Category Theory

Let \(A\) be a ring and let \((X,\mathcal{O}_X)\) be a scheme. Given a morphism \(f:X\rightarrow \text{Spec}(A)\) we have an associated map on sheaves \(f^{\#}:\mathcal{O}_{\text{Spec}(A)}\rightarrow f_* \mathcal{O}_X\). Taking global sections, we obtain a homomorphism \(A\rightarrow \mathcal{O}_X(X)\). Thus there is a natural map \(\alpha: \text{Hom}_{\text{Schemes}}(X,\text{Spec}(A))\rightarrow \text{Hom}_{\text{Rings}}(A,\mathcal{O}_X(X)).\) Then, \(\alpha\) is bijective. We will try to understand this adjointness of Global section functor and Spec functor. Suppose we are given a scheme \((X,\mathcal{O}_X)\) and a ring homomoprhism \(\varphi: A\rightarrow \mathcal{O}_X(X)\). We construct a morphism of schemes \((f,f^{\#}):(X,\mathcal{O}_X)\rightarrow (\text{Spec}(A),\mathcal{O}_{\text{Spec}(A)})\). We first define morphism of topological spaces \(f:X\rightarrow \text{Spec}(A)\). Let \(x\in X\), we want to assign a prime ideal \(P\) in \(A\). Let \(X=\text{Spec}(B)\) and \(x=\mathfrak{P}\in X=\text{Spec}(B)\), as we have a ring homomorphism \(\varphi : A\rightarrow \mathcal{O}_X(X)=O_{\text{Spec}(B)}(\text{Spec}(B))=B\) ...

Let \(G\) be a group. We are going to construct a category with one element whose morphisms are elements of this group \(G\). Definition : A category \(\mathcal{C}\) consists of a collection of objects, \(\text{Ob}(\mathcal{C})\), for each \(A,B\in \text{Ob}(\mathcal{C})\) a collection of maps from \(A\) to \(B\) denoted by \(\mathcal{C}(A,B)\). for each \(A,B,C \in \text{Ob}(\mathcal{C})\) a function \(\mathcal{C}(B,C)\times \mathcal{C}(A,B)\rightarrow \mathcal{C}(A,C)\) with \((g,f)\mapsto g\circ f\) called the composition for each \(A\in \text{Ob}(\mathcal{C})\) an element \(1_A\in \mathcal{C}(A,A)\) called the identity on \(A\) satisfying the following conditions Associativity : for each \(f\in \mathcal{C}(A,B), g\in \mathcal{C}(B,C), h\in \mathcal{C}(C,D)\) we have \((h\circ g)\circ f=h\circ (g\circ f)\). Identity law : for each \(f\in \mathcal{C}(A,B)\) we have \(f\circ 1_A=f=1_B\circ f\). We are not constructing a categroy \(\mathcal{C}\) with \(\text{Ob}(\mathcal{C})=G\), we are constructing a category \(\mathcal{C}\) with \(\text{Ob}(\mathcal{C})=\{A\}\)(one point set) and \(\mathcal{C}(A,A)=G\). We have defined objects and morphisms collections (here there is only one object so there is only one morphisms collection). We have to define composition \(\mathcal{C}(A,A)\times \mathcal{C}(A,A)\rightarrow \mathcal{C}(A,A)\) i.e., we have to give a map \(G\times G\rightarrow G\). There are two natural ways to give this map \((g,h)\mapsto g.h\) or \((g,h)\mapsto h.g\). We consider the map \((g,h)\mapsto g\circ h:=g.h\) to give composition. Obvious choice of an identity element in \(\mathcal{C}\) is $llatex 1_A=e$ identity element of the group. Associativity of group implies associativity of composition \((g\circ h)\circ f=(g.h)\circ f=(g.h).f=g.(h.f)=g\circ (h\circ f).\) ...

Yoneda lemma : Let \(\mathcal{C}\) be a (locally) small category. Then naturally in \(A\in \mathcal{C}\) and \(X\in [\mathcal{C}^{\rm{op}},\rm{Set}]\). Terminology : \(\mathcal{C}\) is a category mentioned in the lemma, \(\mathcal{C}^{\rm{op}}\) is the opposite category associated to \(\mathcal{C}\). \(\rm{Set}\) is the category with elements as sets and morphisms as functions. \(X:\mathcal{C}^{op}\rightarrow \rm{Set}\) is a functor. Given \(A\in \mathcal{C}\), \(H_A\) is the functor \(H_A:\mathcal{C}^{\rm{op}}\rightarrow \rm{Set}\) given by $B\mapsto \mathcal{C}(B,A)$. \([\mathcal{C}^{\rm{op}},\rm{Set}]\) is the category with functors from \(\mathcal{C}^{\rm{op}}\) to $\rm{Set}$ as elements and natural transformations between these functors as morphisms. \([\mathcal{C},\rm{Set}]\) is the category with functors from \(\mathcal{C}\) to \(rm{Set}\) as elements and natural transformations between these functors as morphisms. Given \(X,A\) as above, \(X(A)\) is a set and \([\mathcal{C}^{\rm{op}},\rm{Set}](H_A,X)\) is a set. Yoneda lemma says that there is a bijection between these sets, natural bijection in both \(A\) and \(X\). Natural transformation : Let \(\mathcal{A},\mathcal{B}\) be two categories and \(F,G:\mathcal{A}\rightarrow \mathcal{B}\) be both contravariant or both covariant functors. A natural transformation \(\eta:F\rightarrow G\) is a family of arrows (morphisms) \(F(A)\xrightarrow{\eta(A)}G(A)\) such that for each \(A\xrightarrow{f}A'\) in \(\mathcal{A}\) the following appropriate diagram commutes. Now that we have defined terminology used in the statement, we will now prove the statement. We will first give a bijectionand then prove that it is natural in \(A\) and \(X\). Construction of Bijective map : Let \(\eta\in [\mathcal{C}^{\rm{op}},\rm{Set}](H_A,X)\) i.e., \(\eta:H_A\rightarrow X\) is a natural transformation. We want to assign an element in \(X(A)\) with this \(\eta\). It is only natural to consider the map \(\eta(A):H_A(A)\rightarrow X(A)\). The set \(H_A(A)\) has a special element namely \(1_A\in H_A(A)\), its image \(\eta(A)(1_A)\in X(A)\). Define \(\Phi(\eta)=\eta(A)(1_A)\). This give a map \(\Phi:[\mathcal{C}^{\rm{op}},\rm{Set}](H_A,X)\rightarrow X(A)\). ...

Let \(A\) be a ring and \(M\) be an \(A\) module. We associate a sheaf of modules \(\widetilde{M}\) on \(X= \text{Spec(A)}\) with this module \(M\). These modules are our models for quasi-coherent sheaves. For an open subset \(U\subseteq \text{Spec(A)}\) we definewhere the condition \(\dagger\) says that given \(p\in U\) we have \(s(p)\in M_p\) and that \(s\) is locally a fraction i.e., given \(p\in U\) there exists an open set \(U(p)\subseteq U\) and \(m\in M, f\in A\) such that \(s(q)=\frac{m}{f}\in M_q\) for all \(q\in U(p)\). With obvious restriction maps this defines a sheaf \(\widetilde{M}\) on \(X= \text{Spec(A)}\) called the sheaf associated with \(M\). This should remind you something similar we have done before. We have defined structure sheaf on \(X=\text{Spec(A)}\) in exactly same way where \(M\) in this definition is replaced by ring \(A\). Just to confirm, we havewhere the condition \(\dagger\) says that given \(p\in U\) we have \(s(p)\in A_p\) and that \(s\) is locally a fraction i.e., given \(p\in U\) there exists an open set \(U(p)\subseteq U\) and \(a\in A, f\in A\) such that \(s(q)=\frac{a}{f}\in A_q\) for all \(q\in U(p)\). This should suggest some relation between \(\widetilde{M}(U)\) and \(\mathcal{O}_X(U)\). It turns out that \(\widetilde{M}(U)\) is an \(\mathcal{O}_X(U)\) module for every open \(U\subseteq \text{Spec(A)}\). So, \(\widetilde{M}\) is a sheaf of \(\mathcal{O}_X\) modules. In case of structure sheaf \(\mathcal{O}_X\) on \(X=\text{Spec (A)}\) we have following proposition. Proposition : Let \(A\) be a ring, and \((\text{Spec A},\mathcal{O})\) its spectrum. For any \(\mathfrak{p}\in \text{Spec A}\), the stalk \(\mathcal{O}_{\mathfrak{p}}\) of the sheaf \(\mathcal{O}_{}\) is isomorphic to the local ring \(A_{\mathfrak{p}}\) i.e., \(\mathcal{O}_{\mathfrak{p}}\cong A_{\mathfrak{p}}\). For any element \(f\in A\), the ring \(\mathcal{O}(D(f))\) is isomorphic to the localized ring \(A_f\) i.e., \(\mathcal{O}(D(f))\cong A_f\). In particular, \(\Gamma(\text{Spec A}, \mathcal{O})\cong A\). In case of associated sheaf \(\widetilde{M}\) on \(X=\text{Spec A}\) we have same results with \(A\) replaced by \(M\) and \(\mathcal{O}\) replaced by \(\widetilde{M}\). Precisely, we have following proposition. Proposition : Let \(A\) be a ring, \(M\) be an \(A\) module and and \(\widetilde{M}\) be the associated sheaf on \(X=\text{Spec A}\). For any \(\mathfrak{p}\in \text{Spec A}\), the stalk \(\widetilde{M}_{\mathfrak{p}}\) of the sheaf \(\widetilde{<}\) is isomorphic to the localization \(M_{\mathfrak{p}}\) i.e., \(\widetilde{M}_{\mathfrak{p}}\cong M_{\mathfrak{p}}\). For any element $f\in A$, the ring \(\widetilde{M}(D(f))\) is isomorphic to the localization \(M_f\) i.e., \(\widetilde{M}(D(f))\cong M_f\). In particular, \(\Gamma(\text{Spec A}, \widetilde{M})\cong M\). Proposition : Let \(A\) be a ring and let \(X=\text{Spec}(A)\). Also let \(A\rightarrow B\) be a ring homomorphism, and let \(f:\text{Spec}(B)\rightarrow \text{Spec}(A)\) be the correpsonding morphism of spectra. Then : the map \(M\mapsto \widetilde{M}\) gives an exact, fully faithful functor from category of \(A\) modules to the category of \(\mathcal{O}_X\) modules. \(\widetilde{M\otimes_A N}=\widetilde{M}\otimes_{\mathcal{O}_X}\widetilde{N}\). \(\widetilde{\bigoplus M_i}=\bigoplus \widetilde{M_i}\). For a \(B\) module \(N\), we have \(f_*(\widetilde{N})=\widetilde{~_A N}\) where \(~_A N\) is \(N\) considered as an \(A\) module. For a \(A\) module \(M\), we have \(f^*(\widetilde{M})=\widetilde{M\otimes_A B}\).

I am Praphulla Koushik, faculty member in department of Mathematics, NIT Calicut. I am interested broadly in Differential geometry and Category theory. I want to share my thoughts as I read the following books. Hartshorne's Algebraic geometry. ... ... I wish to add some articles that seems to be interesting for me on different topics. Most of the times, I start writing a post and leave it in the middle.. I usually write in usual latex before copying it here. It is time consuming to put latex in between dollars. I have to take screen shot of figures, crop into image and then add here which is really time consuming. So, If I feel I am wasting time, I will leave it like that. If you find any of my posts interesting and incomplete, do leave a message. I will continue it.