Resolutions mod I, Golod pairs
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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  which states that if I is generated by a regular sequence of length r then Uâ n=o TorR/ln (M, N) is a graded module over the polynomial ring R/ I [Xl. .. Xr] with deg Xi = -2, 1 ï»¿ï»¿ï»¿ï»¿â ¤ i â ¤ 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 . 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  which states thdt if (M,I) is a Golod pair over R then (Î©nR/I(M),I) is a Golod pair over R where Î©nR/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 (Î©n1/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 Ï : R -> R/1 is a Golod homomorphism ( this means (R/m, I) is a Golod pair over R ,Levin ), then the natural maps Ï l : R -> R/(a) and Ï 2 : R/(a) -> R/1 are Golod homomorphisms.
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