Flat morphism

<h2 id="examplesnon-examples">Examples/non-examples</h2>
Consider the affine scheme morphism

Spec
        ⁡
        
          (
          
            
              
                
                  C
                
                [
                x
                ,
                y
                ,
                t
                ]
              
              
                
                  x
                  
                    2
                  
                
                +
                
                  y
                  
                    2
                  
                
                −
                t
              
            
          
          )
        
        →
        Spec
        ⁡
        (
        
          C
        
        [
        t
        ]
        )
      
    
    {\displaystyle \operatorname {Spec} \left({\frac {\mathbb {C} [x,y,t]}{x^{2}+y^{2}-t}}\right)\to \operatorname {Spec} (\mathbb {C} [t])}

induced from the morphism of algebras

{
            
              
                
                  
                    C
                  
                  [
                  t
                  ]
                  →
                  
                    
                      
                        
                          C
                        
                        [
                        x
                        ,
                        y
                        ,
                        t
                        ]
                      
                      
                        
                          x
                          
                            2
                          
                        
                        +
                        
                          y
                          
                            2
                          
                        
                        −
                        t
                      
                    
                  
                
              
              
                
                  t
                  ↦
                  t
                  .
                
              
            
            
          
        
      
    
    {\displaystyle {\begin{cases}\mathbb {C} [t]\to {\frac {\mathbb {C} [x,y,t]}{x^{2}+y^{2}-t}}\\t\mapsto t.\\\end{cases}}}

Since proving flatness for this morphism amounts to computing<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

Tor
          
            1
          
          
            
              C
            
            [
            t
            ]
          
        
        ⁡
        
          (
          
            
              
                
                  
                    C
                  
                  [
                  x
                  ,
                  y
                  ,
                  t
                  ]
                
                
                  
                    x
                    
                      2
                    
                  
                  +
                  
                    y
                    
                      2
                    
                  
                  −
                  t
                
              
            
            ,
            
              C
            
          
          )
        
        ,
      
    
    {\displaystyle \operatorname {Tor} _{1}^{\mathbb {C} [t]}\left({\frac {\mathbb {C} [x,y,t]}{x^{2}+y^{2}-t}},\mathbb {C} \right),}

we resolve the complex numbers

0
              
              
                →
              
              
                
                  C
                
                [
                t
                ]
              
              
                
                  
                    →
                    
                      ⋅
                      t
                    
                  
                
              
              
                
                  C
                
                [
                t
                ]
              
              
                →
              
              
                0
              
            
            
              
                ↓
              
              
              
                ↓
              
              
              
                ↓
              
              
              
                ↓
              
            
            
              
                0
              
              
                →
              
              
                0
              
              
                →
              
              
                
                  C
                
              
              
                →
              
              
                0
              
            
          
        
      
    
    {\displaystyle {\begin{array}{lcccc}0&\to &\mathbb {C} [t]&{\xrightarrow {\cdot t}}&\mathbb {C} [t]&\to &0\\\downarrow &&\downarrow &&\downarrow &&\downarrow \\0&\to &0&\to &\mathbb {C} &\to &0\\\end{array}}}

and tensor by the module representing our scheme giving the sequence of 
 
 
 
 
 C
 
 [
 t
 ]
 
 
 {\displaystyle \mathbb {C} [t]}
 
-modules

0
        →
        
          
            
              
                C
              
              [
              x
              ,
              y
              ,
              t
              ]
            
            
              
                x
                
                  2
                
              
              +
              
                y
                
                  2
                
              
              −
              t
            
          
        
        
          
            →
            
              ⋅
              t
            
          
        
        
          
            
              
                C
              
              [
              x
              ,
              y
              ,
              t
              ]
            
            
              
                x
                
                  2
                
              
              +
              
                y
                
                  2
                
              
              −
              t
            
          
        
        →
        0
      
    
    {\displaystyle 0\to {\frac {\mathbb {C} [x,y,t]}{x^{2}+y^{2}-t}}{\xrightarrow {\cdot t}}{\frac {\mathbb {C} [x,y,t]}{x^{2}+y^{2}-t}}\to 0}

Because t is not a <a href="/facts/Zero_divisor/5GraAU8o">zero divisor</a> we have a trivial kernel, hence the homology group vanishes.

<h3>Miracle flatness</h3>
Other examples of flat morphisms can be found using "miracle flatness"<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> which states that if you have a morphism 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f\colon X\to Y}
 
 between a <a href="/facts/Cohen%E2%80%93Macaulay_ring/4G8wymtC">Cohen–Macaulay scheme</a> to a regular scheme with equidimensional fibers, then it is flat. Easy examples of this are <a href="/facts/Elliptic_surface/XRDYGhYb">elliptic fibrations</a>, smooth morphisms, and morphisms to <a href="/facts/Topologically_stratified_space/1Ldd6EgY">stratified varieties</a> which satisfy miracle flatness on each of the strata.

<h3>Hilbert schemes</h3>
The universal examples of flat morphisms of schemes are given by <a href="/facts/Hilbert_scheme/kJcbts4e">Hilbert schemes</a>. 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
 :
 X
 →
 S
 
 
 {\displaystyle f\colon X\to S}
 
 is flat, there exists a commutative diagram

X
              
              
                →
              
              
                
                  Hilb
                  
                    S
                  
                
              
            
            
              
                ↓
              
              
              
                ↓
              
            
            
              
                S
              
              
                →
              
              
                S
              
            
          
        
      
    
    {\displaystyle {\begin{matrix}X&\to &\operatorname {Hilb} _{S}\\\downarrow &&\downarrow \\S&\to &S\end{matrix}}}

for the Hilbert scheme of all flat morphisms to 
 
 
 
 S
 
 
 {\displaystyle S}
 
. Since 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is flat, the fibers 
 
 
 
 
 f
 
 s
 
 
 :
 
 X
 
 s
 
 
 →
 s
 
 
 {\displaystyle f_{s}\colon X_{s}\to s}
 
 all have the same Hilbert polynomial 
 
 
 
 Φ
 
 
 {\displaystyle \Phi }
 
, hence we could have similarly written 
 
 
 
 
 
 Hilb
 
 
 S
 
 
 Φ
 
 
 
 
 {\displaystyle {\text{Hilb}}_{S}^{\Phi }}
 
 for the Hilbert scheme above.

<h3>Non-examples</h3>
<h4>Blowup</h4>
One class of non-examples are given by blowup maps

Bl
          
            I
          
        
        ⁡
        X
        →
        X
        .
      
    
    {\displaystyle \operatorname {Bl} _{I}X\to X.}

One easy example is the <a href="/facts/Blowing_up/YejuoWYu">blowup</a> of a point in 
 
 
 
 
 C
 
 [
 x
 ,
 y
 ]
 
 
 {\displaystyle \mathbb {C} [x,y]}
 
. If we take the origin, this is given by the morphism

C
        
        [
        x
        ,
        y
        ]
        →
        
          
            
              
                C
              
              [
              x
              ,
              y
              ,
              s
              ,
              t
              ]
            
            
              x
              t
              −
              y
              s
            
          
        
      
    
    {\displaystyle \mathbb {C} [x,y]\to {\frac {\mathbb {C} [x,y,s,t]}{xt-ys}}}
  
 sending 
  
    
      
        x
        ↦
        x
        ,
        y
        ↦
        y
        ,
      
    
    {\displaystyle x\mapsto x,y\mapsto y,}

where the fiber over a point 
 
 
 
 (
 a
 ,
 b
 )
 ≠
 (
 0
 ,
 0
 )
 
 
 {\displaystyle (a,b)\neq (0,0)}
 
 is a copy of 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} }
 
, i.e.,

C
              
              [
              x
              ,
              y
              ,
              s
              ,
              t
              ]
            
            
              x
              t
              −
              y
              s
            
          
        
        
          ⊗
          
            
              C
            
            [
            x
            ,
            y
            ]
          
        
        
          
            
              
                C
              
              [
              x
              ,
              y
              ]
            
            
              (
              x
              −
              a
              ,
              y
              −
              b
              )
            
          
        
        ≅
        
          C
        
        ,
      
    
    {\displaystyle {\frac {\mathbb {C} [x,y,s,t]}{xt-ys}}\otimes _{\mathbb {C} [x,y]}{\frac {\mathbb {C} [x,y]}{(x-a,y-b)}}\cong \mathbb {C} ,}

which follows from

M
        
          ⊗
          
            R
          
        
        
          
            R
            I
          
        
        ≅
        
          
            M
            
              I
              M
            
          
        
        .
      
    
    {\displaystyle M\otimes _{R}{\frac {R}{I}}\cong {\frac {M}{IM}}.}

But for 
 
 
 
 a
 =
 b
 =
 0
 
 
 {\displaystyle a=b=0}
 
, we get the isomorphism

C
              
              [
              x
              ,
              y
              ,
              s
              ,
              t
              ]
            
            
              x
              t
              −
              y
              s
            
          
        
        
          ⊗
          
            
              C
            
            [
            x
            ,
            y
            ]
          
        
        
          
            
              
                C
              
              [
              x
              ,
              y
              ]
            
            
              (
              x
              ,
              y
              )
            
          
        
        ≅
        
          C
        
        [
        s
        ,
        t
        ]
        .
      
    
    {\displaystyle {\frac {\mathbb {C} [x,y,s,t]}{xt-ys}}\otimes _{\mathbb {C} [x,y]}{\frac {\mathbb {C} [x,y]}{(x,y)}}\cong \mathbb {C} [s,t].}

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

<h4>Infinite resolution</h4>
A simple non-example of a flat morphism is 
 
 
 
 k
 [
 ε
 ]
 =
 k
 [
 x
 ]
 
 /
 
 (
 
 x
 
 2
 
 
 )
 →
 k
 .
 
 
 {\displaystyle k[\varepsilon ]=k[x]/(x^{2})\to k.}
 
 This is because

k
        
          ⊗
          
            k
            [
            ε
            ]
          
          
            
              L
            
          
        
        k
      
    
    {\displaystyle k\otimes _{k[\varepsilon ]}^{\mathbf {L} }k}

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

⋯
         
        
          
            →
            
              
                
                  ⋅
                  
                
              
              ε
            
          
        
         
        k
        [
        ε
        ]
         
        
          
            →
            
              ⋅
              ε
            
          
        
         
        k
        [
        ε
        ]
        
          
            →
            
              ⋅
              ε
            
          
        
        k
        [
        ε
        ]
        →
        k
      
    
    {\displaystyle \cdots ~{\xrightarrow {{\overset {}{\cdot }}\varepsilon }}~k[\varepsilon ]~{\xrightarrow {\cdot \varepsilon }}~k[\varepsilon ]{\xrightarrow {\cdot \varepsilon }}k[\varepsilon ]\to k}

and tensor the resolution with k, we find that

k
        
          ⊗
          
            k
            [
            ε
            ]
          
          
            
              L
            
          
        
        k
        ≃
        
          ⨁
          
            i
            =
            0
          
          
            ∞
          
        
        k
        [
        +
        i
        ]
      
    
    {\displaystyle k\otimes _{k[\varepsilon ]}^{\mathbf {L} }k\simeq \bigoplus _{i=0}^{\infty }k[+i]}

showing that the morphism cannot be flat. Another non-example of a flat morphism is a <a href="/facts/Blowing_up/YejuoWYu">blowup</a> since a flat morphism necessarily has equi-dimensional fibers.

<h2 id="properties-of-flat-morphisms">Properties of flat morphisms</h2>
Let 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f\colon X\to Y}
 
 be a morphism of schemes. For a morphism 
 
 
 
 g
 :
 
 Y
 ′
 
 →
 Y
 
 
 {\displaystyle g\colon Y'\to Y}
 
, let 
 
 
 
 
 X
 ′
 
 =
 X
 
 ×
 
 Y
 
 
 
 Y
 ′
 
 
 
 {\displaystyle X'=X\times _{Y}Y'}
 
 and 
 
 
 
 
 f
 ′
 
 =
 (
 f
 ,
 
 1
 
 
 Y
 ′
 
 
 
 )
 :
 
 X
 ′
 
 →
 
 Y
 ′
 
 .
 
 
 {\displaystyle f'=(f,1_{Y'})\colon X'\to Y'.}
 
 The morphism f is flat if and only if for every g, the pullback 
 
 
 
 
 f
 
 ′
 
 ∗
 
 
 
 
 
 {\displaystyle f'^{*}}
 
 is an exact functor from the category of quasi-coherent 
 
 
 
 
 
 
 O
 
 
 
 
 Y
 ′
 
 
 
 
 
 {\displaystyle {\mathcal {O}}_{Y'}}
 
-modules to the category of quasi-coherent 
 
 
 
 
 
 
 O
 
 
 
 
 X
 ′
 
 
 
 
 
 {\displaystyle {\mathcal {O}}_{X'}}
 
-modules.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>
Assume 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f\colon X\to Y}
 
 and 
 
 
 
 g
 :
 Y
 →
 Z
 
 
 {\displaystyle g\colon Y\to Z}
 
 are morphisms of schemes and f is flat at x in X. Then g is flat at 
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
 if and only if gf is flat at x.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> 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.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

<h3>Fundamental properties</h3>
<ul><li>The composite of two flat morphisms is flat.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a></li>
<li>The fiber product of two flat or faithfully flat morphisms is a flat or faithfully flat morphism, respectively.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a></li>
<li>Flatness and faithful flatness is preserved by base change: If f is flat or faithfully flat and 
 
 
 
 g
 :
 
 Y
 ′
 
 →
 Y
 
 
 {\displaystyle g\colon Y'\to Y}
 
, then the fiber product 
 
 
 
 f
 ×
 g
 :
 X
 
 ×
 
 Y
 
 
 
 Y
 ′
 
 →
 
 Y
 ′
 
 
 
 {\displaystyle f\times g\colon X\times _{Y}Y'\to Y'}
 
 is flat or faithfully flat, respectively.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a></li>
<li>The set of points where a morphism (locally of finite presentation) is flat is open.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a></li>
<li>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.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a></li></ul>
Suppose 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f\colon X\to Y}
 
 is a flat morphism of schemes.

<ul><li>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
 →
 
 
 
 O
 
 
 
 X
 
 
 
 
 {\displaystyle f^{*}J\to {\mathcal {O}}_{X}}
 
, the pullback of the inclusion map, is an injection, and the image of 
 
 
 
 
 f
 
 ∗
 
 
 J
 
 
 {\displaystyle f^{*}J}
 
 in 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 
 
 {\displaystyle {\mathcal {O}}_{X}}
 
 is the annihilator of 
 
 
 
 
 f
 
 ∗
 
 
 F
 
 
 {\displaystyle f^{*}F}
 
 on X.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a></li>
<li>If f is faithfully flat and if G is a quasi-coherent 
 
 
 
 
 
 
 O
 
 
 
 Y
 
 
 
 
 {\displaystyle {\mathcal {O}}_{Y}}
 
-module, then the pullback map on global sections 
 
 
 
 Γ
 (
 Y
 ,
 G
 )
 →
 Γ
 (
 X
 ,
 
 f
 
 ∗
 
 
 G
 )
 
 
 {\displaystyle \Gamma (Y,G)\to \Gamma (X,f^{*}G)}
 
 is injective.<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a></li></ul>
Suppose 
 
 
 
 h
 :
 
 S
 ′
 
 →
 S
 
 
 {\displaystyle h\colon S'\to S}
 
 is flat. Let X and Y be S-schemes, and let 
 
 
 
 
 X
 ′
 
 
 
 {\displaystyle X'}
 
 and 
 
 
 
 
 Y
 ′
 
 
 
 {\displaystyle Y'}
 
 be their base change by h.

<ul><li>If 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f\colon X\to Y}
 
 is quasi-compact and dominant, then its base change 
 
 
 
 
 f
 ′
 
 :
 
 X
 ′
 
 →
 
 Y
 ′
 
 
 
 {\displaystyle f'\colon X'\to Y'}
 
 is quasi-compact and dominant.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a></li>
<li>If h is faithfully flat, then the pullback map 
 
 
 
 
 Hom
 
 S
 
 
 ⁡
 (
 X
 ,
 Y
 )
 →
 
 Hom
 
 
 S
 ′
 
 
 
 ⁡
 (
 
 X
 ′
 
 ,
 
 Y
 ′
 
 )
 
 
 {\displaystyle \operatorname {Hom} _{S}(X,Y)\to \operatorname {Hom} _{S'}(X',Y')}
 
 is injective.<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a></li>
<li>Assume 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f\colon X\to Y}
 
 is quasi-compact and quasi-separated. Let Z be the closed image of X, and let 
 
 
 
 j
 :
 Z
 →
 Y
 
 
 {\displaystyle j\colon Z\to Y}
 
 be the canonical injection. Then the closed subscheme determined by the base change 
 
 
 
 
 j
 ′
 
 :
 
 Z
 ′
 
 →
 
 Y
 ′
 
 
 
 {\displaystyle j'\colon Z'\to Y'}
 
 is the closed image of 
 
 
 
 
 X
 ′
 
 
 
 {\displaystyle X'}
 
.<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a></li></ul>
<h3>Topological properties</h3>
If 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f\colon X\to Y}
 
 is flat, then it possesses all of the following properties:

<ul><li>For every point x of X and every generization y′ of y = f(x), there is a generization x′ of x such that y′ = f(x′).<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a></li>
<li>For every point x of X, 
 
 
 
 f
 (
 Spec
 ⁡
 
 
 
 O
 
 
 
 X
 ,
 x
 
 
 )
 =
 Spec
 ⁡
 
 
 
 O
 
 
 
 Y
 ,
 f
 (
 x
 )
 
 
 
 
 {\displaystyle f(\operatorname {Spec} {\mathcal {O}}_{X,x})=\operatorname {Spec} {\mathcal {O}}_{Y,f(x)}}
 
.<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a></li>
<li>For every irreducible closed subset Y′ of Y, every irreducible component of f−1(Y′) dominates Y′.<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a></li>
<li>If Z and Z′ are two irreducible closed subsets of Y with Z contained in Z′, then for every irreducible component T of f−1(Z), there is an irreducible component T′ of f−1(Z′) containing T.<a class="footnote-ref" id="fnref:21" href="#fn:21">21</a></li>
<li>For every irreducible component T of X, the closure of f(T) is an irreducible component of Y.<a class="footnote-ref" id="fnref:22" href="#fn:22">22</a></li>
<li>If Y is irreducible with generic point y, and if f−1(y) is irreducible, then X is irreducible.<a class="footnote-ref" id="fnref:23" href="#fn:23">23</a></li>
<li>If f is also closed, the image of every connected component of X is a connected component of Y.<a class="footnote-ref" id="fnref:24" href="#fn:24">24</a></li>
<li>For every pro-constructible subset Z of Y, 
 
 
 
 
 f
 
 −
 1
 
 
 (
 
 
 
 Z
 ¯
 
 
 
 )
 =
 
 
 
 
 f
 
 −
 1
 
 
 (
 Z
 )
 
 ¯
 
 
 
 
 {\displaystyle f^{-1}({\bar {Z}})={\overline {f^{-1}(Z)}}}
 
.<a class="footnote-ref" id="fnref:25" href="#fn:25">25</a></li></ul>
If f is flat and locally of finite presentation, then f is universally open.<a class="footnote-ref" id="fnref:26" href="#fn:26">26</a> 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.<a class="footnote-ref" id="fnref:27" href="#fn:27">27</a> Furthermore, no converse to this statement holds: If f is the canonical map from the reduced scheme Xred to X, then f is a universal homeomorphism, but for X non-reduced and noetherian, f is never flat.<a class="footnote-ref" id="fnref:28" href="#fn:28">28</a>
If 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f\colon X\to Y}
 
 is faithfully flat, then:

<ul><li>The topology on Y is the <a href="/facts/Quotient_topology/muE0qQVC">quotient topology</a> relative to f.<a class="footnote-ref" id="fnref:29" href="#fn:29">29</a></li>
<li>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−1(Z) is a locally closed pro-constructible subset of X.<a class="footnote-ref" id="fnref:30" href="#fn:30">30</a></li></ul>
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:<a class="footnote-ref" id="fnref:31" href="#fn:31">31</a>

<ul><li>Serre's condition Sk (for any fixed k).</li>
<li>Geometrically regular.</li>
<li>Geometrically normal.</li></ul>
If in addition f is proper, then the same is true for each of the following properties:<a class="footnote-ref" id="fnref:32" href="#fn:32">32</a>

<ul><li>Geometrically reduced.</li>
<li>Geometrically reduced and having k geometric connected components (for any fixed k).</li>
<li>Geometrically integral.</li></ul>
<h3>Flatness and dimension</h3>
Assume 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are locally noetherian, and let 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f\colon X\to Y}
 
.

<ul><li>Let x be a point of X and y = f(x). If f is flat, then dimx X = dimy Y + dimx f−1(y).<a class="footnote-ref" id="fnref:33" href="#fn:33">33</a> Conversely, if this equality holds for all x, X is <a href="/facts/Glossary_of_scheme_theory/eh7NLhmS">Cohen–Macaulay</a>, and Y is <a href="/facts/Glossary_of_scheme_theory/eh7NLhmS">regular</a>, and furthermore f maps closed points to closed points, then f is flat.<a class="footnote-ref" id="fnref:34" href="#fn:34">34</a></li>
<li>If f is faithfully flat, then for each closed subset Z of Y, codimY(Z) = codimX(f−1(Z)).<a class="footnote-ref" id="fnref:35" href="#fn:35">35</a></li>
<li>Suppose f is flat and F is a quasi-coherent module over Y. If F has projective dimension at most n, then 
 
 
 
 
 f
 
 ∗
 
 
 F
 
 
 {\displaystyle f^{*}F}
 
 has projective dimension at most n.<a class="footnote-ref" id="fnref:36" href="#fn:36">36</a></li></ul>
<h3>Descent properties</h3>
<ul><li>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).<a class="footnote-ref" id="fnref:37" href="#fn:37">37</a> Conversely, if f is also of finite presentation and f−1(y) is reduced or normal, respectively, at x, then X is reduced or normal, respectively, at x.<a class="footnote-ref" id="fnref:38" href="#fn:38">38</a></li>
<li>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.</li>
<li>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).<a class="footnote-ref" id="fnref:39" href="#fn:39">39</a></li>
<li>If f is faithfully flat, X is locally integral, and the topological space of Y is locally noetherian, then Y is locally integral.<a class="footnote-ref" id="fnref:40" href="#fn:40">40</a></li>
<li>If f is faithfully flat and quasi-compact, and if X is locally noetherian, then Y is also locally noetherian.<a class="footnote-ref" id="fnref:41" href="#fn:41">41</a></li>
<li>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−1(f(x)) is regular at x, then X is regular at x.<a class="footnote-ref" id="fnref:42" href="#fn:42">42</a></li>
<li>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−1(f(x)) is normal at x, then X is normal at x.<a class="footnote-ref" id="fnref:43" href="#fn:43">43</a></li></ul>
Let 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′.<a class="footnote-ref" id="fnref:44" href="#fn:44">44</a>
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.<a class="footnote-ref" id="fnref:45" href="#fn:45">45</a>
Suppose f : X → Y is an S-morphism of S-schemes. Let 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.<a class="footnote-ref" id="fnref:46" href="#fn:46">46</a>

<ul><li>Open.</li>
<li>Closed.</li>
<li>Quasi-compact and a homeomorphism onto its image.</li>
<li>A homeomorphism.</li></ul>
Additionally, for each of the following properties P, f has P if and only if f′ has P.<a class="footnote-ref" id="fnref:47" href="#fn:47">47</a>

<ul><li>Universally open.</li>
<li>Universally closed.</li>
<li>A universal homeomorphism.</li>
<li>Quasi-compact.</li>
<li>Quasi-compact and dominant.</li>
<li>Quasi-compact and universally bicontinuous.</li>
<li>Separated.</li>
<li>Quasi-separated.</li>
<li>Locally of finite type.</li>
<li>Locally of finite presentation.</li>
<li>Finite type.</li>
<li>Finite presentation.</li>
<li>Proper.</li>
<li>An isomorphism.</li>
<li>A monomorphism.</li>
<li>An open immersion.</li>
<li>A quasi-compact immersion.</li>
<li>A closed immersion.</li>
<li>Affine.</li>
<li>Quasi-affine.</li>
<li>Finite.</li>
<li>Quasi-finite.</li>
<li>Integral.</li></ul>
It is possible for f′ to be a local isomorphism without f being even a local immersion.<a class="footnote-ref" id="fnref:48" href="#fn:48">48</a>
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.<a class="footnote-ref" id="fnref:49" href="#fn:49">49</a> 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.<a class="footnote-ref" id="fnref:50" href="#fn:50">50</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Fpqc_morphism/dPeLNkoM">fpqc morphism</a></li>
<li><a href="/facts/Relative_effective_Cartier_divisor/JoIxmu2k">Relative effective Cartier divisor</a>, an example of a flat morphism</li>
<li><a href="/facts/Degeneration_(algebraic_geometry)/B0NJbDLR">Degeneration (algebraic geometry)</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/David_Eisenbud/hw2yEnnx">Eisenbud, David</a> (1995), Commutative algebra, Graduate Texts in Mathematics, vol. 150, Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-94268-1, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1322960">1322960</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-94269-8, section 6.</li>
<li><a href="/facts/Jean-Pierre_Serre/1xSr7BCh">Serre, Jean-Pierre</a> (1956), <a href="http://www.numdam.org/numdam-bin/item?id=AIF_1956__6__1_0">"Géométrie algébrique et géométrie analytique"</a>, <a href="/facts/Annales_de_l%27Institut_Fourier/2vyuchLJ">Annales de l'Institut Fourier</a>, 6: 1–42, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.5802%2Faif.59">10.5802/aif.59</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0373-0956">0373-0956</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0082175">0082175</a></li>
<li><a href="/facts/Alexander_Grothendieck/jLQGPY7M">Grothendieck, Alexandre</a>; <a href="/facts/Jean_Dieudonn%C3%A9/gWagzMdu">Dieudonné, Jean</a> (1960). <a href="http://www.numdam.org/item/PMIHES_1960__4__5_0">"Éléments de géométrie algébrique: I. Le langage des schémas"</a>. <a href="/facts/Publications_Math%C3%A9matiques_de_l%27IH%C3%89S/2ur5ofxL">Publications Mathématiques de l'IHÉS</a>. 4. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fbf02684778">10.1007/bf02684778</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0217083">0217083</a>.</li>
<li><a href="/facts/Alexander_Grothendieck/jLQGPY7M">Grothendieck, Alexandre</a>; <a href="/facts/Jean_Dieudonn%C3%A9/gWagzMdu">Dieudonné, Jean</a> (1965). <a href="http://www.numdam.org/articles/PMIHES_1965__24__5_0">"Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Seconde partie"</a>. <a href="/facts/Publications_Math%C3%A9matiques_de_l%27IH%C3%89S/2ur5ofxL">Publications Mathématiques de l'IHÉS</a>. 24. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fbf02684322">10.1007/bf02684322</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0199181">0199181</a>.</li>
<li><a href="/facts/Alexander_Grothendieck/jLQGPY7M">Grothendieck, Alexandre</a>; <a href="/facts/Jean_Dieudonn%C3%A9/gWagzMdu">Dieudonné, Jean</a> (1966). <a href="http://www.numdam.org/articles/PMIHES_1966__28__5_0">"Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Troisième partie"</a>. <a href="/facts/Publications_Math%C3%A9matiques_de_l%27IH%C3%89S/2ur5ofxL">Publications Mathématiques de l'IHÉS</a>. 28. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fbf02684343">10.1007/bf02684343</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0217086">0217086</a>.</li>
<li><a href="/facts/Robin_Hartshorne/mCBLyIW6">Hartshorne, Robin</a> (1977), <a href="/facts/Algebraic_Geometry_(book)/YcAhHWBj">Algebraic Geometry</a>, <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">Graduate Texts in Mathematics</a>, vol. 52, New York: Springer-Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-90244-9, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0463157">0463157</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">EGA IV2, 2.1.1. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">EGA 0I, 6.7.8. <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Sernesi, E. (2010). Deformations of Algebraic Schemes. Springer. pp. 269–279. <a href="https://archive.org/details/deformationsalge00sern_419" target="_blank">https://archive.org/details/deformationsalge00sern_419</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">"Flat Morphisms and Flatness". <a href="https://ayoucis.wordpress.com/2014/03/12/flat-morphisms-and-flatness/" target="_blank">https://ayoucis.wordpress.com/2014/03/12/flat-morphisms-and-flatness/</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">EGA IV2, Proposition 2.1.3. <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">EGA IV2, Corollaire 2.2.11(iv). <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">EGA IV2, Corollaire 2.2.13(iii). <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">EGA IV2, Corollaire 2.1.6. <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">EGA IV2, Corollaire 2.1.7, and EGA IV2, Corollaire 2.2.13(ii). <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">EGA IV2, Proposition 2.1.4, and EGA IV2, Corollaire 2.2.13(i). <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">EGA IV3, Théorème 11.3.1. <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">EGA IV3, Proposition 11.3.16. <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">EGA IV2, Proposition 2.1.11. <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">EGA IV2, Corollaire 2.2.8. <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">EGA IV2, Proposition 2.3.7(i). <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">EGA IV2, Corollaire 2.2.16. <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">EGA IV2, Proposition 2.3.2. <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">EGA IV2, Proposition 2.3.4(i). <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">EGA IV2, Proposition 2.3.4(ii). <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">EGA IV2, Proposition 2.3.4(iii). <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">EGA IV2, Corollaire 2.3.5(i). <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">EGA IV2, Corollaire 2.3.5(ii). <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">EGA IV2, Corollaire 2.3.5(iii). <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
<li id="fn:24">EGA IV2, Proposition 2.3.6(ii). <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn:25">EGA IV2, Théorème 2.3.10. <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
<li id="fn:26">EGA IV2, Théorème 2.4.6. <a href="#fnref:26" class="footnote-back-ref">↩</a></li>
<li id="fn:27">EGA IV2, Remarques 2.4.8(i). <a href="#fnref:27" class="footnote-back-ref">↩</a></li>
<li id="fn:28">EGA IV2, Remarques 2.4.8(ii). <a href="#fnref:28" class="footnote-back-ref">↩</a></li>
<li id="fn:29">EGA IV2, Corollaire 2.3.12. <a href="#fnref:29" class="footnote-back-ref">↩</a></li>
<li id="fn:30">EGA IV2, Corollaire 2.3.14. <a href="#fnref:30" class="footnote-back-ref">↩</a></li>
<li id="fn:31">EGA IV3, Théorème 12.1.6. <a href="#fnref:31" class="footnote-back-ref">↩</a></li>
<li id="fn:32">EGA IV3, Théorème 12.2.4. <a href="#fnref:32" class="footnote-back-ref">↩</a></li>
<li id="fn:33">EGA IV2, Corollaire 6.1.2. <a href="#fnref:33" class="footnote-back-ref">↩</a></li>
<li id="fn:34">EGA IV2, Proposition 6.1.5. Note that the regularity assumption on Y is important here. The extension 
 
 
 
 
 C
 
 [
 
 x
 
 2
 
 
 ,
 
 y
 
 2
 
 
 ,
 x
 y
 ]
 ⊂
 
 C
 
 [
 x
 ,
 y
 ]
 
 
 {\displaystyle \mathbb {C} [x^{2},y^{2},xy]\subset \mathbb {C} [x,y]}
 
 gives a counterexample with X regular, Y normal, f finite surjective but not flat. <a href="#fnref:34" class="footnote-back-ref">↩</a></li>
<li id="fn:35">EGA IV2, Corollaire 6.1.4. <a href="#fnref:35" class="footnote-back-ref">↩</a></li>
<li id="fn:36">EGA IV2, Corollaire 6.2.2. <a href="#fnref:36" class="footnote-back-ref">↩</a></li>
<li id="fn:37">EGA IV2, Proposition 2.1.13. <a href="#fnref:37" class="footnote-back-ref">↩</a></li>
<li id="fn:38">EGA IV3, Proposition 11.3.13. <a href="#fnref:38" class="footnote-back-ref">↩</a></li>
<li id="fn:39">EGA IV2, Proposition 2.1.13. <a href="#fnref:39" class="footnote-back-ref">↩</a></li>
<li id="fn:40">EGA IV2, Proposition 2.1.14. <a href="#fnref:40" class="footnote-back-ref">↩</a></li>
<li id="fn:41">EGA IV2, Proposition 2.2.14. <a href="#fnref:41" class="footnote-back-ref">↩</a></li>
<li id="fn:42">EGA IV2, Corollaire 6.5.2. <a href="#fnref:42" class="footnote-back-ref">↩</a></li>
<li id="fn:43">EGA IV2, Corollaire 6.5.4. <a href="#fnref:43" class="footnote-back-ref">↩</a></li>
<li id="fn:44">EGA IV2, Proposition 2.5.1. <a href="#fnref:44" class="footnote-back-ref">↩</a></li>
<li id="fn:45">EGA IV2, Proposition 2.5.2. <a href="#fnref:45" class="footnote-back-ref">↩</a></li>
<li id="fn:46">EGA IV2, Proposition 2.6.2. <a href="#fnref:46" class="footnote-back-ref">↩</a></li>
<li id="fn:47">EGA IV2, Corollaire 2.6.4 and Proposition 2.7.1. <a href="#fnref:47" class="footnote-back-ref">↩</a></li>
<li id="fn:48">EGA IV2, Remarques 2.7.3(iii). <a href="#fnref:48" class="footnote-back-ref">↩</a></li>
<li id="fn:49">EGA IV2, Corollaire 2.7.2. <a href="#fnref:49" class="footnote-back-ref">↩</a></li>
<li id="fn:50">EGA IV2, Remarques 2.7.3(ii). <a href="#fnref:50" class="footnote-back-ref">↩</a></li>
</ol>

Flat morphism open-in-new

Flat morphism