# 2021-10-08

Tags: #web/quick-notes

# 2021-10-08

## 21:03

• comonads in $$\mathsf{C}$$: coalgebra object in $$[\mathsf{C}, \mathsf{C}]$$.

• Comodules over a comonad $$T$$: an object $$X$$, a map $$a^\sharp: X\to TX$$, and some coherence conditions. Often called $$T{\hbox{-}}$$algebras, called the category of $$T{\hbox{-}}$$comodules $${\mathsf{T}{\hbox{-}}\mathsf{coMod}}(\mathsf{C})$$.

• A fun but non-obvious consequence of https://stacks.math.columbia.edu/tag/06WS: for $G\in{\mathsf{Grp}}{\mathsf{Sch}}_{/ {R}}$ faithfully flat, there is an equivalence of categories \begin{align*} {\mathsf{QCoh}}({\mathbf{B}}G) { \, \xrightarrow{\sim}\, }{\mathsf{Rep}}(G) ,\end{align*} the category of regular $$G{\hbox{-}}$$representations, i.e. $${\mathsf{\Gamma(G)}{\hbox{-}}\mathsf{coMod}}$$. See regular representation.

• Why this is true:

• $${\mathsf{QCoh}}(\operatorname{Spec}R) { \, \xrightarrow{\sim}\, }{\mathsf{R}{\hbox{-}}\mathsf{Mod}}$$
• Use $$p: \operatorname{Spec}R\to {\mathbf{B}}G$$ induced by $$G\to R$$ to induce a pullback functor $$p^*: {\mathsf{QCoh}}({\mathbf{B}}G)\to {\mathsf{QCoh}}(\operatorname{Spec}R) \cong {\mathsf{R}{\hbox{-}}\mathsf{Mod}}$$.
• Set up a adjunction that yields a comonad equivalent to $$F: ({-})\otimes_R \Gamma(G)$$.
• Apply Barr-Beck :
• Given an adjunction $$\adjunction{L}{R}{\mathsf{D}}{\mathsf{C}}$$, get a comonad $$LR\in [\mathsf{C}, \mathsf{C}]$$.
• Then every $$X\in \mathsf{D}$$ yields $$L(X) \in {\mathsf{LR}{\hbox{-}}\mathsf{coMod}}$$, and $$\mathsf{D} \xrightarrow{L} \mathsf{C}$$ factors:

• Barr-Beck says $$\tilde L$$ is an equivalence under suitable conditions ($$L, R$$ conservative with $$L$$ preserving equalizers).

• Set up the Unsorted/adjoint (categorical) \begin{align*} \mathsf{D} \coloneqq\adjunction{p^*}{p_*}{{\mathsf{QCoh}}({\mathbf{B}}G)}{{\mathsf{QCoh}}(\operatorname{Spec}R)} \coloneqq\mathsf{C} .\end{align*} Then $$LR \coloneqq p^*p_*$$, and Barr-Beck yields \begin{align*} {\mathsf{QCoh}}({\mathbf{B}}G)\underset{\tilde{p^*}}{ { \, \xrightarrow{\sim}\, }} {\mathsf{(p^*p_*)}{\hbox{-}}\mathsf{coMod}}({\mathsf{QCoh}}(\operatorname{Spec}R)) .\end{align*}

• Use that if $G\in{\mathsf{Aff}}{\mathsf{Grp}}{\mathsf{Sch}}_{/ {R}}$ then $$\Gamma(G) \in \mathsf{Hopf}{\mathsf{Alg}_{/R}}$$. Set $$\mathsf{C} \coloneqq{\mathsf{R}{\hbox{-}}\mathsf{Mod}}$$, and $$F\in [\mathsf{C}, \mathsf{C}]$$ to be $$F({-}) \coloneqq({-})\otimes_R \Gamma(G)$$. Then there is an equivalence of categories \begin{align*} {\mathsf{F}{\hbox{-}}\mathsf{coMod}}(\mathsf{C}) { \, \xrightarrow{\sim}\, }{\mathsf{Rep}}(G) .\end{align*}

• Then show that $$F$$ is equivalent to $$p^*p_*$$.

## 22:52

Refs: stacks vector bundle Unsorted/descent

• Vector bundles as descent data: consider describing $$E\to X$$; one needs the cocycle condition. This means choosing $${\mathcal{U}}\rightrightarrows X$$ and bundle automorphisms $$\phi_{ij}: (U_i \cap U_j)\times {\mathbb{R}}^n {\circlearrowleft}$$ of the trivial bundle.
• We then want to glue up to obtain some $$E$$ over $$X$$: finding local bundle isomorphisms $$\phi_i: U_i \times {\mathbb{R}}^n { \, \xrightarrow{\sim}\, }{ \left.{{E}} \right|_{{U_i}} }$$ with $$\phi_{ij} = \phi_i \circ \phi_j^{-1}$$ on $$U_i \cap U_j$$. The cocycle condition is necessary, and for topological vector bundles, also sufficient.
• How to glue: set $$E \coloneqq{\textstyle\coprod}_{i} (U_i \times {\mathbb{R}}^n)/\sim$$ where $$(x, \mathbf{v})\sim (x, \phi_{ij}(\mathbf{v}))$$ with the quotient topology.
• Alternative formulation:
• Let $${\mathcal{U}}\rightrightarrows X$$ and define $$Y\coloneqq\displaystyle\coprod_i U_i$$, which induces $$Y \xrightarrow{\pi} X$$ by the inclusions $$U_i \hookrightarrow X$$.
• Then \begin{align*} Y{ {}^{ \scriptscriptstyle{ \underset{\scriptscriptstyle {X} }{\times} }^{2} } } = \displaystyle\coprod_{(i, j)\in I{ {}^{ \scriptscriptstyle\times^{2} } }} U_i \cap U_j .\end{align*} The cocycle condition becomes the existence of an isomorphism of bundles over $$Y{ {}^{ \scriptscriptstyle{ \underset{\scriptscriptstyle {X} }{\times} }^{2} } }$$:

• Note that pullbacks of trivial bundles are trivial, so this is an automorphism of the trivial bundle on $$Y{ {}^{ \scriptscriptstyle{ \underset{\scriptscriptstyle {X} }{\times} }^{2} } }$$
• The cocycle condition becomes an identity among bundle isomorphisms on $$Y{ {}^{ \scriptscriptstyle{ \underset{\scriptscriptstyle {X} }{\times} }^{3} } }$$: \begin{align*} p_{12}^* \phi \circ p_{23}^* \phi = p_{13}^*\phi \end{align*} 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 $${\mathsf{{\mathbb{R}}}{\hbox{-}}\mathsf{Mod}} { \, \xrightarrow{\sim}\, }\mathsf{Tw}{\mathsf{{\mathbb{C}}}{\hbox{-}}\mathsf{Mod}}$$ where the latter consists of objects which are pairs $$(V, f:V\to V)$$ where $$f(\lambda \mathbf{v}) = \mkern 1.5mu\overline{\mkern-1.5mu\lambda\mkern-1.5mu}\mkern 1.5mu \mathbf{v}$$ is a structure map and $$f^2 = \operatorname{id}_V$$ and morphisms $$\phi:V\to W$$ that commute with the structure maps.

• The forward map is $$V\mapsto (V\otimes_{\mathbb{R}}{\mathbb{C}}, f)$$ with $$f$$ the generator $$f\in { \mathsf{Gal}} ({\mathbb{C}}_{/ {{\mathbb{R}}}} )$$, and the inverse is $$(V, f)\mapsto V^f$$, the $$f{\hbox{-}}$$invariant subspace.
• For field extensions $L_{/ {k}}$, the ring morphism $$k\hookrightarrow L$$ yields $$\operatorname{Spec}L \to \operatorname{Spec}k$$, which behaves like a covering space with $$\mathop{\mathrm{Deck}}(\operatorname{Spec}L _{/ {\operatorname{Spec}k}} ) \cong { \mathsf{Gal}} (L_{/ {k}} )$$.

• Vector bundles on $$\operatorname{Spec}k$$ correspond to $${\mathsf{k}{\hbox{-}}\mathsf{Mod}}$$, and Galois-equivariant vector bundles on $$\operatorname{Spec}L$$ will correspond to vector bundles on the quotient $$\operatorname{Spec}k$$.

• $R\in {\mathsf{Alg}}_{/ {A}}$: a ring morphism $$A\to R$$.

• Given $$f\in {\mathbb{Z}}[x_1,\cdots, x_n]$$, taking the zero locus in a ring $$R$$ yields a functor $$\mathsf{CRing}\to {\mathsf{Set}}$$. To do this with $$f\in A[x_1,\cdots, x_n]$$ for $$A\in \mathsf{CRing}$$, one needs $R\in {\mathsf{Alg}}_{/ {A}}$, so this yields a functor $${\mathsf{Alg}}_{/ {A}} \to {\mathsf{Set}}$$.

• Think of spaces as functors $$X\in [\mathsf{CRing}, {\mathsf{Set}}]$$, then $$\operatorname{Spec}R \coloneqq\mathsf{CRing}(R, {-})$$, so $$R$$ corepresents $$\operatorname{Spec}R$$ in $$\mathsf{CRing}$$.

• Can represent $$R \left[ { \scriptstyle { {f}^{-1}} } \right] = R[t]/\left\langle{tf-1}\right\rangle$$.

• Standard open subfunctors: $$\operatorname{Spec}R \left[ { \scriptstyle { {f_i}^{-1}} } \right] \to \operatorname{Spec}R$$. These form an open cover if $$\left\langle{f_i}\right\rangle = \left\langle{1}\right\rangle$$.

• If $$k\in \mathsf{Field}$$, there is an equivalence $$\operatorname{Spec}R(k) \cong Z_f(k)$$, the zeros of $$f$$ in $$k$$. Then $$\operatorname{Spec}R \left[ { \scriptstyle { {h}^{-1}} } \right](k) = Z_f(k)\setminus Z_h(k)$$ for $$R = {\mathbb{Z}}[x_1,\cdots, x_n]/\left\langle{f}\right\rangle$$.

• Analog of 2-dimensional $${\mathbb{C}}{\hbox{-}}$$module over a ringer ring: the free $$R{\hbox{-}}$$module $$R{ {}^{ \scriptscriptstyle\times^{2} } }$$ of rank 2.

• $${\mathbb{P}}^1_{{\mathbb{Z}}}: \mathsf{CRing}\to{\mathsf{Set}}$$ is the functor sending $$R$$ to the set of direct summands $$M \leq R{ {}^{ \scriptscriptstyle\times^{2} } }$$ for which there’s an open covering corresponding to $$\left\{{h_i}\right\}$$ where $$M \left[ { \scriptstyle { {h_i}^{-1}} } \right] = M\otimes_R R \left[ { \scriptstyle { {h_i}^{-1}} } \right]$$ is a free $$R{\hbox{-}}$$module of rank 1 for all $$i$$.

• This recovers ${\mathbb{P}}^1_{{\mathbb{Z}}}({\mathbb{C}}) = {\mathbb{P}}^1_{/ {{\mathbb{C}}}}$ classically, since sub-vector spaces are direct summands.
• $${\mathbb{P}}^1_{\mathbb{Z}}({\mathbb{Z}}[t])$$ induces a continuously varying family of 1-dimensional subspaces of $${\mathbb{C}}^2$$? Somehow, even though $${\mathbb{C}}$$ isn’t in the definition..
• For $$S\in{\mathsf{Alg}_{/R}}$$, we have $$\alpha: R\to S$$ and for $$N\in {\mathsf{S}{\hbox{-}}\mathsf{Mod}}$$ we can forget the module structure along this map by defining \begin{align*} R\times N &\to N \\ (r, n) &\mapsto \alpha(r) \cdot n .\end{align*} This induces a restriction functor $$\operatorname{res}_{\alpha}: {\mathsf{S}{\hbox{-}}\mathsf{Mod}} \to {\mathsf{R}{\hbox{-}}\mathsf{Mod}}$$.

• Conversely we can tensor $$R{\hbox{-}}$$modules up to $$S{\hbox{-}}$$modules to get a functor $$S\otimes_R({-})$$, where the interesting bit is $$s\otimes(rm) \coloneqq\alpha(r) (s\otimes m) = (\alpha(r)s)\otimes m$$.

• Any reasonable property of modules should be preserved by base change!

• Descent for modules: when does $$M\otimes_R S$$ having property $$P$$ as an $$S{\hbox{-}}$$module descend to $$M$$ having property $$P$$ has an $$R{\hbox{-}}$$module?

• Left adjoints are right exact (LARE). In particular, base change is right exact, but not always left exact: take $$\alpha: {\mathbb{Z}}\to {\mathbb{Z}}/2$$, take the SES $$0 \to {\mathbb{Z}}\xrightarrow{2} {\mathbb{Z}}\to {\mathbb{Z}}/2\to 0$$, and tensor with $${\mathbb{Z}}/2$$. So an $$R{\hbox{-}}$$algebra $$S$$ is flat precisely when the base change $$S\otimes_R({-})$$ is exact.

• Free implies flat, and every module over a field is free.
• $$S$$ is faithfully flat when $$S\otimes_R M = 0\implies M=0$$. 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 $$T\to S\otimes_R T$$ is faithfully flat.

• General idea: $$R{\hbox{-}}$$modules $$M$$ can be specified by $$S\otimes_R M$$ along with Unsorted/descent.

• faithfully flat descent : there is an equivalence of categories $${\mathsf{R}{\hbox{-}}\mathsf{Mod}} \to {\mathsf{Desc}}(R\searrow S)$$,

• descent: pairs $$(M, \phi)$$ where $$M\in {\mathsf{S}{\hbox{-}}\mathsf{Mod}}$$ and $$\phi: M\otimes_RS { \, \xrightarrow{\sim}\, }S\otimes_R M$$ is a twist isomorphism.
• Given $$F\in [\mathsf{A}, \mathsf{B}]$$ and $$G\in [\mathsf{A}, \mathsf{C}]$$, the left Kan extension of $$G$$ along $$F$$ is a functor $$L\in [B, C]$$ and a sufficiently universal natural transformation $$\alpha\in [G, LF]$$.

• Example: $$G:{\mathsf{A}{\hbox{-}}\mathsf{Mod}}\to \mathsf{A}$$ into some abelian category. Here simplicial resolution by projective objects for projective resolutions, and $${\mathbb{L}}G$$ is the left Kan extension of $$G:\mathsf{C} \to K^-(\mathsf{A})$$ along the inclusion $$\mathsf{C} \to K^-(\mathsf{A})$$, where $$\mathsf{C} \leq K^-(\mathsf{A})$$ are complexes of projective modules. So this replaces cofibrant replacement.