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Resolutions of Finite Length Modules Over Complete Intersections
The structure of free resolutions of finite length modules over regular local rings has long been a topic of interest in commutative algebra. Conjectures by Buchsbaum-Eisenbud-Horrocks and Avramov-Buchweitz predict that in this setting the minimal free resolution of the residue field should give, in some sense, the smallest possible free resolution of a finite length module. Results of Tate and Shamash describing the minimal free resolution of the residue field over a local hypersurface ring, together with the theory of matrix factorizations developed by Eisenbud and Eisenbud-Peeva, suggest analogous lower bounds for the size of free resolutions of finite length modules of infinite projective dimension over such rings. In this dissertation we describe both positive and negative results pertaining to these lower bounds. By refining an argument of Charalambous, we show that the lower bounds hold in certain multigraded settings. We are also able to obtain results for finite free resolutions of multigraded modules, recovering results of Charalambous and Santoni. For the local case, however, we use a construction of Iyengar-Walker to provide examples showing that the lower bounds do not always hold. In order to accomplish this, we make use of the theory of higher matrix factorizations developed by Eisenbud-Peeva to investigate the structure of free resolutions over complete intersections of arbitrary codimension.
Lindokken, Seth, "Resolutions of Finite Length Modules Over Complete Intersections" (2018). ETD collection for University of Nebraska-Lincoln. AAI10792659.