## Final Activity Report Summary - COMBCOMMALG (Combinatorics in Commutative Algebra)

A key interactions between Algebraic Topology, Commutative Algebra and Combinatorics is the translation of an abstract finite simplicial complex into a square-free monomial ideal in a set of variables corresponding to the vertices of the simplicial complex. Together with the scientist in charge, the researcher translated a monomial ideal (not necessarily square-free) in some variables into a so called a multicomplex. Cleanness is the algebraic counterpart of shellability for simplicial complexes. Now the so-called shellable multicomplexes correspond to pretty clean algebras. As in the clean case, the multigraded pretty clean modules are sequentially Cohen-Macaulay modules. One conjecture of Stanley is established for multigraded pretty clean modules.

It is well known that a monomial ideal I and its generic ideal Gin(I) with respect to the reverse lexicographical order, have the same regularity and extremal Betti numbers. The generic ideal is a so-called p-Borel if the characteristic p of the basis field is non-zero. Actually one can define p-Borel ideals combinatorially and one can speak about p-Borel ideals independently of the characteristic of the field. The researcher showed that the regularity and the extremal Betti numbers of p-Borel ideals do not depend on the characteristic of the field. Moreover he studied when the Koszul homology modules of a principal p-Borel have monomial cyclic bases extending a theorem of Aramova-Herzog.

The strong Lefschetz property and the conjecture of Froberg have important applications in cryptography. Using a theorem of Harima and Watanabe, the researcher showed that the strong Lefschetz property preserves on complete intersection extensions of a standard graded Artinian Gorenstein algebra over a field K of characteristic zero if the fiber in the closed point has strong Lefschetz property. In particular, strong Lefschetz property preserves by 2-complete intersection extensions of such algebras.

One important conjecture says that if I is a complete intersection ideal of S=K[x_1,x_2,x_3], then S/I has the strong Lefschetz property. Together with M. Vladoiu, the researcher proved that if S/I has the strong Lefschetz property then Gin(I) depends only on the Hilbert function of S/I.

It is well known that a monomial ideal I and its generic ideal Gin(I) with respect to the reverse lexicographical order, have the same regularity and extremal Betti numbers. The generic ideal is a so-called p-Borel if the characteristic p of the basis field is non-zero. Actually one can define p-Borel ideals combinatorially and one can speak about p-Borel ideals independently of the characteristic of the field. The researcher showed that the regularity and the extremal Betti numbers of p-Borel ideals do not depend on the characteristic of the field. Moreover he studied when the Koszul homology modules of a principal p-Borel have monomial cyclic bases extending a theorem of Aramova-Herzog.

The strong Lefschetz property and the conjecture of Froberg have important applications in cryptography. Using a theorem of Harima and Watanabe, the researcher showed that the strong Lefschetz property preserves on complete intersection extensions of a standard graded Artinian Gorenstein algebra over a field K of characteristic zero if the fiber in the closed point has strong Lefschetz property. In particular, strong Lefschetz property preserves by 2-complete intersection extensions of such algebras.

One important conjecture says that if I is a complete intersection ideal of S=K[x_1,x_2,x_3], then S/I has the strong Lefschetz property. Together with M. Vladoiu, the researcher proved that if S/I has the strong Lefschetz property then Gin(I) depends only on the Hilbert function of S/I.