A study of homological invariants modulo an exact zero divisor
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Abstract
Let Q be a commutative local ring and R a quotient ring of Q by an ideal generated by an exact zero divisor. We use differential graded structures to describe a construction that produces a minimal free resolution of an R-module from the free resolution of the same module, considered as a Q-module, and the free resolution of R as a Q-module. We provide an explicit algorithm for this construction, written for the computer algebra system Macaulay2. Given two R-modules, we then use a mapping cone construction to relate homology of the two modules over Q to homology over R, and we give applications of this construction to the study of several homological invariants, such as complexity, curvature and generalized Poincaré series.
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Introduction -- Background -- DG module structures and minimal free resolutions modulo an exact zero divisor -- The mapping cone of an eisenbud operator -- Applications
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Ph.D. (Doctor of Philosophy)