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Abstract. It is proved that if G is a finite group, then the order of G is a proper upper bound for the phantom number of G. This answers a question of Benson and Gnacadja. More specifically, if k is a field whose characteristic divides the order of G, and Φ is the ideal of phantom morphisms in the stable category k[G]-Mod of modules over the group algebra k[G], then Φ n−1 = 0, where n is the nilpotency index of the Jacobson radical J of k [G]. If R is a semiprimary ring, with J n = 0, and Φ denotes the phantom ideal in the module category R-Mod, then Φ n is the ideal of morphisms that factor through a projective module. If R is a right coherent ring and every cotorsion left R-module has a coresolution of length n by pure injective modules, then Φ n+1 is the ideal of morphisms that factor through a flat module.These results are obtained by introducing the mono-epi (ME) exact structure on the morphisms of an exact category (A; E ). This exact structure (Arr(A); ME) is used to develop further the ideal approximation theory of (A; E ), by proving new versions of Salce's Lemma, the Ghost Lemma of Christensen, and Wakamatsu's Lemma. Salce's Lemma states that if (A; E ) has enough injective morphisms and projective morphisms, then the map I → I ⊥ on ideals is a bijective correspondence between the class of special precovering ideals of (A; E ) and that of its special preenveloping ideals. The exact category (Arr(A); ME) of morphisms allows us to introduce the notion of an extension i ⋆ j of morphisms in an exact category (A; E ), and the notion of an extension I ⋄ J of ideals of A. The Ghost Lemma, instrumental in proving the consequences above, asserts that the class of special precovering (resp., special preenveloping) ideals is closed under products and extensions and that the bijective correspondence of Salce's Lemma satisfies (IJ )Wakamatsu's Lemma is the statement that if a covering ideal I is closed under extensions I ⋄I = I, then I is a special precovering ideal that possesses a syzygy ideal Ω(I) ⊆ I ⊥ generated by objects.
Abstract. We prove first (Proposition 3) that, over any ring R, an acyclic complex of projective modules is totally acyclic if and only if the cycles of every acyclic complex of Gorenstein projective modules are Gorenstein projective. The dual result for injective and Gorenstein injective modules also holds over any ring R (Proposition 4). And, when R is a GF-closed ring, the analogue result for flat/Gorenstein flat modules is also true (Proposition 5). Then we show (Theorem 2) that over a left noetherian ring R, a third equivalent condition can be added to those in Proposition 4, more precisely, we prove that the following are equivalent: 1. Every acyclic complex of injective modules is totally acyclic. 2. The cycles of every acyclic complex of Gorenstein injective modules are Gorenstein injective. 3. Every complex of Gorenstein injective modules is dgGorenstein injective. Theorem 3 shows that the analogue result for complexes of flat and Gorenstein flat modules holds over any left coherent ring R. We prove (Corollary 1) that, over a commutative noetherian ring R, the equivalent statements in Theorem 3 hold if and only if the ring is Gorenstein. We also prove (Theorem 4) that when moreover R is left coherent and right n-perfect (that is, every flat right R-module has finite projective dimension ≤ n) then statements 1, 2, 3 in Theorem 2 are also equivalent to the following: 4. Every acyclic complex of projective right R-modules is totally acyclic. 5. Every acyclic complex of Gorenstein projective right R-modules is in GP. 6. Every complex of Gorenstein projective right R-modules is dg-Gorenstein projective. Corollary 2 shows that when R is commutative noetherian of finite Krull dimension, the equivalent conditions (1)-(6) from Theorem 4 are also equivalent to those in Theorem 3 and hold if and only if R is an Iwanaga-Gorenstein ring. Thus we improve slightly on a result of Iyengar's and Krause's; in [22] they proved that for a commutative noetherian ring R with a dualizing complex, the class of acyclic complexes of injectives coincides with that of totally acyclic complexes of injectives if and only if R is Gorenstein. We are able to remove the dualizing complex hypothesis and add more equivalent conditions. In the second part of the paper we focus on two sided noetherian rings that satisfy the Auslander condition. We prove (Theorem 7) that for such a ring R that also has finite finitistic flat dimension, every complex of injective (left and respectively right) R-modules is totally acyclic if and only if R is an Iwanaga-Gorenstein ring.
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