Module-Theoretic Characterizations of Prufer $v$-Multiplication Domains
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918142542872576 |
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| author | Zhang, Xiaolei Kim, Hwankoo |
| author_facet | Zhang, Xiaolei Kim, Hwankoo |
| contents | We present unified $w$-theoretic characterizations of Prüfer $v$-multiplication domains (P$v$MDs).
A module-theoretic perspective shows that torsion submodules are $w$-pure, and for $(w$-)$\,$finitely generated modules $M$, the canonical sequence
$0\to T(M)\to M\to M/T(M)\to 0$ $w$-splits, resolving an open question of Geroldinger--Kim--Loper.
In a $w$-version of Hattori-Davis theory, these conditions are equivalent to $Tor^R_2(M,N)$ being $GV$-torsion for all $R$-modules $M,N$, equivalently $w$-w.gl.dim$(R)\leq 1$, or $Tor^R_1(X,A)$ being $GV$-torsion for all $X$ and torsion-free $A$, or the Davis map $A\otimes_R B \to \mathcal T\otimes_K \mathcal S$ having $GV$-torsion kernel.
From an overring viewpoint, $R$ is a P$v$MD if and only if for every $R\subseteq T\subseteq K$ and every $w$-maximal ideal $m$, the localization $R_{m}\to T_{\m}$ is a flat epimorphism, so that each overring is $w$-flat and the inclusion is $w$-epimorphic.
Finally, $R$ is a P$v$MD if and only if every pure $w$-injective divisible $R$-module is injective. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_13617 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Module-Theoretic Characterizations of Prufer $v$-Multiplication Domains Zhang, Xiaolei Kim, Hwankoo Commutative Algebra We present unified $w$-theoretic characterizations of Prüfer $v$-multiplication domains (P$v$MDs). A module-theoretic perspective shows that torsion submodules are $w$-pure, and for $(w$-)$\,$finitely generated modules $M$, the canonical sequence $0\to T(M)\to M\to M/T(M)\to 0$ $w$-splits, resolving an open question of Geroldinger--Kim--Loper. In a $w$-version of Hattori-Davis theory, these conditions are equivalent to $Tor^R_2(M,N)$ being $GV$-torsion for all $R$-modules $M,N$, equivalently $w$-w.gl.dim$(R)\leq 1$, or $Tor^R_1(X,A)$ being $GV$-torsion for all $X$ and torsion-free $A$, or the Davis map $A\otimes_R B \to \mathcal T\otimes_K \mathcal S$ having $GV$-torsion kernel. From an overring viewpoint, $R$ is a P$v$MD if and only if for every $R\subseteq T\subseteq K$ and every $w$-maximal ideal $m$, the localization $R_{m}\to T_{\m}$ is a flat epimorphism, so that each overring is $w$-flat and the inclusion is $w$-epimorphic. Finally, $R$ is a P$v$MD if and only if every pure $w$-injective divisible $R$-module is injective. |
| title | Module-Theoretic Characterizations of Prufer $v$-Multiplication Domains |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2509.13617 |