--- sort: 031 title: O_X Modules flashcard: "Orals::O_X Modules" --- # O_X Modules - [x] What is an $O_X\dash$modules? : #syllabus ✅ 2022-12-05 - [ ] $\mcf \in \Sh(X)$ where $\mcf(U)\in \mods{\OO_X(U)}$ with compatible restrictions $(rm)\mid_V = \ro r V \ro m V$. ![](attachments/Pasted%20image%2020221128124229.png) - [x] What is the **tensor product** of two $\OO_X\dash$modules? ✅ 2022-11-25 - [ ] Sheafify $U\mapsto \mcf(U) \tensor_{\OO_X(U)} \mcg(U)$. - [x] What is a **free** $\OO_X\dash$module? ✅ 2022-11-25 - [ ] Isomorphic to $\OO_X\sumpower{n}$ for some $n$. - [x] What is a **locally free** ${\mathcal{O}}_X$ module? ✅ 2022-12-08 - [ ] Admits an open cover where $\ro{\mcf}{U_i}$ is free. - [x] What is the **rank** of an $\OO_X\dash$module? ✅ 2022-11-25 - [ ] If locally free, on some open set, the $n$ appearing in $\ro\mcf U_i \cong \OO_X\sumpower{n}$. - [x] What is a **sheaf of ideals** on $X$? #syllabus ✅ 2022-12-05 - [ ] An $\OO_X\dash$module $\mci$ where $\mci \leq \OO_X$ is a subsheaf, so each $\mci(U) \normal \OO(U)$. - [x] What is the **sheaf hom**? #syllabus ✅ 2022-11-25 - [ ] For $\mcf, \mcg\in \Sh(X, \cat C)$, the sheaf $U\mapsto \Hom_{\ro{\OO_X} U}(\ro\mcf U, \ro\mcg U)$ (no need to sheafify). - [x] What is the **direct image** in $\mods{\OO_X}$? #syllabus ✅ 2022-12-05 - [ ] For $f: X\to Y$ in ringed spaces and $\mcf\in \mods{\OO_X}$, take $f_* \mcf \in \mods{\OO_Y}$ - [x] What is the **inverse image** in $\mods{\OO_X}$? #syllabus ✅ 2022-12-08 - [ ] For $f:X\to Y$ and $\mcg\in \mods{\OO_Y}$, $f^* \mcg \da f\inv \mcg \tensor_{f\inv \OO_Y}\OO_X$. I.e. base change $f\inv G$ from $f\inv \OO_Y$ to $\OO_X$. - [x] What is the $\OO_X\dash$module associated to $M\in\amod$? #syllabus ✅ 2022-12-05 - [ ] $\tilde M \in \Sh(\spec A)$ where $\tilde M(U)$ is the set of functions $s: U\to \Disjoint_{p\in U} M_p$ such that $s(p)\in M_p$ and $s$ is locally a fraction $m/f$ with $m\in M, f\in A$. The stalks are the localizations, $\tilde M_p = M_p$, and the global sections are $M$. - [ ] More precisely: for each $p\in U$ there exists $p\in V \subset U$ and $m\in M, f\in A$ such that for each $q\in V$, $f\not \in q$ and $s(q) = m/f$ in $M_q$. - [x] What is a **quasicoherent sheaf**/$\OO_X\dash$module? : #syllabus ✅ 2022-12-05 - [ ] An $\OO_X\dash$module which is locally of the form $\tilde M$, where $\tilde M$ is the sheaf associated to $M$. - [ ] Admits an affine open cover $\spec A_i\covers X$ where $\exists M_i\in\mods{A_i}$ with $\ro{\mcf} {\tilde M_i}$. - [ ] Equivalently, by a theorem of Serre, **locally** $\exists \OO_X\sumpower{J} \to \OO_X\sumpower{I}\surjects \mcf$. - [x] What is a **coherent sheaf**/ $\OO_X\dash$module? : #syllabus ✅ 2022-12-05 - [ ] Quasicoherent where $\spec A_i\covers X$ and each $M_i \in \mods{A_i}^\fg$. - [ ] Equivalently, by a theorem of Serre, $\exists \OO_X\sumpower{J} \to \OO_X\sumpower{I}\surjects \mcf$ with $\size I, \size J < \infty$. - [x] Give an example of a sheaf that is not quasicoherent. ✅ 2022-11-25 - [ ] For $Y\leq X$ a closed subscheme, $\ro{\OO_X}{Y}$. - [x] Give an example of a sheaf that is not coherent. ✅ 2022-12-08 - [ ] Take $f:\AA^1\slice k\to \spec k$ and $f_* \OO_{\AA^1\slice k}$ has global sections $k[t]\not\in \kmod^\fg$. - [x] Give an example of a quasicoherent but not coherent $\OO_X\dash$module. ✅ 2022-11-25 - [ ] The constant sheaf $\ul{\mck}$ where $U\mapsto K(X)$, the rational function field of $X$. - [x] What is a consequence of an $\OO_X\dash$module being quasicoherent? ✅ 2022-11-25 - [ ] For $X$ affine, global sections is exact on sequences $\mcf \injects\bullet\surjects\bullet$ when $\mcf\in \QCoh(X)$. - [ ] Kernels, cokernels, images, extensions, pullbacks, and pushforwards along qcs are again quasicoherent. - [x] What is the **ideal sheaf**? #syllabus ✅ 2022-12-08 - [ ] For $Y\leq X$ a closed subscheme with $\iota: Y\injects X$, defined by $\mci_Y \da \ker\qty{\OO_X \mapsvia{i^\sharp}i_* \OO_Y}$. Always a qcoh sheaf of ideals on $X$, and all such ideals arise this way. - [x] What is $\OO_X(n)$? ✅ 2022-12-05 - [ ] For $S$ a graded ring and $X \da \Proj S$, $\OO_X(n)\da S(n)\tilde{}$ (the sheaf associated to a module). - [ ] $\OO_X(1)$ is Serre's twisting sheaf, and if $\mcf\in \mods{\OO_X}$ then $\mcf(n) \da \mcf \tensor_{\OO_X} \OO_X(n)$. - [x] For $S$ a graded ring, $X = \Proj S$, and $\mcf\in \mods{\OO_X}$, what is the graded $S\dash$module associated to $\mcf$? ✅ 2022-11-25 - [ ] $\globsec{X; \mcf} \da \bigoplus_{n\in \ZZ} \globsec{X; \mcf(n)}$. - [x] What is the **twisting sheaf** $\OO(1)$ on $\PP^n\slice Y$? ✅ 2022-11-25 - [ ] Take $g^*(\OO(1))$ where $g:\PP^n\slice Y \to \PP^n\slice\ZZ$. - [x] What is a **very ample** invertible sheaf? ✅ 2022-11-25 - [ ] For $X$ over $Y$, $\mcl\in \Pic(X)$ is very ample iff $\exists \iota: X\to \PP^n\slice Y$ such that $\iota^* \OO(1) \cong \mcl$. - [x] Why is a scheme $X$ over $Y$ projective iff proper? ✅ 2022-11-25 - [ ] Projective implies proper: take a closed immersion $\iota: X\to \PP^n\slice Y$ to get $\iota^* \OO(1)\in \Pic(X)^\mathrm{va}$. - [ ] Conversely if proper, take $\mcl\in \Pic(X)^{\mathrm{va}}$ to get $\mcl \cong \iota^* \OO(1)$ for some immersion $\iota: X\to \PP^n\slice Y$ for some $n$ -- but $\iota(X)$ is closed, so $\iota$ is a *closed* immersion, making $X$ projective. - [x] What does it mean for an ${\mathcal{O}}_X$ module to be **globally generated** (or **generated by global sections**)? ✅ 2022-11-25 - [ ] There exists a family of sections $\ts{s_i}_{i\in I} \subseteq \globsec{X; \mcf}$ where for each $x\in X$ the images of $s_i$ in $\mcf_x$ generate that stalk as an $\OO_{X, x}\dash$module. - [ ] Equivalently, $\mcf$ is the quotient of a free sheaf: $\exists \bigoplus_{i\in I}\OO_X\surjects \mcf$. - [x] Give examples of globally generated sheaves. ✅ 2022-11-25 - [ ] Example: any $\mcf \in \QCoh(\spec A)$ -- write $\mcf \cong M\tilde{}$ and take any set of generators for $M\in\amod$. - [x] What is a major result regarding global generation? ✅ 2022-12-08 - [ ] For $X$ projective over a Noetherian scheme $A$, $\OO(1)$ a very ample invertible sheaf on $X$, $\mcf\in \Coh(X)$, there exists some $N_0$ such that $\mcf(n)$ is globally generated by a finite number of sections for all $n\geq N_0$. - [x] What is the **Euler characteristic** of an ${\mathcal{O}}_X$ module? ✅ 2022-11-25 - [ ] $\chi(X; \mcf) \da\sum_i (-1)^i h^i(X; \mcf)$. - [x] What is the **Hilbert polynomial** of an ${\mathcal{O}}_X$ module? ✅ 2022-11-25 - [ ] Using that $\chi(X; \mcf(n))$ is some polynomial $p(n)$ in $n$, take $p$ to be the Hilbert polynomial.