Editing Flat morphism

{{Short description|Scheme theory concept}}
In [[mathematics]], in particular in [[algebraic geometry]], a '''flat morphism''' ''f'' from a [[scheme (mathematics)|scheme]] ''X'' to a scheme ''Y'' is a [[morphism]] such that the induced map on every [[Stalk (sheaf)|stalk]] is a flat map of rings, i.e.,

:<math>f_P\colon \mathcal{O}_{Y, f(P)} \to \mathcal{O}_{X, P}</math>

is a flat map for all ''P'' in ''X''.<ref>EGA IV<sub>2</sub>, 2.1.1.</ref> A map of rings <math>A\to B</math> is called '''flat''' if it is a homomorphism that makes ''B'' a [[flat module|flat]] ''A''-module. A morphism of schemes is called '''faithfully flat''' if it is both surjective and flat.<ref>EGA 0<sub>I</sub>, 6.7.8.</ref>

Two basic intuitions regarding flat morphisms are: 
*flatness is a [[generic property]]; and 
*the failure of flatness occurs on the jumping set of the morphism.

The first of these comes from [[commutative algebra]]: subject to some [[finiteness condition on a morphism of schemes|finiteness conditions]] on ''f'', it can be shown that there is a non-empty open subscheme <math>Y'</math> of ''Y'', such that ''f'' restricted to <math>Y'</math> is a flat morphism ([[generic flatness]]). Here 'restriction' is interpreted by means of the [[fiber product of schemes]], applied to ''f'' and the [[inclusion map]] of <math>Y'</math> into ''Y''.

For the second, the idea is that morphisms in algebraic geometry can exhibit discontinuities of a kind that are detected by flatness. For instance, the operation of [[blowing up|blowing down]] in the [[birational geometry]] of an [[algebraic surface]] can give a single [[fiber product of schemes|fiber]] that is of dimension 1 when all the others have dimension 0. It turns out (retrospectively) that flatness in morphisms is directly related to controlling this sort of [[semicontinuity]], or one-sided jumping.

Flat morphisms are used to define (more than one version of) the [[flat topos]], and [[flat cohomology]] of sheaves from it. This is a deep-lying theory, and has not been found easy to handle. The concept of [[étale morphism]] (and so [[étale cohomology]]) depends on the flat morphism concept: an étale morphism being flat, of [[morphism of finite type|finite type]], and [[Ramification (mathematics)|unramified]].

== Examples/non-examples ==
Consider the affine scheme morphism

:<math>\operatorname{Spec}\left(\frac{\Complex[x,y,t]}{x^2 + y^2 - t}\right) \to \operatorname{Spec}(\Complex[t])</math>

induced from the morphism of algebras

:<math>\begin{cases} \Complex[t] \to \frac{\Complex[x,y,t]}{x^2 + y^2 - t} \\ t \mapsto t. \\ \end{cases}</math>

Since proving flatness for this morphism amounts to computing the [[Tor group]]<ref>{{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 Science+Business Media|Springer]]|pages=[https://archive.org/details/deformationsalge00sern_419/page/n274 269]–279}}</ref>

:<math>\operatorname{Tor}_1^{\Complex [t]}\left(\frac{\Complex [x,y,t]}{x^2 + y^2 -t}, \Complex \right),</math>

we resolve the complex numbers

:<math>\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}</math>

and tensor by the module representing our scheme giving the sequence of <math>\Complex [t]</math>-modules

:<math>0 \to \frac{\Complex[x,y,t]}{x^2 + y^2 - t} \xrightarrow{\cdot t} \frac{\Complex[x,y,t]}{x^2 + y^2 - t} \to 0</math>

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

=== Miracle flatness ===
Other examples of flat morphisms can be found using "miracle flatness"<ref>{{Cite web| url= https://ayoucis.wordpress.com/2014/03/12/flat-morphisms-and-flatness/ |title=Flat Morphisms and Flatness }}</ref> which states that if a morphism <math>f\colon X \to Y</math> between a [[Cohen–Macaulay ring#Cohen–Macaulay schemes|Cohen–Macaulay scheme]] to a [[regular scheme]] has [[Equidimensional scheme|equidimensional]] fibers, then it is flat. Easy examples of this are [[Elliptic surface|elliptic fibrations]], [[smooth morphism]]s, and morphisms to [[Topologically stratified space|stratified varieties]] which satisfy miracle flatness on each of the strata.

===Hilbert schemes===
The universal examples of flat morphisms of schemes are given by [[Hilbert scheme]]s. 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 <math>f\colon X\to S</math> is flat, there exists a commutative diagram

: <math>\begin{matrix}
X & \to & \operatorname{Hilb}_{S} \\
\downarrow & & \downarrow \\
S & \to & S
\end{matrix}</math>

for the Hilbert scheme of all flat morphisms to <math>S</math>. Since <math>f</math> is flat, the fibers <math>f_s\colon X_s \to s</math> all have the same Hilbert polynomial <math>\Phi</math>, hence we could have similarly written <math>\text{Hilb}_S^\Phi</math> for the Hilbert scheme above.

===Non-examples===

====Blowup====
One class of non-examples are given by [[Blowing up|blowup]] maps

:<math>\operatorname{Bl}_I X \to X.</math>

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

:<math>\Complex[x,y] \to \frac{\Complex[x,y,s,t]}{xt - ys}</math> sending <math>x \mapsto x, y \mapsto y ,</math>

where the fiber over a point <math>(a,b) \neq (0,0)</math> is a copy of <math>\Complex</math>, i.e.,

:<math>\frac{\Complex[x,y,s,t]}{xt - ys}\otimes_{\Complex[x,y]} \frac{\Complex[x,y]}{(x-a,y-b)} \cong \Complex ,</math>

which follows from

:<math>M\otimes_R \frac{R}{I} \cong \frac{M}{IM} .</math>

But for <math>a=b=0</math>, we get the isomorphism

:<math>\frac{\Complex[x,y,s,t]}{xt - ys}\otimes_{\Complex[x,y]} \frac{\Complex[x,y]}{(x,y)} \cong \Complex[s,t].</math>

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

====Infinite resolution====
A simple non-example of a flat morphism is <math>k[\varepsilon] = k[x]/(x^2) \to k.</math> This is because

:<math>k \otimes^\mathbf{L}_{k[\varepsilon]} k</math>

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

:<math>\cdots ~\xrightarrow{\overset{} \cdot \varepsilon}~ k[\varepsilon] ~\xrightarrow{\cdot \varepsilon}~ k[\varepsilon] \xrightarrow{\cdot \varepsilon} k[\varepsilon] \to k</math>

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

:<math>k\otimes^\mathbf{L}_{k[\varepsilon]} k \simeq \bigoplus_{i=0}^\infty k[+i]</math>

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

==Properties of flat morphisms==
Let <math>f\colon X \to Y</math> be a morphism of schemes. For a morphism <math>g\colon Y' \to Y</math>, let <math>X' = X\times_{Y} Y'</math> and <math>f' = (f, 1_{Y'}) \colon X' \to Y'.</math> The morphism ''f'' is flat if and only if for every ''g'', the [[Inverse image functor|pullback]] <math>f'^*</math> is an [[exact functor]] from the [[category (mathematics)|category]] of [[Quasi-coherent sheaf|quasi-coherent]] <math>\mathcal{O}_{Y'}</math>-modules to the category of quasi-coherent <math>\mathcal{O}_{X'}</math>-modules.<ref>EGA IV<sub>2</sub>, Proposition 2.1.3.</ref>

Assume <math>f\colon X \to Y</math> and <math>g\colon Y \to Z</math> are morphisms of schemes and ''f'' is flat at ''x'' in ''X''. Then ''g'' is flat at <math>f(x)</math> if and only if ''gf'' is flat at ''x''.<ref>EGA IV<sub>2</sub>, Corollaire 2.2.11(iv).</ref> 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.<ref>EGA IV<sub>2</sub>, Corollaire 2.2.13(iii).</ref>

===Fundamental properties===
* The composite of two flat morphisms is flat.<ref>EGA IV<sub>2</sub>, Corollaire 2.1.6.</ref>
* The fiber product of two flat or faithfully flat morphisms is a flat or faithfully flat morphism, respectively.<ref>EGA IV<sub>2</sub>, Corollaire 2.1.7, and EGA IV<sub>2</sub>, Corollaire 2.2.13(ii).</ref>
* Flatness and faithful flatness is preserved by base change: If ''f'' is flat or faithfully flat and <math>g\colon Y' \to Y</math>, then the fiber product <math>f\times g\colon X\times_Y Y' \to Y'</math> is flat or faithfully flat, respectively.<ref>EGA IV<sub>2</sub>, Proposition 2.1.4, and EGA IV<sub>2</sub>, Corollaire 2.2.13(i).</ref>
* The set of points where a morphism (locally of finite presentation) is flat is open.<ref>EGA IV<sub>3</sub>, Théorème 11.3.1.</ref>
* 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.<ref>EGA IV<sub>3</sub>, Proposition 11.3.16.</ref>

Suppose <math>f\colon X \to Y</math> 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 <math>f^*J \to \mathcal{O}_X</math>, the pullback of the inclusion map, is an injection, and the image of <math>f^*J</math> in <math>\mathcal{O}_X</math> is the annihilator of <math>f^*F</math> on ''X''.<ref>EGA IV<sub>2</sub>, Proposition 2.1.11.</ref>
* If ''f'' is faithfully flat and if ''G'' is a quasi-coherent <math>\mathcal{O}_Y</math>-module, then the pullback map on global sections <math>\Gamma(Y, G) \to \Gamma(X, f^*G)</math> is injective.<ref>EGA IV<sub>2</sub>, Corollaire 2.2.8.</ref>

Suppose <math>h\colon S' \to S</math> is flat. Let ''X'' and ''Y'' be ''S''-schemes, and let <math>X'</math> and <math>Y'</math> be their base change by ''h''.

* If <math>f\colon X \to Y</math> is [[Quasi-compact morphism|quasi-compact]] and dominant, then its base change <math>f'\colon X' \to Y'</math> is quasi-compact and dominant.<ref>EGA IV<sub>2</sub>, Proposition 2.3.7(i).</ref>
* If ''h'' is faithfully flat, then the pullback map <math>\operatorname{Hom}_S(X,Y) \to \operatorname{Hom}_{S'}(X',Y')</math> is injective.<ref>EGA IV<sub>2</sub>, Corollaire 2.2.16.</ref>
* Assume <math>f\colon X \to Y</math> is quasi-compact and [[Quasi-separated morphism|quasi-separated]]. Let ''Z'' be the closed image of ''X'', and let <math>j\colon Z \to Y</math> be the canonical injection. Then the closed subscheme determined by the base change <math>j'\colon Z' \to Y'</math> is the closed image of <math>X'</math>.<ref>EGA IV<sub>2</sub>, Proposition 2.3.2.</ref>

===Topological properties===
If <math>f\colon X \to Y</math> 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''}})}}.<ref>EGA IV<sub>2</sub>, Proposition 2.3.4(i).</ref>
*For every point ''x'' of ''X'', <math>f(\operatorname{Spec} \mathcal{O}_{X,x}) = \operatorname{Spec} \mathcal{O}_{Y,f(x)}</math>.<ref>EGA IV<sub>2</sub>, Proposition 2.3.4(ii).</ref>
*For every irreducible closed subset {{′|''Y''}} of ''Y'', every [[irreducible component]] of ''f''{{i sup|−1}}({{′|''Y''}}) dominates {{′|''Y''}}.<ref>EGA IV<sub>2</sub>, Proposition 2.3.4(iii).</ref>
*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''.<ref>EGA IV<sub>2</sub>, Corollaire 2.3.5(i).</ref>
*For every irreducible component ''T'' of ''X'', the closure of ''f''({{itco|''T''}}) is an irreducible component of ''Y''.<ref>EGA IV<sub>2</sub>, Corollaire 2.3.5(ii).</ref>
*If ''Y'' is irreducible with generic point ''y'', and if ''f''{{i sup|−1}}(''y'') is irreducible, then ''X'' is irreducible.<ref>EGA IV<sub>2</sub>, Corollaire 2.3.5(iii).</ref>
*If ''f'' is also closed, the image of every connected component of ''X'' is a connected component of ''Y''.<ref>EGA IV<sub>2</sub>, Proposition 2.3.6(ii).</ref>
*For every pro-constructible subset ''Z'' of ''Y'', <math>f^{-1}(\bar Z) = \overline{f^{-1}(Z)}</math>.<ref>EGA IV<sub>2</sub>, Théorème 2.3.10.</ref>

If ''f'' is flat and locally of finite presentation, then ''f'' is universally open.<ref>EGA IV<sub>2</sub>, Théorème 2.4.6.</ref> 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 scheme|noetherian]].<ref>EGA IV<sub>2</sub>, Remarques 2.4.8(i).</ref> Furthermore, no converse to this statement holds: If ''f'' is the canonical map from the reduced scheme ''X''<sub>red</sub> to ''X'', then ''f'' is a universal homeomorphism, but for ''X'' non-reduced and noetherian, ''f'' is never flat.<ref>EGA IV<sub>2</sub>, Remarques 2.4.8(ii).</ref>

If <math>f\colon X \to Y</math> is faithfully flat, then:
*The topology on ''Y'' is the [[quotient topology]] relative to ''f''.<ref>EGA IV<sub>2</sub>, Corollaire 2.3.12.</ref>
*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''.<ref>EGA IV<sub>2</sub>, Corollaire 2.3.14.</ref>

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:<ref>EGA IV<sub>3</sub>, Théorème 12.1.6.</ref>
*Serre's condition S<sub>''k''</sub> (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:<ref>EGA IV<sub>3</sub>, Théorème 12.2.4.</ref>
*Geometrically reduced.
*Geometrically reduced and having ''k'' geometric connected components (for any fixed ''k'').
*Geometrically integral.

===Flatness and dimension===
Assume {{mvar|X}} and {{mvar|Y}} are locally noetherian, and let <math>f\colon X \to Y</math>.

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

===Descent properties===
* 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'').<ref>EGA IV<sub>2</sub>, Proposition 2.1.13.</ref> 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''.<ref>EGA IV<sub>3</sub>, Proposition 11.3.13.</ref>
* 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'').<ref>EGA IV<sub>2</sub>, Proposition 2.1.13.</ref>
* If ''f'' is faithfully flat, ''X'' is locally integral, and the topological space of ''Y'' is locally noetherian, then ''Y'' is locally integral.<ref>EGA IV<sub>2</sub>, Proposition 2.1.14.</ref>
* If ''f'' is faithfully flat and quasi-compact, and if ''X'' is locally noetherian, then ''Y'' is also locally noetherian.<ref>EGA IV<sub>2</sub>, Proposition 2.2.14.</ref>
* 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''.<ref>EGA IV<sub>2</sub>, Corollaire 6.5.2.</ref>
* 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''.<ref>EGA IV<sub>2</sub>, Corollaire 6.5.4.</ref>

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''}}.<ref>EGA IV<sub>2</sub>, Proposition 2.5.1.</ref>

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.<ref>EGA IV<sub>2</sub>, Proposition 2.5.2.</ref>

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'''.<ref>EGA IV<sub>2</sub>, Proposition 2.6.2.</ref>

*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'''.<ref>EGA IV<sub>2</sub>, Corollaire 2.6.4 and Proposition 2.7.1.</ref>
*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 morphism|Finite]].
*[[Quasi-finite morphism|Quasi-finite]].
*Integral.

It is possible for {{′|''f''}} to be a local isomorphism without ''f'' being even a local immersion.<ref>EGA IV<sub>2</sub>, Remarques 2.7.3(iii).</ref>

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.<ref>EGA IV<sub>2</sub>, Corollaire 2.7.2.</ref> 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 morphism|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''.<ref>EGA IV<sub>2</sub>, Remarques 2.7.3(ii).</ref>

== See also ==
* [[fpqc morphism]]
* [[Relative effective Cartier divisor]], an example of a flat morphism
* [[Degeneration (algebraic geometry)]]

== Notes ==
{{Reflist}}

==References==
*{{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}}

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[[Category:Morphisms of schemes]]