Powers of ghost ideals
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912110874722304 |
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| author | Estrada, S. Fu, X. H. Herzog, I. Odabaşı, S. |
| author_facet | Estrada, S. Fu, X. H. Herzog, I. Odabaşı, S. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_05250 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Powers of ghost ideals Estrada, S. Fu, X. H. Herzog, I. Odabaşı, S. Category Theory Rings and Algebras 18E10, 18G25, 18G35, 16D90 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. |
| title | Powers of ghost ideals |
| topic | Category Theory Rings and Algebras 18E10, 18G25, 18G35, 16D90 |
| url | https://arxiv.org/abs/2411.05250 |