diff --git a/F4D1-Analysis-and-Geometry-on-Manifolds/Analysis_on_Manifolds_Notes.tex b/F4D1-Analysis-and-Geometry-on-Manifolds/Analysis_on_Manifolds_Notes.tex index cced28f..965c234 100644 --- a/F4D1-Analysis-and-Geometry-on-Manifolds/Analysis_on_Manifolds_Notes.tex +++ b/F4D1-Analysis-and-Geometry-on-Manifolds/Analysis_on_Manifolds_Notes.tex @@ -63,6 +63,7 @@ \section*{Preliminaries} \include{Lecture 15} \include{Lecture 16} \include{Lecture 17} +\include{Lecture 18} \newpage \printbibliography \end{document} \ No newline at end of file diff --git a/F4D1-Analysis-and-Geometry-on-Manifolds/Lecture 14.tex b/F4D1-Analysis-and-Geometry-on-Manifolds/Lecture 14.tex index 0b5c131..0afb6e1 100644 --- a/F4D1-Analysis-and-Geometry-on-Manifolds/Lecture 14.tex +++ b/F4D1-Analysis-and-Geometry-on-Manifolds/Lecture 14.tex @@ -1,7 +1,7 @@ \section{Lecture 14 -- 26th November 2024}\label{sec: lecture 14} We complete the proof of \Cref{thm: sard} by way of various lemmata. \begin{lemma}\label{lem: Ck for k large have measure zero image} - Let $U\subseteq\RR^{m}$, $F:U\to\RR^{n}$ be a smooth map with critical points $C\subseteq U$, and $C_{k}\subseteq C$ on which the $i$th partial derivatives of $F$ for $1\leq i\leq k$ functions of vanish. If $k>\frac{m}{n}-1$ then $F(C_{k})$ has measure zero. + Let $U\subseteq\RR^{m}$, $F:U\to\RR^{n}$ be a smooth map with critical points $C\subseteq U$, and $C_{k}\subseteq C$ on which the $i$th partial derivatives of $F$ for $1\leq i\leq k$ functions of vanish. If $k>\frac{m}{n}-1$ then $F(C_{k})$ has measure zero. \end{lemma} \begin{proof} For each $p\in U$, there exists a closed cube $E\subseteq U$ containing $p$. By second countability, we can cover $C_{k}$ by countably many such cubes. We show $F(C_{k}\cap E)$ is of measure zero. Let $A>\sup_{q\in E}|\partial_{x}^{\alpha}(q)|$ be a constant and $|\alpha|\leq k+1$. Let $L>0$ be the side length of $E$ and $K>>1$ large a natural number. We can subdivide $E$ into cubes of sidelength $L/K$, of which there are $K^{m}$. Consider an enumeration of these subcubes and an index $i_{0}$ such that $p\in E_{i_{0}}$. diff --git a/F4D1-Analysis-and-Geometry-on-Manifolds/Lecture 17.tex b/F4D1-Analysis-and-Geometry-on-Manifolds/Lecture 17.tex index 6c95686..f9f7cc9 100644 --- a/F4D1-Analysis-and-Geometry-on-Manifolds/Lecture 17.tex +++ b/F4D1-Analysis-and-Geometry-on-Manifolds/Lecture 17.tex @@ -59,4 +59,13 @@ \section{Lecture 17 -- 7th December 2024}\label{sec: lecture 17} Let $\theta$ be the flow for $X$ and for $p\in M$, let $(U,\phi)$ be a smooth chart of $M$ containing $p$. Let $J\subseteq\RR$ be an open interval containing 0 and $U_{0}\subseteq U$ containing $p$ such that $\theta(J_{0}\times U_{0})\subseteq U$. For $(t,q)\in J_{0}\times U_{0}$, the Jacobian matrix of the differential is given by $$(d\Psi_{-t})_{\Psi(t,x)}=(\partial_{x_{i}}\Psi^{j}(-t,x))_{1\leq i,j\leq n}.$$ As such, we have $d\Psi_{-t}Y_{\Psi(t,x)}=(\partial_{x_{i}}\Psi^{j}(-t,x))_{1\leq i,j\leq n}$ giving the claim. +\end{proof} +We will in fact show that the Lie derivative can be computed in terms of the Lie bracket, for which we will require the following lemma. +\begin{lemma}\label{lem: canonical form for a smooth vector field} + Let $M$ be a smooth manifold and let $X\in\Xfrak(M)$ a smooth vector field. If $X_{p}\neq0$ for some $p\in M$ then there exists a chart $(U,\phi)$ around $p$ with respect to which $X=\partial_{x_{1}}$. +\end{lemma} +\begin{proof} + Let $(U,\phi)$ be a chart around $p$ with coordinates $(x_{1},\dots,x_{m})$ and let $S$ be the hypersurface defined by $x_{j}=0$ for $X^{j}_{p}\neq0$ with $X$ the vector field as above. Shrinking $S$ such that $X$ is nowhere tangent to $S$ the flowout theorem \cite[Thm. 9.20]{LeeSM} implies that there is a flow domain $\Ocal_{\delta}\subseteq\RR\times S$ such that the flow of $X$ restricts to a diffeomorphism $\Phi$ from $\Ocal_{\delta}$ to an open subset $W\subseteq M$ containing $S$. + + There is a product neighborhood $(-\varepsilon,\varepsilon)\subseteq W_{0}$ of $(0,p)$ in $\Ocal_{\delta}$. Now taking a smooth local parametrization $X:\Omega\to S$ with image contained in $W_{0}$ and $\Omega$ open in $\RR^{m-1}$ with coordinates $x_{2},\dots,x_{n}$. It follows that the map $\Psi:(-\varepsilon,\varepsilon)\times\Omega\to M$ by $\Psi(t,x_{2},\dots,x_{n})=\Phi(t,X(x_{2},\dots,x_{m}))$ iwhich is a diffeomprhpism onto a neighborhood of $p$ in $M$. Moreover this map pushes $\partial_{t}$ to itself with $\Phi_{*}(\partial_{t})=X$ so $\Psi_{*}(\partial_{t})=X$ with $\Psi$ a smooth coordinate chart with the desired representation. The claim follows by relabeling. \end{proof} \ No newline at end of file diff --git a/F4D1-Analysis-and-Geometry-on-Manifolds/Lecture 18.tex b/F4D1-Analysis-and-Geometry-on-Manifolds/Lecture 18.tex new file mode 100644 index 0000000..89d7ae2 --- /dev/null +++ b/F4D1-Analysis-and-Geometry-on-Manifolds/Lecture 18.tex @@ -0,0 +1,42 @@ +\section{Lecture 18 -- 10th December 2024}\label{sec: lecture 18} +We show the desired result, that Lie derivatives can be computed in terms of Lie brackets. +\begin{proposition}\label{prop: Lie derivatives as Lie brackets} + Let $M$ be a smooth manifold and $X,Y\in\Xfrak(M)$ smooth vector fields. Then $\Lcal_{X}Y=[X,Y]$. +\end{proposition} +\begin{proof} + It suffices to show that $(\Lcal_{Y}X)_{p}=[X,Y]_{p}$. We consider three cases: where $p$ is a nonvanishing point of $X$, $p$ is in the support of $X$, and $p$ is outside the support of $X$. + + In the first case, \Cref{lem: canonical form for a smooth vector field} allows us to choose $\partial_{x_{1}}$ as the coordinate representation for $X$ in which case the flow is given by $\theta_{t}(u)=(u^{1}+t,u^{2},\dots,u^{n})$ and for each fixed $t$, the Jacobian of $d(\theta_{-t})_{\theta_{t}(x)}$ is the identity and an explicit computation gives the pointwise equality. + + In the second case if $p$ is in the support of $X$ the claim holds by the above. Otherwise $X$ is zero at $p$ so $\theta_{t}$ is the identity on a neighborhood of $p$ for all $p$ so the Lie derivative is zero as well, which agrees with the Lie bracket. +\end{proof} +Moreover, we can show the following result for local frames. +\begin{theorem}\label{thm: local frame canonical form} + Let $M$ be a smooth $m$-manifold and let $X^{1},\dots,X^{m}\in\Xfrak(M)$ be a local frame at $p$. There exists a chart $(U,\phi)$ around $p$ such that $X^{i}=\partial_{x_{i}}$ if and only if $[X^{i},X^{j}]=0$ near $p$ for all $i,j$. +\end{theorem} +\begin{proof} + $(\Rightarrow)$ is clear since coordinate vector fields commute. + + $(\Leftarrow)$ is the construction of \Cref{lem: canonical form for a smooth vector field}. +\end{proof} +As a prelude to a discussion of vector bundles, we make some recollections on linear algebra. + +We fix a field $\RR$ and consider the category $\Vect^{\mathsf{fd}}_{\RR}$ of finite dimensional $\RR$-vector spaces and morphisms linear maps. This category has especially nice properties as we now describe. +\begin{theorem}\label{thm: fdVect is Ab closed sym monoidal} + The category $\Vect^{\mathsf{fd}}_{\RR}$ is a closed symmetric monoidal Abelian category. +\end{theorem} +To be more explicit, all finite limits and colimits exist, implying the existence of kernels and cokernels, the category is preserved under finite direct sums which agree with finite coproducts, and admits a tensor product which satisfies the tensor-hom adjunction. +\begin{definition}[Tensor]\label{def: tensor} + Let $V\in \Vect^{\mathsf{fd}}_{\RR}$. A tensor of type $(a,b)$ over $V$ is an element of + $$\underbrace{V\otimes\dots\otimes V}_{a\text{ times}}\otimes\underbrace{V^{\vee}\otimes\dots\otimes V^{\vee}}_{b\text{ times}}.$$ +\end{definition} +This leads us to the definition of alternating and symmetric tensors. +\begin{definition}[Alternating Tensor]\label{def: alternating tensor} + Let $V\in \Vect^{\mathsf{fd}}_{\RR}$ and $\alpha\in T^{0,b}V$. $\alpha$ is an alternating tensor if the induced map $V\times\dots\times V\to\RR$ is such that + $$\alpha(x_{1},\dots,x_{i},\dots,x_{j},\dots,x_{b})=(-1)^{j-i}\alpha(x_{1},\dots,x_{j},\dots,x_{i},\dots,x_{b}).$$ +\end{definition} +\begin{definition}[Symmetric Tensor]\label{def: symmetric tensor} + Let $V\in \Vect^{\mathsf{fd}}_{\RR}$ and $\alpha\in T^{0,b}V$. $\alpha$ is a symmetric tensor if the induced map $V\times\dots\times V\to\RR$ is such that + $$\alpha(x_{1},\dots,x_{b})=\alpha(x_{\sigma(1)},\dots,x_{\sigma(b)})$$ + for all permutations $\sigma\in S_{b}$. +\end{definition} \ No newline at end of file diff --git a/F4D1-Analysis-and-Geometry-on-Manifolds/refs.bib b/F4D1-Analysis-and-Geometry-on-Manifolds/refs.bib index 2fdb073..279f057 100644 --- a/F4D1-Analysis-and-Geometry-on-Manifolds/refs.bib +++ b/F4D1-Analysis-and-Geometry-on-Manifolds/refs.bib @@ -12,4 +12,24 @@ @Book{Warner zbMATH = {3816641}, Zbl = {0516.58001}, shorthand = {War83} +} + +@Book{LeeSM, + Author = {Lee, John M.}, + Title = {Introduction to smooth manifolds}, + Edition = {2nd revised ed}, + FSeries = {Graduate Texts in Mathematics}, + Series = {Grad. Texts Math.}, + ISSN = {0072-5285}, + Volume = {218}, + ISBN = {978-1-4419-9981-8; 978-1-4419-9982-5}, + Year = {2013}, + Publisher = {New York, NY: Springer}, + Language = {English}, + DOI = {10.1007/978-1-4419-9982-5}, + Keywords = {53-01,53-02,58-02,57-02,53Cxx,57Rxx,58Axx}, + URL = {zenodo.org/record/4461500}, + zbMATH = {6034615}, + Zbl = {1258.53002}, + shorthand = {Lee13} } \ No newline at end of file diff --git a/V4A1-Algebraic-Geometry-I/Algebraic_Geometry_I_Notes.tex b/V4A1-Algebraic-Geometry-I/Algebraic_Geometry_I_Notes.tex index 23f0838..b8da15d 100644 --- a/V4A1-Algebraic-Geometry-I/Algebraic_Geometry_I_Notes.tex +++ b/V4A1-Algebraic-Geometry-I/Algebraic_Geometry_I_Notes.tex @@ -45,6 +45,7 @@ \maketitle \section*{Preliminaries} These notes roughly correspond to the course \textbf{V4A1 -- Algebraic Geometry I} taught by Prof. Daniel Huybrechts at the Universit\"{a}t Bonn in the Winter 2024/25 semester. These notes are \LaTeX-ed after the fact with significant alteration and are subject to misinterpretation and mistranscription. Use with caution. Any errors are undoubtedly my own and any virtues that could be ascribed to these notes ought be attributed to the instructor and not the typist. Knowledge of commutative algebra, topology, and category theory will be assumed. +\newpage \tableofcontents \include{Lecture 1} \include{Lecture 2} @@ -61,6 +62,7 @@ \section*{Preliminaries} \include{Lecture 13} \include{Lecture 14} \include{Lecture 15} +\include{Lecture 16} \newpage \printbibliography \end{document} \ No newline at end of file diff --git a/V4A1-Algebraic-Geometry-I/Lecture 12.tex b/V4A1-Algebraic-Geometry-I/Lecture 12.tex index 668cec6..ff5cd26 100644 --- a/V4A1-Algebraic-Geometry-I/Lecture 12.tex +++ b/V4A1-Algebraic-Geometry-I/Lecture 12.tex @@ -135,7 +135,7 @@ \section{Lecture 12 -- 18th November 2024}\label{sec: lecture 12} \arrow[from=3-6, to=4-5] \end{tikzcd}$$ with both rectangles Cartesian so there is a unique map $X_{L}\to Y_{L}$ making the diagram -$$% https://q.uiver.app/#q=WzAsNixbMywyLCJcXHNwZWMoaykiXSxbMSwyLCJcXHNwZWMoTCkiXSxbMywxLCJZIl0sWzEsMSwiWV97TH0iXSxbMSwwLCJYIl0sWzAsMCwiWF97TH0iXSxbMSwwXSxbNSwxXSxbNSw0XSxbNCwyXSxbMiwwXSxbMywyXSxbMywxXSxbNSwzLCJcXGV4aXN0cyEiLDEseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0= +$$% https://q.uiver.app/#q=WzAsNixbMywyLCJcXHNwZWMoaykiXSxbMSwyLCJcXHNwZWMoTCkiXSxbMywxLCJZIl0sWzEsMSwiWV97TH0iXSxbMSwwLCJYIl0sWzAsMCwiWF97TH0iXSxbMSwwXSxbNSwxXSxbNSw0XSxbNCwyXSxbMiwwXSxbMywyXSxbMywxXSxbNSwzLCJcXGV4aXN0cyEiLDEseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbNCwwXV0= \begin{tikzcd} {X_{L}} & X \\ & {Y_{L}} && Y \\ @@ -144,6 +144,7 @@ \section{Lecture 12 -- 18th November 2024}\label{sec: lecture 12} \arrow["{\exists!}"{description}, dashed, from=1-1, to=2-2] \arrow[from=1-1, to=3-2] \arrow[from=1-2, to=2-4] + \arrow[from=1-2, to=3-4] \arrow[from=2-2, to=2-4] \arrow[from=2-2, to=3-2] \arrow[from=2-4, to=3-4] diff --git a/V4A1-Algebraic-Geometry-I/Lecture 16.tex b/V4A1-Algebraic-Geometry-I/Lecture 16.tex new file mode 100644 index 0000000..aacd21a --- /dev/null +++ b/V4A1-Algebraic-Geometry-I/Lecture 16.tex @@ -0,0 +1,137 @@ +\section{Lecture 16 -- 2nd December 2024}\label{sec: lecture 16} +We discuss some ``cancellation'' properties of morphisms of schemes.\marginpar{For this, we follow \cite[\S 11.1]{Vakil}.} +\begin{proposition}[Cancellation for Morphisms of Schemes]\label{prop: cancellation property for morphisms} + Let P be a property of schemes preserved by base change and composition. Consider a diagram of schemes + $$% https://q.uiver.app/#q=WzAsMyxbMCwwLCJYIl0sWzIsMCwiWSJdLFsxLDEsIloiXSxbMCwyLCJnIiwyXSxbMCwxLCJmIl0sWzEsMiwiaCJdXQ== + \begin{tikzcd} + X && Y \\ + & Z + \arrow["f", from=1-1, to=1-3] + \arrow["h"', from=1-1, to=2-2] + \arrow["g", from=1-3, to=2-2] + \end{tikzcd}$$ + where $h$ is in P and the diagonal morphism induced by $g$ is in P. Then $f$ is in P. +\end{proposition} +\begin{proof} + We consider the diagram + $$% https://q.uiver.app/#q=WzAsNyxbMCwwLCJYXFxjb25nIFhcXHRpbWVzX3tZfVkiXSxbNiwwLCJZXFx0aW1lc197Wn1aXFxjb25nIFkiXSxbMCwxLCJZIl0sWzIsMSwiWVxcdGltZXNfe1p9WSJdLFs0LDEsIlgiXSxbNiwxLCJaIl0sWzMsMCwiWFxcdGltZXNfe1p9WSJdLFswLDEsImYiLDAseyJjdXJ2ZSI6LTN9XSxbMCw2XSxbNiwxXSxbMCwyXSxbMiwzXSxbNiwzXSxbNiw0XSxbMSw1XSxbNCw1XV0= + \begin{tikzcd} + {X\cong X\times_{Y}Y} &&& {X\times_{Z}Y} &&& {Y\times_{Z}Z\cong Y} \\ + Y && {Y\times_{Z}Y} && X && Z + \arrow[from=1-1, to=1-4] + \arrow["f", curve={height=-18pt}, from=1-1, to=1-7] + \arrow[from=1-1, to=2-1] + \arrow[from=1-4, to=1-7] + \arrow[from=1-4, to=2-3] + \arrow[from=1-4, to=2-5] + \arrow[from=1-7, to=2-7] + \arrow[from=2-1, to=2-3] + \arrow[from=2-5, to=2-7] + \end{tikzcd}$$ + with Cartesian squares induced by the diagonal base change theorem. Both $Y\to Y\times_{Z}Y$ and $X\to Z$ lie in P and thus so does their composite. +\end{proof} +As a corollary, we can deduce the following results about morphisms. +\begin{corollary}\label{corr: cancellation for separatedness} + If $g\circ f$ is a separated morphism then $f$ is a separated morphism. +\end{corollary} +\begin{proof} + It suffices to show that the diagonals of separated morphisms are separated. But closed immersions are separated giving the claim. +\end{proof} +\begin{corollary}\label{corr: cancellation for properness} + If $g\circ f$ is a proper morphism, $g$ a separated morphism, and $f$ a quasicompact morphism then $f$ is a proper morphism. +\end{corollary} +\begin{proof} + Since $g$ is separated, the diagonal morphism is a closed immersion and hence finite. In particular, $g$ is proper. Applying the proposition yields the claim. +\end{proof} +We consider the valuative criteria for separatedness and properness. Recall the following. +\begin{definition}[Valuation]\label{def: valuation} + Let $K$ be a field and $\Gamma$ a totally ordered Abelian group. A valuation on a field $K$ is a map $\nu:K^{\times}\to\Gamma$ such that + \begin{enumerate}[label=(\roman*)] + \item $\nu(xy)=\nu(x)+\nu(y)$, + \item $\nu(0)=\infty$, and + \item $\nu(x+y)=\min\{\nu(x),\nu(y)\}$. + \end{enumerate} +\end{definition} +A valued field allows us to produce a ring of $\nu$-integers. +\begin{definition}[Valuation Ring]\label{def: valuation ring} + Let $K$ be a field with valuation $\nu$. The valuation ring of $K$ is the ring + $$\Ocal_{\nu}=\{x\in K:\nu(x)\geq0\}.$$ +\end{definition} +\begin{remark} + $\spec(\Ocal_{\nu})=\{\eta,\mfrak_{\nu}\}$ where $\mfrak_{\nu}=\{x\in K:\nu(x)>0\}$. +\end{remark} +The results are as follows. +\begin{theorem}[Valuative Criterion for Separatedness]\label{thm: valuative criterion for separatedness} + Let $f:X\to Y$ be a finite type morphism of schemes and $Y$ locally Noetherian. Then $f$ is separated if and only if for all diagrams + $$% https://q.uiver.app/#q=WzAsNCxbMiwwLCJYIl0sWzIsMSwiWSJdLFswLDAsIlxcc3BlYyhLKSJdLFswLDEsIlxcc3BlYyhBKSJdLFswLDEsImYiXSxbMiwwXSxbMiwzXSxbMywxXSxbMywwLCJcXGxlcTEiLDEseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0= + \begin{tikzcd} + {\spec(K)} && X \\ + {\spec(A)} && Y + \arrow[from=1-1, to=1-3] + \arrow[from=1-1, to=2-1] + \arrow["f", from=1-3, to=2-3] + \arrow["\leq1"{description}, dashed, from=2-1, to=1-3] + \arrow[from=2-1, to=2-3] + \end{tikzcd}$$ + with $A$ a discrete valuation ring with fraction field $K$ there exists at most one map $\spec(A)\to X$ making the diagram commute. +\end{theorem} +This is reflected in the well-examined example of the affine line with doubled origin. +\begin{example} + Let $X$ be the affine line with doubled origin and consider the diagram + $$% https://q.uiver.app/#q=WzAsNCxbMiwwLCJYIl0sWzIsMSwiXFxBXnsxfV97a30iXSxbMCwwLCJcXHNwZWMoayh0KSkiXSxbMCwxLCJcXHNwZWMoa1t0XV97KHQpfSkiXSxbMiwwXSxbMiwzXSxbMywxXSxbMCwxLCJmIl1d + \begin{tikzcd} + {\spec(k(t))} && X \\ + {\spec(k[t]_{(t)})} && {\A^{1}_{k}.} + \arrow[from=1-1, to=1-3] + \arrow[from=1-1, to=2-1] + \arrow["f", from=1-3, to=2-3] + \arrow[from=2-1, to=2-3] + \end{tikzcd}$$ + Writing $\A^{1}_{k}$ as $\spec(k[x])$, there are two maps along the bottom $(x)\mapsto(0)$ and $(x)\mapsto(t)$, each of which induce a lift to $X$. In particular, there is more than one map $\spec(k[t]_{(t)})$ making the diagram commute, which agrees with $X$ not being separated. +\end{example} +In the case of properness, we have the following. +\begin{theorem}[Valuative Criterion for Properness]\label{thm: valuative criterion for properness} + Let $f:X\to Y$ be a finite type morphism of schemes and $Y$ locally Noetherian. Then $f$ is proper if and only if for all diagrams + $$% https://q.uiver.app/#q=WzAsNCxbMiwwLCJYIl0sWzIsMSwiWSJdLFswLDAsIlxcc3BlYyhLKSJdLFswLDEsIlxcc3BlYyhBKSJdLFswLDEsImYiXSxbMiwwXSxbMiwzXSxbMywxXSxbMywwLCJcXGV4aXN0cyEiLDEseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0= + \begin{tikzcd} + {\spec(K)} && X \\ + {\spec(A)} && Y + \arrow[from=1-1, to=1-3] + \arrow[from=1-1, to=2-1] + \arrow["f", from=1-3, to=2-3] + \arrow["{\exists!}"{description}, dashed, from=2-1, to=1-3] + \arrow[from=2-1, to=2-3] + \end{tikzcd}$$ + with $A$ a discrete valuation ring with fraction field $K$ there exists a unqiue map $\spec(A)\to X$ making the diagram commute. +\end{theorem} +\begin{example} + Consider the square + $$% https://q.uiver.app/#q=WzAsNCxbMCwwLCJcXHNwZWMoayh0KSkiXSxbMCwxLCJcXHNwZWMoa1t0XV97KHQpfSkiXSxbMiwwLCJcXEFeezF9X3trfSJdLFsyLDEsIlxcc3BlYyhrKSJdLFswLDJdLFsyLDNdLFswLDFdLFsxLDNdXQ== + \begin{tikzcd} + {\spec(k(t))} && {\A^{1}_{k}} \\ + {\spec(k[t]_{(t)})} && {\spec(k)} + \arrow[from=1-1, to=1-3] + \arrow[from=1-1, to=2-1] + \arrow[from=1-3, to=2-3] + \arrow[from=2-1, to=2-3] + \end{tikzcd}$$ + where taking $\A^{1}_{k}=\spec(k[x])$, the map $(x)\mapsto(t^{-1})$ along the top does not extend to the discrete valuation ring, verifying that $\A^{1}_{k}$ is not proper. +\end{example} +We now turn to a discussion of sheaves of modules, which we discuss in the generality of ringed spaces. +\begin{definition}[Modules over the Structure Sheaf]\label{def: OX modules} + Let $X$ be a ringed space. A sheaf of Abelian groups $\Fcal$ on $X$ is a sheaf of $\Ocal_{X}$-modules if for all $U\subseteq V$ the diagram with vertical maps restrictions and horizontal maps the $\Ocal_{X}$ action + $$% https://q.uiver.app/#q=WzAsNCxbMCwwLCJcXE9jYWxfe1h9KFUpXFx0aW1lc1xcRmNhbChVKSJdLFsyLDAsIlxcRmNhbChVKSJdLFsyLDEsIlxcRmNhbChWKSJdLFswLDEsIlxcT2NhbF97WH0oVilcXHRpbWVzXFxGY2FsKFYpIl0sWzAsMV0sWzEsMl0sWzAsM10sWzMsMl1d + \begin{tikzcd} + {\Ocal_{X}(U)\times\Fcal(U)} && {\Fcal(U)} \\ + {\Ocal_{X}(V)\times\Fcal(V)} && {\Fcal(V)} + \arrow[from=1-1, to=1-3] + \arrow[from=1-1, to=2-1] + \arrow[from=1-3, to=2-3] + \arrow[from=2-1, to=2-3] + \end{tikzcd}$$ + commutes. +\end{definition} +Naturally, one defines the morphisms of sheaves of $\Ocal_{X}$-modules to be defined as morphisms of Abelian groups between $\Ocal_{X}$-modules compatible with the action of $\Ocal_{X}$. These form an Abelian category $\Mod_{\Ocal_{X}}$ as shown in \cite[\href{https://stacks.math.columbia.edu/tag/01AF}{Tag 01AF}]{stacks-project}. +\begin{remark} + The forgetful functor $\Mod_{\Ocal_{X}}\to\Sh(X,\AbGrp)$ is faithful but not full -- being a morphism of $\Ocal_{X}$-modules requires compatiblility with the $\Ocal_{X}$ action beyond being a morphism of Abelian groups. +\end{remark} diff --git a/V4A1-Algebraic-Geometry-I/refs.bib b/V4A1-Algebraic-Geometry-I/refs.bib index 9f53813..e8f6ee8 100644 --- a/V4A1-Algebraic-Geometry-I/refs.bib +++ b/V4A1-Algebraic-Geometry-I/refs.bib @@ -53,4 +53,22 @@ @Article{Tohoku zbMATH = {3192990}, Zbl = {0118.26104}, shorthand = {Gro57} +} + +@Book{GWI, + Author = {G{\"o}rtz, Ulrich and Wedhorn, Torsten}, + Title = {Algebraic geometry {I}. {Schemes}. {With} examples and exercises}, + Edition = {2nd edition}, + FSeries = {Springer Studium Mathematik -- Master}, + Series = {Springer Stud. Math. -- Master}, + ISSN = {2509-9310}, + ISBN = {978-3-658-30732-5; 978-3-658-30733-2}, + Year = {2020}, + Publisher = {Wiesbaden: Springer Spektrum}, + Language = {English}, + DOI = {10.1007/978-3-658-30733-2}, + Keywords = {14-01,14A15,14C20,14F06,14L15,14M12,14L30,14B05}, + zbMATH = {7261253}, + Zbl = {1444.14001}, + shorthand = {GW20} } \ No newline at end of file