--- sort: 032 title: Morphisms flashcard: "Orals::Morphisms" --- # Morphisms - [x] How does a morphism of rings induce an algebra structure? ✅ 2022-12-01 - [ ] $f:R\to S$ induces $S\in \rmod$ via $r.s \da f(r)s$. - [x] What is an **open immersion** of schemes? #syllabus ⏫ ✅ 2022-12-05 - [ ] $f:X\to Y$ inducing $X\cong U \subseteq Y$ with $U$ an open subscheme, where $U\subseteq Y$ is an open scheme iff $\abs{U} \subseteq \abs{Y}$ and $\OO_U \cong \ro{\OO_Y}{U}$. - [x] What is a **closed immersion** of schemes? #syllabus ⏫ ✅ 2022-12-05 - [ ] $f:X\to Y$ inducing a homeomorphism $\tilde f: \abs{X} \to \abs{V}$ for $\abs{V} \subseteq \abs{Y}$ a closed subspace of the underlying topological spaces, such that $f^\sharp: \OO_Y \surjects f_* \OO_X$ is a surjective map of sheaves. - [ ] Equivalently, separates points and tangent vectors, i.e. $p\neq q$ closed points in $X \implies \exists s\in \mfm_p \mcl_p \sm\mfm_q \mcl_q$, and $\ts{s\in V \st s_p\in \mfm_p \mcl_p}$ spans $\mfm_p \mcl_p/\mfm_p^2 \mcl_p$, where $\phi: X\to \PP^n\slice k$ corresponds to $\mcl$ and $\ts{s_0, \cdots, s_n}\in H^0(X; \mcl)$. - [x] What is a **closed subscheme**? ✅ 2022-12-05 - [ ] An equivalence class of closed immersions, where $f\sim g$ if they fit into a commuting triangle. - [x] What is the fiber of a morphism $f:X\to Y$? ✅ 2022-11-24 - [ ] $f_y \da X\fiberprod{Y} \spec \kappa(y)$ - [x] What is a **smooth morphism**? ⏫ ✅ 2022-12-08 - [ ] Locally of finite presentation, flat, and regular geometric fibers. (LFP: open cover where $R\to A$ is fp, so $\exists\, R[x_1, \cdots, x_n] \surjects A$). - [ ] Smooth implies regular. - [ ] For ring maps: $f:R\to A$ of finite presentation with $\tau_{\leq 1}\LL_{A/R} \cong M[0]$ with $M\in \mods{A}$ some finite projective (i.e. the naive cotangent complex is quasi-isomorphic to a projective module in degree zero). For schemes: smooth at $x \iff \exists U = \spec A \ni x$ and $V = \spec R \contains f(U)$ such that the induced ring map $R\to A$ is a smooth ring map. - [ ] Sufficient condition: flat, locally of finite presentation, smooth fibers (i.e. for all $a\in A$, the fiber $R\fiberprod{A}\kappa(a)$ is a smooth scheme over $\kappa(a)$). - [ ] Equivalently, $\Omega_{R/A}$ is locally free of finite rank equal to $\reldim R/A$ (roughly: $\dim f\inv(a)$). - [x] What is a smooth morphism of relative dimension $n$? #syllabus ✅ 2022-12-07 - [ ] $f: X\to Y$ flat with $\dim C_X = \dim C_Y + n$ for any irreducible components $C_Y \subseteq f(C_X)$, and $\dim_{\kappa(x)}(\Omega_{X/k}\tensor \kappa(x)) = n$. - [ ] Condition 2 means every irreducible component of every fiber $X_y$ of $f$ has dimension $n$. - [ ] For $X$ integral, flat + irreducible component dimension condition + $\Omega_{X/Y}$ is locally free of rank $n$. - [ ] Closed under base change, composition, fiber products. - [x] How is smoothness related to regularity? ✅ 2022-12-07 - [ ] Over $k = \kbar$, $X$ is smooth of relative dimension $n$ over $\spec k$ $\iff X$ is regular of dimension $n$. - [x] What is generic smoothness? ✅ 2022-12-01 - [ ] For $f:X\to Y\in \Var\slice k, \characteristic k = 0$ with $X$ smooth, then there is a nonempty open $V\subseteq Y$ such that $\ro{f}{f\inv(V)}$ is a smooth morphism. - [x] Give an example a morphism that is not smooth. ✅ 2022-12-08 - [ ] Let $\characteristic k = p, k=\kbar$ and take the Frobenius morphism $f:X\to Y$ to $X=Y=\PP^n\slice k$. - [ ] Since $d(t^p)=0$, the map $f^*\Omega_{Y/k}\to \Omega_{X/k}$ is zero and $\Omega_{X/Y} \cong \Omega_{X/k}$ is locally free of rank 1 but but $\reldim f = 0$, so $f$ is nowhere smooth. - [x] What is a consequence of smoothness? ✅ 2022-11-24 - [ ] Smooth morphisms are flat, and smooth fibers implies this is a flat family of smooth schemes. - [x] What is a **proper morphism**? #syllabus ⏫ ✅ 2022-11-24 - [ ] Separated, finite type, and universally closed. - [ ] Equivalently, use the valuative criterion ($=1$ lift). - [x] How is it related to **completeness**? ✅ 2022-11-24 - [ ] A variety $X$ is complete if $X\to \spec k$ is proper, i.e. a proper variety (integral separated scheme of finite type over a field) is complete. - [x] How is this related to Hausdorff + compactness properties? ✅ 2022-11-24 - [ ] $X/\CC$ is complete iff $X$ is compact in the analytic topology - [x] What is an **etale** morphism $f:X\to Y\in \Sch\slice k$? ✅ 2022-11-24 - [ ] Fully general: flat, finite type, unramified. - [ ] Smooth of relative dimension 0. - [ ] Flat and $\Omega_{X/ Y} = 0$. - [ ] For varieties: flat and unramified. - [x] What is an **unramified** morphism? ✅ 2022-12-08 - [ ] For every $x\in X$, $f^\sharp(\mfm_{f(x)}) \cdot \OO_{X, x} \cong\mfm_x$ where $f^\sharp: \OO_{Y, f(x)}\to \OO_{X, x}$, and $\kappa(x)/\kappa(y)$ is a separable algebraic extension. - [ ] Equivalently, $f:X\to Y$ with $\Omega_{X/Y} = 0$. - [ ] Equivalently, $f$ locally of finite presentation and the diagonal is an open immersion. - [ ] For curves: $\exists e_p\in \ZZ_{\geq 0}$ such that $f^\sharp(\mfm_{f(p)}) \OO_{X, x} = \mfm_{x}^{e_p}$, and unramified iff $e_p = 1$ for all $p\in X$. - [ ] Flat + unramified = etale. - [x] What is a **flat** morphism? ✅ 2022-11-24 - [ ] For projective varieties: the induced maps on stalks are flat ring maps. - [ ] $f: X\to S$ is flat at $x$ iff $\OO_{X, x} \in \mods{\OO_{S, f(x) } }^\flat$ (i.e. the module structure induced by $f$ yields a flat module.) - [ ] Hartshorne's definition: for $f:X\to Y$ and $\mcf\in\mods{\OO_X}$, $\mcf$ is flat over $Y$ at $x\in X$ if $\mcf_x \in \mods{\OO_{Y, f(x)}}^\flat$; flat if flat at every $x\in S$, and $X$ is flat over $Y$ if $\mcf \da \OO_X$ is flat. - [ ] $f\in \CRing(A, B)$ is a flat map iff $B\in \amod^\flat$, and a morphism $f\in \Sch(X, Y)$ is flat iff flat on local rings, i.e. $f_x: \OO_{Y, f(x)}\to \OO_{X, x}$ is flat. - [x] What are some consequences of flatness? ✅ 2022-11-24 - [ ] Numerical invariants remain constant in flat families, e.g. the dimensions of fibers and their hilbert polynomials. - [ ] Defining $\dim_x X \da \dim \OO_{X, x}$, if $f:X\to Y$ is flat then $\dim_x X_{f(x)} = \dim_x X - \dim_{f(x)} Y$. - [ ] If $B$ is smooth 1-dimensional affine, $X\to B$ is a family, and $0\in B$ is a closed point, $\lim_{b\to 0} X_b = X_0$ where $\lim_{b\to 0} X_b \da \bar{X_0}$, the fiber over $0$. - [x] Given an example of a non-flat morphism. ✅ 2022-11-24 - [ ] $\Bl_0 \AA^2 \mapsvia{\pi} \AA^2$, since $\size \pi\inv(p) = 1$ for all $p\neq 0$ and $\pi\inv(0)\cong \PP^1$, contradicting equidimensionality. - [ ] Take a nodal curve $X$ and $f:\tilde X\to X$ its normalization -- if this were flat, $f_* \OO_{\tilde X}$ would be a flat sheaf of $\OO_X\dash$modules, which is coherent and thus locally free, and rank 1 and thus univertible; however the node $q$ has two preimages $p_1, p_2$, so $(f_* \OO_{\tilde X})_q$ needs two generators as an $\OO_X\dash$module. - [ ] Any nontrivial normalization. - [ ] A ramified map of curves: ![](attachments/Pasted%20image%2020221124222411.png) - [x] What are some examples of flat morphisms? ✅ 2022-11-25 - [ ] Smooth morphisms, elliptic fibrations. - [ ] Miracle flatness: any proper morphism between smooth varieties with equidimensional fibers. - [x] What is a **finite ring morphism**? ✅ 2022-11-24 - [ ] A finite ring morphism $f:R\to A$ is one that makes $A\in \rmod^\fg$. - [x] What is a **finite morphism**? #syllabus ✅ 2022-12-05 - [ ] Hartshorne: $f:X\to Y$ where $\spec B_i\covers Y$ with $\spec A_i \da f\inv(B_i)$ with each $A_i\in \algs{B_i}$ such that $A_i \in \mods{B_i}^\fg$. - [ ] Note that being fg as a **module** is **stronger** than being fg as an **algebra**: $k[x] \in \kalg$ has one algebra generator but infinite rank in $\kmod$. - [ ] Thus **finite** implies **finite type** but not conversely. - [ ] Proper and locally quasifinite. - [ ] Equivalently, admits an open cover where it restricts to finite ring morphisms. - [x] What is a consequence of a morphism being finite? ✅ 2022-12-08 - [ ] Closed with finite fibers. - [ ] Finite implies projective, and finite fibers + projective implies proper (not conversely!) - [x] What is a sufficient criterion for a morphism to be finite? ✅ 2022-12-08 - [ ] Proper, locally of finite presentation, finite fibers. - [ ] For locally Noetherian schemes, finite $\iff$ proper + finite fibers. - [x] Give an example of a non-finite morphism. ✅ 2022-11-25 - [ ] The hyperbola projection $V(xy-c)\to \GG_m$, which has finite fibers but is not finite. - [x] What is a **quasifinite ring morphism**? ✅ 2022-11-24 - [ ] **Discrete finite fibers**: for $R\to S$ *finite type* and $\mfq\in \spec S$, the point in the fiber $\bar \mfq\in F \da \spec(S\tensor_R \kappa(\mfp))$ is isolated, i.e. the fiber above that point is zero-dimensional. - [x] What is a **quasifinite** morphism? ✅ 2022-11-24 - [ ] Locally quasifinite and quasicompact. Equivalently, covered my opens where the induced ring maps are finite type and quasifinite at the prime $\mfq$ associated to $x\in X$. - [x] What is a consequence of a morphism of affine schemes being finite? ✅ 2022-11-24 - [ ] If $R\to A$ is finite the $\spec A\to \spec R$ has finite fibers. - [x] What is a **finite type morphism**? #syllabus ✅ 2022-12-05 - [ ] Locally of finite type and quasicompact. - [ ] Equivalently, on an open cover, inducing finitely generated *algebras* and covered by finitely many: $f:X\to Y$ where $\spec B_i\covers Y$ with $\spec A_i \da f\inv(B_i)$ with each $A_i\in \algs{B_i}^\fg$. - [ ] Note that being fg as a module is **stronger** than being fg as an algebra: $k[x] \in \kalg$ has one algebra generator but infinite rank in $\kmod$. Thus finite implies finite type but not conversely. - [x] What is a morphism [locally of finite type](locally%20of%20finite%20type)? ⏫ ✅ 2022-12-05 - [ ] $f:X\to Y$ with open covers $\spec B_i, U_i\da f\inv(\spec B_i)$ where $\spec A_{ij}\covers U_i$ making $A_{ij}\in \algs{B_i}^\fg$. - [ ] Additionally **is of finite type** if each $U_i$ can be covered by finitely many $A_{ij}$. - [ ] A ring map $R\to A$ is of finite type $A$ admits a surjection from $R[x_1, \cdots, x_n]$ for some $n$. Equivalently, $A\in \ralg^\fg$. Being **locally of finite type** means every $x$ admitting a neighborhood $\spec A_i\ni x$ where $f(\spec A_i) \subseteq \spec R_i\ni f(x)$ so that the induced ring map $R\to A$ is finite type. - [x] What is a [finitely presented](finitely%20presented.md) morphism? ✅ 2022-11-25 - [ ] Locally of finite presentation, quasicompact, quasiseparated. - [x] What is a [locally finitely presented](locally%20finitely%20presented) morphism? ✅ 2022-11-25 - [ ] On open covers, the induced rings maps $R_i\to A_i$ are finite presentation, i.e. $R_i[x_1,\cdots, x_n] \surjects A_i$ with fg kernel for some $n$. - [x] A [universally closed morphism](universally%20closed%20morphism.md)? #syllabus ⏫✅ 2022-11-25 - [ ] $f:X\to Y$ closed and every base change $Y'\to Y \leadsto X\fiberprod{Y} Y' \to Y'$ is closed, where maps are closed when images of closed sets are closed. - [x] What is an example of a closed morphism? ✅ 2022-11-25 - [ ] Any finite map of rings, the structure map of any proper scheme. - [x] What is an example of a non-closed morphism? ✅ 2022-11-25 - [ ] Use that $\AA^n$ is not proper: $X\fiberprod{k} X \to X$ is not closed for $X\da \AA^1$ since it is the coordinate projection, and $V(xy-c) \mapsto \GG_m$ which is not closed. - [x] What is a a **projective morphism**? ⏫ #syllabus ✅ 2022-11-25 - [ ] $f:X\to Y$ which factors as $X\injects\PP^N\slice Y \surjects Y$, a closed immersion followed by the natural projection, where $\PP^N\slice Y \da \PP^N\slice \ZZ \fiberprod{\spec \ZZ} Y$. - [x] What is a quasiprojective morphism? ✅ 2022-11-25 - [ ] $f:X\to Y$ which factors as $X\injects X'\to Y'$, a closed immersion followed by a projective morphism. - [x] Give examples and counterexamples of projective morphisms. ✅ 2022-11-25 - [ ] For $A$ any ring and $S$ any graded ring with $S_0 = A$, $\Proj S\to \spec A$ is projective. - [x] Give some consequences of projective morphisms. ✅ 2022-11-25 - [ ] For Noetherian schemes, projective $\implies$ proper. - [x] What is a faithfully flat morphism? ✅ 2022-11-24 - [ ] Flat and surjective. - [x] What is a [rational](rational) morphism? ✅ 2022-11-25 - [ ] An equivalence class of pairs $(U, \ro f U)$ where $U\subseteq X$ is open dense and $\ro f U: U\to Y$ is a morphism, which are identified if they coincide on intersections. - [x] What is a [birational](birational.md) map? ✅ 2022-11-25 - [ ] A rational map with a rational inverse. - [x] What is the [degree of a morphism](degree%20of%20a%20morphism.md)? ✅ 2022-11-25 - [ ] $f:X\to S$ with $f_*\OO_X\in \mods{\OO_s}$ finite and locally free, take its degree as a Coherent sheaf, where $\deg \mcf \da \deg \det \mcf$ since $\det \mcf \in \Pic(X)$, where the degree is the degree of the associated divisor. - [x] What is a **separated morphism**? #syllabus ⏫✅ 2022-12-05 - [ ] A morphism $f:X\to S$ of schemes is **separated** iff $\Delta_{X/S}: X\to X\fiberprod{S} X$ is a **closed immersion**. - [x] How is this separatedness related to Hausdorffness? ✅ 2022-11-24 - [ ] $X\in \Top$ is **Hausdorff** iff the diagonal $\Delta \subseteq X\times X$ is a closed subset. - [x] What is a [quasiseparated morphism](Unsorted/separated.md)? ✅ 2022-11-24 - $f:X\to S$ is quasiseparated iff the diagonal is quasicompact. - [x] What is [qcqs](qcqs)? (Quasi-compact and [quaiseparated](quaiseparated.md)) ✅ 2022-11-25 - [ ] Quasicompact: inverse images of quasicompact sets are quasicompact. Quasiseparated: the diagonal is quasicompact. - [x] What is an [affine morphism](affine%20morphism)? #syllabus ✅ 2022-11-25 - [ ] Inverse images of affines are affine. - [x] What are some examples of affine morphisms? ✅ 2022-11-25 - [ ] Any morphism between affine schemes, closed immersions. - [x] What are some consequences of a morphism being affine? ✅ 2022-11-25 - [ ] Separated and quasicompact. - [x] What is a [quasiaffine morphism](quasiaffine%20morphism)? ✅ 2022-11-25 - [ ] Inverse images of quasiaffines are quasiaffine. - [x] What is a quasiaffine scheme? ✅ 2022-11-25 - [ ] Quasicompact and isomorphic to an open subscheme of an affine scheme. - [x] What is a [normal morphism](normal%20morphism)? ✅ 2022-11-25 - [ ] $f:X\to Y$ with $f$ flat at $x$ and the fiber $X_{f(x)}$ *geometrically normal*, i.e. for every field extension $L/k$ and every $\ell$ over $x$ in $X\fiberprod{k} L$ has a normal local ring, i.e. integrally closed in its field of fractions. - [x] What is a [regular morphism](regular%20morphism)? ✅ 2022-11-25 - [ ] Flat at $x$ and *geometrically regular* at $x$, i.e. for every fg field extension $L/k$ and $\ell$ over $x$ in $X\fiberprod{k} L$, the local ring at $\ell$ is regular, i.e. the minimal number of generators of its maximal ideal equals its Krull dimension. - [x] What is the cotangent complex? ✅ 2022-11-24 - [ ] The nonabelian left-derived functor $\LL_{\wait/A}: \CRing_A\to \derivedcat{\mods{A}}$ of the functor $\Omega^1_{\wait/A}[0]$ - [x] When is a morphism of schemes an immersion? ✅ 2022-11-25 - [ ] $\iota: X\to Z$ making $X$ isomorphic to an open subscheme *of a closed subscheme* of $Z$. - [x] Name some properties preserved by base change. ✅ 2022-11-25 - [ ] Open/closed immersions, affine, finite, integral, locally of ft, qc, ft, quasi-finite, flat, surjective. - [x] What are the three valuative criteria? #syllabus ⏫✅ 2022-12-05 - Universally closed, separated, proper, Generally for $X$ quasi-separated (e.g. is locally Noetherian). In the following diagram, $V$ is a DVR and $K = \ff(V)$: ![](attachments/Pasted%20image%2020221124002619.png) - **Universally closed**: there exists $\geq 1$ lift (**existence** without uniqueness) - **Separated**: there exists $\leq 1$ lift (**uniqueness** without existence) - **Proper**: there exists $=1$ lift (**existence and uniqueness**)