results state that almost all graphs G with β_{i,j}(I_G)=0 can be partitioned into r−2 cliques and one independent set, and in particular for almost all graphs G with β_{i,j}(I_G)=0 the regularity of I_G is r−1.

• G.Fløystad, M.Orlich. Triangulations of polygons and stacked simplicial complexes: separating their Stanley–Reisner ideals. Journal of Algebraic Combinatorics 57 (2023), no. 3, 659–686. [arXiv]

Abstract: A triangulation of a polygon has an associated Stanley--Reisner ideal. We obtain a full algebraic and combinatorial understanding of these ideals, and describe their separated models. More generally we do this for stacked simplicial complexes, in particular for stacked polytopes. 

• A.Engström, L.Jakobsson, M.Orlich. Explicit Boij--Söderberg theory of ideals from a graph isomorphism reduction. Journal of Pure and Applied Algebra 224 (2020), no. 11, 106405, 17 pp. [arXiv]

Abstract: In the origins of complexity theory Booth and Lueker showed that the question of whether two graphs are isomorphic or not can be reduced to the special case of chordal graphs. To prove that, they defined a transformation from graphs G to chordal graphs BL(G). The projective resolutions of the associated edge ideals is manageable and we investigate to what extent their Betti tables also tell non-isomorphic graphs apart. It turns out that the coefficients describing the decompositions of Betti tables into pure diagrams in Boij--Söderberg theory are much more explicit than the Betti tables themselves, and they are expressed in terms of classical statistics of the graph G. 

• M.Orlich. Linearization of monomial ideals. Uploaded on arXiv on 20 June 2020. [arXiv]

Abstract: We introduce a construction, called linearization, that associates to any monomial ideal I an ideal Lin(I) in a larger polynomial ring. The main feature of this construction is that the new ideal Lin(I) has linear quotients. In particular, since Lin(I) is generated in a single degree, it follows that Lin(I) has a linear resolution. We investigate some properties of this construction, such as its interplay with classical operations on ideals, its Betti numbers, functoriality and combinatorial interpretations. We moreover introduce an auxiliary construction, called equification, that associates to any monomial ideal a new monomial ideal generated in a single degree, in a polynomial ring with one more variable. We study some of the homological and combinatorial properties of the equification, which can be seen as a monomial analogue of the well-known homogenization construction. 

• (Combinatorial, computational) commutative algebra.

• Graph theory.