Tags: #quick_notes

2021-10-08

21:03

• [[monad|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 ${{T}{\hbox{-}}\mathsf{coMod}}(\mathsf{C})$.

• 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. ${{\Gamma(G)}{\hbox{-}}\mathsf{coMod}}$. 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 $L(X) \in {{LR}{\hbox{-}}\mathsf{coMod}}$, and $\mathsf{D} \xrightarrow{L} \mathsf{C}$ factors:

Link to diagram

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

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

• 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 $$\begin{align*} {{F}{\hbox{-}}\mathsf{coMod}}(\mathsf{C}) { { \, \xrightarrow{\sim}\, }}{\mathsf{Rep}}(G) .\end{align*}$$

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

22:52

http://individual.utoronto.ca/groechenig/stacks.pdf #references

Refs: [[stack|stacks]] [[vector bundles|vector bundle]] descent data

• 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 $Y{ {}^{ \scriptscriptstyle{ \underset{\scriptscriptstyle {X} }{\times} }^{2} } }$:

Link to diagram

• 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 $s\otimes(rm) \coloneqq\alpha(r) (s\otimes m) = (\alpha(r)s)\otimes m$.

• This yields an adjunction: $$\begin{align*} \adjunction{({-})\otimes_R S}{\operatorname{res}_{\alpha}}{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}{{\mathsf{S}{\hbox{-}}\mathsf{Mod}}} .\end{align*}$$

• 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 descent data.

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

  • Descent data : 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.