flat morphism

Consider the affine scheme morphism

:$\backslash operatorname\{Spec\}\backslash left(\backslash frac\{\backslash Complex[x,y,t]\}\{x^2\; +\; y^2\; -\; t\}\backslash right)\; \backslash to\; \backslash operatorname\{Spec\}(\backslash Complex[t])$

induced from the morphism of algebras

:$\backslash begin\{cases\}\; \backslash Complex[t]\; \backslash to\; \backslash frac\{\backslash Complex[x,y,t]\}\{x^2\; +\; y^2\; -\; t\}\; \backslash \backslash \; t\; \backslash mapsto\; t.\; \backslash \backslash \; \backslash end\{cases\}$

Since proving flatness for this morphism amounts to computing the Tor group{{Cite book|title=Deformations of Algebraic Schemes|year=2010|url=https://archive.org/details/deformationsalge00sern_419|url-access=limited|last=Sernesi|first=E.|publisher=Springer|pages=[https://archive.org/details/deformationsalge00sern_419/page/n274 269]–279}}

:$\backslash operatorname\{Tor\}\_1^\{\backslash Complex\; [t]\}\backslash left(\backslash frac\{\backslash Complex\; [x,y,t]\}\{x^2\; +\; y^2\; -t\},\; \backslash Complex\; \backslash right),$

we resolve the complex numbers

:$\backslash begin\{array\}\{lcccc\}$

0 & \to & \Complex[t] & \xrightarrow{\cdot t} & \Complex [t] & \to & 0 \\

\downarrow & & \downarrow & & \downarrow & & \downarrow\\

0 & \to & 0 & \to & \Complex & \to & 0 \\

\end{array}

and tensor by the module representing our scheme giving the sequence of $\backslash Complex\; [t]$-modules

:$0\; \backslash to\; \backslash frac\{\backslash Complex[x,y,t]\}\{x^2\; +\; y^2\; -\; t\}\; \backslash xrightarrow\{\backslash cdot\; t\}\; \backslash frac\{\backslash Complex[x,y,t]\}\{x^2\; +\; y^2\; -\; t\}\; \backslash to\; 0$

Because {{mvar|t}} is not a zero divisor we have a trivial kernel, hence the homology group vanishes.

Other examples of flat morphisms can be found using "miracle flatness"{{Cite web| url= https://ayoucis.wordpress.com/2014/03/12/flat-morphisms-and-flatness/ |title=Flat Morphisms and Flatness }} which states that if a morphism $f\backslash colon\; X\; \backslash to\; Y$ between a Cohen–Macaulay scheme to a regular scheme has equidimensional fibers, then it is flat. Easy examples of this are elliptic fibrations, smooth morphisms, and morphisms to stratified varieties which satisfy miracle flatness on each of the strata.

The universal examples of flat morphisms of schemes are given by Hilbert schemes. This is because Hilbert schemes parameterize universal classes of flat morphisms, and every flat morphism is the pullback from some Hilbert scheme. I.e., if $f\backslash colon\; X\backslash to\; S$ is flat, there exists a commutative diagram

: $\backslash begin\{matrix\}$

X & \to & \operatorname{Hilb}_{S} \\

\downarrow & & \downarrow \\

S & \to & S

\end{matrix}

for the Hilbert scheme of all flat morphisms to $S$. Since $f$ is flat, the fibers $f\_s\backslash colon\; X\_s\; \backslash to\; s$ all have the same Hilbert polynomial $\backslash Phi$, hence we could have similarly written $\backslash text\{Hilb\}\_S^\backslash Phi$ for the Hilbert scheme above.

One class of non-examples are given by blowup maps

:$\backslash operatorname\{Bl\}\_I\; X\; \backslash to\; X.$

One easy example is the blowup of a point in $\backslash Complex[x,y]$. If we take the origin, this is given by the morphism

:$\backslash Complex[x,y]\; \backslash to\; \backslash frac\{\backslash Complex[x,y,s,t]\}\{xt\; -\; ys\}$ sending $x\; \backslash mapsto\; x,\; y\; \backslash mapsto\; y\; ,$

where the fiber over a point $(a,b)\; \backslash neq\; (0,0)$ is a copy of $\backslash Complex$, i.e.,

:$\backslash frac\{\backslash Complex[x,y,s,t]\}\{xt\; -\; ys\}\backslash otimes\_\{\backslash Complex[x,y]\}\; \backslash frac\{\backslash Complex[x,y]\}\{(x-a,y-b)\}\; \backslash cong\; \backslash Complex\; ,$

which follows from

:$M\backslash otimes\_R\; \backslash frac\{R\}\{I\}\; \backslash cong\; \backslash frac\{M\}\{IM\}\; .$

But for $a=b=0$, we get the isomorphism

:$\backslash frac\{\backslash Complex[x,y,s,t]\}\{xt\; -\; ys\}\backslash otimes\_\{\backslash Complex[x,y]\}\; \backslash frac\{\backslash Complex[x,y]\}\{(x,y)\}\; \backslash cong\; \backslash Complex[s,t].$

The reason this fails to be flat is because of the Miracle flatness lemma, which can be checked locally.

A simple non-example of a flat morphism is $k[\backslash varepsilon]\; =\; k[x]/(x^2)\; \backslash to\; k.$ This is because

:$k\; \backslash otimes^\backslash mathbf\{L\}\_\{k[\backslash varepsilon]\}\; k$

is an infinite complex, which we can find by taking a flat resolution of {{mvar|k}},

:$\backslash cdots\; ~\backslash xrightarrow\{\backslash overset\{\}\; \backslash cdot\; \backslash varepsilon\}~\; k[\backslash varepsilon]\; ~\backslash xrightarrow\{\backslash cdot\; \backslash varepsilon\}~\; k[\backslash varepsilon]\; \backslash xrightarrow\{\backslash cdot\; \backslash varepsilon\}\; k[\backslash varepsilon]\; \backslash to\; k$

and tensor the resolution with {{mvar|k}}, we find that

:$k\backslash otimes^\backslash mathbf\{L\}\_\{k[\backslash varepsilon]\}\; k\; \backslash simeq\; \backslash bigoplus\_\{i=0\}^\backslash infty\; k[+i]$

showing that the morphism cannot be flat. Another non-example of a flat morphism is a blowup since a flat morphism necessarily has equi-dimensional fibers.

Let $f\backslash colon\; X\; \backslash to\; Y$ be a morphism of schemes. For a morphism $g\backslash colon\; Y\text{'}\; \backslash to\; Y$, let $X\text{'}\; =\; X\backslash times\_\{Y\}\; Y\text{'}$ and $f\text{'}\; =\; (f,\; 1\_\{Y\text{'}\})\; \backslash colon\; X\text{'}\; \backslash to\; Y\text{'}.$ The morphism f is flat if and only if for every g, the pullback $f\text{'}^*$ is an exact functor from the category of quasi-coherent $\backslash mathcal\{O\}\_\{Y\text{'}\}$-modules to the category of quasi-coherent $\backslash mathcal\{O\}\_\{X\text{'}\}$-modules.EGA IV₂, Proposition 2.1.3.

Assume $f\backslash colon\; X\; \backslash to\; Y$ and $g\backslash colon\; Y\; \backslash to\; Z$ are morphisms of schemes and f is flat at x in X. Then g is flat at $f(x)$ if and only if gf is flat at x.EGA IV₂, Corollaire 2.2.11(iv). In particular, if f is faithfully flat, then g is flat or faithfully flat if and only if gf is flat or faithfully flat, respectively.EGA IV₂, Corollaire 2.2.13(iii).

• The composite of two flat morphisms is flat.EGA IV₂, Corollaire 2.1.6.
• The fiber product of two flat or faithfully flat morphisms is a flat or faithfully flat morphism, respectively.EGA IV₂, Corollaire 2.1.7, and EGA IV₂, Corollaire 2.2.13(ii).
• Flatness and faithful flatness is preserved by base change: If f is flat or faithfully flat and $g\backslash colon\; Y\text{'}\; \backslash to\; Y$, then the fiber product $f\backslash times\; g\backslash colon\; X\backslash times\_Y\; Y\text{'}\; \backslash to\; Y\text{'}$ is flat or faithfully flat, respectively.EGA IV₂, Proposition 2.1.4, and EGA IV₂, Corollaire 2.2.13(i).
• The set of points where a morphism (locally of finite presentation) is flat is open.EGA IV₃, Théorème 11.3.1.
• If f is faithfully flat and of finite presentation, and if gf is finite type or finite presentation, then g is of finite type or finite presentation, respectively.EGA IV₃, Proposition 11.3.16.

Suppose $f\backslash colon\; X\; \backslash to\; Y$ is a flat morphism of schemes.

• If F is a quasi-coherent sheaf of finite presentation on Y (in particular, if F is coherent), and if J is the annihilator of F on Y, then $f^*J\; \backslash to\; \backslash mathcal\{O\}\_X$, the pullback of the inclusion map, is an injection, and the image of $f^*J$ in $\backslash mathcal\{O\}\_X$ is the annihilator of $f^*F$ on X.EGA IV₂, Proposition 2.1.11.
• If f is faithfully flat and if G is a quasi-coherent $\backslash mathcal\{O\}\_Y$-module, then the pullback map on global sections $\backslash Gamma(Y,\; G)\; \backslash to\; \backslash Gamma(X,\; f^*G)$ is injective.EGA IV₂, Corollaire 2.2.8.

Suppose $h\backslash colon\; S\text{'}\; \backslash to\; S$ is flat. Let X and Y be S-schemes, and let $X\text{'}$ and $Y\text{'}$ be their base change by h.

• If $f\backslash colon\; X\; \backslash to\; Y$ is quasi-compact and dominant, then its base change $f\text{'}\backslash colon\; X\text{'}\; \backslash to\; Y\text{'}$ is quasi-compact and dominant.EGA IV₂, Proposition 2.3.7(i).
• If h is faithfully flat, then the pullback map $\backslash operatorname\{Hom\}\_S(X,Y)\; \backslash to\; \backslash operatorname\{Hom\}\_\{S\text{'}\}(X\text{'},Y\text{'})$ is injective.EGA IV₂, Corollaire 2.2.16.
• Assume $f\backslash colon\; X\; \backslash to\; Y$ is quasi-compact and quasi-separated. Let Z be the closed image of X, and let $j\backslash colon\; Z\; \backslash to\; Y$ be the canonical injection. Then the closed subscheme determined by the base change $j\text{'}\backslash colon\; Z\text{'}\; \backslash to\; Y\text{'}$ is the closed image of $X\text{'}$.EGA IV₂, Proposition 2.3.2.

If $f\backslash colon\; X\; \backslash to\; Y$ is flat, then it possesses all of the following properties:

• For every point x of X and every generization {{′|y}} of {{math|y {{=}} f(x)}}, there is a generization x′ of x such that {{math|y′ {{=}} f({{′|x}})}}.EGA IV₂, Proposition 2.3.4(i).
• For every point x of X, $f(\backslash operatorname\{Spec\}\; \backslash mathcal\{O\}\_\{X,x\})\; =\; \backslash operatorname\{Spec\}\; \backslash mathcal\{O\}\_\{Y,f(x)\}$.EGA IV₂, Proposition 2.3.4(ii).
• For every irreducible closed subset {{′|Y}} of Y, every irreducible component of f{{i sup|−1}}({{′|Y}}) dominates {{′|Y}}.EGA IV₂, Proposition 2.3.4(iii).
• If Z and {{′|Z}} are two irreducible closed subsets of Y with Z contained in {{′|Z}}, then for every irreducible component T of f{{i sup|−1}}(Z), there is an irreducible component {{′|T}} of f{{i sup|−1}}({{′|Z}}) containing T.EGA IV₂, Corollaire 2.3.5(i).
• For every irreducible component T of X, the closure of f({{itco|T}}) is an irreducible component of Y.EGA IV₂, Corollaire 2.3.5(ii).
• If Y is irreducible with generic point y, and if f{{i sup|−1}}(y) is irreducible, then X is irreducible.EGA IV₂, Corollaire 2.3.5(iii).
• If f is also closed, the image of every connected component of X is a connected component of Y.EGA IV₂, Proposition 2.3.6(ii).
• For every pro-constructible subset Z of Y, $f^\{-1\}(\backslash bar\; Z)\; =\; \backslash overline\{f^\{-1\}(Z)\}$.EGA IV₂, Théorème 2.3.10.

If f is flat and locally of finite presentation, then f is universally open.EGA IV₂, Théorème 2.4.6. However, if f is faithfully flat and quasi-compact, it is not in general true that f is open, even if X and Y are noetherian.EGA IV₂, Remarques 2.4.8(i). Furthermore, no converse to this statement holds: If f is the canonical map from the reduced scheme X_red to X, then f is a universal homeomorphism, but for X non-reduced and noetherian, f is never flat.EGA IV₂, Remarques 2.4.8(ii).

If $f\backslash colon\; X\; \backslash to\; Y$ is faithfully flat, then:

• The topology on Y is the quotient topology relative to f.EGA IV₂, Corollaire 2.3.12.
• If f is also quasi-compact, and if Z is a subset of Y, then Z is a locally closed pro-constructible subset of Y if and only if f{{i sup|−1}}({{itco|Z}}) is a locally closed pro-constructible subset of X.EGA IV₂, Corollaire 2.3.14.

If f is flat and locally of finite presentation, then for each of the following properties P, the set of points where f has P is open:EGA IV₃, Théorème 12.1.6.

• Serre's condition S_k (for any fixed k).
• Geometrically regular.
• Geometrically normal.

If in addition f is proper, then the same is true for each of the following properties:EGA IV₃, Théorème 12.2.4.

• Geometrically reduced.
• Geometrically reduced and having k geometric connected components (for any fixed k).
• Geometrically integral.

Assume {{mvar|X}} and {{mvar|Y}} are locally noetherian, and let $f\backslash colon\; X\; \backslash to\; Y$.

• Let $x$ be a point of $X$ and $y=f(x)$. If $f$ is flat, then $\backslash dim\_\{x\}X=\backslash dim\_\{y\}Y+\backslash dim\_\{x\}f^\{-1\}(y)$.EGA IV₂, Corollaire 6.1.2. Conversely, if this equality holds for all x, X is Cohen–Macaulay, and Y is regular, and furthermore f maps closed points to closed points, then f is flat.EGA IV₂, Proposition 6.1.5. Note that the regularity assumption on Y is important here. The extension $\backslash Complex[x^2,\; y^2,\; xy]\backslash subset\; \backslash Complex[x,y]$ gives a counterexample with X regular, Y normal, f finite surjective but not flat.
• If f is faithfully flat, then for each closed subset Z of Y, {{math|codim_Y(Z) {{=}} codim_X(f{{i sup|−1}}(Z))}}.EGA IV₂, Corollaire 6.1.4.
• Suppose f is flat and F is a quasi-coherent module over Y. If F has projective dimension at most n, then $f^*F$ has projective dimension at most n.EGA IV₂, Corollaire 6.2.2.

• Assume f is flat at x in X. If X is reduced or normal at x, then Y is reduced or normal, respectively, at f(x).EGA IV₂, Proposition 2.1.13. Conversely, if f is also of finite presentation and f{{i sup|−1}}(y) is reduced or normal, respectively, at x, then X is reduced or normal, respectively, at x.EGA IV₃, Proposition 11.3.13.
• In particular, if f is faithfully flat, then X reduced or normal implies that Y is reduced or normal, respectively. If f is faithfully flat and of finite presentation, then all the fibers of f reduced or normal implies that X is reduced or normal, respectively.
• If f is flat at x in X, and if X is integral or integrally closed at x, then Y is integral or integrally closed, respectively, at f(x).EGA IV₂, Proposition 2.1.13.
• If f is faithfully flat, X is locally integral, and the topological space of Y is locally noetherian, then Y is locally integral.EGA IV₂, Proposition 2.1.14.
• If f is faithfully flat and quasi-compact, and if X is locally noetherian, then Y is also locally noetherian.EGA IV₂, Proposition 2.2.14.
• Assume f is flat and X and Y are locally noetherian. If X is regular at x, then Y is regular at f(x). Conversely, if Y is regular at f(x) and f{{i sup|−1}}(f(x)) is regular at x, then X is regular at x.EGA IV₂, Corollaire 6.5.2.
• Assume f is flat and X and Y are locally noetherian. If X is normal at x, then Y is normal at f(x). Conversely, if Y is normal at f(x) and f{{i sup|−1}}(f(x)) is normal at x, then X is normal at x.EGA IV₂, Corollaire 6.5.4.

Let {{math|g : {{′|Y}} → Y}} be faithfully flat. Let F be a quasi-coherent sheaf on Y, and let {{′|F}} be the pullback of F to {{′|Y}}. Then F is flat over Y if and only if {{′|F}} is flat over {{′|Y}}.EGA IV₂, Proposition 2.5.1.

Assume f is faithfully flat and quasi-compact. Let G be a quasi-coherent sheaf on Y, and let F denote its pullback to X. Then F is finite type, finite presentation, or locally free of rank n if and only if G has the corresponding property.EGA IV₂, Proposition 2.5.2.

Suppose {{math|f : X → Y}} is an S-morphism of S-schemes. Let {{math|g : {{′|S}} → S}} be faithfully flat and quasi-compact, and let {{′|X}}, {{′|Y}}, and {{′|f}} denote the base changes by g. Then for each of the following properties P, if {{′|f}} has P, then f has P.EGA IV₂, Proposition 2.6.2.

• Open.
• Closed.
• Quasi-compact and a homeomorphism onto its image.
• A homeomorphism.

Additionally, for each of the following properties P, f has P if and only if {{′|f}} has P.EGA IV₂, Corollaire 2.6.4 and Proposition 2.7.1.

• Universally open.
• Universally closed.
• A universal homeomorphism.
• Quasi-compact.
• Quasi-compact and dominant.
• Quasi-compact and universally bicontinuous.
• Separated.
• Quasi-separated.
• Locally of finite type.
• Locally of finite presentation.
• Finite type.
• Finite presentation.
• Proper.
• An isomorphism.
• A monomorphism.
• An open immersion.
• A quasi-compact immersion.
• A closed immersion.
• Affine.
• Quasi-affine.
• Finite.
• Quasi-finite.
• Integral.

It is possible for {{′|f}} to be a local isomorphism without f being even a local immersion.EGA IV₂, Remarques 2.7.3(iii).

If f is quasi-compact and L is an invertible sheaf on X, then L is f-ample or f-very ample if and only if its pullback {{′|L}} is {{′|f}}-ample or {{′|f}}-very ample, respectively.EGA IV₂, Corollaire 2.7.2. However, it is not true that f is projective if and only if {{′|f}} is projective. It is not even true that if f is proper and {{′|f}} is projective, then f is quasi-projective, because it is possible to have an {{′|f}}-ample sheaf on {{′|X}} which does not descend to X.EGA IV₂, Remarques 2.7.3(ii).

• fpqc morphism
• Relative effective Cartier divisor, an example of a flat morphism
• Degeneration (algebraic geometry)

{{Reflist}}

• {{Citation| last1=Eisenbud | first1=David | author1-link=David Eisenbud | title=Commutative algebra | publisher=Springer-Verlag | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-94268-1 | id={{ISBN|978-0-387-94269-8}} | mr= 1322960 | year=1995 | volume=150}}, section 6.
• {{Citation| last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | title=Géométrie algébrique et géométrie analytique | url=http://www.numdam.org/numdam-bin/item?id=AIF_1956__6__1_0 | mr=0082175 | year=1956 | journal=Annales de l'Institut Fourier | issn=0373-0956 | volume=6 | pages=1–42 | doi=10.5802/aif.59| doi-access=free }}
• {{EGA | book=I}}
• {{EGA | book=IV-2}}
• {{EGA | book=IV-3}}
• {{Hartshorne AG}}

{{DEFAULTSORT:Flat Morphism}}

Category:Morphisms of schemes