On minimal graded free resolutions

Minimal graded free resolutions are an important and central topic in algebra. They are a useful tool for studying modules over finitely generated graded K- algebras. Such a resolution determines the Hilbert series, the Castelnuovo-Mumford regularity and other invariants of the module. This thesis is concerned with the structure of minimal graded free resolutions. We relate our results to several recent trends in commutative algebra. The first of these trends deals with relations between properties of the Stanley- Reisner ring associated to a simplicial complex and the Stanley-Reisner ring of its Alexander dual. Another development is the investigation of the linear part of a minimal graded free resolution as defined by Eisenbud and Schreyer. Several authors were interested in the problem to give lower bounds for the Betti numbers of a module. In particular, Eisenbud-Koh, Green, Herzog and Reiner- Welker studied the graded Betti numbers which determine the linear strand of a minimal graded free resolution. Bigraded algebras occur naturally in many research areas of commutative algebra. A typical example of a bigraded algebra is the Rees ring of a graded ideal. Herzog and Trung used this bigraded structure of the Rees ring to study the Castelnuovo- Mumford regularity of powers of graded ideals in a polynomial ring. Conca, Herzog, Trung and Valla dealt with diagonal subalgebras of bigraded algebras. Aramova, Crona and De Negri studied homological properties of bigraded K-algebras

Linear syzygies, hyperbolic Coxeter groups and regularity

We show that the virtual cohomological dimension of a Coxeter group is essentially the regularity of the Stanley--Reisner ring of its nerve. Using this connection between geometric group theory and commutative algebra, as well as techniques from the theory of hyperbolic Coxeter groups, we study the behavior of the Castelnuovo--Mumford regularity of square-free quadratic monomial ideals. We construct examples of such ideals which exhibit arbitrarily high regularity after linear syzygies for arbitrarily many steps. We give a doubly logarithmic bound on the regularity as a function of the number of variables if these ideals are Cohen--Macaulay.Comment: 22 pages, v2: final version as in Compositio Mat

Some relational structures with polynomial growth and their associated algebras II: Finite generation

The profile of a relational structure $R$ is the function $\varphi_R$ which counts for every integer $n$ the number, possibly infinite, $\varphi_R(n)$ of substructures of $R$ induced on the $n$-element subsets, isomorphic substructures being identified. If $\varphi_R$ takes only finite values, this is the Hilbert function of a graded algebra associated with $R$, the age algebra $A(R)$, introduced by P.~J.~Cameron. In a previous paper, we studied the relationship between the properties of a relational structure and those of their algebra, particularly when the relational structure $R$ admits a finite monomorphic decomposition. This setting still encompasses well-studied graded commutative algebras like invariant rings of finite permutation groups, or the rings of quasi-symmetric polynomials. In this paper, we investigate how far the well know algebraic properties of those rings extend to age algebras. The main result is a combinatorial characterization of when the age algebra is finitely generated. In the special case of tournaments, we show that the age algebra is finitely generated if and only if the profile is bounded. We explore the Cohen-Macaulay property in the special case of invariants of permutation groupoids. Finally, we exhibit sufficient conditions on the relational structure that make naturally the age algebra into a Hopf algebra.Comment: 27 pages; submitte

Hilbert series of modules over Lie algebroids

We consider modules $M$ over Lie algebroids ${\mathfrak g}_A$ which are of finite type over a local noetherian ring $A$. Using ideals $J\subset A$ such that ${\mathfrak g}_A \cdot J\subset J$ and the length $\ell_{{\mathfrak g}_A}(M/JM)< \infty$ we can define in a natural way the Hilbert series of $M$ with respect to the defining ideal $J$. This notion is in particular studied for modules over the Lie algebroid of $k$-linear derivations ${\mathfrak g}_A=T_{A/k}(I)$ that preserve an ideal $I\subset A$, for example when $A={\mathcal O}_n$, the ring of convergent power series. Hilbert series over Stanley-Reisner rings are also considered.Comment: 42 pages. This is a substantial revision of the previous versio