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 \(\varphi : A\rightarrow \mathcal{O}_X(X)=O_{\text{Spec}(B)}(\text{Spec}(B))=B\) ...

Definition : Let \(S\) be a scheme. An \(S\) scheme is a scheme \(X\) together with a morphism \(p:X\rightarrow S\). A morphism of \(S\) schemes \((X,p:X\rightarrow S)\) and \((Y,q:Y\rightarrow S)\) is a morphism of schemes \(f:X\rightarrow Y\) such that \(q\circ f=p\).  

let \(\mathcal{F}\) be a presheaf. We define associated presheaf of \(\mathcal{F}\) to be the sheaf given by map \(U\mapsto \widetilde{F}(U)\) where where the condition \(^\dagger\) says that \(s(p)\in \mathcal{F}_p\) for every \(p\in U\) and there exists an open subset \(U(p)\subset U\) containing \(p\) and a section \(t\in \mathcal{F}(U(p))\) such that \(t_q=s(q)\) for every \(q\in U(p)\). Suppose \(\mathcal{F}\) is actually a sheaf then we will see that \(\widetilde{F}(U)\cong \mathcal{F}(U)\) for every open \(U\subseteq X\). Let \(s\in \widetilde{F}(U)\) i.e., \(s:U\rightarrow \bigsqcup_{i\in \Lambda}\mathcal{F}_{p_i}\) (this notation is just for my comfort, we have \(U=\bigcup_{i\in \Lambda}\{p_i\}\)) satisfying some conditions given above. Given such \(s\) we want to assign an element in \(\mathcal{F}(U)\). ...

I will add links of blogs of topics in sequence as in Hartshorne. Morphism of Sheaves – Morphism of Stalks Sheafification of a presheaf that is already a sheaf is itself – Reality check Structure sheaf on spectrum of a ring          

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 \((g\circ h)\circ f=(g.h)\circ f=(g.h).f=g.(h.f)=g\circ (h\circ f).\) ...

Let \(f:X\rightarrow Y\) be an affine morphism and \(\mathcal{F}\) is a quasi coherent sheaf of \(\mathcal{O}_X\) modules. Then, \(f_*\mathcal{F}\) is a quasi coherent sheaf of \(\mathcal{O}_Y\) modules. We have the following result : Let \(X\) be a scheme. Then, an \(\mathcal{O}_X\) module \(\mathcal{F}\) is quasi coherent iff for every open affine subset \(U=\text{Spec}(A)\) of \(X\), there is an \(A\) module \(M\) such that \(\mathcal{F}|_U=\tilde{M}\). Let \(U\subseteq Y\) be an open affine subset say \(U=\text{Spec}(A)\). As \(f\) is affine, \(f^{-1}(U)\) is affine open, say \(f^{-1}(U)=\text{Spec}(B)\subseteq X\). As \(\mathcal{F}\) is quasi coherent sheaf of \(\mathcal{O}_X\) modules and \(f^{-1}(U)=\text{Spec}(B)\) is open affine subset of \(X\), there exists a \(B\) module \(M\) such that \(\mathcal{F}|_{f^{-1}(U)}\cong \widetilde{M}\). As \(f^{-1}(U)=\text{Spec}(B)\) we have \(f:\text{Spec}(B)\rightarrow \text{Spec}(A)\) which also gives a ring morphsim \(A\rightarrow B\). The isomorphism \(\mathcal{F}|_{f^{-1}(U)}\cong \widetilde{M}\) implies \(f_*(\mathcal{F}|_{f^{-1}(U)})\cong f_*\widetilde{M}\). ...

Let \(X=\text{Spec}(R)\) be an affine scheme. Let \(X'\) be an affine \(X\) scheme i.e., \(X'=\text{Spec}(R')\) for some ring \(R'\) with a morphism of schemes \(\pi: X'\rightarrow X\). This \(\pi\) comes with morphism of global sections \(R=\Gamma(X,\mathcal{O}_X)\rightarrow \Gamma(X',\mathcal{O}_X')=R'\) giving \(R'\), structure of an \(R\) algebra. So, any affine scheme over \(X=\text{Spec}(R)\) is simply the specturm of an \(R\) algebra. Conversely, given an \(R\) algebra say \(R'\), we have an affine scheme \(X'=\text{Spec}(R')\) over \(X\) with morphism \(\pi:X'\rightarrow X\). Being a morphism of affine schemes, \(\pi: X\rightarrow X\) is an affine morphism. Let \(X\) be an arbitrary scheme. We want to associate an \(X\) scheme \(X'\) such that the structure morphism \(\pi:X'\rightarrow X\) is an affine morphism. To do this in case of \(X=\text{Spec}(R)\) we have fixed an \(R\) algebra and then associated an affine scheme for this. In case of an arbitrary scheme \(X\) unlike the case of affine scheme \(X=\text{Spec}(R)\) there is no single ring that has all information about the scheme \(X\). It is only natural to consider the collection \(\{\mathcal{O}_X(U): U\subseteq X\}\) varying over all open subsets of \(X\) to get information about the scheme \(X\). Choosing an \(\mathcal{O}_X(U)\) algebra \(\mathcal{F}(U)\) for each open \(U\subseteq X\) we associate an \(X\) scheme \(X'\) for this collection \(\{\mathcal{F}(U)\}\) of \(\mathcal{O}_X(U)\) algebras. It is only natural to put a condition that this collection \(\{\mathcal{F}(U)\}\) to be compatible with structure sheaf \(\mathcal{O}_X\) i.e., we want \(U\mapsto \mathcal{F}(U)\) to give a structure of sheaf of \(\mathcal{O}_X\) algebras on \(X\). So, given an arbitrary scheme \(X\) and a sheaf \(\mathcal{F}\) of \(\mathcal{O}_X\) algebras we associate an \(X\) scheme \(X'\) such that the structure map \(X'\rightarrow X\) is an affine morphism. It is not obvious at this point but we also want \(\mathcal{F}\) to be a quasicoherent sheaf of \(\mathcal{O}_X\) modules. We call this \(X'\), Global spec or Relative spec of sheaf \(\mathcal{F}\) of \(\mathcal{O}_X\) algebras over \(X\) denoted by \(\textbf{Spec} (\mathcal{F})\). Here we make two important remarks : The \(\textbf{Spec}\) construction gives an important way to understand affine morphisms. Note that \(\textbf{Spec}(\mathcal{F})\rightarrow X\) is an affine morphism. The converse is also true. If \(f:X\rightarrow Y\) is an affine morphism then \(\mathcal{A}=f_*\mathcal{O}_X\) is a quasi coherent sheaf of \(\mathcal{O}_Y\) algebras and \(X\cong \text{Spec} (\mathcal{A})\). The \(\textbf{Spec}\) construction is used to assign a geometric vector bundle on a scheme \(Y\) to each locally free sheaf \(\mathcal{E}\) of rank \(n\) on a scheme \(Y\) which gives a bijection between isomorphism classes of locall free sheaves of rank \(n\) on \(Y\), and isomorphism classes of vector bundles of rank \(n\) on \(Y\). To define \(\textbf{Spec}(\mathcal{F})\) we do not need \(\mathcal{F}\) to be quasi coherent, but \(\mathcal{F}\) quasi coherent implies the structure map \(\textbf{Spec}(\mathcal{F})\rightarrow X\) is an affine morphism. Now, we try to construct \(\textbf{Spec}(\mathcal{F})\) and structure map \(\textbf{Spec}(\mathcal{F})\rightarrow X\). One way to do this is gluing the schemes \(\text{Spec}(\mathcal{F}(U))\) over all open subsets \(U\subseteq X\). Another way is to use universal property of \(\text{Spec}\) of a ring. We have following result : 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 ring homomorphism \(A\rightarrow \mathcal{O}_X(X)\). Thus there is a natural map \(\alpha: \text{Hom}_{\text{Schemes}}(X,\text{Spec}(A))\rightarrow\text{Hom}_{\text{Rings}}(A,\mathcal{O}_X(X)).\) ...

Here I will add links of pages of interesting questions/answers from Math Stack Exchange and Math over flow. Learning Algebraic Geometry 1 transitive Lie groupoid is Morita equivalent to the isotropy group