Algebraic theory of affine monoids
This thesis treats several aspects of affine monoids. First, we consider the structure of the set of holes of an affine monoid Q. This set is the difference between Q and its normalization. We find connections to algebraic properties of the monoi algebra K[Q] and in particular to its local cohom...
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Format:  Dissertation 
Language:  English 
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PhilippsUniversität Marburg
2013
Reine und Angewandte Mathematik 
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Summary:  This thesis treats several aspects of affine monoids. First, we consider the structure of the set of holes of an affine monoid Q. This set is the difference between Q and its normalization. We find connections to algebraic properties of the monoi algebra K[Q] and in particular to its local cohomology. In special cases, this allows to compute the depth of K[Q]. We then specialize to simplicial and seminormal affine monoids and reprove or extend several known results using our theory. Moreover, we consider the dependency of algebraic properties of K[Q] upon the field K. We can show that Serre's (S3) does not depend on K. Further, we prove a special case of a conjecture by Eisenbud and Goto. In the next chapter, we construct a fammily of nonnormal lattice simplices, such that the holes have an arbitrary lattice distance from the facets. In the next chapter, we consider toric edge rings and give a criterion for the toric egde ring to satisfy Serre's (R1) and for being seminormal. In the last chapter, we consider the affine monoid of the linear ordering polytope. Using graph theory, we give a combinatorial description of the degree 2 generators of its toric ideal. 

DOI:  https://doi.org/10.17192/z2013.0237 