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1. Rigidity properties of the cotangent complex

   https://doi.org/10.1090/jams/1000  

   Briggs, Benjamin; Iyengar, Srikanth (January 2023, Journal of the American Mathematical Society)

   This work concerns a map φ : R → S \varphi \colon R\to S of commutative noetherian rings, locally of finite flat dimension. It is proved that the André-Quillen homology functors are rigid, namely, if D n ( S / R ; − ) = 0 \mathrm {D}_n(S/R;-)=0 for some n ≥ 1 n\ge 1 , then D i ( S / R ; − ) = 0 \mathrm {D}_i(S/R;-)=0 for all i ≥ 2 i\ge 2 and φ {\varphi } is locally complete intersection. This extends Avramov’s theorem that draws the same conclusion assuming D n ( S / R ; − ) \mathrm {D}_n(S/R;-) vanishes for all n ≫ 0 n\gg 0 , confirming a conjecture of Quillen. The rigidity of André-Quillen functors is deduced from a more general result about the higher cotangent modules which answers a question raised by Avramov and Herzog, and subsumes a conjecture of Vasconcelos that was proved recently by the first author. The new insight leading to these results concerns the equivariance of a map from André-Quillen cohomology to Hochschild cohomology defined using the universal Atiyah class of φ \varphi . 

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2. Models of G–spectra as presheaves of spectra

   https://doi.org/10.2140/agt.2024.24.1225  

   Guillou, Bertrand J; May, J Peter (January 2024, Algebraic & Geometric Topology)

   Greenlees, John (Ed.)

   Let G be a finite group. We give Quillen equivalent models for the category of G–spectra as categories of spectrally enriched functors from explicitly described domain categories to nonequivariant spectra. Our preferred model is based on equivariant infinite loop space theory applied to elementary categorical data. It recasts equivariant stable homotopy theory in terms of point–set-level categories of G–spans and nonequivariant spectra. We also give a more topologically grounded model based on equivariant Atiyah duality. 

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3. A Tannakian Framework for G-displays and Rapoport–Zink Spaces

   https://doi.org/10.1093/imrn/rnz311  

   Daniels, Patrick (December 2019, International Mathematics Research Notices)

   Abstract We develop a Tannakian framework for group-theoretic analogs of displays, originally introduced by Bültel and Pappas, and further studied by Lau. We use this framework to define Rapoport–Zink functors associated to triples $$(G,\{\mu \},[b])$$, where $$G$$ is a flat affine group scheme over $${\mathbb{Z}}_p$$ and $$\mu$$ is a cocharacter of $$G$$ defined over a finite unramified extension of $${\mathbb{Z}}_p$$. We prove these functors give a quotient stack presented by Witt vector loop groups, thereby showing our definition generalizes the group-theoretic definition of Rapoport–Zink spaces given by Bültel and Pappas. As an application, we prove a special case of a conjecture of Bültel and Pappas by showing their definition coincides with that of Rapoport and Zink in the case of unramified EL-type local Shimura data. 

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4. Flat line bundles and the Cappell-Miller torsion in Arakelov geometry

   https://doi.org/10.24033/asens.2409  

   Freixas i Montplet, Gerard; Wentworth, Richard A. (January 2019, Annales scientifiques de lÉcole normale supérieure)

   null (Ed.)

   In this paper, we extend Deligne’s functorial Riemann-Roch isomorphism for Hermitian holomorphic line bundles on Riemann surfaces to the case of flat, not necessarily unitary connections. The Quillen metric and ?-product of Gillet-Soulé are replaced with complex valued logarithms. On the determinant of cohomology side, we show that the Cappell-Miller torsion is the appropriate counterpart of the Quillen metric. On the Deligne pairing side, the logarithm is a refinement of the intersection connections considered in a previous work. The construction naturally leads to an Arakelov theory for flat line bundles on arithmetic surfaces and produces arithmetic intersection numbers valued in C/πi Z. In this context we prove an arithmetic Riemann-Roch theorem. This realizes a program proposed by Cappell-Miller to show that their holomorphic torsion exhibits properties similar to those of the Quillen metric proved by Bismut, Gillet and Soulé. Finally, we give examples that clarify the kind of invariants that the formalism captures; namely, periods of differential forms. 

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5. Program adverbs and Tlön embeddings

   https://doi.org/10.1145/3547632  

   Li, Yao; Weirich, Stephanie (August 2022, Proceedings of the ACM on Programming Languages)

   Free monads (and their variants) have become a popular general-purpose tool for representing the semantics of effectful programs in proof assistants. These data structures support the compositional definition of semantics parameterized by uninterpreted events, while admitting a rich equational theory of equivalence. But monads are not the only way to structure effectful computation, why should we limit ourselves? In this paper, inspired by applicative functors, selective functors, and other structures, we define a collection of data structures and theories, which we call program adverbs, that capture a variety of computational patterns. Program adverbs are themselves composable, allowing them to be used to specify the semantics of languages with multiple computation patterns. We use program adverbs as the basis for a new class of semantic embeddings called Tlön embeddings. Compared with embeddings based on free monads, Tlön embeddings allow more flexibility in computational modeling of effects, while retaining more information about the program's syntactic structure. 

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