Sheaves of Modules on Geometry and some category theory

Global Spec Or Relative Spec of a Scheme

Thu, 13 Jul 2017 09:03:44 +0000

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

$\alpha: \text{Hom}_{\text{Schemes}}(X,\text{Spec}(A))\rightarrow\text{Hom}_{\text{Rings}}(A,\mathcal{O}_X(X)).$

Quasi coherent/coherent sheaf of Modules

Sat, 08 Jul 2017 03:55:50 +0000

Sheaf associated to a Module over a ring

Sat, 08 Jul 2017 03:46:15 +0000

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 $f^*(\widetilde{M})=\widetilde{M\otimes_A B}$.