---
sort: 050
title: Cohomology of Schemes
flashcard: "Orals::Cohomology of Schemes"
---

# Cohomology of schemes

- [ ] Compute $H^*(\PP^n; \OO_{\PP^n})$. 
- [x] What is a **limit**? A **colimit**? ✅ 2022-12-07
	- [ ] How to remember: limits are like sequences and live above (sequences project), colimits are like gluing and live below (include into disjoint unions)
	      ![<https://q.uiver.app/?q=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>https://q.uiver.app/?q=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](attachments/Pasted%20image%2020221207215352.png)
- [x] What is an **abelian category**? #syllabus ✅ 2022-12-07
	- [ ] $\cat C(A, B) \in \Ab\Grp$ and composition is a group morphism,
	- [ ] All Finite direct sums
	- [ ] All kernels and cokernels
	- [ ] Mono = $\ker(\coker)$.
	- [ ] Epi $= \coker(\ker)$.
	- [ ] Epi-mono factorization for all morphisms
- [x] What is a **delta functor**? #syllabus ✅ 2022-12-07
	- [ ] A collection $T^n$ ($T^{> 0}$ are *satellite functors*) and for each $A\injects B\surjects C$ a collection of morphisms $\delta^n: T^n(C) \to T^n(A)$ fitting into a LES, and for each morphism of SESs $\xi_1\to \xi_1$ the $\delta^n$ fit into commuting squares.
	      ![](attachments/Pasted%20image%2020221207141027.png)
- [x] What is a **universal delta functor**? ✅ 2022-12-07
	- [ ] $\delta = (T^n)$ is universal iff for any other delta functor $(S^n)$ and any morphism $f^0: T^0\to S^0$ of functors there exist lifts $f^i: T^i \to S^i$ which commute with the $\delta^i$.
- [x] What is a **homotopy** between maps $f,g: A\to B$ of complexes? ✅ 2022-12-07
	- [ ] A morphism $k: A \to B[1]$, so $k^i: A^i\to B^{i-1}$, with $f-g = dk + kd$.
- [x] What is an **additive** functor? ✅ 2022-12-07
	- [ ] $F$ is additive iff $\cat C(A, B)\to \cat D(FA, FB)$ is a group morphism.
- [x] What is a **left exact functor**? ✅ 2022-12-07
	- [ ] Additive and $A\injects B\surjects C \leadsto FA \injects FB \to FC \to \cdots$.
- [x] What are the variance and exactness of hom functors? ✅ 2022-12-07
	- [ ] Both are left-exact (just remember $\RHom$), $\cat C(A, \wait)$ is covariant and $\cat C(\wait, A)$ is contravariant.
- [x] What is an **injective resolution** of $A$? ✅ 2022-12-07
	- [ ] $A \mapsvia{\eps} I^0 \to I^1 \to \cdots$ an exact complex with $I^j$ injective objects
- [x] What is an **injective object**? ✅ 2022-12-07
	- [ ] Remembering injectives and projectives: 
	      ![<https://q.uiver.app/?q=WzAsNSxbMCwyLCJBIl0sWzIsMiwiQiJdLFs0LDIsIkMiXSxbMCw0LCJJIl0sWzQsMCwiUCJdLFswLDEsIiIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzEsMiwiIiwwLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzQsMl0sWzAsM10sWzQsMSwiXFxleGlzdHMgXFxpZmYgQyBcXHRleHR7IHByb2plY3RpdmV9IiwyLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzEsMywiXFxleGlzdHMgXFxpZmYgQSBcXHRleHR7IGluamVjdGl2ZX0iLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0=>](attachments/Pasted%20image%2020221207140632.png)
- [x] What is a **(co)effaceable functor**? ✅ 2022-12-07
	- [ ] For each $A$ there is a mono $u: A\to I$ with $F(u) = 0$.
	      Coeffaceable if there is an epi $u: P\to A$ with $F(u) = 0$.
- [x] Discuss **effaceable functors**. ✅ 2022-12-07
	- [ ] If $T$ is a delta functor and each $T^i$ is effaceable then $T$ is universal.
	      Derived functors computed via injective resolutions form a universal $\delta$ functor.
- [x] Why does $\mods{\OO_X}$ have enough injectives? #syllabus ✅ 2022-12-07
	- [ ] For $\mcf\in \mods{\OO_X}$, the stalk $\mcf_{x} \in \mods{\OO_{X, x}}$ and module categories have enough injectives.
	      So find $\mcf_x \injects I_x$, let $j^x: \ts{x} \injects X$ and set $\mci \da \prod_{x\in x} j^x_*(I_x)$; claim this is injective.
	      Check $$\Hom_{\OO_X}(\mcg, \mci) = \prod_{x\in X} \Hom_{\OO_X}(\mcg, j^x_* I_x) = \prod_{x\in X} \Hom_{\OO_{X, x}}(\mcg_x, I_x)$$
	      So one gets a sheaf morphism $\mcf \to \mci$ induced by $\mcf_x \to \mci_x$ which is injective.
	      Claim: $\Hom_{\OO_X}(\wait, \mci)$ is exact. Use that it is $\prod_{x\in X}(\wait)_x$, the product of stalk functors (which is exact) followed by $\Hom_{\OO_{X, x}}(\wait, I_x)$ (which is exact).
	      I.e. it's a composition of exact functors:
	      $$\Hom_{\OO_X}(\wait, \mci) = \Hom_{\OO_{X, x}}(\wait, I_x) \circ \prod_{x\in X} (\wait)_x$$
- [x] What is the **higher direct image**? #syllabus ✅ 2022-12-07
	- [ ] Idea: for a family $f:X\to Y$, a form of relative cohomology of $X$ over $Y$ or cohomology along the fibers of $f$.
	      $f_*: \Sh(X; \Ab\Grp)\to\Sh(Y; \Ab\Grp)$ left exact, so take right-derived functors.
	      Can be computed using flasque resolutions.   
- [x] Discuss **higher direct images**. ✅ 2022-12-07 #syllabus 
	- [ ] If $\RR^i f_* \mcf = 0$ for $i>0$ then $H^i(X; \mcf)\iso H^i(Y, f_* \mcf)$.
	      $\RR^if_*(\mcf)$ is the sheafification of $V\mapsto H^i(f\inv(V); \ro \mcf {f\inv(V)})$.
- [x] Discuss higher direct images for morphisms $f:X\to \spec A$. ✅ 2022-12-07
	- [ ] Let $M_i \da H^i(X; \mcf)$, then $\RR^i f_* \mcf = \tilde  M_i$.
- [x] What is a **flasque** or **flabby** sheaf? #syllabus ✅ 2022-12-07
	- [ ] $V\subseteq U \implies F(U) \surjects F(V)$.
	      Any flasque is injective and thus $\Gamma\dash$acyclic.
- [x] Give examples of flasque sheaves. ✅ 2022-12-07
	- [ ] $\tilde M \in \QCoh(\spec A)$ for $M\in \amod$ an injective object.
- [x] What is a **fine** sheaf? #syllabus ✅ 2022-12-07
	- [ ] Sheaves admitting a partition of unity: for any $\mcu \covers X$ there exist $\phi_i: \mcf(U_i) \to \mcf(U)$ where for all $s\in\mcf(U)$ one has $\supp(s(U_i)) \subseteq U_i$ and $\sum_i \phi_i(\ro s {U_i}) = s$.
	- [ ] Equivalently: for every two disjoint $A, B\subseteq X$, there exists $f\in \Endo(\mcf)$ such that $\ro{f}{A} = \id$ and $\ro{f}{B} = 0$. 
	- [ ] More generally, for all $\mcu \covers X, \exists 1=\sum f_i$ subordinate to the cover.
	- [ ] Fine implies soft.
- [x] What is the **tangent sheaf**? #syllabus ✅ 2022-12-07
	- [ ] $\T_X \da \Omega_{X/k}\dual \da \shom_{\OO_X}(\Omega_{X/k}, \OO_X)$, locally free of rank $\dim X$.
- [x] What is the **geometric genus** in full generality? ✅ 2022-12-07
	- [ ] $p_g(X) \da h^{n, 0}(X) \da H^0(\omega_X)$, the number of linearly-independent top forms.
- [x] What are the **conormal** sheaf and the **normal sheaf**? #syllabus ✅ 2022-12-07
	- [ ] For $Y\embeds X$ a closed embedding of smooth varieties, $N_{Y/X}\dual \da \mci_Y/\mci_Y^2$ is the conormal sheave and $N_{Y/X} = (\mci_Y/\mci_Y^2)\dual$ is the normal sheaf, which is locally free of rank $\codim_X(Y)$.
- [x] What is a **derivation**? #syllabus ✅ 2022-11-28
	- [ ] $d\in \Der_A(B, M) \iff d$ is additive and $d(b_1b_2) = d(b_1)b_2 + b_1d(b_2)$ where $d(a) = 0$ for all $a\in A$.
- [x] What is the module of relative differentials $\Omega_{B/A}$? #syllabus ✅ 2022-11-28
	- [ ] $\Omega_{B/A}\in \mods{B}$ with $d\in \Der_A(B, \Omega_{B/A})$ satisfying a universal property: for any $M\in \mods{B}$ and $d'\in \Der_A(B, M)$, it factors through $d$.
	      Can be construct as $\Free_B(db \st b\in B)/R$ where $R$ are relations $d(b_1+b_2)-d(b_1)-d(b_2), da, d(b_1b_2)-d(b_1)b_2-b_1d(b_2)$.
- [x] What is the **sheaf of differentials** or **relative differential forms**? #syllabus ✅ 2022-12-07
	- [ ] For $f:X\to Y$, embed $X$ as a locally closed subscheme $X\injects W \da \Delta_{X/Y}$ with sheaf of ideals $\mci$ and define $\Omega_{X/Y} \da \Delta^*(\mci/\mci^2) \in \QCoh(X)$.
	      The local derivations glue to $d:\OO_X\to \Omega_{X/Y}$.
- [x] What are the two SESs for sheaves of differentials? ✅ 2022-11-28
	- [ ] For $X \mapsvia{f} Y \mapsvia{g} Z$, $$f^* \Omega_{Y / Z} \rightarrow \Omega_{X / Z} \rightarrow \Omega_{X / Y} \rightarrow 0$$.
	      For $f:X\to Y$ and $Z\leq X$ a closed subscheme with ideal sheaf $\mci$,
	      $$\mathscr{I} / \mathscr{I}^2 \stackrel{\delta}{\rightarrow} \Omega_{X / Y} \otimes \mathcal{O}_Z \rightarrow \Omega_{Z / Y} \rightarrow 0$$
- [x] What is the **geometric genus** and the **arithmetic genus** for curves and for general schemes? #syllabus ✅ 2022-11-28
	- [ ] For a smooth projective curve, $p_a(X) = h^1(\OO_X)$ and $p_g(X) =h^0(\omega_X)$.
	      More generally, $p_a(X) \da (-1)^{\dim X} (\chi(\OO_X)-1)$ and $p_g(X) = h^0(\Omega_{X/k})$
- [x] What is **Cech cohomology**? #syllabus ✅ 2022-11-28
	- [ ] For an open cover $\mcu\covers X$, define 
	      $$C^p(\mcu; \mcf) \da \prod_{I \da i_0 < \cdots < i_p} \mcf(U_{I}),\qquad U_{I} \da U_{i_1}\intersect \cdots \intersect U_{i_p}$$
	      with boundary map
      $$(d\alpha)_{I} \da\sum_{0\leq k\leq p+1} (-1)^k \alpha_{I\sm i_k}\mid_{U_I}$$
      More explicitly,
      $$C^p(\mcu; \mcf) = \prod_{i_1 < \cdots < i_p} \mcf(U_{i_1} \intersect \cdots \intersect U_{i_p})$$
      So a $p\dash$chain is a collection of elements $\alpha_{i_1,\cdots, i_p} \in \mcf(U_{i_1\cdots i_p})$ for each $(p+1)\dash$tuple of elements $i_1 < \cdots < i_p$ in $I$ where $\mcu \da \ts{U_i}_{i\in I}$. The boundary map is 
      $$(d \alpha)_{i_0, \ldots, i_{p+1}}=\sum_{k=0}^{p+1}(-1)^k \alpha_{i_0}, \ldots, \hat{i}_k, \ldots, i_{p+1} \mid_{U_{i_0, \ldots, i_{p+1}}}$$
- [x] Give an example of a computation with Cech cohomology. #syllabus ✅ 2022-12-07
	- [ ] See example 4.0.3 on page 219 and example 4.0.4 for $S^1$.
	- [ ] $H^*(\PP^1; \Omega_{\PP^1/k})$ over $k$: let $\mcu = \ts{H_0, H_\infty}$ with coordinates $x$ and $1/x$, then: $$\begin{align} 0 \to C^0 &\to C^1\to 0 \\ 0 \to \globsec{H_0; \Omega} \times \globsec{H_\infty;\Omega} &\to \globsec{H_0\intersect H_\infty;\Omega} \\ 0 \to k[x]\dx \times k[y]\dy &\mapsvia{d: x\mapsto x, y\mapsto 1/x, dy\mapsto -x^{-2}\dx} k[x, x\inv]\dx \to 0\end{align}$$
      - [ ] Then $\ker d = \gens{f(x)\dx, g(y)\dy}$ such that $f(x) = -x^{-2} g(x\inv) \implies f=g=0$ since one side is a polynomial in $x$ and the other is a polynomial in $1/x$ with no constant term, so $H^0 = 0$.
      - [ ] For $H^1$, check that $\im d = \gens{f(x) + x^{-2} g(x\inv)\dx}$ where $f,g$ are polynomials. This is the subspace of $k[x,x\inv]\dx$ generated by $x^n\dx$ for $n\in \ZZ\smts{-1}$, so $H^1 = \gens{x^{-1}\dx}$ is 1-dimensional.
      - [ ] The circle: ![](attachments/Pasted%20image%2020221207150823.png)
- [x] When does Cech cohomology compute sheaf cohomology? #syllabus ✅ 2022-11-28
	- [ ] For $X$ a Noetherian separated scheme and $\mcf\in \QCoh(X)$. (Thm 4.5)
	      Alternatively, whenever $\mcf$ has no cohomology on any open in a cover $\mcu$.
- [x] What is a **regular immersion**? #syllabus ✅ 2022-11-28
	- [ ] An immmersion whose sheaf of ideals is regular, where $\mci \leq \OO_X$ is regular iff for every $x\in \supp(\OO_X/\mci)$ admits an open neighborhood $U$ and a regular sequence $f_1,\cdots, f_r\in \OO_X(U)$ generating $\ro\mci U$.
- [x] What are consequences of a morphism being flat? #syllabus ✅ 2022-11-28
	- [ ] The Hilbert polynomials of all fibers are the same.
	- [ ] If $f:X\to Y$ with $X,Y$ finite type over a field, $\dim_x X_y = \dim_x X - \dim_y Y$ where $\dim_x X \da \dim \OO_{X, x}$.
	- [ ] For $\mcx\to C^\circ$ a flat family of closed subschemes of $\PP^n$ over a punctured curve $C^\circ$, one can "pass to the limit": given $Y$ regular integral of dimension 1 and $p\in Y$ a closed point, let $Y^\circ \da Y\smts{p}$ and $X\subseteq \PP^n_{Y^\circ}$ be a closed subscheme which is flat over $Y^\circ$; then $\exists ! \tilde X \subseteq \PP^n_{Y}$ a closed subscheme flat over $Y$ which restricts to $X$.
- [x] Give an example and a non-example of a flat morphism. #syllabus ✅ 2022-11-28
	- [ ] Example: any open immersion, $\tilde M$ over $A$ for any $M \in \amod^\flat$. Any smooth morphism by miracle flatness
	      Non-example: any blowup, since the fibers jump in dimension.
- [x] What is an **associated point**? ✅ 2022-11-28
	- [ ] $x\in X$ where every element of $\mfm_x$ is a zero divisor.
- [x] What is a **flat family**? #syllabus ✅ 2022-11-28
	- [ ] Any morphism $\mcx\to X$ of schemes to $X$ which is flat.
- [x] How can one check is a morphism is flat? ✅ 2022-11-28
	- [ ] For $f: X\to Y$ with $Y$ integral and regular of dimension 1, $f$ is flat iff every *associated point* maps to $\eta_Y$.
	      For $X$ reduced, this says every irreducible component of $X$ dominates $Y$.
- [x] What are some properties not preserved in a flat family? ✅ 2022-11-28
	- [ ] Being irreducible or reduced, Picard numbers, generally most things that aren't related to Hilbert polynomials.
- [x] What is **Serre's vanishing** characterization of affine schemes among Noetherian schemes? #syllabus ✅ 2022-12-07
	- [ ] For $X\in \Sch$ Noetherian, $X$ is affine $\iff H^{i\geq 1}(X; \mcf) = 0$ for all $\mcg \in \QCoh(X)$ $\iff H^1(X; \mci) = 0$ for all coherent sheaves of ideals $\mci$.
	- [ ] Equivalently, $X$ is affine $\iff \mathrm{cd}(X) = 0$; this is the *cohomological dimension*, the least $n$ such that $H^i(\mcf) = 0$ for all $i> n$ and for all $\mcf \in \QCoh(X)$.
	- [ ] Another **Serre vanishing**: for $X \in \Sch\slice A$ projective and $A$ Noetherian with $\OO_X(1)$ very ample over $\spec A$, if $\mcf \in \Coh(X)$ then $H^i(\mcf(n)) = 0$ for $n\gg 0$ and every $i>0$.
- [x] Discuss cohomology for $\QCoh(\spec A)$. #syllabus ✅ 2022-12-07
	- [ ] $H^{i\geq 1}(\spec A, \mcf) = 0$ for any $\mcf \in \QCoh(\spec A)$.
	      Why: let $M = \globsec{X; \mcf}$ and take $M\injects I$ an injective resolution in $\amod$ to get an exact sequence $\tilde M \injects \tilde I \in \QCoh(X)$.
	      Check $\mcf \cong \tilde M$ by an earily result and each $\tilde I^i$ is flasque thus $\Gamma\dash$acyclic, so use it to compute cohomology.
	      Apply $\globsec{X; \wait}$ to $0\to \tilde M\to \tilde I$ to get $0\to M\to I\in \amod$ and thus $H^0(\mcf) = M$ and $H^i(\mcf) = 0$ for $i\geq 1$.
- [x] What is $H^i(X; \OO_X(n))$ for $X \da \PP^r\slice A$? ✅ 2022-12-07
	- [ ] Write $\Gamma^*(\mcf ) \da \bigoplus_{n\geq 0} H^0(X; \OO_X(n))$ for the section ring, then $A[x_0,\cdots, x_r] \iso \Gamma(\OO_X)$ as graded rings. So $H^0(X; \OO_X(n)) \cong A[x_0, \cdots, x_r]_n$, degree $n$ homogeneous polynomials over $A$.
	- [ ] $H^i(X; \OO_X(n)) = 0$ for $0<i<r$ and every $n$.
	- [ ] $H^r(X; \OO_X(-(r+1))) = A$.
- [x] What is the **Hilbert polynomial** of a sheaf? ✅ 2022-12-07
	- [ ] A polynomial $p\in \QQ[z]$ such that $p(n) = \chi(\mcf(n))$ for every $n$.
- [x] When is an invertible sheaf ample? ✅ 2022-12-07
	- [ ] For $A$ Noetherian and $X\in \Sch\slice A$ proper, $\mcl$ is ample $\iff \forall \mcf\in \Coh(X)$ one has $H^i(\mcl\tensor \mcf^n) = 0$ for $n\gg 0$ and every $i$.
- [x] What is **Serre duality**? ✅ 2022-12-07
	- [ ] For $X$ smooth proper of dimension $n$ and $\mcl\in \Coh(X)$ there is a perfect pairing
	      $$ H^i(X;\mcl) \tensor H^{n-i}(X; \mck\tensor \mcl\dual) \to H^n(X;\mck) \cong \CC \implies H^i(X;\mcl) \cong H^{n-i}(X; \mck\tensor \mcl\dual)\dual$$
	- [ ] For curves
	      $$H^0(C; \mcl) \cong H^1(C; \mck \tensor \mcl\dual)\dual$$
	- [ ] Consequence for Hodge numbers:
	      $$h^i(X; \mcl) = h^{n-i}(X; \mck \tensor \mcl\dual),\qquad \chi(X; \mcl) = (-1)^n \chi(X; \mck\tensor \mcl\dual)$$
- [x] How is $\Pic(X)$ related to cohomology? ✅ 2022-12-07
	- [ ] $\Pic(X) \cong H^1(X; \OO_X\units)$. Idea: use Cech cohomology, pick $\mcl\in \Pic(X)$, restrict to opens where $\ro \mcl {U_i}$ is free, fix isos $\phi_i: \OO_{U_i}\iso \ro\mcl{U_i}$, check that on $U_{ij}$ one gets isomorphisms $\phi_i\inv \phi_j: \OO_{U_{ij}}\selfmap$ which give an element in $\Hc(\mcu; \OO_X\units)$, take a limit over all coverings.

Exact sequences

- [x] What is the **exponential exact sequence**? ✅ 2022-12-07
	- [ ] $$2\pi i \ZZ \injects \OO_X \surjects \OO_X\units$$
	      Since $H^1(\OO_X) = \Pic(X)$, $\delta^1 = c_1(\wait) \in H^2(2\pi i \ZZ)$ is the first Chern class.
- [x] What is the **conormal exact sequence**? ✅ 2022-12-07
	- [ ] How to derive: $$\T_X \injects \T_Y \surjects N_{X/Y} \quad\underset{\text{dualize}}{\leadsto}\quad N_{X/Y}\dual \to j^*\Omega_{Y/Z} \surjects\Omega_{X/Z}.$$![](attachments/Pasted%20image%2020221207204617.png)
- [x] What is the **Euler exact sequence**? ✅ 2022-12-07
	- [ ] For $X \da \PP^n \slice A$, dualize $\OO_X\injects \OO_X(1)\sumpower{n+1}\surjects \T_X$ to get the sequence $$\Omega^1_{X}\injects\OO_X(-1)\sumpower{n+1}\surjects\OO_X$$
- [x] What is $\omega_X$ for $X = \PP^n\slice A$? Derive it. #syllabus
	- [ ] Take determinants in the Euler exact sequence $\Omega^1_{X}\injects\OO_X(-1)\sumpower{n+1}\surjects\OO_X$ to get $\det \OO_X(-1)\sumpower{n+1} = \omega_X \tensor \OO_X = \omega_X$ and so $\omega_X = \OO_X(-(n+1))$.
- [x] What are the two main exact sequences for modules of algebraic differentials? ✅ 2022-12-07
	- [ ] The composition exact sequence: for $A\to B\to C$, 
	      $$\Omega_{B / A} \otimes_B C \rightarrow \Omega_{C / A} \rightarrow \Omega_{C / B} \rightarrow 0 \in \mods{C}$$
	- [ ] The "conormal" exact sequence: for $B\in \Alg_{A}, I\normal B, C\da B/I$, $$I / I^2 \stackrel{\delta}{\rightarrow} \Omega_{B / A} \otimes_B C \rightarrow \Omega_{C / A} \rightarrow 0\in \mods{C}$$	
- [x] What is the composition exact sequence for $X\to Y\to Z$? ✅ 2022-12-07
	- [ ] At the level of sheaves, $$f^* \Omega_{Y/Z} \to \Omega_{X/Z} \to \Omega_{X/Y} \to 0 \qquad \in \Sh_X$$How to remember:
      ![](attachments/Pasted%20image%2020221207200258.png)
- [x] What is the conormal exact sequence for an immersion $C\injects S$? ✅ 2022-12-07
	- [ ] At the level of sheaves on $C$, for $\iota: C\injects S$,
	      $$N_{C/S}\dual \to \iota^*\Omega_{S/k} \surjects \Omega_{C/k}\to 0 \qquad \in \Sh(C)$$
	      where $N_{C/S}\dual \da I_C/I_C^2$ which is a sheaf version of: if $C,S\in \rmod$ with $\iota: C\surjects S$ a surjection with kernel $I$
	      $$I/I^2 \to \Omega_{C/R}\tensor_R S \surjects \Omega_{S/R}\to 0 \qquad\in\mods{S}$$
	      For $C \subseteq S$ a closed subscheme and $f:S\to T$, can rewrite this as
	      $$ \mci_C / \mci_C^2 \to \Omega_{S/T} \tensor \OO_C \surjects \Omega_{C/T}\to 0 \qquad \in \Sh_C$$
- [x] Compute the canonical sheaf for $Y$ a smooth hypersurface of degree $d$ in $\PP^n$ for $n\geq 2$. ✅ 2022-12-07
	- [ ] Use that for $Y\injects X$ one has $\omega_Y = \omega_X \tensor\det N_{Y/X} = \omega_X \tensor \OO_X(Y) \tensor \OO_Y$ for codimension 1 and $X = \PP^n \implies \omega_X = \OO_X(-n-1)$ and so 
	      $$\omega_Y = \OO_X(-n-1) \tensor \OO_{\PP^n}(Y) \tensor \OO_Y = \OO_Y(-n-1+d)$$
- [x] Compute the canonical sheaf of a smooth plane curve of degree $d$ and its degree. ✅ 2022-12-07
	- [ ] For $X\subseteq \PP^n$ a hypersurface one has $\omega_X = \OO_X(d-n-1)$, here $n=2$ so $\omega_X = \OO_X(d-3)$.
	      Since $\deg \OO_X(1) = d$, we have $\deg \omega_X = d(d-3)$.
- [x] Show that a smooth plane curve of degree $d\geq 4$ is not rational. ✅ 2022-12-07
	- [ ] $\omega_X = \OO_X(d-3)$ and $d-3>0$ so $p_g(X) > 0$, and a curve is rational iff $p_g(X) = 0$.