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By Melvin Hochster, C. Huneke

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Phantom Homology

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N are sequences of modules and ideals, respectively, and that Mj has c-depth > i on /,- compatibly for 1 < i < j < n. Consider the sequences M * + i , . . ,iVn_jfc and Ji,... , xn-k satisfy the requirement in the definition of compatibility for the new pair of sequences. , Ih for 1 < h < n. e) The compatible c-depth requirement for a pair of sequences is evidently preserved by localization. e. $ *' 5s i < H> then this is true compatibly. We recall from §9 of [HH4] that the function • is defined recursively by D(0) = 1 and D(n + 1) = (D(0) + • • • 4- D(n)) + D(n) + n + 2, n > 1.

Of the Koszul complex K*(TL\ R) into N. ) The 28 MELVIN HOCHSTER AND CRAIG HUNEKE fact that . is a map of complexes implies that it is a cycle in Tm. We shall show that it is a boundary after multiplying by c n . 44. One of the spectral sequences of the double complex has 0j_j = fcU*(x;iV , j) as the degree k term of E\ (where # ' ( x ; ) indicates Koszul cohomology: recall that H*(x; N) = # n _ i ( x ; N) by virtue of the self-duality of the Koszul complex). ;Nj), which is killed by c if j < n and vanishes if j = n because Nn is 0.

14) for a discussion concerning complexes of projective modules. Let (G # , a # ) be a complex of finitely generated projective R-modules. 14): however, the ranks may vary with the connected component in the case where Spec(-R) is disconnected and so the additivity condition must be checked locally. If a complex of projectives has the property that Hi(FeG*) is phantom for all i > 1 and all e G N, this condition is also true in each localization. It follows that for each situation where we know that the standard conditions on rank and height (respectively, minheight) must then hold in the free case, we get the same result in the projective case.

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