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Algebraic Geometry lecture 16 and smooth manifolds lecture 18
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| \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 $X$. | ||
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| 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. | ||
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| 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. | ||
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| $(\Leftarrow)$ 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. | ||
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| 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} |
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| \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} |
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