--- date: 2023-01-25 19:42 title: Notes Lectures on Etale Cohomology (Litt) aliases: - Notes Lectures on Etale Cohomology (Litt) annotation-target: status: Started page_current: 0 page_total: 100000 flashcard: created: 2023-01-25T19:42 updated: 2024-01-29T17:51 --- Last modified: `=this.file.mday` # Videos ## Video 1 - [ ] What is a mixed Hodge structure? - [ ] What is a cycle class? - [ ] What is singular cohomology with twisted coefficients? - [ ] What is a supersingular elliptic curve? - [ ] Why is there no functorial cohomology theory on schemes over $\bar\FF_p$ satisfying functoriality, Kunneth, and $F(E; \QQ) = \QQ^2$? - [ ] How do zeros/poles of a meromorphic function $f$ control the growth rate of a Laurent expansion of $\dlog f$? - [ ] What are the Weil conjectures? - [ ] What is the Euler characteristic of $X\in \Sch\slice{\FF_p}$? - [ ] What is the Lefschetz trace formula? - [ ] What is the Euler product for $\zeta_X(s)$? - [ ] Why is $h^0(\OO(D)) = 0$ true when $\deg D < 0$? - [ ] What is $\size \PP^n(\FF_q)$? - [ ] What is the Lefschetz fixed point formula? - [ ] What is the absolute Frobenius? - [ ] What is the geometric Frobenius? - [ ] How are hyperplane classes related to $c_1$? ## Video 2 - [x] What is an etale morphism? ✅ 2023-01-25 - [ ] Locally of finite presentation, flat, unramified. - [ ] Smooth of relative dimension zero. - [ ] Locally finite presentation and formally etale. - [ ] Locally standard etale: $U \ni x$ and $V\ni f(x)$ with $f(U) \subseteq V$ and $V=\spec R, U = \spec R[x]_h/g$ where $g$ is monic and $g'\in R[x]_h\units$ - [ ] Idea: $g$ is a curve in $\spec R \times \AA^1$, localizing at $h$ deletes a locus including the double roots of $g$. ![](attachments/2023-01-25-pic.png) - [x] Is an isogeny etale? ✅ 2023-01-25 - [ ] Not always - [x] What are some examples of etale morphisms? ✅ 2023-01-25 - [ ] $\cdot n$ on an elliptic curve if $n$ is invertible in the base field. - [ ] $t\mapsto t^m$ as a map $\GG_m\selfmap$ if $m$ is prime to the characteristic. - [ ] $\GG_m\injects \AA^1$: lfp since $k[t, t\inv] = k[t][t\inv]$, flat since a Zariski open embedding, and $\Omega^1_{\AA^1/\GG_m} = 0$ for the same reason. - [ ] Any Zariski open embedding - [x] Give an example of an etale morphism that is not finite onto its image. ✅ 2023-01-25 - [ ] $t\mapsto t^2$ on $\GG_m\smts{1}\to \GG_m$, not proper so not finite etale. ![](attachments/2023-01-25-loc-et.png) - [x] Give an example of a non-etale morphism ✅ 2023-01-25 - [ ] The normalization of $k[x,y]/xy$, since it is not flat. - [x] Give an example of a finite flat map that is not etale ✅ 2023-01-25 - [ ] $t\mapsto t^2$ on $\AA^1\selfmap$ since it's ramified at zero: $\Omega^1_f = k[t]\dt/d(t^2)$ but $2t\dt$ doesn't generate $k[t]\dt$. Not the relative differentials are torsion. - [x] Give an example of a morphism that is not etale, but is finite flat where $\Omega^1_{X/Y}$ is not torsion? ✅ 2023-01-25 - [ ] Take the relative Frobenius $\Frob:\AA^1\to \AA^1$ over $\FF_p$, then $\Omega^1_{\Frob} = k[t]\dt/d(t^p) = k[t]\dt$ which is a line bundle and thus not torsion. - [x] Where is a map $F:\AA^m\selfmap$ etale? ✅ 2023-01-25 - [ ] $F$ is given by $f_1,\cdots, f_n$, and the etale locus is where $\Jac_f$ is a unit. - [x] What is an unramified morphism? ✅ 2023-01-25 - [ ] Relative Kahler differentials vanish - [ ] All residue field extensions are separable. - [x] What is a formally etale morphism? ✅ 2023-01-25 - [ ] Uniqueness of lifts through nilpotent quotients: for $I$ nilpotent, ![](attachments/2023-01-25.png) - [ ] Idea: tangent vectors lift ![](attachments/2023-01-25-1.png) - [ ] - [x] Discuss the image of a flat morphism. ✅ 2023-01-25 - [ ] Always open. - [x] What are the preservation properties of etale morphisms? ✅ 2023-01-25 - [ ] Any open immersion is etale - [ ] Compositions of etale are etale - [ ] Base change of etale are etale - [ ] 2 out of 3 property: $\phi \circ \psi, \phi$ etale $\implies \psi$ etale. - [x] What is a site? ✅ 2023-01-25 - [ ] $\ts{X_\alpha\to X}\covers X$ in $\cat C$ a covering family where - [ ] Intersections exist: $X_\alpha \fiberprod{X} Y$ exists in $\cat C$ for any $Y\to X$ - [ ] Intersecting with a cover is a cover: $\ts{X_\alpha\fiberprod{X} Y}\covers Y$. - [ ] Closed under compositions: if $\ts{X_{\alpha\beta} \to X_\alpha}_{\alpha, \beta} \covers X_\alpha$ then $\ts{X_{\alpha_\beta} \to X_\alpha \to X}_{\alpha, \beta} \covers X$. - [ ] Isos are always covers. - [x] What are the small and big etale sites? ✅ 2023-01-25 - [ ] **Small**: $X_\et \subset \Sch\slice{X}$ consisting of all etale morphisms over $X$, and families $\ts{f_\alpha}$ cover if $\union \im f_\alpha = X$. - [ ] **Big**: $X_\Et = \Sch\slice X$ all schems over $X$ where covering families $f_\alpha$ all etale with $\union \im f_\alpha = X$. - [x] What is the fppf topology? ✅ 2023-01-25 - [ ] $X_\fppf \subset \Sch\slice X$ with faithfuly flat and finite presentation structure morphisms, covers are covers as in $X_\et$. - [x] What are $X_\zar$ and $X_\Zar$? ✅ 2023-01-25 - [ ] Small: the site associated to $\abs{X}\in\Top$. - [ ] Big: $X_\Zar = \Sch\slice X$ with Zariski-open embeddings and jointly surjective images. - [x] What is a sheaf on a site? ✅ 2023-01-25 - [ ] $F\in \Fun(\cat C, \cat D)$ where $F(U) \injects \prod_i F(U_i) \covers \prod_{i, j} F(U_{ij})$ is an equalizer for all covering families. - [ ] Exactness in the middle is gluing, exactness at the beginning is uniqueness. - [ ] So $F(U) = \lim \prod F(U_i) \surjects \prod F(U_{ij})$. - [x] Give an example of a sheaf on $X_\Et$ ✅ 2023-01-25 - [ ] Any representable functor. - [ ] $\mu_n$, since represented by $k[t]/t^n-1$ with sections $U\mapsto \ts{f\in H^0(\OO_U) \st f^n=1}$. - [ ] $U\mapsto H^0(\OO_U)$, since it's represented by $\AA^1\slice X$. - [ ] $\ul{C_\ell}$ represented by the $C_\ell \fiberprod{k} X$ where $C_\ell$ is the constant group scheme, so the underlying set is $X\disjointpower{\ell}$. This has sections $U\mapsto \Top(\abs{U}, C_\ell)$. - [ ] $\GG_m$ represented by $\GG_{m\slice X} = \spec \ZZ[t, t\inv]\fiberprod{\ZZ} X$, sending $U\mapsto H^0(\OO_U)\units$. - [ ] $\PP^n$ represented by $\Hom(\wait, \PP^n\slice X)$, written as a functor sending $U$ to $\mcl\in \Pic(U)$ with $f_U: \OO(U)\sumpower{n} \surjects \mcl$. - [x] What is a morphism of sites? ✅ 2023-01-25 - [ ] $f\in \Hom(T_1, T_2) \iff f\in \Fun(T_2, T_1)$ which preserves fiber products and covering families. - [ ] Compare to $f\in \Top(X, Y)$ inducing $f: \Open(Y)\to \Open(X)$ where $U\mapsto f\inv(U)$. - [ ] What is the Nisnevich site? - [ ] What is the crystalline site? - [ ] What is the cdh site? - [ ] What is the arc site? - [ ] What is a relative dimension zero morphism? - [ ] What is Henselization? - [ ] What is the cotangent exact sequence? - [ ] What is the etale site? - [ ] What is the Hard Lefschetz theorem? - [ ] What is primitive cohomology? - [ ] What is the Hodge index theorem? - [ ] What is a definite form? - [ ] What is intersection cohomology? ## Video 3 - [x] How are the various sites for a scheme related? ✅ 2023-01-26 - [ ] $X_\fppf \to X_\Et \to X_\et \to X_\zar$. - [x] Give examples of sheaves on $X_\fppf$ ✅ 2023-01-26 - [ ] $\Sch\slice X(\wait, Y)$ for any $Y\in \Sch\slice X$. - [ ] $(Z\mapsvia{f} X) \mapsto H^0(Z, f^* \mcf)$ for any $\mcf\in \Coh(X)$. - [x] What is $\fpqc$? ✅ 2023-01-26 - [ ] Finitely presented and quasicompact. - [x] What is descent data for $\mcf\in \QCoh(U\slice X)$? ✅ 2023-01-26 - [ ] $\mcf \in \QCoh(U)$ and $f: U\to X$ - [ ] Gluing data: $\proj_1^* \mcf \isovia{\phi} \proj_2^* \mcf$ where $\proj_i: U\fiberprod{X} U\to U$. - [ ] Cocyle condition: for $\proj_{ij}U\fiberpower{X}{3}\to U\fiberpower{X}{2}$, $\proj_{23}^* \phi \circ \proj_{12}^* \phi = \proj_{13}^*\phi$. - [x] What is effective descent? ✅ 2023-01-26 - [ ] The pullback $f^*$ induces an equivalence of categories. - [x] What is descent for $\QCoh(X)$? ✅ 2023-01-26 - [ ] If $U\to X$ is fppf, there is an equivalence of categories between $\QCoh(X)$ and descent data on $U\slice X$ induced by $f^*$, where $\mcf\in \QCoh(X)$ is sent to $f^* \mcf \in \QCoh(U)$ with descent data $(f\circ \proj_1)^* \mcf \iso (f\circ \proj_2)^*\mcf$ an isomorphism on $U\fiberprod{X} U$ induced by pulling back the identity. - [x] What is a quasicoherent sheaf on $\spec L$ for $L$ a field? ✅ 2023-01-26 - [ ] An $L\dash$vector space. - [x] What is the sheaf-theoretic definition of a scheme? ✅ 2023-01-26 - [ ] A sheaf on the Zariski site which is locally representable. - [x] What is the sheaf-theoretic definition of an algebraic space? ✅ 2023-01-26 - [ ] A sheaf on the Zariski site which are is not representable by schemes but are locally representable in the etale topology. - [x] Give a sufficient condition for the Amitsur complex to be exact. ✅ 2023-01-26 - [ ] $R\to S$ faithfully flat, $N\in \rmod$, and forming the complex from $(\wait)\tensor_R S$. - [ ] What is a sieve? - [ ] What is a topos? - [ ] What is Galois descent? - [ ] What is faithfully flat descent for $\rmod$? - [ ] What is the Amitsur complex? - [ ] What is a faithfully flat ring morphism? - [ ] How is $f^*$ defined for $\OO_X\dash$modules? - [ ] Describe $\ro\mcf U$ in terms of pullbacks. - [ ] Why is descent data for $\spec L\to \spec k$ for $L/k$ a Galois extension the same as Galois descent? - [ ] What is a fully faithful functor? Essentially surjective? - [ ] Describe global sections in terms of maps in $\oxmods$. ## Video 5 - [x] What is fppf descent for $\QCoh(X)$? ✅ 2023-01-26 - [ ] If $f: U\to X$ is an fppf cover, then $f^*: \QCoh(X) \iso \mathsf{Descent}(U\slice X)$ is an equivalence of categories. - [x] For $\mcf \in \QCoh(X)$, describe $\mcf^\et$. ✅ 2023-01-26 - [ ] $\mcf^\et(U\mapsvia{\pi} X) \da \pi^* \mcf(U)$. - [x] Why is the presheaf $\mcf^\et$ a sheaf? ✅ 2023-01-26 - [ ] For $U\to V$ an etale cover, need $\mcf(V) \to \mcf(U) \covers \mcf(U\fiberprod V U)$. - [ ] Use the equalizer $\Hom_X(\mcf_1, \mcf_2) \to \Hom_U(f^* \mcf_1, f^* \mcf_2) \covers \Hom_{U\fiberprod X U}(\proj_1 f^* \mcf_1, \proj_1 f^* \mcf_2)$ with $\mcf_2 = \mcf, \mcf_1 = \OO$. - [x] Describe $\OO_X^\et$. ✅ 2023-01-26 - [ ] $U\to X$ an etale cover maps to $H^0(U; \OO_U)$. - [x] Describe faithfully flat descent along $\Frob: X\to X^{(p)}$ over $\FF_p$. ✅ 2023-01-26 - [ ] Vector bundles on $X^{(p)}$ are equivalent to descent date for vector bundles on $X^{(p)}$, described as vector bundles $\mce$ on $X$ with a flat connection $\nabla:\mce \to \mce\tensor \Omega^1_{X}$ with $p\dash$curvature zero. - [x] What is an algebraic space? ✅ 2023-01-26 - [ ] Descent data relative to an etale cover $U\slice X$. - [ ] What is a separable morphism? - [ ] What is a morphism of descent data? - [ ] What is the $p\dash$curvature of a connection? - [ ] What is a $\Dmod$ on a scheme? - [ ] What is the $p\dash$curvature conjecture? - [ ] What is relative spec? - [ ] What is a polarized scheme? - [ ] $(X, \mcl)$ a scheme with an ample line bundle. ## Video 6 - [x] What is an etale cover of $\spec k$? ✅ 2023-01-26 - [ ] $Y \da \disjoint_{i\in I} \spec k$, corresponding to an etale $k\dash$algebra. - [x] Describe $\Sh(\spec k)$ for $k=\kbar$. ✅ 2023-01-26 - [ ] $\Sh(\spec k) = \Sets$. - [x] What is the (etale) skyscraper sheaf for $\mcf \in \Sh(\spec k)$? ✅ 2023-01-26 - [ ] For $\bar x = \spec k \injectsvia{\iota_{\bar x}} X$, take $(\iota_{\bar x})_* \mcf(U\to X) \mcf(U\fiberprod{X}\bar x) = \mcf(\disjoint_i \spec k) = \prod_i \mcf(\spec k)$ where $i\in I$ with $\size I$ the number of preimages of $\bar x$. - [x] What is the pullback of $\mcf\in \Sh(X_\et)$ to a geometric point $\bar x$? ✅ 2023-01-26 - [ ] $\mcf_{\bar x} = \iota_{\bar x}^*\mcf =\colim_{(U, \bar u)} \mcf(U)$ where $\bar u\to U$ is a cartesian over $\bar x\to X$ and $U\to X$ is etale. - [x] What is the stalk of an etale sheaf $\mcf$? ✅ 2023-01-26 - [ ] $\mcf_{\bar x} \da (\iota_{\bar x})^* \mcf$, desribed as a colimit over pointed etale covers of $\bar x\to X$. - [x] What is $(\iota_{\bar x})^*\OO_X^\et$? ✅ 2023-01-26 - [ ] The strict henselization $\OO_{X, x}^{\mathrm{sh}}$ of the local ring at $\bar x$. - [x] What is a Henselian ring? Strictly Henselian? ✅ 2023-01-26 - [ ] Where Hensel's lemma is true: given a monic polynomial that factors in the residue field, that factorization lifts to the entire ring. - [ ] Strict: Henselian with algebraically closed residue field. - [x] Show that $\mcf \surjects \mcg \implies \mcf_{x} \surjects \mcg_x$. ✅ 2023-01-26 - [ ] If not, let $\Lambda = \coker(\mcf_x \to \mcg_x) \neq 0$ at an $x$ where it fails, then let $\mch \da (\iota_x)_* \Lambda$ be the skyscraper sheaf. - [ ] Get an equalizer $\mcf \to \mcg \covers^{0}_f \mch$ where $f$ sends a section to its stalk; the composition is zero, forcing $\Lambda = 0$, a contradiction. - [x] Describe global sections as a pushforward ✅ 2023-01-26 - [ ] Let $\pi:X\to \spec k$ be the structure map, then if $k=\kbar$ one has $H^0(\spec k; \pi_* \mcf) = H^0(X; \mcf)$. - [x] What is the espace etale for $\mcf$? ✅ 2023-01-26 - [ ] Write $\Et(\mcf) \da \prod_{\bar x}(\iota_{\bar x})_*\mcf_{\bar x}$ where $\bar x$ is a geometric point lying over every $x\in X$. - [ ] Define $\mcf^a \leq \Et(\mcf)$ the subsheaf generated by $\mcf$, where $\mcf^a(U)$ are sections $s\in \Et(\mcf)(U)$ where are locally in the image of the natural map $\mcf \to \Et(\mcf)$ sending sections to their germs at all points. - [x] When do colimits exist in $\Presh(\cat C, \cat D)$? ✅ 2023-01-26 - [ ] A sufficient condition: colimits exist in $\cat D$, since you can compute them pointwise. - [x] Why do colimits exist in $\Sh(X_\et)$? ✅ 2023-01-26 - [ ] Presheaves over cocomplete categories are cocomplete, and LAPC (left adjoints preserve colimits, left adjoints are easy to map *out* of). - [x] Discuss adjointness of sheafification. ✅ 2023-01-26 - [ ] Left adjoint to the forgetful functor $\Sh(X_\et)\to \Presh(X_\et)$ (left adoints are a mapping out property). - [x] Why is $\Sh(X_\et)$ abelian? ✅ 2023-01-26 - [ ] Limits: computed pointwise, over a complete category - [ ] Cokernels: a colimit. - [ ] $\im = \coim$ can be checked on stalks. - [ ] What is a cofinal diagram? ## Video 7 - [ ] Mathematics: What is the Godemont resolution? - [x] Characterize injective objects in $\zmod$. ✅ 2023-01-27 - [ ] Divisible groups. Generally need Baer's criterion. - [ ] Why does $\zmod$ have enough injectives? - [x] How do you map into or out of a product? ✅ 2023-01-27 - [ ] Into: map into each component. - [x] Describe pullbacks in terms of homs for representable functors. ✅ 2023-01-27 - [ ] If $\mcf = \Hom_X(\wait, Z)$, then $f^* \mcf = \Hom_Y(\wait, Y\fiberprod {X} Z)$. - [ ] What is an effaceable delta functor? - [ ] When does Cech cohomology not compute sheaf cohomology? - [ ] What is the Cech complex of $X_\et$? - [ ] Discuss exactness of limits and colimits. - [x] When does Cech cohomology compute etale cohomology? ✅ 2023-01-27 - [ ] If $X$ is qc and every finite subset is contained in an affine, e.g. if $X$ is quasiprojective. - [ ] What is the Cech-to-derived spectral sequence? - [ ] What is the Mayer-Vietoris sequence for sheaf cohomology? - [ ] Why is a 2-column spectral sequence the same as a LES? - [x] Write global sections as a hom and cohomology as an ext ✅ 2023-01-27 - [ ] $\Globsec{X;\mcf} = \Sh(X)(\ul\ZZ \to \mcf)$, so $H^i(\mcf) = \Ext_{\Sh(X)}(\ul\ZZ, \mcf)$. - [ ] For $\mcf\in \oxmods$, $\globsec{X;\mcf} = \QCoh(X)(\OO_X, \mcf)$. - [ ] What is a flasque sheaf? - [ ] What is a coarser/finer topology? - [ ] How are nullhomotopies related to sections of a complex? (E.g. for the Amitsur complex). - [x] What is $H^*_\et(\PP^n; \OO)$? ✅ 2023-01-27 - [ ] $k\cdot t^0$. - [x] What is $H^*(X; \ul{\FF_p})$ for $X\in \Sch\slice{\FF_p}$ quasiprojective? ✅ 2023-01-27 - [ ] Use $\GG_a \da \Hom(\wait, \AA^1)$ where $U\mapsto \OO_U(U)$ and the Artin-Schreir exact sequence $$\ul{\FF_p} \injects \GG_a \surjectsvia{x^p-x}\GG_a$$ - [ ] What is the Artin-Shreir exact sequence? - [x] Is $\GG_a \mapsvia{x^p-x}\GG_a$ an etale morphism over $\FF_p$? ✅ 2023-01-27 - [ ] Yes: $\dd{}{x}(x^p-x) = -1$ which is invertible. - [x] Why is $H_\et(X; \FF_p)$ poorly behaved? ✅ 2023-01-27 - [ ] Expect a functorial assignment to $\zmod^\fg$, but e.g. even on affines on gets $H^0(X;\FF_p)\injects \OO_X(X) \mapsvia{\mathrm{AS}: t^p-t} \OO_X(X)\surjects H^1(X; \FF_p)$ where $\coker \mathrm{AS}$ is not finitely generated. - [ ] What is proper pushforward for coherent cohomology? - [ ] What is a supersingular curve? - [x] Give an example of an etale cover of $\spec k$. ✅ 2023-01-27 - [ ] $\spec L$ where $L/k$ is a separable extension. - [ ] Defne Galois cohomology in terms of etale cohomology. - [ ] What is an Artin neighborhood? - [ ] What is an etale $K(\pi, 1)$? - [x] What is a $G\dash$torsor? ✅ 2023-01-27 - [ ] Idea: $\mcf\in \Sh(X_\et,\Set)$ where $G\actson \mcf$ simply transitively on every fiber. - [ ] Precisely: for $G\in \Sh(X, \Grp)$ and $\mcf\in \Sh(X, \Set)$, an action $G\times \mcf \mapsvia{\alpha} \mcf$ which is simply transitive, i.e. an isomorphism $G\times \mcf \mapsvia{\alpha \times \proj_2}\mcf \times \mcf$ - [ ] What is a connected curve? - [x] What is $A\times B$ in terms of fiber products? ✅ 2023-01-27 - [ ] $A\fiberprod{T} B$ where $T$ is the terminal object. - [x] Describe $\torsors{G}$ for $G\in \Fin\Grp$. ✅ 2023-01-27 - [ ] A finite etale cover with Galois group $G$. - [ ] E.g. for $X$ a smooth curve: an extension $L/K(X)$ with $\Gal(L/K(X)) = G$ which is everywhere unramified. - [x] What is the representing object for $\GG_m$? ✅ 2023-01-27 - [ ] $\Hom_X(\wait, \spec k[t, t\inv])$. - [x] Describe $\torsors{\GG_m}$ ✅ 2023-01-27 - [ ] A line bundle $\mcl\smz$ with the zero section deleted. - [ ] Send $\mcl\to \relspec_X \bigoplus_{n\in \ZZ}\mcl\tensorpower{}{n}$. - [x] Descibre $\torsors{\GL_n}$? ✅ 2023-01-27 - [ ] Vector bundles of rank $n$, sending a bundle $\mce \to \Frame(\mce) \da \Isom_{X_\et}(\OO\sumpower{n}, \mce)$ with inverse $T\mapsto (T\times \OO\sumpower{n})/ \GL_n$, using that $\GL_n = \ul{\Aut}(\OO_X\sumpower{n})$. - [x] What is a splitting of a $G\dash$torsor? ✅ 2023-01-27 - [ ] Locally trivial: $T\in\torsors{G}$ is split by an etale cover $U\to X$ if $\ro T U \iso \ro G U$ as a torsor. - [x] Describe $\Aut(T)$ for $T\in\torsors{G}$ trivial. ✅ 2023-01-27 - [ ] An element of $G$ -- look at $G\dash$torsors over a point. - [x] Describe $H^1_\et(\tau; \mcf)$ for $\tau$ a site. ✅ 2023-01-27 - [ ] Locally trivial $\mcf\dash$torsors. - [x] What is Hilbert 90? How is it proved? ✅ 2023-01-27 - [ ] $$\Hc^1(X_\zar; \ul{\GL_n}) \iso \Hc^1(X_\et; \ul{\GL_n}) \iso \Hc^1(X_\fppf; \ul{\GL_n})$$ - [ ] Use bijection with locally split $\GL_n\dash$torsors, use this to get descent data for a vector bundle, use fppf descent. - [ ] Alternative statement: $H^1(\Gal(k^s/k), \bar{k}\units)$. - [x] Describe how to parameterize $n\dash$dimensional vector bundles on $X$. ✅ 2023-01-27 - [ ] $H^1_\et(X_\tau; \GL_n)$ where $\tau$ is any site appearing in Hilbert 90. - [x] Are all $G\dash$torsors represented by an $X\dash$scheme? ✅ 2023-01-27 - [ ] Yes. - [x] If $T\in\torsors{G}$, does fppf-locally trivial imply etale locally trivial? ✅ 2023-01-27 - [ ] No in general, yes if $G$ is smooth. - [ ] Take $\ker \Frob$ on $\GG_a$ or $\GG_m$, so $\alpha_p$ or $\mu_p$, in characteristic $p$. - [ ] In general, any positive dimensional affine group scheme. - [x] What is $H^1_\et(\spec k; \GG_m)?$ ✅ 2023-01-27 - [ ] Line bundles on a point are trivial: $H^1_\et(\spec k, \GG_m) = 0 = H^1(\Gal(k^s/k), \bar{k}\units)$. - [x] What is $H^1(X_\et; \GG_m)$? ✅ 2023-01-27 - [ ] $\Pic(X)$. - [x] What is $H^1(X_\et; \mu_\ell)$ with $\ell$ invertible in $X$. ✅ 2023-01-27 - [ ] Use the Kummer SES $\mu_\ell \injects \GG_m \surjectsvia{z\mapsto z^\ell} \GG_m$. - [ ] Prove that if $X$ is proper, $H^0(\OO_X) = k$. - [x] How do you explicitly write down a cover corresponding to an element in $H^1(X_\et; \FF_p)$? ✅ 2023-01-27 - [ ] Write $H^1 = \coker(\OO_X \mapsvia{x^p - x}\OO_X) \ni f$ and take $Y = \ts{y^p-y = f}$. - [x] How do you explicitly write down a cover corresponding to an element in $H^1(X_\et; \mu_\ell)$? ✅ 2023-01-27 - [ ] Write $H^1 = \coker(\OO_X\units \mapsvia{x\mapsto x^\ell}\OO_X\units)\ni f$ and take $Y = \ts{z^\ell = f}$. - [ ] Needs $\Pic(X) = 0$. - [x] Describe $H_\et(X; \GG_m)$ for $X$ a smooth curve over $k=\kbar$. ✅ 2023-01-27 - [ ] $\OO_X\units + \Pic(X) t$. - [x] Describe $H_\et(X; \mu_\ell)$ for $X$ a smooth proper curve over $k=\kbar$. ✅ 2023-01-27 - [ ] $C_{\ell^n} + \Pic[\ell^n]t + C_\ell t^2$ where $\Pic(\ell^n) \cong C_{\ell^n}^{2g}$, using the Kummer sequence and that $\Pic^0(X) \cong \Jac(X)$ is divisible and $\NS(X) = \ZZ$. - [ ] What is the cyclotomic character? - [x] For $f\in \Sch(X, Y)$, how is $f_* \in [\Sh(X_\et), \Sh(Y_\et)]$ defined? ✅ 2023-01-27 - [ ] $f_*\mcf(U\to X)\da \mcf(U\fiberprod{Y} X)$. - [x] What is the interpretation of $\RR^* f_*$? ✅ 2023-01-27 - [ ] Cohomology of fibers, sometimes -- obstructed by non-commuting with base change. - [ ] Sheafification of $V\mapsto H^i(f\inv(V); \mcf)$. - [x] Give an example where $\tau_{\geq 1}\RR f_* = 0$. ✅ 2023-01-27 - [ ] $f$ a finite morphism, e.g. a closed immersion. Show $f_*$ is right exact by checking on stalks $\bigoplus_{\bar x\in f\inv(\bar y)}\mcf_{\bar x}$. - [x] Why does $f_*$ preserve injectives? ✅ 2023-01-27 - [ ] True for any functor with an exact left adjoint, here take $f^*$, exact since filtered colimits and sheafification are exact (or check on stalks). - [x] What is the Leray spectral sequence? ✅ 2023-01-27 - [ ] $f:X\to Y$ and $g:Y\to Z$ yields $\RR^i g_* \RR^j f_* \mcf \abuts \RR^{i+j}(g\circ f)_* \mcf$. - [ ] Take $Z=\spec k$ to get global sections: $H^i(Y; \RR^i f_* \mcf) \abuts H^{i+j}(X; \mcf)$. ## Video 8 - [x] What is $H_\Gal(k; V)$ for $k$ a finite field and $V\in \kmod$? ✅ 2023-01-28 - [ ] $V^G t^0 + V_G t^1$. - [x] Is a sequence of presheaves exact iff exact on all sections? ✅ 2023-01-28 - [ ] True! Just not true for sheaves. - [x] For $X$ integral and $\iota:\eta \injects X$ a generic point, what are the stalks of $\RR \iota_* \mcf$? ✅ 2023-01-28 - [ ] Write $K_X \da \ff(\OO_X)$, use that stalks agree on sheaves vs presheaves to get $$\colim_{u\injects U} H^i(U_\eta; \ro \mcf {U_\eta}) = H^i(K_{X, \bar x}; \ro\mcf {K_{X, \bar x}})$$ at $\bar x\injects X$ a geometric point. - [ ] What is a regular variety? - [x] What is the point pushforward SES? ✅ 2023-01-28 - [ ] If $\iota:\eta \injects X$ for $X$ a regular variety, $$\GG_m \injects \eta_*\GG_m\surjects \bigoplus_{Z\leq X \,\, \codim_X(Z) = 1} \iota_* \ul{\ZZ}$$ where the last term is the divisor sheaf. - [ ] The first map is restriction to $\eta$, the second is taking the associated divisor. - [ ] What are the sections of $\GG_m$ and $\eta_* \GG_m$ for $\eta$ a generic point? - [ ] How is an integral domain related to all of its localizations? - [ ] $A = \Intersect_{p\in \spec(A)^{\height = 1}} A_p$. - [ ] What is a regular scheme? - [ ] What is a normal scheme? - [ ] What is a locally factorial scheme? - [ ] What is regularity in codimension 1? - [ ] Define the cohomological Brauer group. - [ ] $\Br(X_\et; \GG_m)_\tors$. ## Video 9: Brauer Groups - [x] What is $H^i(C; \CC_m)$? ✅ 2023-01-29 - [ ] $\OO_C(C)\units t^0 + \Pic(C) t$. - [ ] Why do we care about $H^i(k(C); \GG_m)$ and $H^i(K_{\bar x}; \GG_m)$? - [ ] Describe $K_{\bar x}$. - [x] What does $H^1$ classify? $H^2$? ✅ 2023-01-29 - [ ] $H^1$: torsors. $H^2$: gerbes. - [ ] What is a gerbe? - [x] What is $\Br^\coh(X)$? ✅ 2023-01-29 - [ ] $H^i_\et(X; \GG_m)_\tors$. - [ ] Equivalently, $\im\qty{\Union_n \mcp_n \mapsvia{\delta} H^2_\et(X; \GG_m)}$ where $\mcp_n$ is the set of all etale locally trivial $\PGL_n\dash$torsors over $X$, which is the boundary map coming from $\GG_m\injects \GL_n\surjects \PGL_n$. - [ ] What is the formula for $\delta$ in the snake lemma? - [x] What is the interpretation of $\delta([T])$? ✅ 2023-01-29 - [ ] For $T\in H^i(X_\et; \PGL_n)$, the obstruction to lifting $T$ to a $\GL_n\dash$torsor. - [ ] Equivalently: the Brauer class of $T$. - [x] Describe how to relate $G\dash$torsors and forms. ✅ 2023-01-29 - [ ] For $T = \Sh(X_\et, \Set)$ and $G = \Aut(X)$, locally split $G\dash$torsors $\mapstofrom$ forms of $T$. - [ ] Forms $F$ map to $\Isom(F, T)$. - [ ] Torsors $\tau$ map to the sheaf quotient $(T\times \tau)/G$. - [x] What is a Severi-Brauer X\dash$schemes? ✅ 2023-01-29 - [ ] Forms of $\PP^n_X$; for $X=\spec k$ these are Severi-Brauer varieties. - [ ] What is the Noether-Skoler theorem? - [x] What is an Azumaya algebra? ✅ 2023-01-29 - [ ] Forms of $\Mat_{n\times n} = \Endo_\OO(\OO\sumpower n)$. - [x] What is $\QCoh(X, \alpha)$? ✅ 2023-01-29 - [ ] For $U\to X$ an etale cover, $\alpha\in H^0(U\fiberpower{X}3; \GG_m)$ representing $[\alpha]\in H^2_\et(X; \GG_m)$, a sheaf $\mcf\in \QCoh(X)$ with an isomorphism $\phi: \proj_1 \mcf \iso \proj_2\mcf$ for $\proj_i: U\fiberpower{X}{2}\to U$ satisfying a twisted cocycle condition $$\pi_{23}^* \phi \circ \pi_{12}^* \phi = \alpha \cdot \pi_{13}^* \phi$$ - [ ] Morphisms are sheaf morphisms commuting with $\phi$. - [x] State etale descent in terms of $\alpha\dash$twisted sheaves. ✅ 2023-01-29 - [ ] $\QCoh(X, 1) = \QCoh(X)$ canonically. - [x] Describe homs, tensors, and symmetric powers on $\QCoh(X, \alpha)$. ✅ 2023-01-29 - [ ] $$\begin{align*}\Hom: \QCoh(X,\alpha_1) \times \QCoh(X,\alpha_2) &\to \QCoh(X,\alpha_1 \cdot \alpha_2\inv) \\ \tensor: \QCoh(X,\alpha_1) \times \QCoh(X,\alpha_2) &\to \QCoh(X,\alpha_1\cdot \alpha_2) \\ \Sym^n, \Wedge^n: \QCoh(X,\alpha_1) &\to \QCoh(X,n \cdot \alpha_1)\end{align*}$$ - [x] Discuss when a 2-cocycle $[\alpha]$ for $\GG_m$ relates to Brauer classes. ✅ 2023-01-29 - [ ] $[\alpha]\in \Br(X) \iff \exists \mce$ and $\alpha\dash$twisted vector bundle. - [ ] The forward direction corresponds to $\delta([\alpha])$, the reverse is projectiving a bundle to get a form of $\PP^n$. - [x] Why is $\Br(X)$ a group? ✅ 2023-01-29 - [ ] For $\alpha_1\cdot \alpha_2$, take $\mce_1\tensor \mce_2$ for the corresponding twisted vector bundles. - [ ] For $\alpha_1\inv$, take $\mce_1\dual$. - [x] How can one check when $[\alpha]$ is trivial? ✅ 2023-01-29 - [ ] $[\alpha] = 1 \iff \exists \mcl$ an $\alpha\dash$twisted line bundle. - [ ] Forward: take $\mcl = \OO_X$, reverse: use descent data to get $\beta$ with $\delta(\beta) = \alpha$. - [x] What does $\mce$ an $\alpha\dash$twisted vector bundle of rank $n$ correspond to in $\Br(X)$? ✅ 2023-01-29 - [ ] $\alpha\in H^2(X_\et; \GG_m)[n]$, $n\dash$torsion. - [ ] Proof: take $\det \mce$ to get an $\alpha^n\dash$twisted line bundle, implies $1 = [\alpha^n] = [\alpha]^n$. - [ ] Warning: $n\dash$torsion elements do not generally yield rank $n$ bundles. - [x] Give an example of a twisted form of $\PP^1$. ✅ 2023-01-29 - [ ] $x^2+y^2+z^2 = 0$ over $\RR$, a smooth conic with no rational points. By projection, a rational point induces an isomorphism to $\PP^1$, and this acquires points after base-changing to $\CC$. - [ ] What is the generator of $\Br(\RR)$? - [ ] Hamilton quaternions: central division algebra, hence Azumaya. - [ ] How are twisted sheaves related to Severi-Brauers and Azumaya algebras? - [ ] $\PP(\mce)$ is SB. - [ ] $\Endo(\mce)$ is Azumaya. - [x] What is $\Br(\QQpadic)$? ✅ 2023-01-29 - [ ] $\Br(\QQpadic) = \QQ/\ZZ$. - [x] What is the SES involving $\Br(\QQ)$? ✅ 2023-01-29 - [ ] $$\Br(\QQ) \injects \bigoplus_{p\in \Places{\QQ}} \Br(\QQpadic) \surjects \QQ/\ZZ$$ - [ ] What is $\Br(F)$ for $F$ a number field? - [ ] What is the period-index question? - [ ] Given $\alpha\in \Br(X)$, what is the minimal rank of an $\alpha\dash$twisted vector bundle? - [ ] Over a field, for 2-torsion one can always find $\rank \mce = 2$. ## Video 10 ## Video 11 ## Video 12