On the exactness of products in the localization of (Ab.4^⁎) Grothendieck categories

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• 作者：Simone Virili¹ ; ^{virili@math.unipd.it" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 18E15 ; 18E40 ; 13D45 ; secondary ; 18G25 ; 13D09 ; 18E35
• 刊名：Journal of Algebra
• 出版年：2017
• 出版时间：15 January 2017
• 年：2017
• 卷：470
• 期：Complete
• 页码：37-67
• 全文大小：717 K

文摘

Let cc3b372832d00e006ab6c9f19890cec3" title="Click to view the MathML source">C$\mathfrak{C}$ be an (Ab.4^⁎) Grothendieck category, that is, products are exact in cc3b372832d00e006ab6c9f19890cec3" title="Click to view the MathML source">C$\mathfrak{C}$. Given a hereditary torsion class T⊆C$\mathcal{T}\subseteq \mathfrak{C}$, we study the exactness of products in the Gabriel localization C/T$\mathfrak{C}/\mathcal{T}$ of cc3b372832d00e006ab6c9f19890cec3" title="Click to view the MathML source">C$\mathfrak{C}$. We show that, under suitable assumptions on cc3b372832d00e006ab6c9f19890cec3" title="Click to view the MathML source">C$\mathfrak{C}$, the k+1$k+1$-th derived functor of the product vanishes, provided the Gabriel dimension of C/T$\mathfrak{C}/\mathcal{T}$ is smaller than k  . As a consequence, we deduce that, under suitable hypotheses, the derived category D(C/T)$\mathbf{D}(\mathfrak{C}/\mathcal{T})$ is left-complete.