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 Kalgebras.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 (see [13,22,33,34,49]) 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 by Eisenbud and Schreyer in [26].Several authors were interested in the problem to give lower bounds for the Betti numbers of a module.In particular, Eisenbud-Koh [24], Green [31], Herzog [32] and Reiner-Welker [42] 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.In [21] Cutkosky, 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 in [20].Aramova, Crona and De Negri studied homological properties of bigraded K-algebras in [3].This thesis is divided in 6 chapters.Chapter 1 introduces definitions, notation and gives a short survey on those facts which are relevant in the following chapters.Recently Yanagawa [53] introduced the category of square-free modules over a polynomial ring S = K[x 1 , . . ., x n ].This concept generalizes Stanley-Reisner rings associated to simplicial complexes.In Chapter 2 we define the generalized Alexander dual for square-free S-modules.This definition is a natural extension of the well-known Alexander duality for simplicial complexes.Miller [40] studied Alexander duality in a more general situation.In the case of square-free S-modules his definition and ours coincide.We extend homological theorems on Stanley-Reisner rings to square-free Smodules.Bayer, Charalambous and S. Popescu introduced in [13] the extremal Betti numbers, which are a refinement of the Castelnuovo-Mumford regularity and of the projective dimension of a finitely generated graded S-module.Theorem 2.2.9 states that there is a 1-1 correspondence between the extremal Betti numbers of a

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