The acyclicity of the Frobenius functor for modules of finite flat dimension

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• 作者：Thomas Marley^a ; ^{tmarley1@unl.edu" class="auth_mail" title="E-mail the corresponding author} ; Marcus Webb^b ; ^{webbm@sfasu.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Primary ; 13D07 ; secondary ; 13A35
• 刊名：Journal of Pure and Applied Algebra
• 出版年：2016
• 出版时间：August 2016
• 年：2016
• 卷：220
• 期：8
• 页码：2886-2896
• 全文大小：377 K

文摘

Let R be a commutative Noetherian local ring of prime characteristic p   and f:R⟶R$f:R⟶R$ the Frobenius ring homomorphism. For e≥1$e\ge 1$ let R^(e)$R^{(e)}$ denote the ring R viewed as an R  -module via f^e$f^e$. Results of Peskine, Szpiro, and Herzog state that for finitely generated modules M, M   has finite projective dimension if and only if ${\mathrm{Tor}}_i^R(R^{(e)},M)=0$ for all i>0$i>0$ and all (equivalently, infinitely many) e≥1$e\ge 1$. We prove this statement holds for arbitrary modules using the theory of flat covers and minimal flat resolutions.