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Let <i>R</i> be a commutative ring, <i>I</i> be an ideal in <i>R</i> and let <i>M</i> be a <i>R/ I</i> -module. In this thesis we construct a <i>R/ I</i> -projective resolution of <i>M</i> using given <i>R</i>-projective resolutions of <i>M</i> and <i>I</i>. As immediate consequences of our construction we give descriptions of the canonical maps Ext<sub>R/I</sub><i>(M,N)</i> -> Ext<sub>R</sub><i>(M,N)</i> and Tor<sup>R</sup><sub>N</sub><i>(M, N)</i> -> Tor<sup>R/I</sup><sub>n</sub><i>(M, N)</i> for a <i>R/I</i> module <i>N</i> and we give a new proof of a theorem of Gulliksen [6] which states that if <i>I</i> is generated by a regular sequence of length r then ∐∞<sub>n=o</sub> Tor<sup>R/I</sup><sub>n</sub> <i>(M, N)</i> is a graded module over the polynomial ring </i>R/ I</i> [X₁. .. X<sub>r</sub>] with deg X<sub>i</sub> = -2, 1 ≤ i ≤ r. If <i>I</i> is generated by a regular element and if the <i>R</i>-projective dimension of <i>M</i> is finite, we show that <i>M</i> has a <i>R/ I</i>-projective resolution which is eventually periodic of period two.
This generalizes a result of Eisenbud [3]. In the case when <i>R</i> = (<i>R</i>, m) is a Noetherian local ring and <i>M</i> is a finitely generated <i>R/ I</i> -module, we discuss the minimality of the constructed resolution. If it is minimal we call (<i>M, I</i>) a Golod pair over <i>R</i>. We give a direct proof of a theorem of Levin [10] which states thdt if (<i>M,I</i>) is a Golod pair over <i>R</i> then (Ω<sup>n</sup><sub>R/I</sub>R/I(M),I) is a Golod pair over <i>R</i> where Ω<sup>n</sup><sub>R/I</sub>R/I(M) is the nth syzygy of the constructed <i>R/ I</i> -projective resolution of <i>M</i>. We show that the converse of the last theorem is not true and if (Ω¹<sub>R/I</sub>R/I(M),I) is a Golod pair over <i>R</i> then we give a necessary and sufficient condition for (<i>M, I</i>) to be a Golod pair over <i>R</i>.
Finally we prove that if (<i>M, I</i>) is a Golod pair over <i>R</i> and if a ∈ <i>I</i> - m<i>I</i> is a regular element in </i>R</i> then (<i>M</i>, (a)) and (1/(a), (a)) are Golod pairs over <i>R</i> and (<i>M,I</i>/(a)) is a Golod pair over <i>R</i>/(a). As a corrolary of this result we show that if the natural map π : <i>R</i> → <i>R/1</i> is a Golod homomorphism ( this means (<i>R</i>/m, <i>I</i>) is a Golod pair over <i>R</i> ,Levin [8]), then the natural maps π₁ : <i>R</i> → <i>R</i>/(a) and π₂ : <i>R</i>/(a) → <i>R/1</i> are Golod homomorphisms. |
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