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(UMI)AAI9026154 http://hdl.handle.net/2142/19766 eng Copyright 1990 Charalambous, Hara Mathematics On lower bounds for the betti numbers of finite length modules text Griffith, Phillip A. Charalambous, Hara 2011-05-07T12:17:51Z 2011-05-07T12:17:51Z 10000-01-01 1990 In this manuscript we consider multigraded modules. Chapter 1 gives the necessary definitions and examples that develop the theory of multigraded modules. A multigraded module has a multigraded minimal resolution. We give the necessary conditions for a matrix to correspond to a multigraded map and show that all associated primes of a multigraded module are multigraded. In Chapter 2 we show that the betti numbers of multigraded modules satisfy the bound: $\beta\sbsp{i}{R}(M) \geq (\sbsp{i}{d})$. The multigraded modules that are not isomorphic to R modulo an R-sequence can be divided into two categories according to their betti numbers. Either for all i $\beta\sbsp{i}{R}(M) \geq (\sbsp{i}{d}) + (\sbsp{i-1}{d-1})$ or for all i $\beta\sbsp{i}{R}(M) \geq (\sbsp{i}{d}) + (\sbsp{\ i}{d-1})$. Thus if $\beta\sbsp{i}{R}(M)$ = $(\sbsp{i}{d})$ then M must be R modulo a maximal R-sequence. In case M is an almost complete intersection we derive its betti numbers. In Chapter 3 we give better bounds for the betti numbers of cyclic multigraded modules using a deformation argument combined with localization. Lastly we point out that the sum of the betti numbers for all modules of finite length is greater than or equal to 2$\sp{d}$ + 2$\sp{d-1}$ where d is the dimension of the ring up to dimension 4. Made available in DSpace on 2011-05-07T12:17:51Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9026154.pdf: 1903811 bytes, checksum: ac17ec844ea5bb9444970f23bb333c61 (MD5) Previous issue date: 1990 Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:39:17Z Item is restricted indefinitely. Restriction data tranferred 2014-07-01T11:16:30-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission ETDs are only available to UIUC Users without author permission U of I Only AAI9026154 Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.