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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*{Tor} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra} [[!include homological algebra - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{ExistenceAndBalancing}{Existence and balancing}\dotfill \pageref*{ExistenceAndBalancing} \linebreak \noindent\hyperlink{RespectForDirectSumsAndFilteredColimits}{Respect for direct sums and filtered colimits}\dotfill \pageref*{RespectForDirectSumsAndFilteredColimits} \linebreak \noindent\hyperlink{RelationToTorsionGroups}{Relation to torsion groups}\dotfill \pageref*{RelationToTorsionGroups} \linebreak \noindent\hyperlink{SymmetryInTheTwoArguments}{Symmetry in the two arguments}\dotfill \pageref*{SymmetryInTheTwoArguments} \linebreak \noindent\hyperlink{Localization}{Localization}\dotfill \pageref*{Localization} \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} In the context of [[homological algebra]], the \emph{$Tor$-functor} is the [[derived tensor product]]: the [[left derived functor]] of the [[tensor product of modules|tensor product of]] $R$-[[modules]], for $R$ a [[commutative ring]]. Together with the [[Ext-functor]] it constitutes one of the central operations of interest in [[homological algebra]]. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} Given a [[ring]] $R$ the [[bifunctor]] $\otimes_R : Mod_{R} \times {}_{R}Mod\to Ab$ from two copies of $R$-[[Mod]] to [[Ab]] is a [[right exact functor]]. Its [[left derived functors]] are the \textbf{Tor-functors} \begin{displaymath} Tor(-,B) : Mod_R \to Ab \end{displaymath} and \begin{displaymath} Tor(A,-) : Mod_R \to Ab \end{displaymath} with respect to one argument with fixed another, if they exist, are parts of a bifunctor \begin{displaymath} Tor : Mod_{R}\times {}_{R}Mod\to Ab \,. \end{displaymath} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{ExistenceAndBalancing}{}\subsubsection*{{Existence and balancing}}\label{ExistenceAndBalancing} Given a right $R$-module \begin{displaymath} A \in Mod_R \end{displaymath} and a left $R$-module \begin{displaymath} B \in {}_R Mod \end{displaymath} there are in principle three different ways to compute their derived tensor product $Tor_\bullet(A,B)$: \begin{enumerate}% \item keeping $B$ fixed and deriving the functor \begin{displaymath} (-) \otimes_R B : Mod_R \to Ab \end{displaymath} \item keeping $A$ fixed and deriving the functor \begin{displaymath} A \otimes_R (-) : {}_R Mod \to Ab \end{displaymath} \item deriving the functor \begin{displaymath} (-) \otimes_R (-) : Mod_R \times {}_R Mod \to Ab \end{displaymath} in both arguments \end{enumerate} \begin{theorem} \label{}\hypertarget{}{} If both $Mod_{R}$ and $_{R}Mod$ have [[projective object|enough projectives]], then all these three derived functors exist and all give the same result. \end{theorem} \begin{proof} Existence is clear from the very definition of [[derived functor in homological algebra]]. So we show that deriving in the left argument gives the same result as deriving in the right argument. Let $Q^A_\bullet \stackrel{\simeq_{qi}}{\to} A$ and $Q^B_\bullet \stackrel{\simeq_{qi}}{\to} B$ be [[projective resolutions]] of $A$ and $B$, respectively. The corresponding [[tensor product of chain complexes]] $Tot (Q^A_\bullet\otimes Q^B_\bullet)$, hence the [[total complex]] of the degreewise [[tensor product of modules]] [[double complex]] carries the [[filtered chain complex|filtration]] by horizontal degree as well as that by vertical degree. Accordingly there are the corresponding two [[spectral sequences of a double complex]], to be denoted here $\{{}^{A}E^r_{p,q}\}_{r,p,q}$ (for the filtering by $A$-degree) and $\{{}^{B}E^r_{p,q}\}_{r,p,q}$ (for the filtering by $B$-degree). By the discussion there, both converge to the chain homology of the total complex. We find the value of both spectral sequences on low degree pages according to the general discussion at \emph{\href{spectral+sequence+of+a+double%20complex#LowDegreePages}{spectral sequence of a double complex - low degree pages}}. The 0th page for both is \begin{displaymath} {}^A E^0_{p,q} = {}^B E^0_{p,q} \coloneqq Q^A_p \otimes_R Q^B_q \,. \end{displaymath} For the first page we have \begin{displaymath} \begin{aligned} {}^A E^1_{p,q} & \simeq H_q(C_{p,\bullet}) \\ & \simeq H_q( Q^A_p \otimes Q^B_\bullet ) \end{aligned} \end{displaymath} and \begin{displaymath} \begin{aligned} {}^B E^1_{p,q} & \simeq H_q(C_{\bullet,p}) \\ & \simeq H_q( Q^A_\bullet \otimes Q^B_p ) \end{aligned} \,. \end{displaymath} Now using the [[universal coefficient theorem]] \href{universal%20coefficient%20theorem#InHomology}{in homology} and the fact that $Q^A_\bullet$ and $Q^B_\bullet$ is a [[resolution]] by [[projective objects]], by construction, hence of tensor [[acyclic objects]] for which all [[Tor]]-modules vanish, this simplifies to \begin{displaymath} \begin{aligned} {}^A E^1_{p,q} & \simeq Q^A_p \otimes H_q(Q^B_\bullet) \\ & \simeq \left\{ \itexarray{ Q^A_p \otimes_R B & if\; q = 0 \\ 0 & otherwise } \right. \end{aligned} \end{displaymath} and similarly \begin{displaymath} \begin{aligned} {}^B E^1_{p,q} & \simeq H_q(Q^A_\bullet) \otimes_R Q^B_p \\ & \simeq \left\{ \itexarray{ A \otimes_R Q^B_p & if\; q = 0 \\ 0 & otherwise } \right. \end{aligned} \,. \end{displaymath} It follows for the second pages that \begin{displaymath} \begin{aligned} {}^A E^2_{p,q} & \simeq H_p(H^{vert}_q(Q^A_\bullet \otimes Q^B_\bullet)) \\ & \simeq \left\{ \itexarray{ (L_p( (-)\otimes_R B ))(A) & if \; q = 0 \\ 0 & otherwise } \right. \end{aligned} \end{displaymath} and \begin{displaymath} \begin{aligned} {}^B E^2_{p,q} & \simeq H_p(H^{hor}_q(Q^A_\bullet \otimes Q^B_\bullet)) \\ & \simeq \left\{ \itexarray{ (L_p ( A \otimes_R (-) ))(B) & if \; q = 0 \\ 0 \; otherwise } \right. \end{aligned} \,. \end{displaymath} Now both of these second pages are concentrated in a single row and hence have converged on that page already. Therefore, since they both converge to the same value: \begin{displaymath} L_p((-)\otimes_R B)(A) \simeq {}^A E^2_{p,0} \simeq {}^A E^\infty_{p,0} \simeq {}^B E^2_{p,0} \simeq L_p(A \otimes_R (-))(B) \,. \end{displaymath} \end{proof} \hypertarget{RespectForDirectSumsAndFilteredColimits}{}\subsubsection*{{Respect for direct sums and filtered colimits}}\label{RespectForDirectSumsAndFilteredColimits} \begin{prop} \label{Tor1RespectsDirectSum}\hypertarget{Tor1RespectsDirectSum}{} Each $Tor_n^R(-,N)$ respects [[direct sums]]. \end{prop} \begin{proof} Let $S \in$ [[Set]] and let $\{N_s\}_{s \in S}$ be an $S$-family of $R$-[[modules]]. Observe that \begin{enumerate}% \item if $\{(F_s)_\bullet\}_{s \in S}$ is an family of [[projective resolutions]], then their degreewise [[direct sum]] $(\oplus_{s \in S} F)_\bullet$ is a projective resolution of $\oplus_{s \in S} N_s$. \item the tensor product functor distributes over direct sums (this is discussed at \emph{\href{module#MonoidalCategoryStructure}{tensor product of modules -- monoidal category structure}}) \item the [[chain homology]] functor preserves direct sums (this is discussed at \emph{\href{http://ncatlab.org/nlab/show/chain+homology+and+cohomology#RespectForDirectSum}{chain homology - respect for direct sums}}). \end{enumerate} Using this we have \begin{displaymath} \begin{aligned} Tor_n^R(\oplus_{s \in S} N_s, N) & \simeq H_n\left( \left(\oplus_{s \in S} F\right) \otimes N \right) \\ & \simeq H_n\left( \oplus_{s \in S} \left(F_s \otimes N \right) \right) \\ & \simeq \oplus_{s \in S} H_n( F_s \otimes N ) \\ & \simeq \oplus_{s \in S} Tor_n(N_s, N) \end{aligned} \,. \end{displaymath} \end{proof} \begin{prop} \label{TorPreservesFilteredColimits}\hypertarget{TorPreservesFilteredColimits}{} Each $Tor_n^R(-,N)$ respects [[filtered colimits]]. \end{prop} \begin{proof} Let hence $A \colon I \to R Mod$ be a [[filtered category|filtered]] [[diagram]] of modules. For each $A_i$, $i \in I$ we may find a [[nLab:projective resolution]] and in fact a [[nLab:free resolution]] $(Y_i)_\bullet \stackrel{\simeq_{qi}}{\to} A$. Since [[chain homology]] commutes with filtered colimits (this is discussed at \emph{\href{http://ncatlab.org/nlab/show/chain+homology+and+cohomology#RespectForDirectSum}{chain homology - respect for filtered colimits}}), this means that \begin{displaymath} (\underset{\to_i}{\lim} Y_i)_\bullet \to A \end{displaymath} is still a [[quasi-isomorphism]]. Moreover, by [[Lazard's criterion]] the degreewise filtered colimits of [[free modules]] $\underset{\to_i}{\lim} (Y_i)_n$ for each $n \in \mathbb{N}$ are [[flat modules]]. This means that $\underset{\to_i}{\lim} (Y_i)_\bullet \to A$ is [[flat resolution]] of $A$. By the very definition or else by the basic properties of [[flat modules]], this means that it is a $(-)\otimes N$-[[acyclic resolution]]. By the discussion there it follows that \begin{displaymath} Tor_n^\mathbb{Z}(A,N) \simeq H_n( (\underset{\to_i}{\lim} Y_i) \otimes N ) \,. \end{displaymath} Now the [[tensor product of modules]] is a [[left adjoint]] [[functor]] (the [[right adjoint]] being the [[internal hom]] of modules) and so it commutes over the filtered colimit to yield, using again that [[chain homology]] commutes with filtered colimits, \begin{displaymath} \begin{aligned} \cdots & \simeq H_n( \underset{\to_i}{\lim} (Y_i \otimes N) ) \\ & \simeq \underset{\to_i}{\lim} H_n( Y_i \otimes N ) \\ & \simeq \underset{\to_i}{\lim} Tor_n( A_i, N) \end{aligned} \,. \end{displaymath} \end{proof} \hypertarget{RelationToTorsionGroups}{}\subsubsection*{{Relation to torsion groups}}\label{RelationToTorsionGroups} An [[abelian group]] is called \emph{[[torsion]]} if its elements are ``nilpotent'', hence if all its elements have finite [[order]]. \begin{defn} \label{}\hypertarget{}{} For $A \in$ [[Ab]] and $p \in \mathbb{N}$, write \begin{displaymath} {}_p A \coloneqq \{ a \in A | p \cdot a = 0 \} \end{displaymath} for the \textbf{$p$-[[torsion subgroup]]} consisting of all those elements whose $p$-fold sum with themselves gives 0. \end{defn} For $n \in \mathbb{N}$ with $n \geq 1$, write $\mathbb{Z}_n = \mathbb{Z}/n\mathbb{Z}$ for the [[cyclic group]] of [[order]] $n$, as usual. \begin{prop} \label{TorOutOfCyclicGroup}\hypertarget{TorOutOfCyclicGroup}{} For $p \in \mathbb{N}$, $p \geq 1$, and $A \in$ [[Ab]] $\simeq \mathbb{Z}$[[Mod]] any [[abelian group]], we have an [[isomorphism]] \begin{displaymath} Tor_1^\mathbb{Z}(\mathbb{Z}_p, A) \simeq {}_p A \end{displaymath} of the $Tor_1$-group with the $p$-[[torsion subgroup]] of $A$. For $p = 0$ we have \begin{displaymath} Tor_1^{\mathbb{Z}}(\mathbb{Z}, A) \simeq 0 \,. \end{displaymath} \end{prop} \begin{proof} For the first statement, the [[short exact sequence]] \begin{displaymath} 0 \to \mathbb{Z} \stackrel{\cdot p}{\to} \mathbb{Z} \stackrel{mod\, p}{\to} \mathbb{Z}_p \to 0 \end{displaymath} constitutes a [[projective resolution]] (even a [[free resolution]]) of $\mathbb{Z}_p$. Accordingly we have \begin{displaymath} \begin{aligned} Tor_1^\mathbb{Z}(\mathbb{Z}_p, A) &\simeq H_1( [\cdots\to 0 \to \mathbb{Z}\otimes A \stackrel{(\cdot p) \otimes A}{\to} \mathbb{Z} \otimes A ) \\ & \simeq ker( (\cdot p) \otimes A ) \\ & \simeq \{ a\in A | p\cdot a = 0 \} \end{aligned} \,. \end{displaymath} Here in the last step we use that $(\cdot p)\otimes A$ acts as \begin{displaymath} \begin{aligned} (1, a) &\mapsto (p,a) \\ & = p \cdot (1,a) \\ & = (1, p \cdot a) \end{aligned} \,. \end{displaymath} For the second statement, $\mathbb{Z}$ is already free hence $[\cdots \to 0 \to 0 \to \mathbb{Z}]$ is already a projective resolution and hence $Tor_1(\mathbb{Z}, A) \simeq H_1(0) \simeq 0$. \end{proof} \begin{prop} \label{}\hypertarget{}{} Let $A$ be a [[finite abelian group]] and $B$ any abelian group. Then $Tor_1(A,B)$ is a [[torsion group]]. Specifically, $Tor_1(A,B)$ is a [[direct sum]] of [[torsion subgroups]] of $A$. \end{prop} \begin{proof} By a fundamental fact about [[finite abelian groups]] (see \emph{\href{finite+abelian+group#FiniteAbelianGroupIsDirectSumOfCyclics}{this theorem}}), $A$ is a [[direct sum of abelian groups|direct sum]] of [[cyclic group]] $A \simeq \oplus_k \mathbb{Z}_{p_k}$. By prop. \ref{Tor1RespectsDirectSum} $Tor_1$ respects this direct sum, so that \begin{displaymath} Tor_1(A,B) \simeq \oplus_k Tor_1(\mathbb{Z}_{p_k}, B) \,. \end{displaymath} By prop. \ref{TorOutOfCyclicGroup} every direct summand on the right is a torsion group and hence so is the whole direct sum. \end{proof} More generally we have: \begin{prop} \label{}\hypertarget{}{} Let $A$ and $B$ be [[abelian groups]]. Write $Tor^\mathbb{Z}$ for the [[left derived functor]] of tensoring over $R = \mathbb{Z}$. Then \begin{enumerate}% \item $Tor^\mathbb{Z}_1(A,B)$ is a [[torsion group]]. Specifically it is a [[filtered colimit]] of torsion subgroups of $B$. \item $Tor^{\mathbb{Z}}_1(\mathbb{Q}/\mathbb{Z}, A)$ is the [[torsion subgroup]] of $A$. \item $A$ is a [[torsion subgroup|torsion-free group]] precisely if $Tor^\mathbb{Z}_1(A,-) = 0$, equivalently if $Tor^\mathbb{Z}_1(-,A) = 0$. \end{enumerate} \end{prop} For instance (\hyperlink{Weibel}{Weibel, prop. 3.1.2, prop. 3.1.3, cor. 3.1.5}). \begin{proof} The group $A$ may be expressed as a [[filtered colimit]] \begin{displaymath} A \simeq \underset{\to_i}{\lim} A_i \end{displaymath} of finitely generated [[subgroups]] (this is discussed at \emph{\href{http://ncatlab.org/nlab/show/Mod#LimitsAndColimits}{Mod - Limits and colimits}}). Each of these is a [[direct sum]] of [[cyclic groups]]. By prop. \ref{TorPreservesFilteredColimits} $Tor_1^\mathbb{Z}(-,B)$ preserves these colimits. By prop. \ref{TorOutOfCyclicGroup} every cyclic group is sent to a torsion group (of either $A$ or $B). Therefore by prop. \ref{TorOutOfCyclicGroup}$Tor\_1(A,B)\$ is a filtered colimit of direct sums of torsion groups. This is itself a torsion group. \end{proof} \begin{remark} \label{}\hypertarget{}{} Analogous results fail, in general, for $\mathbb{Z}$ replaced by another ring $R$. \end{remark} \begin{cor} \label{}\hypertarget{}{} An [[abelian group]] is [[torsion subgroup|torsion free]] precisely if regarded as a $\mathbb{Z}$-[[module]] it is a [[flat module]]. \end{cor} See at \emph{\href{flat+module#Examles}{flat module - Examples}} for more. \hypertarget{SymmetryInTheTwoArguments}{}\subsubsection*{{Symmetry in the two arguments}}\label{SymmetryInTheTwoArguments} \begin{prop} \label{}\hypertarget{}{} For $N_1, N_2 \in R Mod$ and $n \in \mathbb{N}$ there is a [[natural isomorphism]] \begin{displaymath} Tor_n(A,B) \simeq Tor_n(B,A) \,. \end{displaymath} \end{prop} We first give a proof for $R$ a [[principal ideal domain]] such as $\mathbb{Z}$. \begin{proof} Let $R$ be a [[principal ideal domain]] such as $\mathbb{Z}$ (in the latter case $R$[[Mod]]$\simeq$ [[Ab]]). Then by the discussion at \emph{\href{projective%20resolution#Lenght1ResolutionsOfAbelianGroups}{projective resolution -- length-1 resolutions}} there is always a [[short exact sequence]] \begin{displaymath} 0 \to F_1 \to F_0 \to N \to 0 \end{displaymath} exhibiting a [[projective resolution]] of any module $N$. It follows that $Tor_{n \geq 2}(-,-) = 0$. Let then $0 \to F_1 \to F_2 \to N_2 \to 0$ be such a short resolution for $N_2$. Then by the \href{derived+functor+in+homological+algebra#LongExactSequence}{long exact sequence of a derived functor} this induces an [[exact sequence]] of the form \begin{displaymath} 0 \to Tor_1(N_1, F_1) \to Tor_1(N_1, F_0) \to Tor_1(N_1, N_2) \to N_1 \otimes F_1 \to N_1 \otimes F_0 \to N_1 \otimes N_2 \to 0 \,. \end{displaymath} Since by construction $F_0$ and $F_1$ are already [[projective modules]] themselves this collapses to an exact sequence \begin{displaymath} 0 \to Tor_1(N_1, N_2) \hookrightarrow N_1 \otimes F_1 \to N_1 \otimes F_0 \to N_1 \otimes N_2 \to 0 \,. \end{displaymath} To the last three terms we apply the natural [[braided monoidal category|symmetric braiding]] [[isomorphism]] in $(R Mod, \otimes_R)$ to get \begin{displaymath} \itexarray{ 0 &\to& Tor_1(N_1, N_2) &\hookrightarrow& N_1 \otimes F_1 &\to& N_1 \otimes F_0 &\to& N_1 \otimes N_2 &\to& 0 \\ && \downarrow && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} && \\ 0 &\to& Tor_1(N_2, N_1) &\hookrightarrow& F_1 \otimes N_1 &\to& F_0 \otimes N_1 &\to& N_2 \otimes N_1 &\to& 0 } \,. \end{displaymath} This exhibits a morphism $Tor_1(N_1,N_2) \to Tor_1(N_2, N_1)$ as the morphism induced on [[kernels]] from an isomorphism between two morphisms. Hence this is itself an isomorphism. (This is just by the [[universal property]] of the [[kernel]], but one may also think of it as a simple application of the the [[four lemma]]/[[five lemma]].) \end{proof} \hypertarget{Localization}{}\subsubsection*{{Localization}}\label{Localization} (\ldots{}) For instance (\hyperlink{Weibel}{Weibel, cor. 3.2.13}). \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[Ext]] \item [[Cotor]] \item [[flat resolution lemma]] \item [[universal coefficient theorem]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} Standard textbook accounts include the following: \begin{itemize}% \item [[Charles Weibel]], \emph{[[An Introduction to Homological Algebra]]}, Cambridge Studies in Adv. Math. 38, CUP 1994 \end{itemize} \begin{itemize}% \item [[Henri Cartan]], [[Samuel Eilenberg]], \emph{Homological algebra}, Princeton Univ. Press 1956. \item M. Kashiwara and P. Schapira, \emph{[[Categories and Sheaves]]}, Springer (2000) \item S. I . Gelfand, Yu. I. Manin, \emph{Methods of homological algebra} \end{itemize} Lecture notes include \begin{itemize}% \item Daniel Murfet, \emph{Tor} (\href{http://therisingsea.org/notes/Tor.pdf}{pdf}) \end{itemize} section 3 of \begin{itemize}% \item [[Peter May]], \emph{Notes on Tor and Ext} (\href{http://www.math.uchicago.edu/~may/MISC/TorExt.pdf}{pdf}) \end{itemize} and specifically for [[symmetric smash product of spectra|symmetric]] [[model categories of spectra]] \begin{itemize}% \item [[Anthony Elmendorf]], [[Igor Kriz]], [[Peter May]], section 7 of \emph{[[Modern foundations for stable homotopy theory]]} (\href{http://hopf.math.purdue.edu/Elmendorf-Kriz-May/modern_foundations.pdf}{pdf}) \end{itemize} Original articles include \begin{itemize}% \item Patrick Keef, \emph{On the Tor functor and some classes of abelian groups}, Pacific J. Math. Volume 132, Number 1 (1988), 63-84. (\href{http://projecteuclid.org/euclid.pjm/1102689795}{Euclid}) \end{itemize} [[!redirects Tor-functor]] [[!redirects Tor functor]] [[!redirects Tor-functors]] [[!redirects Tor functors]] \end{document}