Annetta Aramova

Resolutions of monomial ideals and cohomology over exterior algebras

This paper studies the homology of finite modules over the exterior algebra E of a vector space V . To such a module M we associate an algebraic set VE(M) ⊆ V , consisting of those v ∈ V that have a non-minimal annihilator in M . A cohomological description of its defining ideal leads, among other things, to complementary expressions for its dimension, linked by a ‘depth formula’. Explicit results are obtained for M = E/J , when J is generated by products of elements of a basis e1, . . . , en of V . A (infinite) minimal free resolution of E/J is constructed from a (finite) minimal resolution of S/I, where I is the squarefree monomial ideal generated by ‘the same’ products of the variables in the polynomial ring S = K[x1, . . . , xn]. It is proved that VE(E/J) is the union of the coordinate subspaces of V , spanned by subsets of { e1, . . . , en } determined by the Betti numbers of S/I over S.

Shifting Operations and Graded Betti Numbers

The behaviour of graded Betti numbers under exterior and symmetric algebraic shifting is studied. It is shown that the extremal Betti numbers are stable under these operations. Moreover, the possible sequences of super extremal Betti numbers for a graded ideal with given Hilbert function are characterized. Finally it is shown that over a field of characteristic 0, the graded Betti numbers of a squarefree monomial ideal are bounded by those of the corresponding squarefree lexsegment ideal.

Bigeneric initial ideals, diagonal subalgebras and bigraded Hilbert functions

Abstract In this paper we study some problems concerning bigraded ideals. By introducing the concept of bigeneric initial ideal, we answer an open question about diagonal subalgebras and we give a necessary condition for a function to be the bigraded Hilbert function of a bigraded algebra. Moreover, we give an upper bound for the regularity of a bistable ideal in terms of the degrees of its generators.

Finite Lattices and Lexicographic Gröbner Bases

By means of combinatorics on finite distributive lattices, lexicographic quadratic Grobner bases of certain kinds of subrings of an affine semigroup ring arising from a finite distributive lattice will be studied.

Free resolutions and Koszul homology

Abstract Let (R, m) be a regular local ring, and M an R-module. The minimal free resolution F of M has a natural m-adic filtration. In this paper we establish an isomorphism of the spectral sequence which is associated to the filtered complex F, and the spectral sequence associated to a double complex derived from the Koszul complex of M with respect to a minimal system of generators of m. We use this result to describe explicitly the maps in the resolution of terms of Koszul cycles whenever the resolution of the module is pure.

When do the symmetric tensors of a commutative algebra form a Frobenius algebra

For a commutative k-algebra B, consider the subalgebra (B?n)Sn of the nth tensor power of B, formed by the tensors invariant under arbitrary permutations of the indices. Necessary and sufficient conditions are found for (B?n)Sn to be Frobenius. When dimk B

Squarefree lexsegment ideals

2, these say that B is Frobenius and n! is invertible in k, unless B is separable. Some additional cases occur for two-dimensional algebras in positive characteristic, depending on the divisibility of n + 1.