The squaring operation for commutative DG rings

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• 作者：Amnon Yekutieli ^{amyekut@math.bgu.ac.il" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 16E45 ; secondary ; 16E35 ; 18G10 ; 13D03 ; 18E30
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：1 March 2016
• 年：2016
• 卷：449
• 期：Complete
• 页码：50-107
• 全文大小：810 K

文摘

Let e05cd3a5da7" title="Click to view the MathML source">A→B$A\to B$ be a homomorphism of commutative rings. The squaring operation   is a functor Sq_B/A${\mathrm{Sq}}_{B/A}$ from the derived category D(B)$\mathsf{D}(B)$ of complexes of B-modules into itself. This operation is needed for the definition of rigid complexes (in the sense of Van den Bergh), that in turn leads to a new approach to Grothendieck duality for rings, schemes and even DM stacks.

In our paper with J.J. Zhang from 2008 we introduced the squaring operation, and explored some of its properties. Unfortunately some of the proofs in that paper had severe gaps in them.

In the present paper we reproduce the construction of the squaring operation. This is done in a more general context than in the first paper: here we consider a homomorphism e05cd3a5da7" title="Click to view the MathML source">A→B$A\to B$ of commutative DG rings. Our first main result is that the square  Sq_B/A(M)${\mathrm{Sq}}_{B/A}(M)$of a DG B-module M is independent of the resolutions used to present it. Our second main result is on the trace functoriality of the squaring operation. We give precise statements and complete correct proofs.

In a subsequent paper we will reproduce the remaining parts of the 2008 paper that require fixing. This will allow us to proceed with the other papers, mentioned in the bibliography, on the rigid approach to Grothendieck duality.

The proofs of the main results require a substantial amount of foundational work on commutative and noncommutative DG rings, including a study of semi-free DG rings, their lifting properties, and their homotopies. This part of the paper could be of independent interest.