Phantom Homology by Melvin Hochster, C. Huneke

By Melvin Hochster, C. Huneke

Ebook by way of Hochster, Melvin, Huneke, C.

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For all i > 1 and all e > 0. b) Conversely, let R be an arbitrary Noetherian ring of characteristic p > 0, and let G9 be a free complex over R with a,-, r,-, iV, 9$, R" defined as above. )) forl 0. 40 MELVIN HOCHSTER AND CRAIG HUNEKE Then the rank of a" — r; for 1 < i < n, and the minheight of N on Iri{ot") is at least h 1 < * < ™> which is equivalent to the statement that G'l satisfies the standard conditions for rank and minheight on R" <8>RN. Proof: We first prove that if J?

AND CRAIG HUNEKE These follow from the fact that they hold for M n _ i , hence for ( M „ _ i ) x , and from the fact that the sequence 0 —> (Mn)x —> ( M w _ i ) , —> (Coker an)x —+ 0 is split exact, so that (Coker a „ ) x is a direct summand of ( M n _ i ) x . (The fact that all the 3V for v > i expand to the unit ideal in Rx when x = X{ is harmless. )X is any of x\y . . , x n . It follows that 2, which imphes that c D ( n _ 1 ) kills 2:.

22). 45 Suppose that G. satisfies the phantom Let c G R be such that Rc is weakly Cohen-Macaulay let S be a regular ring (or a weakly F-regular ring with a completely stable test Suppose h : R —* S with h(c) G 5 ° . Then the induced map Hi(G9) and element). —» Hi(G. ®R S) is 0 for i > 1, and the same is true for the map # t ( G # ) —• #j(G' # ) for any complex G'm of S-modules to which G # maps. Proof: If we have a counterexample we can replace S by its localization at a suitable maximal ideal and we will still have a counterexample.

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