--- date: 2021-10-08 tags: [ web/quick-notes ] --- Tags: #web/quick-notes # 2021-10-08 ## 21:03 - [comonads](monad.md) in $\cat{C}$: [coalgebra object](coalgebra%20object) in $[\cat{C}, \cat{C}]$. - [Comodules](Comodules) over a comonad $T$: an object $X$, a map $a^\sharp: X\to TX$, and some coherence conditions. Often called $T\dash$algebras, called the category of $T\dash$comodules $\comods{T}(\cat C)$. - A fun but non-obvious consequence of [https://stacks.math.columbia.edu/tag/06WS](https://stacks.math.columbia.edu/tag/06WS): for $G\in\Grp\Sch\slice R$ faithfully flat, there is an equivalence of categories \[ \QCoh(\B G) \iso \Rep(G) ,\] the category of *regular* $G\dash$representations, i.e. $\comods{\Gamma(G)}$. See [regular representation](regular%20representation). - Why this is true: - $\QCoh(\spec R) \iso \mods{R}$ - Use $p: \spec R\to \B G$ induced by $G\to R$ to induce a pullback functor $p^*: \QCoh(\B G)\to \QCoh(\spec R) \cong \mods{R}$. - Set up a adjunction that yields a comonad equivalent to $F: (\wait)\tensor_R \Gamma(G)$. - Apply [Barr-Beck](Barr-Beck.md) : - Given an adjunction $\adjunction{L}{R}{\cat D}{\cat C}$, get a comonad $LR\in [\cat C, \cat C]$. - Then every $X\in \cat D$ yields $L(X) \in \comods{LR}$, and $\cat D \mapsvia{L} \cat C$ factors: \begin{tikzcd} {\cat{D}} && {\cat{C}} \\ \\ && {\comods{LR}(\cat C)} \arrow["L", from=1-1, to=1-3] \arrow["{\exists \tilde L}"', from=1-1, to=3-3] \arrow[hook, from=3-3, to=1-3] \end{tikzcd} > [Link to diagram](https://q.uiver.app/?q=WzAsMyxbMCwwLCJcXGNhdHtEfSJdLFsyLDAsIlxcY2F0e0N9Il0sWzIsMiwiXFxjb21vZHN7TFJ9KFxcY2F0IEMpIl0sWzAsMSwiTCJdLFswLDIsIlxcZXhpc3RzIFxcdGlsZGUgTCIsMl0sWzIsMSwiIiwyLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XV0=) - [Barr-Beck](Barr-Beck.md) says $\tilde L$ is an equivalence under suitable conditions ($L, R$ [conservative](conservative) with $L$ preserving [equalizers](equalizers)). - Set up the [Unsorted/adjoint (categorical)](Unsorted/adjoint%20(categorical).md) \[ \cat D \da \adjunction{p^*}{p_*}{\QCoh(\B G)}{\QCoh(\spec R)} \da \cat C .\] Then $LR \da p^*p_*$, and Barr-Beck yields \[ \QCoh(\B G)\isovia{\tilde{p^*}} \comods{(p^*p_*)}(\QCoh(\spec R)) .\] - Use that if $G\in\Aff\Grp\Sch\slice R$ then $\Gamma(G) \in \Hopf\ralg$. Set $\cat{C} \da \mods{R}$, and $F\in [\cat C, \cat C]$ to be $F(\wait) \da (\wait)\tensor_R \Gamma(G)$. Then there is an [equivalence of categories](equivalence%20of%20categories.md) \[ \comods{F}(\cat C) \iso \Rep(G) .\] - Then show that $F$ is equivalent to $p^*p_*$. ## 22:52 > [http://individual.utoronto.ca/groechenig/stacks.pdf](http://individual.utoronto.ca/groechenig/stacks.pdf) #resources Refs: [stacks](Unsorted/stacks%20MOC.md) [vector bundle](vector%20bundles.md) [Unsorted/descent](Unsorted/descent.md) - Vector bundles as descent data: consider describing $E\to X$; one needs the [cocycle condition](cocycle%20condition). This means choosing $\mcu \covers X$ and bundle automorphisms $\phi_{ij}: (U_i \intersect U_j)\times \RR^n \selfmap$ of the trivial bundle. - We then want to glue up to obtain some $E$ over $X$: finding local bundle isomorphisms $\phi_i: U_i \times \RR^n \iso \ro{E}{U_i}$ with $\phi_{ij} = \phi_i \circ \phi_j\inv$ on $U_i \intersect U_j$. The cocycle condition is necessary, and for topological vector bundles, also sufficient. - How to glue: set $E \da \disjoint_{i} (U_i \times \RR^n)/\sim$ where $(x, \vector v)\sim (x, \phi_{ij}(\vector v))$ with the quotient topology. - Alternative formulation: - Let $\mcu \covers X$ and define $Y\da \Disjoint_i U_i$, which induces $Y \mapsvia{\pi} X$ by the inclusions $U_i \injects X$. - Then \[ Y\fiberpower{X}{2} = \Disjoint_{(i, j)\in I\cartpower{2}} U_i \intersect U_j .\] The cocycle condition becomes the existence of an isomorphism of bundles over $Y\fiberpower{X}{2}$: \begin{tikzcd} {p_1^* \tilde E} && {\tilde E \da Y\times \RR^n} & {p_2^* \tilde E} \\ \\ {Y\fiberpower{X}{2}} && {Y \da \Disjoint_i U_i} & {Y\fiberpower{X}{2}} \\ \\ Y && {X = \Union_i U_i} & Y \arrow["{p_1}", from=3-1, to=3-3] \arrow[from=1-3, to=3-3] \arrow["\pi"', from=3-3, to=5-3] \arrow[from=5-1, to=5-3] \arrow[from=3-1, to=5-1] \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=3-1, to=5-3] \arrow["{p_2}"', from=3-4, to=3-3] \arrow[from=1-1, to=3-1] \arrow[from=1-1, to=1-3] \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=3-3] \arrow[from=1-4, to=1-3] \arrow[from=1-4, to=3-4] \arrow["\lrcorner"{anchor=center, pos=0.125, rotate=-90}, draw=none, from=1-4, to=3-3] \arrow["{\exists \phi \cong}", curve={height=-30pt}, dashed, hook, two heads, from=1-1, to=1-4] \arrow[from=3-4, to=5-4] \arrow[from=5-4, to=5-3] \arrow["\lrcorner"{anchor=center, pos=0.125, rotate=-90}, draw=none, from=3-4, to=5-3] \end{tikzcd} > [Link to diagram](https://q.uiver.app/?q=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) - Note that pullbacks of trivial bundles are trivial, so this is an automorphism of the trivial bundle on $Y\fiberpower{X}{2}$ - The cocycle condition becomes an identity among bundle isomorphisms on $Y\fiberpower{X}{3}$: \[ p_{12}^* \phi \circ p_{23}^* \phi = p_{13}^*\phi \] as maps $p_3^*\tilde E\to p_1^* \tilde E$. Local trivializations translate to $\pi^* E \cong \tilde E$, the trivial bundle. ## 23:25 - There is an equivalence of categories $\mods{\RR} \iso \Tw\mods{\CC}$ where the latter consists of objects which are pairs $(V, f:V\to V)$ where $f(\lambda \vector v) = \bar{\lambda} \vector v$ is a structure map and $f^2 = \id_V$ and morphisms $\phi:V\to W$ that commute with the structure maps. - The forward map is $V\mapsto (V\tensor_\RR \CC, f)$ with $f$ the generator $f\in \Gal(\CC\slice \RR)$, and the inverse is $(V, f)\mapsto V^f$, the $f\dash$invariant subspace. - For field extensions $L\slice k$, the ring morphism $k\injects L$ yields $\spec L \to \spec k$, which behaves like a [covering space](covering%20space.md) with $\Deck(\spec L \slice {\spec k}) \cong \Gal(L\slice k)$. - Vector bundles on $\spec k$ correspond to $\mods{k}$, and Galois-equivariant vector bundles on $\spec L$ will correspond to vector bundles on the quotient $\spec k$. - $R\in \Alg\slice A$: a ring morphism $A\to R$. - Given $f\in \ZZ[x_1,\cdots, x_n]$, taking the zero locus in a ring $R$ yields a functor $\CRing\to \Set$. To do this with $f\in A[x_1,\cdots, x_n]$ for $A\in \CRing$, one needs $R\in \Alg\slice A$, so this yields a functor $\Alg\slice A\to \Set$. - Think of spaces as functors $X\in [\CRing, \Set]$, then $\spec R \da \CRing(R, \wait)$, so $R$ corepresents $\spec R$ in $\CRing$. - Can represent $R\localize{f} = R[t]/\gens{tf-1}$. - Standard open subfunctors: $\spec R\localize{f_i} \to \spec R$. These form an open cover if $\gens{f_i} = \gens{1}$. - If $k\in \Field$, there is an equivalence $\spec R(k) \cong Z_f(k)$, the zeros of $f$ in $k$. Then $\spec R\localize{h}(k) = Z_f(k)\sm Z_h(k)$ for $R = \ZZ[x_1,\cdots, x_n]/\gens{f}$. - Analog of 2-dimensional $\CC\dash$module over a ringer ring: the free $R\dash$module $R\cartpower{2}$ of rank 2. - $\PP^1_{\ZZ}: \CRing\to\Set$ is the functor sending $R$ to the set of direct summands $M \leq R\cartpower{2}$ for which there's an open covering corresponding to $\ts{h_i}$ where $M\localize{h_i} = M\tensor_R R\localize{h_i}$ is a free $R\dash$module of rank 1 for all $i$. - This recovers $\PP^1_{\ZZ}(\CC) = \PP^1\slice{\CC}$ classically, since sub-vector spaces are direct summands. - $\PP^1_\ZZ(\ZZ[t])$ induces a continuously varying family of 1-dimensional subspaces of $\CC^2$? Somehow, even though $\CC$ isn't in the definition.. - For $S\in\ralg$, we have $\alpha: R\to S$ and for $N\in \mods{S}$ we can forget the module structure along this map by defining \[ R\times N &\to N \\ (r, n) &\mapsto \alpha(r) \cdot n .\] This induces a [restriction functor](restriction%20functor) $\res_{\alpha}: \mods{S} \to \mods{R}$. - Conversely we can tensor $R\dash$modules up to $S\dash$modules to get a functor $S\tensor_R(\wait)$, where the interesting bit is $s\tensor(rm) \da \alpha(r) (s\tensor m) = (\alpha(r)s)\tensor m$. - This yields an adjunction: \[ \adjunction{(\wait)\tensor_R S}{\res_{\alpha}}{\mods{R}}{\mods{S}} .\] - Any reasonable property of modules should be preserved by base change! - Descent for modules: when does $M\tensor_R S$ having property $P$ as an $S\dash$module descend to $M$ having property $P$ has an $R\dash$module? - Left adjoints are right exact (LARE). In particular, [base change](base%20change.md) is right exact, but not always left exact: take $\alpha: \ZZ\to \ZZ/2$, take the SES $0 \to \ZZ \mapsvia{2} \ZZ \to \ZZ/2\to 0$, and tensor with $\ZZ/2$. So an $R\dash$algebra $S$ is flat precisely when the base change $S\tensor_R(\wait)$ is exact. - Free implies flat, and every module over a field is free. - $S$ is [faithfully flat](faithfully%20flat.md) when $S\tensor_R M = 0\implies M=0$. Allows checking things after base-changing to $S$: - Exactness of any sequence, so in particular injectivity/surjectivity - Finite generation (over $R$ vs $S$) - Projectivity, - Flatness - If $R\to S$ is faithfully flat and $R\to T$ is an arbitrary ring morphism, the co-base change $T\to S\tensor_R T$ is faithfully flat. - General idea: $R\dash$modules $M$ can be specified by $S\tensor_R M$ along with [Unsorted/descent](Unsorted/descent.md). - [faithfully flat descent](Unsorted/descent.md) : there is an equivalence of categories $\mods{R} \to \Desc(R\searrow S)$, - [descent](Unsorted/descent.md): pairs $(M, \phi)$ where $M\in \mods{S}$ and $\phi: M\tensor_RS \iso S\tensor_R M$ is a twist isomorphism. - Given $F\in [\cat A, \cat B]$ and $G\in [\cat A, \cat C]$, the left [Kan extension](Kan%20extension.md) of $G$ along $F$ is a functor $L\in [B, C]$ and a sufficiently universal natural transformation $\alpha\in [G, LF]$. - Example: $G:\mods{A}\to \cat A$ into some [abelian category](abelian%20category.md). Here [simplicial resolution](simplicial%20resolution) by projective objects for projective resolutions, and $\LL G$ is the left [Kan extension](Kan%20extension.md) of $G:\cat{C} \to K^-(\cat{A})$ along the inclusion $\cat{C} \to K^-(\cat A)$, where $\cat{C} \leq K^-(\cat A)$ are complexes of projective modules. So this replaces [cofibrant replacement](cofibrant%20replacement).