# Wednesday, January 13 Reference: - The course text is Weibel [@weibel_2011]. - See the many corrections/errata: - Sections we'll cover: - 1.1-1.5, - 2.2-2.7, - 3.4, - 3.6, - 6.1, - 5.1-5.2, - 5.4-5.8, - 6.8, - 6.7, - 6.3, - 7.1-7.5, - 7.7-7.8, - Appendix A (when needed) - Course Website: ## Overview :::{.definition title="Exact complexes"} A **complex** is given by \[ \cdots \mapsvia{d_{i-1}} M_{i-1} \mapsvia{d_i} M_i \mapsvia{d_{i+1}}M_{i+1} \to \cdots .\] where $M_i \in \rmod$ and $d_i \circ d_{i-1} = 0$, which happens if and only if $\im d_{i-1} \subseteq \ker d_i$. If $\im d_{i-1} = \ker d_i$, this complex is **exact**. ::: :::{.example title="?"} We can apply a functor such as $\tensor_R N$ to get a new complex \[ \cdots \mapsvia{d_{i-1} \tensor 1_N} M_{i-1} \tensor_R N \mapsvia{d_i \tensor 1} M_i \tensor N \to M_{i+1} \mapsvia{d_{i+1} \tensor 1} \cdots .\] ::: :::{.example title="?"} Applying $\Hom(N, \wait)$ similarly yields \[ \Hom_R(N, M_{i}) \mapsvia{d_{i-1}^*} \Hom_R(N, M_{i+1}) ,\] where $d_i^* = d_i \circ (\wait)$ is given by composition. ::: :::{.example title="?"} Applying $\Hom(\wait, N)$ yields \[ \Hom_R(M_i, N) \mapsvia{d_{i}^*} \Hom_R(M_{i+1}, N) \] where $d_i^* = (\wait) \circ d_i$. ::: :::{.remark} Note that we can also take complexes with arrows in the other direction. For $F$ a functor, we can rewrite these examples as \[ d_i^* \circ d_{i-1}^* = F(d_i) \circ F(d_{i-1}) = F(d_i \circ d_{i-1}) = F(0) = 0 ,\] provided $F$ is nice enough and sends zero to zero. This follows from the fact that functors preserve composition. Even if the original complex is exact, the new one may not be, so we can define the following: ::: :::{.definition title="Cohomology"} \[ H^i(M^*) = \ker d_i^* / \im d_{i-1}^* .\] ::: :::{.remark} These will lead to **$i$th derived functors**, and category theory will be useful here. See appendix in Weibel. For a category \( \mathcal{C} \) we'll define - \( \mathrm{Obj}(\mathcal{C} ) \) as the objects - \( \Hom_{\mathcal{C}}(A, B) \) a set of morphisms between them, where a more modern notation might be \( \mathrm{Mor}(A, B) \). - Morphisms compose: $A \mapsvia{f} B \mapsvia{g} C$ means that \( g\circ f \in \Hom_{\mathcal{C}}(A, C) \) - Associativity - Identity morphisms See the appendix for diagrams defining zero objects and the zero map, which we'll need to make sense of exactness. We'll also needs notions of kernels and images, or potentially cokernels instead of images since they're closely related. ::: :::{.remark} In the examples, we had \( \ker d_i \subseteq M_i \), but this need not be true since the objects in the category may not be sets. Such an example is the category of complexes of \(R\dash\)modules: \( \Cx (\rmod) \). In this setting, kernels will be subcomplexes but not subsets. ::: :::{.definition title="Functors"} Recall that **functors** are "functions" between categories \( F: \mathcal{C}\to \mathcal{D} \) such that - Objects are sent to objects, - Morphisms are sent to morphisms, so $A \mapsvia{f} B \leadsto F(A) \mapsvia{F(f)} F(B)$, - $F$ respects composition and identities ::: :::{.example title="Hom"} $\Hom_R(N, \wait): \rmod \to \Ab$, noting that the hom set may not have an \(R\dash\)module structure. ::: :::{.remark} Taking cohomology yields the $i$th derived functors of $F$, for example $\Ext^i, \Tor_i$. Recall that functors can be *covariant* or contravariant. See section 1 for formulating simplicial and singular homology (from topology) in this language. ::: ## Chapter 1: Chain Complexes ### Complexes of \(R\dash\)modules :::{.definition title="Exactness"} Let $R$ be a ring with 1 and define \( \rmod\) to be the category of *right* \(R\dash\)modules. $A \mapsvia{f} B \mapsvia{g} C$ is **exact** if and only if $\ker g = \im f$, and in particular $g\circ f = 0$. ::: :::{.definition title="Chain Complex"} A **chain complex** is \[ C_\wait \da (C_\wait, d_\wait) \da \qty{ \cdots \to C_{n+1} \mapsvia{d_{n+1}} C_n \mapsvia{d_n} C_{n-1} \to \cdots } \] for $n \in \ZZ$ such that $d_n \circ d_{n+1} = 0$. We drop the $n$ from the notation and write $d^2 \da d\circ d = 0$. ::: :::{.definition title="Cycles and boundaries"} \envlist - \( Z_n = Z_n(C_\wait) = \ker d_n \) are referred to as **$n\dash$cycles**. - $B_n = B_n(C_\wait) = \im d_{n+1}$ are the **$n\dash$boundaries**. ::: :::{.definition title="Homology of a chain complex"} Note that if $d^2 = 0$ then $B_n \leq Z_n \leq C_n$. In this case, it makes sense to define the quotient module $H^n(C_\wait) \da Z_n / B_n$, the **$n$th homology** of $C_\wait$. ::: :::{.definition title="Maps of chain complexes"} A map $u: C_\wait \to D_\wait$ of chain complexes is a sequence of maps $u_n: C_n \to D_n$ such that all of the following squares commute: \begin{tikzcd} {\cdots} & {C_{n+1}} & {C_n} & {C_{n-1}} & {\cdots} \\ \\ {\cdots} & {D_{n+1}} & {D_n} & {D_{n-1}} & {\cdots} \arrow[from=1-1, to=1-2] \arrow[from=1-2, to=1-3] \arrow[from=1-3, to=1-4] \arrow[from=3-1, to=3-2] \arrow[from=3-2, to=3-3] \arrow[from=3-3, to=3-4] \arrow[from=3-4, to=3-5] \arrow[from=1-4, to=1-5] \arrow["{u_{n+1}}", from=1-2, to=3-2] \arrow["{u_n}", from=1-3, to=3-3] \arrow["{u_{n-1}}", from=1-4, to=3-4] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsMTAsWzEsMCwiQ197bisxfSJdLFsyLDAsIkNfbiJdLFszLDAsIkNfe24tMX0iXSxbMSwyLCJEX3tuKzF9Il0sWzIsMiwiRF9uIl0sWzMsMiwiRF97bi0xfSJdLFswLDAsIlxcYnVsbGV0Il0sWzAsMiwiXFxidWxsZXQiXSxbNCwyLCJcXGJ1bGxldCJdLFs0LDAsIlxcYnVsbGV0Il0sWzYsMF0sWzAsMV0sWzEsMl0sWzcsM10sWzMsNF0sWzQsNV0sWzUsOF0sWzIsOV0sWzAsMywidV97bisxfSIsMV0sWzEsNCwidV9uIiwxXSxbMiw1LCJ1X3tuLTF9IiwxXV0=) ::: :::{.remark} We can thus define a category \( \mathrm{Ch}(\rmod) \) where - The objects are chain complexes, - The morphisms are chain maps. ::: :::{.exercise title="Weibel 1.1.2"} A chain complex map $u: C_\wait \to D_\wait$ restricts to \[ u_n: Z_n(C_\wait) \to Z_n(D_\wait) \\ u_n: B_n(D_\wait) \to B_n(D_\wait) \] and thus induces a well-defined map $u_{n, *}: H_n(C_\wait) \to H_n(D_\wait)$. ::: :::{.remark} Each $H_n$ thus becomes a functor \( \mathrm{Ch}(\rmod) \to \rmod \) where $H_n(u) \da u_{*. n}$. :::