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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{fracture theorem} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory} [[!include homotopy - contents]] \hypertarget{arithmetic_geometry}{}\paragraph*{{Arithmetic geometry}}\label{arithmetic_geometry} [[!include arithmetic geometry - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak \noindent\hyperlink{Statement}{Statement}\dotfill \pageref*{Statement} \linebreak \noindent\hyperlink{ArithmeticFractureSquares}{In number theory and arithmetic geometry}\dotfill \pageref*{ArithmeticFractureSquares} \linebreak \noindent\hyperlink{InHomotopyTheory}{In homotopy theory}\dotfill \pageref*{InHomotopyTheory} \linebreak \noindent\hyperlink{InStableHomotopyTheory}{In stable homotopy theory}\dotfill \pageref*{InStableHomotopyTheory} \linebreak \noindent\hyperlink{TheArithmeticFractureSquare}{The arithmetic fracture square for spectra}\dotfill \pageref*{TheArithmeticFractureSquare} \linebreak \noindent\hyperlink{CompletionAndTorsionOnDerivedCategories}{The arithmetic fracture square for chain complexes}\dotfill \pageref*{CompletionAndTorsionOnDerivedCategories} \linebreak \noindent\hyperlink{GeneralFractureSquares}{General fracture squares of spectra}\dotfill \pageref*{GeneralFractureSquares} \linebreak \noindent\hyperlink{for_modules}{For $E_\infty$-modules}\dotfill \pageref*{for_modules} \linebreak \noindent\hyperlink{InCohesiveHomotopyTheory}{In cohesive (stable) homotopy theory}\dotfill \pageref*{InCohesiveHomotopyTheory} \linebreak \noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{Idea}{}\subsection*{{Idea}}\label{Idea} A basic fact in [[number theory]] is that the [[natural numbers]] may be decomposed into the [[rational numbers]] and the [[p-adic integers]] for all [[prime numbers]] $p$. Dually in [[arithmetic geometry]] this says that [[Spec(Z)]] has a [[cover]] by all its [[formal disks]] and the complements of finitely many points, a fact that is crucial in the geometric interpretation of the [[function field analogy]] and which motivates for instance the [[geometric Langlands correspondence]]. (See \hyperlink{ArithmeticFractureSquares}{below}.) Lifting this statement to [[stable homotopy theory]] and ``[[higher arithmetic geometry]]'' the \emph{arithmetic fracture theorem} says that [[stable homotopy types]] (and suitably tame plain [[homotopy types]]) canonically decompose into their [[rationalization]] and their [[p-completion]] for all primes $p$, hence into their images in [[rational homotopy theory]] and [[p-adic homotopy theory]]. Since these images are typically simpler than the original homotopy type itself, this decomposition is a fundamental computational tool in stable homotopy theory, often known under the slogan of ``working one prime at a time''. (See \hyperlink{TheArithmeticFractureSquare}{below}.) One finds that this arithmetic fracturing in stable homotopy theory is really a statement about the [[Bousfield localization of spectra]] with respect to the [[Moore spectrum]] for $\mathbb{Q}$ and that of $\mathbb{Q}/\mathbb{Z}$. Viewed this way there is a more general fracture theorem which says that for any suitable pair $E,F$ of [[spectra]]/[[homology theories]] the [[Bousfield localization of spectra|Bousfield localization]] at their [[coproduct]] decomposes into the separate Bousfield localizations. This generalized fracture theorem appears for instance in [[chromatic homotopy theory]] for localization at [[Morava K-theory]] and [[Morava E-theory]]. (See \hyperlink{GeneralFractureSquares}{below}.) In [[cohesive homotopy theory]] every [[stable homotopy type]] canonically sits in a fracture square formed from the localizations exhibited by the [[shape modality]] and the [[flat modality]]. For [[differential cohesion]] over [[infinitesimal cohesion]] this is a [[higher geometry|higher geometric]] analog of the classical artihmetic fracture. (See \hyperlink{InCohesiveHomotopyTheory}{below}.) \hypertarget{Statement}{}\subsection*{{Statement}}\label{Statement} \hypertarget{ArithmeticFractureSquares}{}\subsubsection*{{In number theory and arithmetic geometry}}\label{ArithmeticFractureSquares} The statement in [[number theory]]/[[arithmetic geometry]] is the following: \begin{prop} \label{ArithmeticFractureSquare}\hypertarget{ArithmeticFractureSquare}{} The [[integers]] $\mathbb{Z}$ are the [[fiber product]] of all the [[p-adic integers]] $\underset{p\;prime}{\prod} \mathbb{Z}_p$ with the [[rational numbers]] $\mathbb{Q}$ over the [[rationalization]] of the former, hence there is a [[pullback]] [[diagram]] in [[CRing]] of the form \begin{displaymath} \itexarray{ && \mathbb{Q} \\ & \swarrow && \nwarrow \\ \mathbb{Q}\otimes_{\mathbb{Z}}\left(\underset{p\;prime}{\prod} \mathbb{Z}_p \right) && && \mathbb{Z} \\ & \nwarrow && \swarrow \\ && \underset{p\;prime}{\prod} \mathbb{Z}_p } \,. \end{displaymath} Equivalently this is the [[fiber product]] of the rationals with the [[integral adeles]] $\mathbb{A}_{\mathbb{Z}}$ over the [[ring of adeles]] $\mathbb{A}_{\mathbb{Q}}$ \begin{displaymath} \itexarray{ && \mathbb{Q} \\ & \swarrow && \nwarrow \\ \mathbb{A}_{\mathbb{Q}} && && \mathbb{Z} \\ & \nwarrow && \swarrow \\ && \mathbb{A}_{\mathbb{Z}} } \,, \end{displaymath} Since the [[ring of adeles]] is the [[rationalization]] of the integral adeles $\mathbb{A}_{\mathbb{Q}} = \mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{A}_{\mathbb{Z}}$, this is also (by the discussion \href{category+of+monoids#PushoutOfCommutativeMonoids}{here}) a [[pushout]] diagram in [[CRing]], and in fact in topological [[commutative rings]] (for $\mathbb{Q}$ with the [[discrete topology]] and $\mathbb{A}_{\mathbb{Z}}$ with its profinite/[[completion of a ring|completion]] topology). \end{prop} An original discussion is (\hyperlink{Sullivan05}{Sullivan 05, prop. 1.18}). Review includes (\hyperlink{Riehl14}{Riehl 14, lemma 14.4.2}). \begin{remark} \label{GeometricMeaning}\hypertarget{GeometricMeaning}{} Under the [[function field analogy]] we may think of \begin{itemize}% \item $Spec(\mathbb{Z})$ as an [[arithmetic curve]] over [[F1]]; \item $\mathbb{A}_{\mathbb{Z}}$ as the [[ring of functions]] on the [[formal disks]] around all the points in this curve; \item $\mathbb{Q}$ as the ring of functions on the complement of a finite number of points in the curve; \item $\mathbb{A}_{\mathbb{Q}}$ is the ring of functions on punctured formal disks around all points, at most finitely many of which do not extend to the unpunctured disk. \end{itemize} Under this [[analogy]] the arithmetic fracture square of prop. \ref{ArithmeticFractureSquare} says that the curve $Spec(\mathbb{Z})$ has a [[cover]] whose patches are the complement of the curve by some points, and the formal disks around these points. This kind of cover plays a central role in [[number theory]], see for instance the following discussions: \begin{itemize}% \item \emph{\href{moduli+space+of+bundles#OverCurvesAndTheLanglandsCorrespondence}{Moduli stack of bundles over curves}}; \item \emph{[[geometric Langlands correspondence]]}; \item \emph{[[Weil conjecture on Tamagawa numbers]]}. \end{itemize} \end{remark} \hypertarget{InHomotopyTheory}{}\subsubsection*{{In homotopy theory}}\label{InHomotopyTheory} In [[homotopy theory]] the corresponding statement is that [[homotopy types]] may be decomposed into that of [[rational homotopy types]] and [[p-complete homotopy types]] of [[p-local homotopy types]]. \begin{remark} \label{FractureForSpectra}\hypertarget{FractureForSpectra}{} Let $p$ be a [[prime number]]. Let $X$ be a [[homotopy type]]/[[∞-groupoid]] satisfying at least one of the following sufficient conditions \begin{itemize}% \item $X$ is a [[connected]], [[nilpotent space]] with finitely generated [[homotopy groups]]; \item $X$ is [[p-local homotopy type]]; \end{itemize} Then $X$ is the [[homotopy fiber product]] \begin{displaymath} X \simeq X_{\mathbb{Q}} \underset{(X_p^\wedge)_{\mathbb{Q}}}{\times} X_p^\wedge \end{displaymath} of its [[rationalization]] $X_{\mathbb{Q}}$ with its [[p-completion]] $X_p^\wedge$ over the rationalization $(X_p^\wedge)_{\mathbb{Q}}$of the $p$-completions. \end{remark} This originates around (\hyperlink{BousfieldKan72}{Bousfield-Kan 72, VI.8.1}). A detailed more modern account is in (\hyperlink{MayPonto}{May-Ponto, theorem 13.1.4}). A quick survey is in (\hyperlink{Riehl14}{Riehl 14, theorem 14.4.14}). \hypertarget{InStableHomotopyTheory}{}\subsubsection*{{In stable homotopy theory}}\label{InStableHomotopyTheory} Similar statements hold in [[stable homotopy theory]] for [[spectra]]. There is a stable version of \begin{itemize}% \item \hyperlink{TheArithmeticFractureSquare}{The arithmetic fracture square} \end{itemize} but more generally there are fracture squares for the [[coproduct]] homology theory $E \vee F$ whenever $F$-localization is $E$-acyclic: \begin{itemize}% \item \hyperlink{GeneralFractureSquares}{General fracture squares} \end{itemize} \begin{quote}% warning: a condition missing in the following, see the comment section of \href{https://mathoverflow.net/a/91057/381}{this MO reply}. Somebody should add the relevant clause here\ldots{} \end{quote} \hypertarget{TheArithmeticFractureSquare}{}\paragraph*{{The arithmetic fracture square for spectra}}\label{TheArithmeticFractureSquare} For $p$ a [[prime number]] write \begin{itemize}% \item $L_p$ for [[Bousfield localization of spectra]] at the [[Moore spectrum]] $S \mathbb{F}_p$, hence for [[p-completion]] $(-)_p^\wedge$; \item $L_{\mathbb{Q}}$ for the [[Bousfield localization of spectra]] at the [[Moore spectrum]]/[[Eilenberg-MacLane spectrum]] $S \mathbb{Q} \simeq H \mathbb{Q}$, hence for [[rationalization]]. \end{itemize} \begin{prop} \label{SullivanArithmeticFracture}\hypertarget{SullivanArithmeticFracture}{} \textbf{(Sullivan arithmetic square)} For every [[spectrum]] $X$ the canonical square \begin{displaymath} \itexarray{ && L_{\mathbb{Q}}X \\ & \swarrow && \nwarrow \\ L_{\mathbb{Q}} \left( \prod_p L_p X \right) && && X \\ & \nwarrow && \swarrow \\ && \prod_p L_p X } \end{displaymath} is a [[homotopy pushout]] (hence also a [[homotopy pullback]]). \end{prop} Original statements of this include (\hyperlink{Bousfield79}{Bousfield 79}, \hyperlink{Sullivan05}{Sullivan 05, prop. 3.20}). Review includes (\hyperlink{VanKoughnett13}{van Koughnett 13, prop. 4.5}, \hyperlink{Bauer11}{Bauer 11, lemma 2.1}). \begin{remark} \label{}\hypertarget{}{} \textbf{(``one prime at a time'')} The impact of prop. \ref{FractureForSpectra} is that it decomposes the study of ([[stable homotopy theory|stable]]) [[homotopy theory]] into that of \begin{enumerate}% \item [[rational homotopy theory]] and \item [[p-adic homotopy theory]] for each prime $p$. \end{enumerate} Both the [[rationalization]] $X_{\mathbb{Q}}$ and the [[p-completion]] $X_{p}^\wedge$ are typically much easier to analyze than $p$ itself and so the fracture theorem gives a way to decompose the remaining hard part of study of [[homotopy types]] into that of $p$-local/$p$-complete spaces. This procedure is known in homotopy theory as working ``one prime at a time''. \end{remark} More generally: \begin{prop} \label{ReformulationOfProdOverPComletionByLocalizationAtCoproduct}\hypertarget{ReformulationOfProdOverPComletionByLocalizationAtCoproduct}{} The product of all [[p-completions]] is equivalently the [[Bousfield localization of spectra]] at the [[wedge sum]] $\vee_p S \mathbb{F}_p$ of all [[Moore spectra]] \begin{displaymath} \prod_p L_p X \simeq L_{\vee_p S \mathbb{F}_p} X \,. \end{displaymath} Moreover there is a [[Bousfield equivalence]] \begin{displaymath} S (\mathbb{Q}/\mathbb{Z}) \simeq_{Bousf} \vee_p S \mathbb{F}_p \,, \end{displaymath} and therefore also an equivalence \begin{displaymath} \prod_p L_p X \simeq L_{S (\mathbb{Q}/\mathbb{Z})} X \,. \end{displaymath} \end{prop} The first statement originates around (\href{Bousfield+localization+of+spectra#Bousfield79}{Bousfield 79, prop. 2.6}), review includes (\href{Bousfield+localization+of+spectra#VanKoughnett13}{van Koughnett 13, prop. 4.4}, \hyperlink{Bauer11}{Bauer 11, below prop. 2.2}); the second is highlighted in (\href{http://mathoverflow.net/a/91057/381}{Strickland 12, MO comment}). \begin{remark} \label{TwoThirdHexagon}\hypertarget{TwoThirdHexagon}{} By the discussion at \emph{[[Bousfield localization of spectra]]} and at \emph{[[localization of a space]]}, the [[rationalization]] and the [[p-completion]] maps on spectra are [[homotopy cofibers]] of $E$-acyclifications $G_E(X) \to X$, for $E = S \mathbb{Q} \simeq H \mathbb{Q}$ and $E = S \mathbb{F}_p$ the [[Moore spectra]] of $\mathbb{Q}$ and of the [[cyclic group]]/[[finite field]] $\mathbb{F}_p = \mathbb{Z}/(p)$, respectively (e.g. \href{localization+of+a+space#Lurie}{Lurie 10, lecture 20}). Including this into the statement of prop. \ref{FractureForSpectra} says that for spectra $X$ satisfying sufficient conditions as above, then the canonical diagram \begin{displaymath} \itexarray{ && X_{\mathbb{Q}} && \longleftarrow && G_{S (\mathbb{Q}/\mathbb{Z})}(X) \\ & \swarrow && \nwarrow && \swarrow \\ (\prod_p X_p^\wedge)_{\mathbb{Q}} && && X \\ & \nwarrow && \swarrow && \nwarrow \\ && \prod_p X_p^\wedge && \longleftarrow && G_{H\mathbb{Q}}(X) } \end{displaymath} is homotopy exact, in that \begin{enumerate}% \item the square is a [[homotopy pullback]] and hence also a [[homotopy pushout]] (this is prop. \ref{ReformulationOfProdOverPComletionByLocalizationAtCoproduct}); \item the diagonals are [[homotopy cofiber sequences]] and hence also [[homotopy fiber sequences]] (by \href{Bousfield+localization+of+spectra#LocalizationCofiber}{this proposition} at \emph{[[Bousfield localization of spectra]]}); \item the top and bottom outer composite sequences are [[homotopy fiber sequences]] (and hence [[homotopy cofiber sequences]]) (by applying the [[pasting law]] to the previous two items). \end{enumerate} Together this is like two-thirds of a [[differential cohomology hexagon]] -- to the extent that $p$-[[adic completion]] is adjoint to $p$-[[torsion approximation]]. This is indeed the case, as the next proposition asserts Notice that in view of remark \ref{GeometricMeaning} then $X_p^\wedge$ is like the restriction of $X$ from [[Spec(Z)]] to all [[formal disks]] around the points $(p)$, and hence $G_{S\mathbb{F}_p}$ is like the restriction to the ``complement of all formal disks''. Finally $X_{\mathbb{Q}}$ may be understood as the restriction to the [[Ran space]] of $Spec(\mathbb{Z})$ (\hyperlink{Gaitsgory11}{Gaitsgory 11}), roughly the colimit of the restriction of $X$ to the complement of finitely many points, as this set of points ranges through all points. \end{remark} In view of remark \ref{TwoThirdHexagon} we may regard the following fact as an refinement of the traditional arithmetic fracture theorem. \begin{prop} \label{CompletionTorsionAdjointModalityForModuleSpectra}\hypertarget{CompletionTorsionAdjointModalityForModuleSpectra}{} Let $A$ be an [[E-∞ ring]] and let $\mathfrak{a} \subset \pi_0 A$ be a [[generators and relations|finitely generated]] ideal in its underlying [[commutative ring]]. Then there is an [[adjoint triple]] of [[adjoint (∞,1)-functors]] \begin{displaymath} \itexarray{ \underoverset{ A Mod_{\mathfrak{a}comp}^{op}} {A Mod_{\mathfrak{a}tors}^{op}} {\simeq} &\stackrel{\overset{\Pi_{\mathfrak{a}}}{\longleftarrow}}{\stackrel{\hookrightarrow}{\underset{\flat_{\mathfrak{a}}}{\longleftarrow}}}& A Mod^{op} } \end{displaymath} where \begin{itemize}% \item $A Mod$ is the [[stable (∞,1)-category|stable]] [[(∞,1)-category of modules]], i.e. of [[∞-modules]] over $A$; \item $A Mod_{\mathfrak{a}tors}$ and $A Mod_{\mathfrak{a} comp}$ are the [[full sub-(∞,1)-categories]] of $\mathfrak{a}$-[[torsion approximation|torsion]] and of $\mathfrak{a}$-[[completion of a module|complete]] $A$-[[∞-modules]], respectively; \item $(-)^{op}$ denotes the [[opposite (∞,1)-category]]; \item the [[equivalence of (∞,1)-categories]] on the left is induced by the restriction of $\Pi_{\mathfrak{a}}$ and $\flat_{\mathfrak{a}}$. \end{itemize} \end{prop} \begin{proof} This is effectively the content of (\hyperlink{LurieProper}{Lurie ``Proper morphisms'', section 4}): \begin{itemize}% \item the existence of $\Pi_{\mathfrak{a}}$ is corollary 4.1.16 and remark 4.1.17 \item the existence of $\flat_{\mathfrak{a}}$ is lemma 4.2.2 there; \item the equivalence of sub-$\infty$-categories is proposition 4.2.5 there. \end{itemize} \end{proof} \begin{cor} \label{FractureFromCohesion}\hypertarget{FractureFromCohesion}{} The traditional arithmetic fracture square of prop. \ref{SullivanArithmeticFracture}, regarded as in remark \ref{TwoThirdHexagon}, is the left part of the ``\href{differential+cohomology+diagram#TheHexagonDiagram}{differential cohomology diagram}'' induced by the [[adjoint modality]] $(\Pi_{\mathfrak{a}} \dashv \flat_{\mathfrak{a}} )$ of prop. \ref{CompletionTorsionAdjointModalityForModuleSpectra}, for the special case that $X = \mathbb{S}$ is the [[sphere spectrum]] and $\mathfrak{a} = (p)$ a [[prime ideal]] \begin{displaymath} \itexarray{ && \Pi_{\mathfrak{a}dR} X && \stackrel{\mathbf{d}}{\longrightarrow} && \flat_{\mathfrak{a}dR} X \\ & \nearrow & & \searrow & & \nearrow && \searrow \\ \Pi_{\mathfrak{a}dR} \flat X && \Downarrow && X && \Downarrow && \Pi_{\mathfrak{a}} \flat_{\mathfrak{a}dR} X \\ & \searrow & & \nearrow & & \searrow && \nearrow \\ && \flat_{\mathfrak{a}} X && \longrightarrow && \Pi_{\mathfrak{a}} X } \,, \end{displaymath} [[!include arithmetic cohesion -- table]] \end{cor} \begin{remark} \label{}\hypertarget{}{} The special case of prop. \ref{CompletionTorsionAdjointModalityForModuleSpectra} where $A$ is the [[Eilenberg-MacLane spectrum]] of a plain [[commutative ring]], and hence -- by the [[stable Dold-Kan correspondence]] -- the case where $A$-[[∞-modules]] are equivalently [[chain complexes]], has a longer tradition in the existing literature. This we highlight separately \hyperlink{CompletionAndTorsionOnDerivedCategories}{below}. \end{remark} \begin{remark} \label{FracturingOnEInfinityAlgebras}\hypertarget{FracturingOnEInfinityAlgebras}{} By the discussion at \emph{[[completion of modules]]} in the section \emph{\href{completion+of+a+module#Monoidalness}{Monoidalness}}, the [[adjoint modality]] of prop. \ref{CompletionTorsionAdjointModalityForModuleSpectra} is a [[monoidal (∞,1)-functor]] without, possibly, respect the [[tensor unit]] in $A Mod$. This means that $(\Pi_{\mathfrak{a}}\dashv \flat_{\mathfrak{a}})$ passes to ``[[commutative monoid in a symmetric monoidal (∞,1)-category|commutative ∞-monoids]]-without unit'' in $A Mod$, hence to ([[Isbell duality|formal duals of]]) [[nonunital E-∞ algebras]]. By \href{nonunital+Ek-algebra#RelationToAugmentedEkAlgebras}{this proposition} (\hyperlink{LurieAlgebra}{Lurie ``Algebra'', prop. 5.2.3.15}) [[nonunital E-∞ rings]] are [[equivalence of (∞,1)-categories|equivalent]] to [[augmented E-∞ rings]] over the [[sphere spectrum]], hence this is [[E-∞ arithmetic geometry]] under $Spec(\mathbb{S})$. Notice that in addition $\Pi_{\mathfrak{a}}$ here should preserve [[finite products]] (because by the discussion at \href{completion%20of%20a%20module#Monoidalness}{completion of a module -- monoidalness} the underlying $\Pi_{\mathfrak{a}} \colon A Mod \to A Mod$ preserves all small [[(∞,1)-colimits]] and because by \href{commutative+monoid+in+a+symmetric+monoidal+%28infinity%2C1%29-category#LimitsInCRing}{this proposition} finite coproducts in $CRng(A Mod)$ are computed in the underlying $A Mod$. Therefore we may think of $\Pi_{\mathfrak{a}}$ as a [[shape modality]] and of $\flat_{\mathfrak{a}}$ as a [[sharp modality]] on affine [[E-∞ geometry|E-∞]]-[[arithmetic geometry]] under $Spec(\mathbb{S})$ -- namely on [[Isbell duality|formal duals]] of [[nonunital E-∞ rings]] . (It may be entertaining to note that on the level of [[∞-groups of units]] then [[E-∞ arithmetic geometry]] under $Spec(\mathbb{S})$ translates to [[abelian ∞-groups]] of [[twisted cohomology|twists]] over the [[sphere spectrum]] -- which has been argued to be the homotopy-theoretic incarnation of [[superalgebra]], see at \emph{\href{super+algebra#AbstractIdea}{superalgebra -- abstract idea}} for more on this.) \end{remark} For more discussion of this see also \emph{[[differential cohesion and idelic structure]]}. \hypertarget{CompletionAndTorsionOnDerivedCategories}{}\paragraph*{{The arithmetic fracture square for chain complexes}}\label{CompletionAndTorsionOnDerivedCategories} We discuss here arithmetic fracturing on [[chain complexes]] of modules. Under the [[stable Dold-Kan correspondence]] this is a special case of prop. \ref{CompletionTorsionAdjointModalityForModuleSpectra}, but this special case has a longer tradition in the literature -- going back to \emph{Grothendieck local duality} and sometimes called \emph{Greenlees-May duality} due to (\hyperlink{GreenlessMay92}{Greenlees-May 92}) -- and we point to these original proofs. \begin{defn} \label{}\hypertarget{}{} Let $A$ be a [[commutative ring]], let $\mathfrak{a} \subset A$ be be an ideal generated by a single regular element (i.e. not a zero divisor). Write $A Mod_{\infty}^{op}$ for the [[opposite (∞,1)-category]] of the [[(∞,1)-category of modules]] over $A$. Write \begin{itemize}% \item $\flat_{\mathfrak{a}}\colon A Mod_\infty^{op} \to A Mod_{\infty}^{op}$ for the [[derived functor]] of [[formal completion]] (adic completion) of modules at $\mathfrak{a}$; with canonical natural transformation $\epsilon_{\mathfrak{a}} \colon \flat_{\mathfrak{a}} \longrightarrow id$ \item $\Pi_{\mathfrak{a}} \colon A Mod_\infty^{op} \to A Mod_\infty^{op}$ for the total [[derived functor]] of the $\mathfrak{a}$-[[torsion approximation]]-functor; with canonical natural transformation $\eta_{\mathfrak{a}}\colon id \longrightarrow \Pi_{\mathfrak{a}}$ \end{itemize} Finally write \begin{displaymath} (A Mod_\infty^{op})^{\mathfrak{a}com}, (A Mod_\infty^{op})^{\mathfrak{a}tor} \hookrightarrow A Mod_\infty \end{displaymath} for the [[full (∞,1)-subcategories]] of objects $X$ for which, $\epsilon_{\mathfrak{a}}(X)$ or $\eta_{\mathfrak{a}}(X)$ is an [[equivalence in an (∞,1)-category]], respectively. \end{defn} \begin{prop} \label{}\hypertarget{}{} \begin{enumerate}% \item The transformation $\epsilon_{\mathfrak{a}}$ exhibits $(A Mod_\infty^{op})^{\mathfrak{a}com}\hookrightarrow A Mod_\infty$ as a [[reflective (∞,1)-subcategory]], hence $\flat_{\mathfrak{a}}$ as an [[idempotent (∞,1)-monad]]. \item The transformation $\eta_{\mathfrak{a}}$ exhibits $(A Mod_\infty^{op})^{\mathfrak{a}tor}\hookrightarrow A Mod_\infty$ as a co-reflective $(\infty,1)$-category, hence $\Pi_{\mathfrak{a}}$ as an idempotent $(\infty,1)$-comonad. \item Restricted to these sub-$(\infty,1)$-categories both $\flat_{\mathfrak{a}}$ as well as $\Pi_{\mathfrak{a}}$ become [[equivalences of (∞,1)-categories]], hence exhibiting $(\Pi_{\mathfrak{a}} \dashv \flat_{\mathfrak{a}})$ as a higher [[adjoint modality]]. \end{enumerate} \end{prop} \begin{proof} This is a paraphrase of the results in (\hyperlink{DwyerGreenlees99}{Dwyer-Greenlees 99}) and (\hyperlink{PortaShaulYekutieli10}{Porta-Shaul-Yekutieli 10}) from the language of [[derived categories]] to [[(∞,1)-category theory]]. First of all, by our simplifying assumption that $\mathfrak{a}$ is generated by a single regular element, the running assumption of ``weak proregularity'' in (\hyperlink{PortaShaulYekutieli10}{Porta-Shaul-Yekutieli 10, def.3.21}) is satisfied. Then in view of (\hyperlink{PortaShaulYekutieli10}{Porta-Shaul-Yekutieli 10, corollary 3.31}) the statement of (\hyperlink{PortaShaulYekutieli10}{Porta-Shaul-Yekutieli 10, theorem 6.12}) is the \href{reflective+sub-(infinity,1}{characterization of reflectors}-category\#CharacterizationOfReflectors) as discussed at \emph{[[reflective sub-(∞,1)-category]]}, and formally dually so for the coreflection. With the fully faithfulness that goes with this the equivalence of the two inclusions on the level of [[homotopy categories]] given by (\hyperlink{HoveyPalieriStrickland97}{Hovey-PalieriS-trickland 97, 3.3.5}, \hyperlink{DwyerGreenlees99}{Dwyer-Greenlees 99, theorem 2.1} \hyperlink{PortaShaulYekutieli10}{Porta-Shaul-Yekutieli 10, theorem 6.11}) implies the canonical equivalence of the two [[sub-(∞,1)-categories]] and this means that $\Pi_{\mathfrak{a}}$ and $\flat_{\mathfrak{a}}$ are the [[adjoint pair]] induced from the reflection/coreflection [[adjoint triple]]. This adjoint triple is stated more explicitly in (\hyperlink{DwyerGreenlees99}{Dwyer-Greenlees 99, section 4}), see also (\hyperlink{PortaShaulYekutieli10}{Porta-Shaul-Yekutieli 10, end of remark 6.14}). \end{proof} Therefore arithmetic fracture squares in the homotopy theory of chain complexes are induced by this as in corollary \ref{FractureFromCohesion} above. \hypertarget{GeneralFractureSquares}{}\paragraph*{{General fracture squares of spectra}}\label{GeneralFractureSquares} By prop. \ref{ReformulationOfProdOverPComletionByLocalizationAtCoproduct} the arithmetic fracture square of prop. \ref{SullivanArithmeticFracture} is equivalently of the form \begin{displaymath} \itexarray{ && L_{H\mathbb{Q}}X \\ & \swarrow && \nwarrow \\ L_{H\mathbb{Q}} L_{S \mathbb{Q}/\mathbb{Z}} X && && X \\ & \nwarrow && \swarrow \\ && L_{S \mathbb{Q}/\mathbb{Z}} X } \,. \end{displaymath} In this form the statement holds much more generally: \begin{prop} \label{GeneralFractureSquare}\hypertarget{GeneralFractureSquare}{} Let $E, F, X$ be [[spectra]] such that the $F$-[[Bousfield localization of spectra|localization]] of $X$ is $E$-acyclic, i.e. $E_\bullet(L_F X) \simeq 0$, then the canonical square [[diagram]] \begin{displaymath} \itexarray{ && L_F X \\ & \swarrow && \nwarrow \\ L_F L_E X && && L_{E\vee F} X \\ & \nwarrow && \swarrow \\ && L_E X } \end{displaymath} is a [[homotopy pullback]] (and hence by stability also a [[homotopy pushout]]). \end{prop} e.g. (\hyperlink{Bauer11}{Bauer 11, prop. 2.2}) \begin{remark} \label{ExampleOfChromaticFracturing}\hypertarget{ExampleOfChromaticFracturing}{} The general version of the fracture statement in prop. \ref{GeneralFractureSquare} is used frequently in [[chromatic homotopy theory]] for decomposition in [[Morava K-theory]] and [[Morava E-theory]]-localizations. For example there is a \textbf{chromatic} fracture square: \begin{displaymath} \itexarray{ && L_{E(n-1)} X \\ & \swarrow && \nwarrow \\ L_{E(n-1)} L_{K(n)} X && && L_{E(n)} X \\ & \nwarrow && \swarrow \\ && L_{K(n)} X } \end{displaymath} In particular it is used for instance in the construction of [[tmf]], see example \ref{ConstructionOfTmf} below. \end{remark} From another perspective: \begin{prop} \label{}\hypertarget{}{} Suppose that $L \colon Spectra \to Spectra$ is a [[smashing localization]] given by [[smash product]] with some [[spectrum]] $T$. Write $F$ for the [[homotopy fiber]] \begin{displaymath} F \longrightarrow \mathbb{S} \longrightarrow T \,. \end{displaymath} Then there is a [[fracture diagram]] of operations \begin{displaymath} \itexarray{ T \wedge (-) && \longleftarrow && [T,-] \\ & \nwarrow && \swarrow \\ && \mathbb{S} \\ & \swarrow & & \nwarrow \\ [F,-] && \longleftarrow && F \wedge (-) } \end{displaymath} where $[F,-]$ and $T \wedge (-) \colon Spectra \to Spectra$ are [[idempotent (∞,1)-monads]] and $[T,-]$, $F \wedge (-)$ are idempotent $\infty$-comonads, the diagonals are [[homotopy fiber sequences]]. \end{prop} ([[Charles Rezk]], \href{http://mathoverflow.net/a/178316/381}{MO comment,August 2014}) \begin{example} \label{}\hypertarget{}{} For $T = S \mathbb{Z}[p^{-1}]$ the [[Moore spectrum]] of the [[integers]] [[localization of a ring|localized away from]] $p$, then \begin{displaymath} F = \Sigma^{-1} S (\mathbb{Z}[p^{-1}]/\mathbb{Z}) \to \mathbb{S} \to S \mathbb{Z}[p^{-1}] \end{displaymath} and hence \begin{itemize}% \item $\Sigma^{-1} S (\mathbb{Z}[p^{-1}]/\mathbb{Z}) \wedge (-)$ is $p$-[[torsion approximation]]; \item $[\Sigma^{-1} S (\mathbb{Z}[p^{-1}]/\mathbb{Z}),-]$ is $p$-[[completion of a module|completion]]; \item $S \mathbb{Z}[p^{-1}] \wedge (-)$ is $p$-[[rationalization]]; \item $[T,-]$ is forming $p$-[[adic residual]]. \end{itemize} \begin{displaymath} \itexarray{ && localization\;away\;from\;\mathfrak{a} && \stackrel{}{\longrightarrow} && \mathfrak{a}\;adic\;residual \\ & \nearrow & & \searrow & & \nearrow && \searrow \\ && && X && && \\ & \searrow & & \nearrow & & \searrow && \nearrow \\ && formal\;completion\;at\;\mathfrak{a}\; && \longrightarrow && \mathfrak{a}\;torsion\;approximation } \,, \end{displaymath} \end{example} With (\hyperlink{Bousfield79}{Bousfield 79, prop.2.5}) \hypertarget{for_modules}{}\paragraph*{{For $E_\infty$-modules}}\label{for_modules} \begin{prop} \label{CompletionTorsionAdjointModalityForModuleSpectra}\hypertarget{CompletionTorsionAdjointModalityForModuleSpectra}{} Let $A$ be an [[E-∞ ring]] and let $\mathfrak{a} \subset \pi_0 A$ be a [[generators and relations|finitely generated]] ideal in its underlying [[commutative ring]]. Then there is an [[adjoint triple]] of [[adjoint (∞,1)-functors]] \begin{displaymath} \itexarray{ \underoverset{ A Mod_{\mathfrak{a}comp}^{op}} {A Mod_{\mathfrak{a}tors}^{op}} {\simeq} &\stackrel{\overset{\Pi_{\mathfrak{a}}}{\longleftarrow}}{\stackrel{\hookrightarrow}{\underset{\flat_{\mathfrak{a}}}{\longleftarrow}}}& A Mod^{op} } \end{displaymath} where \begin{itemize}% \item $A Mod$ is the [[stable (∞,1)-category|stable]] [[(∞,1)-category of modules]], i.e. of [[∞-modules]] over $A$; \item $A Mod_{\mathfrak{a}tor}$ and $A Mod_{\mathfrak{a} comp}$ are the [[full sub-(∞,1)-categories]] of $\mathfrak{a}$-[[torsion approximation|torsion]] and of $\mathfrak{a}$-[[completion of a module|complete]] $A$-[[∞-modules]], respectively; \item $(-)^{op}$ denotes the [[opposite (∞,1)-category]]; \item the [[equivalence of (∞,1)-categories]] on the left is induced by the restriction of $\Pi_{\mathfrak{a}}$ and $\flat_{\mathfrak{a}}$. \end{itemize} \end{prop} This is effectively the content of (\hyperlink{LurieProper}{Lurie ``Completions'', section 4}): \hypertarget{InCohesiveHomotopyTheory}{}\subsubsection*{{In cohesive (stable) homotopy theory}}\label{InCohesiveHomotopyTheory} In [[cohesive homotopy theory]] every [[stable homotopy type]] $X$ sits in a fracture square of the form \begin{displaymath} \itexarray{ && \Pi_{dR} X && \longrightarrow && \flat_{dR} X \\ & \nearrow & & \searrow && \nearrow \\ \Pi_{dR} \flat X && && X \\ & \searrow & & \nearrow && \searrow \\ && \flat X && \longrightarrow && \Pi X } \end{displaymath} where $\flat$ is the [[flat modality]] and $\Pi_{dR}$ the [[homotopy fiber]] of the [[unit of a monad|unit]] $X\to \Pi X$ of the [[shape modality]]. This is the left part of the [[differential cohomology hexagon]] for $X$, see there for details. Here $\Pi_{dR} X$ is such that for any other stable cohesive homotopy type $\hat E$ then functions $\Pi_{dR} X \to \hat E$ are equivalent to functions $X \to \flat_{dR} \hat E$, where $\hat E \to \flat_{dR} \hat E$ is a generalized form of rationalization in the sense discussed at \emph{[[differential cohomology hexagon]]}. In particular if $\hat E$ is a \href{differential%20cohomology%20diagram#HopkinsSingerCoefficients}{Hopkins-Singer-type} [[differential cohomology]] refinement of a plain [[spectrum]] $E$, then $E\to \flat_{dR} E$ is its ordinary [[rationalization]] given by the [[Chern character]] and $\hat E \to \flat_{dR} \hat E$ is the corresponding map on Chern [[curvature forms]]. Moreover, if the ambient [[cohesion]] is [[differential cohesion]] over a base of [[infinitesimal cohesion]], then the [[flat modality]] $\flat$ takes any space $X$ to the union of all its [[formal disks]]. (See at \emph{[[differential cohesion and idelic structure]]}.) Accordingly the collection of functions $\flat X \to \hat E$ in this case behave like the product of all [[formal power series]] of $\hat E$-valued functions around all global points of $X$, analogous to remark \ref{GeometricMeaning}. An example of this are [[synthetic differential ∞-groupoids]] regarded as cohesive over their [[formal moduli problems]], as its its complex analytic incarnation by synthetic differential [[complex analytic ∞-groupoids]]. In this context if $X = \Sigma$ is a [[complex curve]] then $\flat \Sigma$ is precisely the analog of the [[integral adeles]] as it is predicted by the [[function field analogy]]. \hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples} \begin{example} \label{ConstructionOfTmf}\hypertarget{ConstructionOfTmf}{} The construction of the [[tmf]]-spectrum -- the spectrum of [[global sections]] of the [[derived Deligne-Mumford stack]] of [[derived elliptic curves]] -- as described in (\href{tmf#Behrens13}{Behrens 13}) proceeds by first applying the arithmetic fracture square of prop. \ref{ArithmeticFractureSquare}, prop. \ref{SullivanArithmeticFracture} to decompose the [[moduli stack of elliptic curves]] into rational and $p$-adic curves, and then in a second step in applying in turn the general fracture square of prop. \ref{GeneralFractureSquare} for [[Morava K-theory]] to the remaining $p$-adic pieces. See at \emph{\href{tmf#DecomopositionViaArithmeticSquares}{tmf -- Decomposition via arithmetic fracture squares}} for more on this. \end{example} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[p-adic homotopy theory]], [[p-completion]] \item [[rational homotopy theory]], [[rationalization]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \begin{itemize}% \item [[Aldridge Bousfield]], [[Daniel Kan]], \emph{[[Homotopy limits, completions and localizations]]}, Lecture Notes in Mathematics, Vol 304, Springer 1972 \item [[Aldridge Bousfield]], \emph{The localization of spectra with respect to homology} , Topology vol 18 (1979) (\href{http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/bousfield-topology-1979.pdf}{pdf}) \item [[Dennis Sullivan]], \emph{Geometric topology: localization, periodicity and Galois symmetry}, volume 8 of K- Monographs in Mathematics. Springer, Dordrecht, 2005. The 1970 MIT notes, Edited and with a preface by [[Andrew Ranicki]] (\href{http://www.maths.ed.ac.uk/~aar/books/gtop.pdf}{pdf}) \item [[Tilman Bauer]], \emph{Bousfield localization and the Hasse square}, 2011 (\href{http://math.mit.edu/conferences/talbot/2007/tmfproc/Chapter09/bauer.pdf}{pdf}) \item [[Paul VanKoughnett]], \emph{Spectra and localization}, 2013 ([[VanKoughnettLocalization.pdf:file]]) \item [[Emily Riehl]], \emph{Categorical homotopy theory}, new mathematical monographs 24, Cambridge University Press 2014 (published version) \item [[Peter May]], [[Kate Ponto]], chapters 7 and 8 of \emph{More concise algebraic topology: Localization, completion, and model categories} (\href{http://www.maths.ed.ac.uk/~aar/papers/mayponto.pdf}{pdf}) \item [[Michael Shulman]], \emph{The Propositional Fracture Theorem}, (\href{http://golem.ph.utexas.edu/category/2013/05/the_propositional_fracture_the.html}{blog post}) \end{itemize} Related MO-discussion: \begin{itemize}% \item \emph{\href{http://mathoverflow.net/a/91057/381}{Fracture squares of Bousfield Localizations of Spectra}} \end{itemize} Discussion of rational functions as functions on the [[Ran space]] is in \begin{itemize}% \item [[Dennis Gaitsgory]], \emph{Contractibility of the space of rational maps} (\href{http://arxiv.org/abs/1108.1741}{arXiv:1108.1741}) \end{itemize} Discussion of $\mathfrak{a}$-[[adic completion]] and $\mathfrak{a}$-[[torsion approximation]] as derived idempotent (co-)monads on a [[derived category]] of chain complexes of modules -- \emph{Greenlees-May duality} -- is in \begin{itemize}% \item Marco Porta, [[Liran Shaul]], [[Amnon Yekutieli]], \emph{On the Homology of Completion and Torsion} (\href{http://arxiv.org/abs/1010.4386}{arXiv:1010.4386}) \end{itemize} building on \begin{itemize}% \item [[John Greenlees]], [[Peter May]], \emph{Derived functors of I-adic completion and local homology}, J. Algebra 149 (1992), 438--453 (\href{http://math.uchicago.edu/~may/PAPERS/73.pdf}{pdf}) \item Leovigildo Alonso, Ana Jerem\'i{}as, [[Joseph Lipman]], \emph{Local Homology and Cohomology on Schemes} (\href{http://arxiv.org/abs/alg-geom/9503025}{arXiv:alg-geom/9503025}) \item [[Mark Hovey]], [[John Palmieri]], [[Neil Strickland]], \emph{Axiomatic stable homotopy theory}, Mem. Amer. Math. Soc. 128 (1997), no. 610, x+114. \item [[William Dwyer]], [[John Greenlees]], \emph{Complete modules and torsion modules}, Amer. J. Math. 124, No. 1, (1999) (\href{https://www3.nd.edu/~wgd/Dvi/Complete.And.Torsion.pdf}{pdf}) \end{itemize} Discussion of this in [[stable homotopy theory]] and the full generality of [[higher algebra]] is in \begin{itemize}% \item [[Jacob Lurie]], section 4 of \emph{[[Proper Morphisms, Completions, and the Grothendieck Existence Theorem]]} \end{itemize} And in the context of commutative DG-rings in \begin{itemize}% \item [[Liran Shaul]], \emph{Completion and torsion over commutative DG rings}, \href{http://arxiv.org/abs/1605.07447}{arXiv:1605.07447} \end{itemize} This and further generalization is in \begin{itemize}% \item [[Tobias Barthel]], Drew Heard, Gabriel Valenzuela \emph{Local duality in algebra and topology} (\href{http://arxiv.org/abs/1511.03526}{arXiv:1511.03526}) \end{itemize} Discussion in [[homotopy type theory]]: \begin{itemize}% \item Luis Scoccola, \emph{Nilpotent Types and Fracture Squares in Homotopy Type Theory} (\href{https://arxiv.org/abs/1903.03245}{arXiv:1903.03245}) \end{itemize} See also \begin{itemize}% \item [[John Greenlees]], \emph{Tate cohomology in axiomatic stable homotopy theory} (\href{http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.490.322&rep=rep1&type=pdf}{pdf}) \end{itemize} [[!redirects fracture theorems]] [[!redirects arithmetic square]] [[!redirects arithmetic squares]] [[!redirects fracture square]] [[!redirects fracture squares]] [[!redirects arithmetic fracture square]] [[!redirects arithmetic fracture squares]] [[!redirects fracture diagram]] [[!redirects fracture diagrams]] [[!redirects Grothendieck local duality]] [[!redirects Greenlees-May duality]] \end{document}