Algebraic Properties of Generic Tropical Varieties (2009)
We show that the algebraic invariants multiplicity and depth of a graded ideal in the polynomial ring are closely connected to the fan structure of its generic tropical variety in the constant...
Generic Tropical Varieties (2009)
We show that in the constant coefficient case the generic tropical variety of a graded ideal exists. This can be seen as the analogon to the existence of the generic initial ideal in Groebner basis...
Koszul homology and syzygies of Veronese subalgebras (2009)
Bruns, Winfried, Conca, Aldo, Roemer, Tim
A graded K-algebra R has property N_p if it is generated in degree 1, has relations in degree 2 and the syzygies of order less or equal to p on the relations are linear. The Green-Lazarsfeld index of...
Linearity defects of modules over commutative rings (2008)
Iyengar, Srikanth B., Roemer, Tim
This paper concerns linear parts of minimal resolutions of finitely generated modules over commutative local, or graded rings. The focus is on the linearity defect of a module, which marks the point...
Homological properties of Orlik-Solomon algebras (2008)
The Orlik-Solomon algebra of a matroid can be considered as a quotient ring over the exterior algebra E. At first we study homological properties of E-modules as e.g. complexity, depth and...
Gr\"obner bases and Betti numbers of monoidal complexes (2007)
Bruns, Winfried, Koch, Robert, Roemer, Tim
In this note we consider monoidal complexes and their associated algebras, called toric face rings. These rings generalize Stanley-Reisner rings and affine monoid algebras. We compute initial ideals...
Generic initial ideals and fibre products (2007)
We study the behavior of generic initial ideals with respect to fibre products. In our main result we determine the generic initial ideal of the fibre product with respect to the reverse...
Glicci simplicial complexes (2007)
One of the main open questions in liaison theory is whether every homogeneous Cohen-Macaulay ideal in a polynomial ring is glicci, i.e. if it is in the G-liaison class of a complete intersection. We...
Betti numbers and shifts in minimal graded free resolutions (2007)
Let S be a polynomial ring and R=S/I where I is a graded ideal of S. The Multiplicity Conjecture of Herzog, Huneke, and Srinivasan states that the multiplicity of R is bounded above by a function of...
On canonical modules of toric face rings (2006)
Generalizing the concepts of Stanley-Reisner and affine monoid algebras, one can associate to a rational pointed fan the toric face ring. Assuming that this ring is Cohen-Macaulay, the main result of...
On the regularity over positively graded algebras (2006)
We study the relationship between the Tor-regularity and the local-regularity over a positively graded algebra defined over a field which coincide if the algebra is a standard graded polynomial ring....
Following a construction of Stanley we consider toric face rings associated to rational pointed fans. This class of rings is a common generalization of the concepts of Stanley--Reisner and affine...
Algebraic shifting and exterior and symmetric algebra methods (2005)
Nagel, Uwe, Roemer, Tim, Vinai, Natale Paolo
We establish results about algebraic shifting of simplicial complexes and use them to compare different shifting operations. In particular, we show that each shifting operation does not decrease the...
h-vectors of Gorenstein polytopes (2005)
We show that the Ehrhart h-vector of an integer Gorenstein polytope with a regular unimodular triangulation satisfies McMullen's g-theorem; in particular, it is unimodal. This result generalizes a...
On algebras associated to partially ordered sets (2005)
We continue the study on sheaves of rings on finite posets. We present examples where the ring of global sections coincide with toric faces rings, quotients of a polynomial ring by a monomial ideal...
On seminormal monoid rings (2005)
Bruns, Winfried, Li, Ping, Roemer, Tim
Given a seminormal affine monoid M we consider several monoid properties of M and their connections to ring properties of the associated affine monoid ring K[M] over a field K. We characterize when...
Extensions of the Multiplicity Conjecture (2005)
Migliore, Juan, Nagel, Uwe, Roemer, Tim
The Multiplicity conjecture of Herzog, Huneke, and Srinivasan states an upper bound for the multiplicity of any graded $k$-algebra as well as a lower bound for Cohen-Macaulay algebras. In this note...
Cohomology of partially ordered sets and local cohomology of section rings (2005)
Brun, Morten, Bruns, Winfried, Roemer, Tim
We study local cohomology of rings of global sections of sheafs on the Alexandrov space of a partially ordered set. We give a criterion for a splitting of the local cohomology groups into summands...
Extended degree functions and monomial modules (2004)
The arithmetic degree, the smallest extended degree, and the homological degree are invariants that have been proposed as alternatives of the degree of a module if this module is not Cohen-Macaulay....
Subdivisions of toric complexes (2004)
We introduce toric complexes as polyhedral complexes consisting of rational cones together with a set of integral generators for each cone, and we define their associated face rings. Abstract...
Note on bounds for multiplicities (2003)
Let S=K[x_1,...,x_n] be a polynomial ring and R=S/I be a graded K-algebra where I is a graded ideal in S. Herzog, Huneke and Srinivasan have conjectured that the multiplicity of R is bounded above by...
Initial algebras of determinantal rings, Cohen-Macaulay and Ulrich ideals (2003)
Bruns, Winfried, Roemer, Tim, Wiebe, Attila
We study initial algebras of determinantal rings, defined by minors of generic matrices, with respect to their classical generic point. This approach leads to very short proofs for the structural...
Betti numbers of Z^n-graded modules (2003)
Let S=K[X_1,...,X_n] be the polynomial ring over a field K. For bounded below Z^n-graded S-modules M and N we show that if Tor^S_p(M,N) is nonzero, then for every i between 0 and p, the dimension of...
Homological properties of bigraded algebras (2000)
We study the x- and y-regularity of a bigraded K-algebra R. These notions are used to study asymptotic properties of certain finitely generated bigraded modules. As an application we get for any...
Bounds for Betti numbers (2000)
In this paper we prove parts of a conjecture of Herzog giving lower bounds on the rank of the free modules appearing in the linear strand of a graded $k$-th syzygy module over the polynomial ring. If...