# 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 Ext_{R/I}*(M,N)* -> Ext_{R}*(M,N)* and Tor^{R}_{N}*(M, N)* -> Tor^{R/I}_{n}*(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 ∐∞_{n=o} Tor^{R/I}_{n} *(M, N)* is a graded module over the polynomial ring R/ I [X₁. .. X_{r}] with deg X_{i} = -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 (Ω^{n}_{R/I}R/I(M),I) is a Golod pair over *R* where Ω^{n}_{R/I}R/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 (Ω¹_{R/I}R/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 ∈ *I* - m*I* 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 π₁ : *R* → *R*/(a) and π₂ : *R*/(a) → *R/1* are Golod homomorphisms.