Title page for ETD etd-09202005-091014

Type of Document Dissertation
Author Gokhale, Dhananjay R.
URN etd-09202005-091014
Title Resolutions mod I, Golod pairs
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Green, Edward L. Committee Chair
Arnold, J. T. Committee Member
Farkas, Daniel R. Committee Member
McCoy, Robert A. Committee Member
Thomson, James E. Committee Member
  • Commutative rings.
  • Projective modules (Algebra)
  • Homomorphisms (Mathematics)
Date of Defense 1992-04-05
Availability restricted

Let R be a commutative ring, I be an ideal in R and let M be a R/ I -module. In this thesis we construct a R/ I -projective resolution of M using given R-projective resolutions of M and I. As immediate consequences of our construction we give descriptions of the canonical maps ExtR/I(M,N) -> ExtR(M,N) and TorRN(M, N) -> TorR/In(M, N) for a R/I module N and we give a new proof of a theorem of Gulliksen [6] which states that if I is generated by a regular sequence of length r then U∞ n=o TorR/ln (M, N) is a graded module over the polynomial ring R/ I [Xl. .. Xr] with deg Xi = -2, 1 ≤ i ≤ r. If I is generated by a regular element and if the R-projective dimension of M is finite, we show that M has a R/ I -projective resolution which is eventually periodic of period two.

This generalizes a result of Eisenbud [3]. In the case when R = (R, m) is a Noetherian local ring and M is a finitely generated R/ I -module, we discuss the minimality of the constructed resolution. If it is minimal we call (M, I) a Golod pair over R. We give a direct proof of a theorem of Levin [10] which states thdt if (M,I) is a Golod pair over R then (ΩnR/I(M),I) is a Golod pair over R where ΩnR/I(M) is the nth syzygy of the constructed R/ I -projective resolution of M. We show that the converse of the last theorem is not true and if (Ωn1/I(M), I) is a Golod pair over R then we give a necessary and sufficient condition for (M, I) to be a Golod pair over R.

Finally we prove that if (M, I) is a Golod pair over R and if a E I - mI is a regular element in R then (M, (a)) and (1/(a), (a)) are Golod pairs over R and (M,I/(a)) is a Golod pair over R/(a). As a corrolary of this result we show that if the natural map π: R -> R/1 is a Golod homomorphism ( this means (R/m, I) is a Golod pair over R ,Levin [8]), then the natural maps πl : R -> R/(a) and π2 : R/(a) -> R/1 are Golod homomorphisms.

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