Powers of ghost ideals
S. Estrada, X. H. Fu, I. Herzog, S. Odabaşı
Abstract
A theory of ordinal powers of the ideal $\mathfrak{g}_{\mathcal{S}}$ of $\mathcal{S}$-ghost morphisms is developed by introducing for every ordinal $λ$, the $λ$-th inductive power $\mathcal{J}^{(λ)}$ of an ideal $\mathcal{J}.$ The Generalized $λ$-Generating Hypothesis ($λ$-GGH) for an ideal $\mathcal J$ of an exact category $\mathcal{A}$ is the proposition that the $λ$-th inductive power ${\mathcal{J}}^{(λ)}$ is an object ideal. It is shown that under mild conditions every inductive power of a ghost ideal is an object-special preenveloping ideal. When $λ$ is infinite, the proof is based on an ideal version of Eklof's Lemma. When $λ$ is an infinite regular cardinal, the Generalized $λ$-Generating Hypothesis is established for the ghost ideal $\mathfrak{g}_{\mathcal{S}}$ for the case when $\mathcal A$ a locally $λ$-presentable Grothendieck category and $\mathcal{S}$ is a set of $λ$-presentable objects in $\mathcal A$ such that $^\perp (\mathcal{S}^\perp)$ contains a generating set for $\mathcal A.$ As a consequence of $λ$-GGH for the ghost ideal $\mathfrak{g}_{R\mbox{-}\mathrm{mod}}$ in the category of modules $R\mbox{-}\mathrm{Mod}$ over a ring, it is shown that if the class of pure projective left $R$-modules is closed under extensions, then every left FP-projective module is pure projective. A restricted version $n$-GGH($\mathfrak{g}(\mathbf{C}(R))$) for the ghost ideal in $\mathbf{C}(R))$ is also considered and it is shown that $n$-GGH($\mathfrak{g}(\mathbf{C}(R))$) holds for $R$ if and only if the $n$-th power of the ghost ideal in the derived category $\mathbf{D}(R)$ is zero if and only if the global dimension of $R$ is less than $n.$ If $R$ is coherent, then the Generating Hypothesis holds for $R$ if and only if $R$ is von Neumann regular.