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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*{Mod} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra} [[!include higher algebra - contents]] \hypertarget{the_category__of_modules}{}\section*{{The category $Mod$ of modules}}\label{the_category__of_modules} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition_of_}{Definition of $Mod$}\dotfill \pageref*{definition_of_} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{ModAsBifibration}{$Mod$ as a bifibration}\dotfill \pageref*{ModAsBifibration} \linebreak \noindent\hyperlink{TangentsAndDeformationTheory}{Tangents and deformation theory}\dotfill \pageref*{TangentsAndDeformationTheory} \linebreak \noindent\hyperlink{RModIsAbelian}{$R Mod$ is an abelian category}\dotfill \pageref*{RModIsAbelian} \linebreak \noindent\hyperlink{RModIsClosedMonoidalCategory}{$R Mod$ is a closed monoidal category}\dotfill \pageref*{RModIsClosedMonoidalCategory} \linebreak \noindent\hyperlink{exact_functors_between_categories_of_modules}{Exact functors between categories of modules}\dotfill \pageref*{exact_functors_between_categories_of_modules} \linebreak \noindent\hyperlink{LimitsAndColimits}{Limits and colimits}\dotfill \pageref*{LimitsAndColimits} \linebreak \noindent\hyperlink{tiny_objects}{Tiny objects}\dotfill \pageref*{tiny_objects} \linebreak \noindent\hyperlink{tannaka_duality}{Tannaka duality}\dotfill \pageref*{tannaka_duality} \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} Given a [[monoid]] $R$ in a [[monoidal category]] $(\mathcal{C}, \otimes)$, $R$[[Mod]] is the [[category]] whose [[objects]] are $R$-[[modules]] in $\mathcal{C}$ and whose morphisms are module homomorphisms. Specifically if $(\mathcal{C}, \otimes)$ is the category [[Ab]] of [[abelian groups]] and $\otimes$ the [[tensor product of abelian groups]], then $R$ is a [[ring]]. We write just $Mod$ for the category whose objects are pairs $(R,N)$ consisting of a monoid $R$ and an $R$-module, and whose morphisms may also map between different monoids. \hypertarget{definition_of_}{}\subsection*{{Definition of $Mod$}}\label{definition_of_} We assume that the ambient [[monoidal category]] is [[Ab]] with the [[tensor product of abelian groups]]. But the definition works more generally \begin{defn} \label{ModSpelledOut}\hypertarget{ModSpelledOut}{} An [[object]] in $Mod$ is a pair $(R,N)$ consisting of a commutative [[ring]] $R$ and an $R$-[[module]] $N$. A [[morphism]] \begin{displaymath} (\phi,\kappa) : (R,N) \to (R',N') \end{displaymath} is a pair consisting of a ring [[homomorphism]] $\phi : R \to R'$ and a morphism $\kappa : N \to \phi^* N'$ of $R$-modules, where $\phi^* N'$ is the [[tensor product]] $\phi^* N' := R \otimes_{\phi} N$. \end{defn} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{ModAsBifibration}{}\subsubsection*{{$Mod$ as a bifibration}}\label{ModAsBifibration} Projecting out the first items in the pairs appearing in def. \ref{ModSpelledOut} yields a canonical functor \begin{displaymath} p :Mod \to CRing \end{displaymath} \begin{displaymath} (R,N) \mapsto R \,. \end{displaymath} that exhibits $Mod$ as a [[bifibration]] over $R$. The fiber of this projection over a ring $R$ is $Mod_R$, the category of $R$-modules. In particular the fiber over the initial commutative ring $R = \mathbb{Z}$ is \begin{displaymath} Mod_{\mathbb{Z}} = Ab \end{displaymath} the category [[Ab]] of abelian groups. \hypertarget{TangentsAndDeformationTheory}{}\subsubsection*{{Tangents and deformation theory}}\label{TangentsAndDeformationTheory} By an old observation of Quillen -- reviewed at [[module]] -- the bifibration $Mod \to CRing$ this is [[equivalence of categories|equivalent]] to the category of fiberwise abelian [[group object]] in the [[codomain fibration]] $[I,CRing] \to CRing$: \begin{displaymath} (Mod \to CRing) \simeq Ab([I,CRing]] \to CRing) \,. \end{displaymath} For a fixed ring $R$, the category $Mod_R$ of $R$-modules is canonically equivalent to $Ab(CRing/R)$, the category of abelian [[group object]]s in the [[overcategory]] $CRing/R$: \begin{displaymath} Mod_R \simeq Ab(CRing/R) \,. \end{displaymath} This says that $Mod \to Ring$ is the [[tangent category]] of $CRing$: the above equivalence regards an $R$-module $N$ equivalently as the [[square-0 extension]] ring $R \oplus N$ (with multiplication $(r_1,n_1) \cdots (r_2,n_2) = (r_1 r_2, r_1 n_2 + r_2 n_1)$), which may be thought of as the ring of functions on the infinitesimal neighbourhood of the 0-section of the [[vector bundle]] (or rather [[quasicoherent sheaf]]) over $Spec R$ that is given by $N$. There is thus another natural projection from $Mod$ to rings, namely the functor that remembers these [[square-0 extensions]] \begin{displaymath} f : Mod \to CRing \end{displaymath} \begin{displaymath} (R,N) \mapsto R \oplus N \,. \end{displaymath} This functor has a [[left adjoint]] $\Omega : CRing \to Mod$ which is also a [[section]]: this is the functor that sends a ring to its module of [[Kähler differential]]s. \begin{displaymath} (\Omega \dashv f) : Mod \stackrel{\overset{\Omega}{\leftarrow}}{\underset{f}{\to}} CRing \,. \end{displaymath} \hypertarget{RModIsAbelian}{}\subsubsection*{{$R Mod$ is an abelian category}}\label{RModIsAbelian} Let the ambient [[monoidal category]] be [[Ab]] equipped with the [[tensor product of abelian groups]]. \begin{theorem} \label{RModIsAbelian}\hypertarget{RModIsAbelian}{} Let $R$ be a [[commutative ring]]. Then $R Mod$ is an [[abelian category]]. In fact $R Mod$ is a [[Grothendieck category]]. \end{theorem} We discuss now all the ingredients of this statement in detail. Let $U : R Mod \to Set$ be the [[forgetful functor]] to the underlying sets. \begin{prop} \label{RModHasZeroObject}\hypertarget{RModHasZeroObject}{} $R Mod$ has a [[zero object]], given by the 0-module, the [[trivial group]] equipped with trivial $R$-action. \end{prop} \begin{proof} Clearly the 0-module $0$ is a [[terminal object]], since every morphism $N \to 0$ has to send all elements of $N$ to the unique element of $0$, and every such morphism is a [[homomorphism]]. Also, 0 is an [[initial object]] because a morphism $0 \to N$ always exists and is unique, as it has to send the unique element of 0, which is the neutral element, to the neutral element of $N$. \end{proof} \begin{prop} \label{RModHasKernelsAndCokernels}\hypertarget{RModHasKernelsAndCokernels}{} \begin{enumerate}% \item $R Mod$ has all [[kernels]]. The kernel of a homomorphism $f : N_1 \to N_2$ is the set-theoretic [[preimage]] $U(f)^{-1}(0)$ equipped with the induced $R$-module structure. \item $R Mod$ has all [[cokernels]]. The cokernel of a homomorphism $f : N_1 \to N_2$ is the [[quotient]] abelian group \begin{displaymath} coker f = \frac{N_2}{im(f)} \end{displaymath} of $N_2$ by the [[image]] of $f$. \end{enumerate} \end{prop} \begin{proof} The defining [[universal property]] of kernel and cokernels is immediately checked. \end{proof} \begin{prop} \label{}\hypertarget{}{} $U : R Mod \to Set$ preserves and reflects [[monomorphisms]] and [[epimorphisms]]: A homomorphism $f : N_1 \to N_2$ in $R Mod$ is a [[monomorphism]] / [[epimorphism]] precisely if $U(f)$ is an [[injection]] / [[surjection]]. \end{prop} \begin{proof} Suppose that $f$ is a [[monomorphism]], hence that $f : N_1 \to N_2$ is such that for all morphisms $g_1, g_2 : K \to N_1$ such that $f \circ g_1 = f \circ g_2$ already $g_1 = g_2$. Let then $g_1$ and $g_2$ be the inclusion of [[submodules]] generated by a single element $k_1 \in K$ and $k_2 \in K$, respectively. It follows that if $f(k_1) = f(k_2)$ then already $k_1 = k_2$ and so $f$ is an [[injection]]. Conversely, if $f$ is an injection then its image is a [[submodule]] and it follows directly that $f$ is a monomorphism. Suppose now that $f$ is an [[epimorphism]] and hence that $f : N_1 \to N_2$ is such that for all morphisms $g_1, g_2 : N_2 \to K$ such that $f \circ g_1 = f \circ g_2$ already $g_1 = g_2$. Let then $g_1 : N_2 \to \frac{N_2}{im(f)}$ be the natural projection. and let $g_2 : N_2 \to 0$ be the [[zero morphism]]. Since by construction $f \circ g_1 = 0$ and $f \circ g_2 = 0$ we have that $g_1 = 0$, which means that $\frac{N}{im(f)} = 0$ and hence that $N = im(f)$ and so that $f$ is surjective. The other direction is evident on elements. \end{proof} \begin{defn} \label{AbelianGroupStructureOnHoms}\hypertarget{AbelianGroupStructureOnHoms}{} For $N_1, N_2 \in R Mod$ two modules, define on the [[hom set]] $Hom_{R Mod}(N_1,N_2)$ the structure of an [[abelian group]] whose addition is given by argumentwise addition in $N_2$: $(f_1 + f_2) : n \mapsto f_1(n) + f_2(n)$. \end{defn} \begin{prop} \label{RModIsAbEnriched}\hypertarget{RModIsAbEnriched}{} With def. \ref{AbelianGroupStructureOnHoms} $R Mod$ composition of morphisms \begin{displaymath} \circ : Hom(N_1, N_2) \times Hom(N_2, N_3) \to Hom(N_1,N_3) \end{displaymath} is a [[bilinear map]], hence is equivalently a morphism \begin{displaymath} Hom(N_1, N_2) \otimes Hom(N_2,N_3) \to Hom(N_1, N_3) \end{displaymath} out of the [[tensor product of abelian groups]]. This makes $R Mod$ into an [[Ab-enriched category]]. \end{prop} \begin{proof} Linearity of composition in the second argument is immediate from the pointwise definition of the abelian group structure on morphisms. Linearity of the composition in the first argument comes down to linearity of the second module homomorphism. \end{proof} \begin{remark} \label{}\hypertarget{}{} In fact $R Mod$ is even a [[closed category]], see prop. \ref{RModIsClosedMonoidal} below, but this we do not need for showing that it is abelian. \end{remark} Prop. \ref{RModHasZeroObject} and prop. \ref{RModIsAbEnriched} together say that: \begin{cor} \label{RModIsAdditive}\hypertarget{RModIsAdditive}{} $R Mod$ is an [[pre-additive category]]. \end{cor} \begin{prop} \label{RModHasProductsAndCoproducts}\hypertarget{RModHasProductsAndCoproducts}{} $R Mod$ has all [[products]] and [[coproducts]], being [[direct products]] $\prod_{i \in I} N_i$ and [[direct sums]] $\oplus_{i \in I} N_i$. The products are given by [[cartesian product]] of the underlying sets with componentwise addition and $R$-action. The direct sum is the [[submodule]] of the direct product consisting of tuples of elements such that only finitely many are non-zero. \end{prop} \begin{proof} The defining [[universal properties]] are directly checked. Notice that the direct product $\prod_{i \in I} N_i$ consists of arbitrary tuples because it needs to have a projection map \begin{displaymath} p_j : \prod_{i \in I} N_i \to N_j \end{displaymath} to each of the modules in the product, reproducing all of a possibly infinite number of non-trivial maps $\{K \to N_j\}$. On the other hand, the direct sum just needs to contain all the modules in the sum \begin{displaymath} \iota_j : N_j \to \oplus_{i \in I} N_i \end{displaymath} and since, being a module, it needs to be closed only under addition of \emph{finitely} many elements, so it consists only of [[linear combinations]] of the elements in the $N_j$, hence of finite formal sums of these. \end{proof} Together cor. \ref{RModIsAdditive} and prop. \ref{RModHasProductsAndCoproducts} say that: \begin{cor} \label{RModIsAdditive}\hypertarget{RModIsAdditive}{} $R Mod$ is an [[additive category]]. \end{cor} \begin{prop} \label{InRModMonosAreKernelOfTheirCokernel}\hypertarget{InRModMonosAreKernelOfTheirCokernel}{} In $R Mod$ \begin{itemize}% \item every [[monomorphism]] is the [[kernel]] of its [[cokernel]]; \item every [[epimorphism]] is the [[cokernel]] of its [[kernel]]. \end{itemize} \end{prop} \begin{proof} Using prop. \ref{RModHasKernelsAndCokernels} this is directly checked on the underlying sets: given a monomorphism $K \hookrightarrow N$, its cokernel is $N \to \frac{N}{K}$, The kernel of that morphism is evidently $K \hookrightarrow N$. \end{proof} Now cor. \ref{RModIsAdditive} and prop. \ref{InRModMonosAreKernelOfTheirCokernel} imply theorem \ref{RModIsAbelian}, by definition. \begin{prop} \label{FilteredColimitsInRModAreExact}\hypertarget{FilteredColimitsInRModAreExact}{} The operation of forming [[filtered colimits]] in $R Mod$ is an [[exact functor]]. \end{prop} (e.g. \hyperlink{Kiersz}{Kiersz, prop. 4}) \hypertarget{RModIsClosedMonoidalCategory}{}\subsubsection*{{$R Mod$ is a closed monoidal category}}\label{RModIsClosedMonoidalCategory} Let $R$ be a [[commutative ring]]. \begin{defn} \label{HomModule}\hypertarget{HomModule}{} For $N_1, N_2 \in R Mod$, equip the [[hom-set]] $Hom_{R Mod}(N_1, N_2)$ with the structure of an $R$-module as follows: for all $f,g \in Hom_{R Mod}(N_1, N_2)$, all $n_1 \in N_1$ and all $r \in R$ set \begin{itemize}% \item $(f + g) \colon n_1 \mapsto f(n_1) + g(n_2)$ \item $r \cdot f \colon n_1 \mapsto r\cdot (f(n_1))$. \end{itemize} Write $[N_1,N_2] \in R Mod$ for the resulting $R$-module structure. \end{defn} \begin{prop} \label{RModIsClosedMonoidal}\hypertarget{RModIsClosedMonoidal}{} Equipped with the [[tensor product of modules]], $R Mod$ becomes a [[monoidal category]] (in fact a [[distributive monoidal category]]). The tensor unit is $R$ regarded canonically as an $R$-module over itself. This is a [[closed monoidal category]], the [[internal hom]] is given by the hom-modules of def. \ref{HomModule}. \end{prop} \begin{proof} Either by definition or by a basic property of the [[tensor product of modules]], a module [[homomorphism]] $\phi \colon N_1 \otimes_R N_2 \to N_3$ is precisely an $R$-[[bilinear function]] of the underlying sets. For fixed elements $n_1 \in N_1$ and $n_2 \in N_2$ write \begin{displaymath} \overline{\phi}(n_1) \coloneqq \phi(n_1, -) \colon N_2 \to N_3 \end{displaymath} and \begin{displaymath} \phi(-,n_2) \colon N_1 \to N_3 \end{displaymath} for the hom-[[adjuncts]] on the underlying sets. By the bilinearity of $\phi$ both of these are $R$-linear maps. The first being linear means that $\overline{\phi}$ is a function $\overline{\phi} \colon N_1 \to [N_2, N_3]$ to the set of module homomorphisms, and the second being linear says that it is itself a mododule homomorphisms by def. \ref{HomModule}, since \begin{displaymath} \overline{\phi}(r\cdot n_1) = (n_2 \mapsto \phi(r\cdot n_1, n_2) = r \phi(n_1, n_2)) = r \cdot \left(\overline{\phi}(n_1)\right) \,. \end{displaymath} The map $\phi \mapsto \overline{\phi}$ establishes a [[natural transformation]] \begin{displaymath} Hom_{R Mod}(N_1 \otimes_R N_2, N_3) \stackrel{}{\to} Hom_{R Mod}(N_1, [N_2, N_3]) \,. \end{displaymath} Conversely, every element of $Hom_{R Mod}(N_1, [N_2, N_3])$ defines [[bilinear map]], hence a homomorphism $N_1 \otimes_R N_2 \to N_3$ and this construction is inverse to the above, showing that it is a [[natural isomorphism]]. This exhibits the [[internal hom]]-[[adjunction]] $(-) \otimes_R N_2 \vdash [N_2,-]$. \end{proof} \hypertarget{exact_functors_between_categories_of_modules}{}\subsubsection*{{Exact functors between categories of modules}}\label{exact_functors_between_categories_of_modules} The \emph{[[Eilenberg-Watts theorem]]} says that sufficiently [[exact functors]] between categories of modules are necessarily given by forming [[tensor products of modules]]. \hypertarget{LimitsAndColimits}{}\subsubsection*{{Limits and colimits}}\label{LimitsAndColimits} Let $R$ be a [[ring]]. \begin{prop} \label{ModuleIsFilteredColimitOverFinitelyGeneratedSubmodules}\hypertarget{ModuleIsFilteredColimitOverFinitelyGeneratedSubmodules}{} Every $R$-module is the [[filtered colimit]] over its [[finitely generated module|finite generated]] [[submodules]]. \end{prop} See for instance (\hyperlink{Kiersz}{Kiersz, prop. 3}). \hypertarget{tiny_objects}{}\subsubsection*{{Tiny objects}}\label{tiny_objects} For discussion of [[tiny objects]] in $Mod$, see at \emph{\href{tiny+object#InCategoriesOfModulesOverRings}{Tiny object -- In categories of modules over rings}}. \hypertarget{tannaka_duality}{}\subsubsection*{{Tannaka duality}}\label{tannaka_duality} [[!include structure on algebras and their module categories - table]] \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[Ring]], [[CRing]] \item [[Vect]] \item [[(∞,1)Mod]] \item [[2Mod]] \item [[nMod]] \item [[2-category of module categories]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} Discussion of $R Mod$ in $(Ab, \otimes)$ being an [[abelian category]] is for instance in \begin{itemize}% \item Rankeya Datta, \emph{The category of modules over a commutative ring and abelian categories} (\href{http://www.math.columbia.edu/~ums/pdf/Rankeya_R-mod_and_Abelian_Categories.pdf}{pdf}) \end{itemize} Discussion of limits and colimits in $R Mod$ is in \begin{itemize}% \item Andy Kiersz, \emph{Colimits and homological algebra}, 2006 (\href{http://www.math.uchicago.edu/~may/VIGRE/VIGRE2006/PAPERS/Kiersz.pdf}{pdf}) \end{itemize} Discussion of homotopy theoretic modules via [[stabilization]] of [[slice model structures]] is in \begin{itemize}% \item [[Stefan Schwede]], \emph{Spectra in model categories and applications to the algebraic cotangent complex}, Journal of Pure and Applied Algebra 120 (1997) 104 \end{itemize} A summary of the discussion in \hyperlink{ModAsBifibration}{Mod as a bifibration} and \hyperlink{TangentsAndDeformationTheory}{Tangents and deformation theory} together with their embedding into the bigger picture of [[tangent (∞,1)-category|tangent (∞,1)-categories]] is in \begin{itemize}% \item [[Jacob Lurie]], \emph{[[Deformation Theory]]} \end{itemize} category: category [[!redirects Mod]] [[!redirects RMod]] [[!redirects R Mod]] [[!redirects R-Mod]] [[!redirects KMod]] [[!redirects K Mod]] [[!redirects K-Mod]] [[!redirects kMod]] [[!redirects k Mod]] [[!redirects k-Mod]] [[!redirects category of modules]] [[!redirects categories of modules]] \end{document}