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The complexity of equivalence for commutative rings

H.B. Hunt, R.E. Stearns
1990 Journal of symbolic computation  
We study the deterministic time complexity of the equivalence problems for formulas and for straight-line programs on commutative rings.  ...  As corollaries of this theorem, we characterize the deterministic time complexity of these two equivalence problems, for atl finite commutative rings and for all commutative unitary rings of zero or prime  ...  Preliminary versions of some of these results were presented at the Fifth International Conference on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes, Menorca, Spain, June 1987.  ... 
doi:10.1016/s0747-7171(08)80053-3 fatcat:yn76kp2gabfubkwdrlmt6ej6ha

The homotopy theory of complete modules [article]

Luca Pol, Jordan Williamson
2020 arXiv   pre-print
Given a ring homomorphism R → S, we then give necessary and sufficient conditions for the categories of complete R-complexes and the categories of complete S-complexes to have equivalent homotopy theories  ...  Given a commutative ring R and finitely generated ideal I, one can consider the classes of I-adically complete, L_0^I-complete and derived I-complete complexes.  ...  For a commutative ring R and an ideal I, we fix the following notation: • Ch(R) denotes the category of (unbounded) chain complexes of R-modules; • Ch(R, I) denotes the category of complexes of I-adically  ... 
arXiv:2011.06989v1 fatcat:vyqsy44l7rf6roxqfduwdfxv4y

Genuine-commutative ring structure on rational equivariant K-theory for finite abelian groups [article]

Anna Marie Bohmann, Christy Hazel, Jocelyne Ishak, Magdalena Kędziorek, Clover May
2021 arXiv   pre-print
This means that every genuine-commutative ring spectrum whose homotopy groups are those of KU_ℚ,G is weakly equivalent, as a genuine-commutative ring spectrum, to KU_ℚ,G.  ...  In contrast, the connective rational equivariant K-theory spectrum does not have this type of uniqueness of genuine-commutative ring structure.  ...  We also thank Mike Hill for many useful conversations and Christian Wimmer for sharing a draft of his work. The first author thanks Spencer Dowdall for acting as a notational soundingboard.  ... 
arXiv:2104.01079v1 fatcat:c22a5kmqfjgfjnyftqeensg7ye

Cochains and homotopy type

Michael A. Mandell
2006 Publications mathématiques (Bures-sur-Yvette)  
The cochain functor from the homotopy category of finite type nilpotent spaces to the homotopy category of E-infinity algebras is faithful but not full.  ...  Finite type nilpotent spaces are weakly equivalent if and only if their singular cochains are quasi-isomorphic as E-infinity algebras.  ...  For commutative ring coefficients, expect some kind of ring structure. For commutative ring coefficients, expect some kind of ring structure.  ... 
doi:10.1007/s10240-006-0037-6 fatcat:x3rgb7ydh5dirj7l5hlms4q7hm

An algebraic model for commutative Hℤ–algebras

Birgit Richter, Brooke Shipley
2017 Algebraic and Geometric Topology  
We show that the homotopy category of commutative algebra spectra over the Eilenberg-Mac Lane spectrum of the integers is equivalent to the homotopy category of E-infinity-monoids in unbounded chain complexes  ...  We also provide a Quillen equivalence to commutative monoids in the category of functors from the category of finite sets and injections to unbounded chain complexes.  ...  Mike Mandell showed in [M03, 7.11 ] that for every commutative ring R the homotopy categories of E ∞ -HR-algebra spectra and of E ∞ monoids in the category of unbounded R-chain complexes are equivalent  ... 
doi:10.2140/agt.2017.17.2013 fatcat:jqjigera4bbfznbsyzwzcda4n4

Totally acyclic complexes

Sergio Estrada, Xianhui Fu, Alina Iacob
2017 Journal of Algebra  
Thus we improve slightly on a result of Iyengar's and Krause's; in [22] they proved that for a commutative noetherian ring R with a dualizing complex, the class of acyclic complexes of injectives coincides  ...  We prove (Corollary 1) that, over a commutative noetherian ring R, the equivalent statements in Theorem 3 hold if and only if the ring is Gorenstein.  ...  Thus R is an Iwanaga-Gorenstein ring.  ... 
doi:10.1016/j.jalgebra.2016.09.009 fatcat:gtrlapwu7ngibeuijrnvqttwni

Local Gorenstein duality in chromatic group cohomology [article]

Luca Pol, Jordan Williamson
2021 arXiv   pre-print
We consider local Gorenstein duality for cochain spectra C^*(BG;R) on the classifying spaces of compact Lie groups G over complex orientable ring spectra R.  ...  We show that it holds systematically for a large array of examples of ring spectra R, including Lubin-Tate theories, topological K-theory, and various forms of topological modular forms.  ...  By a complex oriented commutative ring spectrum we always mean a commutative ring spectrum with an E 1 -complex orientation.  ... 
arXiv:2106.08669v2 fatcat:ixkbfs2llbbtfmtgzqa37dmf3q


2011 Glasgow Mathematical Journal  
In this paper we investigate the computational complexity of the equivalence problem for finite rings.  ...  We investigate the complexity of the equivalence problem over a finite ring when the input polynomials are written as sum of monomials.  ...  Theorem 3 establishes polynomial time complexity for an abundant class of rings for which R/J is commutative. COROLLARY 4.  ... 
doi:10.1017/s001708951100053x fatcat:wsaerkkpjndjffq35h543ylfla

Derived equivalences between associative deformations

Amnon Yekutieli
2010 Journal of Pure and Applied Algebra  
The main result of the paper (Theorem 2.7) says that if T is a two-sided tilting complex over B-A relative to K, then T ∼ = P [n] for some invertible bimodule P and integer n.  ...  Introduction Let K be a commutative ring, and let A and B be associative unital K-algebras. We denote by Mod A and Mod B the corresponding categories of left modules.  ...  I wish to thank Bursztyn and Waldmann for explaining their work to me, and the organizers of the conference for providing the background for this interaction.  ... 
doi:10.1016/j.jpaa.2009.11.011 fatcat:yenvdgnzyzbjthwdmfyjpdo4oi

Derived completions in stable homotopy theory [article]

Gunnar Carlsson
2007 arXiv   pre-print
We construct a notion of derived completion which applies to homomorphisms of commutative S-algebras.  ...  We study the relationship of the construction with other constructions of completions, and prove various invariance properties.  ...  The case of rings For any ring A, we may construct the Eilenberg-MacLane spectrum H(A).  ... 
arXiv:0707.2585v1 fatcat:6yrjbo37nnh6rnbm4f7v7ri5ty

Commutative ring spectra [article]

Birgit Richter
2017 arXiv   pre-print
The notion of etale extensions in the spectral world is tricky and we explain why. We define Picard groups and Brauer groups of commutative ring spectra and present examples.  ...  As a first interesting class of examples of such ring spectra we focus on (commutative) algebra spectra over commutative Eilenberg-MacLane ring spectra.  ...  Then there is a chain of Quillen equivalences between the model category of commutative HR-algebra spectra and E ∞monoids in the category of unbounded R-chain complexes.  ... 
arXiv:1710.02328v1 fatcat:wvemp5mb6fa5fjzg6ub424mhvy

Formal groups and stable homotopy of commutative rings

Stefan Schwede
2004 Geometry and Topology  
We show that formal group laws account for all such ring spectrum maps, and we identify the space of ring spectrum maps between HZ and DB.  ...  We study a ring spectrum denoted DB which depends on a commutative ring B and is closely related to the topological Andre-Quillen homology of B.  ...  Something similar happens for Gamma-rings. Suppose R is a Gamma-ring which is stably equivalent to an algebra over the commutative Gamma-ring k .  ... 
doi:10.2140/gt.2004.8.335 fatcat:jhu57cpgwfhfpep56x2nyth5ei

Duality in algebra and topology

W.G. Dwyer, J.P.C. Greenlees, S. Iyengar
2006 Advances in Mathematics  
rings of finite groups, (4) Poincaré duality for groups and (5) Gross-Hopkins duality in chromatic stable homotopy theory as examples of a single phenomenon.  ...  This allows us to prove new duality results in algebra and topology, and to view (1) Poincaré duality for manifolds, (2) Gorenstein duality for commutative rings, (3) Benson-Carlson duality for cohomology  ...  From this point of view S plays the role of the ground ring for the category of spectra, just as Z is the ground ring for the category of chain complexes. Mapping spectra.  ... 
doi:10.1016/j.aim.2005.11.004 fatcat:ad7v54ahlrgkjmgnwzxheatgga

On the relationship between logarithmic TAQ and logarithmic THH [article]

Tommy Lundemo
2021 arXiv   pre-print
The latter is analogous to results of Weibel-Geller for Hochschild homology of discrete rings, and of McCarthy-Minasian and Mathew for topological Hochschild homology.  ...  We provide a new description of logarithmic topological Andr\'e-Quillen homology in terms of the indecomposables of an augmented ring spectrum.  ...  For example, the above result applies for the augmented commutative symmetric ring spectrum A → THH(A) → A for A connective.  ... 
arXiv:2004.03524v2 fatcat:u72pnpy3t5h5hl7n75wrzo2vrm

Spectra for commutative algebraists [article]

2006 arXiv   pre-print
The article is designed to explain to commutative algebraists what spectra (in the sense of algebraic topology) are, why they were originally defined, and how they can be useful for commutative algebra  ...  Just as any derived equivalence of rings is given by tensoring with a complex of bimodules, any Quillen equivalence between ring spectra is given by smashing with a bimodule spectrum [28] .  ...  The best known example is that of Morita equivalence, showing that a ring is derived equivalent to the ring of n × n matrices over it.  ... 
arXiv:math/0609452v1 fatcat:xlizx6fc3fh7lofdao4ujynyrm
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