Graded Free Resolutions
The study of free resolutions is a core and beautiful area in Commutative Algebra. The idea to associate a free resolution to a finitely generated module was introduced in two famous papers by Hilbert in 1890, 1893. Free resolutions provide a method for describing the structure of modules. There are several challenging and exciting conjectures involving resolutions. A number of open problems on graded syzygies and Hilbert functions are listed in a paper by Peeva and Stillman (2009).
We are using a grading on the polynomial ring S=k[x 1,…,x n ] and on the objects which we are interested to study: ideals, quotient rings, modules, complexes, and free resolutions. The grading is a powerful tool. The general principle using that tool is the following: in order to understand the properties of a graded object X, we consider X as a direct sum of vector spaces (its graded components) and we study the properties of each of these vector spaces.
KeywordsSimplicial Complex Short Exact Sequence Betti Number Hilbert Series Hilbert Function
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- [Hilbert 1]
- [Hilbert 2]