Derivation (differential algebra)

<h2 id="properties">Properties</h2>
If A is a K-algebra, for K a ring, and D: A → A is a K-derivation, then

<ul><li>If A has a unit 1, then D(1) = D(12) = 2D(1), so that D(1) = 0. Thus by K-linearity, D(k) = 0 for all k ∈ K.</li>
<li>If A is commutative, D(x2) = xD(x) + D(x)x = 2xD(x), and D(xn) = nxn−1D(x), by the Leibniz rule.</li>
<li>More generally, for any x1, x2, …, xn ∈ A, it follows by <a href="/facts/Mathematical_induction/KxAPqNrH">induction</a> that

D
 (
 
 x
 
 1
 
 
 
 x
 
 2
 
 
 ⋯
 
 x
 
 n
 
 
 )
 =
 
 ∑
 
 i
 
 
 
 x
 
 1
 
 
 ⋯
 
 x
 
 i
 −
 1
 
 
 D
 (
 
 x
 
 i
 
 
 )
 
 x
 
 i
 +
 1
 
 
 ⋯
 
 x
 
 n
 
 
 
 
 {\displaystyle D(x_{1}x_{2}\cdots x_{n})=\sum _{i}x_{1}\cdots x_{i-1}D(x_{i})x_{i+1}\cdots x_{n}}
 
</li></ul>
which is 
 
 
 
 
 ∑
 
 i
 
 
 D
 (
 
 x
 
 i
 
 
 )
 
 ∏
 
 j
 ≠
 i
 
 
 
 x
 
 j
 
 
 
 
 {\textstyle \sum _{i}D(x_{i})\prod _{j\neq i}x_{j}}
 
 if for all i, D(xi) commutes with 
 
 
 
 
 x
 
 1
 
 
 ,
 
 x
 
 2
 
 
 ,
 …
 ,
 
 x
 
 i
 −
 1
 
 
 
 
 {\displaystyle x_{1},x_{2},\ldots ,x_{i-1}}
 
.
<ul><li>For n > 1, Dn is not a derivation, instead satisfying a higher-order Leibniz rule:</li></ul>

D
          
            n
          
        
        (
        u
        v
        )
        =
        
          ∑
          
            k
            =
            0
          
          
            n
          
        
        
          
            
              (
            
            
              n
              k
            
            
              )
            
          
        
        ⋅
        
          D
          
            n
            −
            k
          
        
        (
        u
        )
        ⋅
        
          D
          
            k
          
        
        (
        v
        )
        .
      
    
    {\displaystyle D^{n}(uv)=\sum _{k=0}^{n}{\binom {n}{k}}\cdot D^{n-k}(u)\cdot D^{k}(v).}

Moreover, if M is an A-bimodule, write

Der
          
            K
          
        
        ⁡
        (
        A
        ,
        M
        )
      
    
    {\displaystyle \operatorname {Der} _{K}(A,M)}

for the set of K-derivations from A to M.
<ul><li>DerK(A, M) is a <a href="/facts/Module_(mathematics)/jP0LIhFL">module</a> over K.</li>
<li>DerK(A) is a <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> with Lie bracket defined by the <a href="/facts/Commutator/eYxzFmFY">commutator</a>:</li></ul>

[
        
          D
          
            1
          
        
        ,
        
          D
          
            2
          
        
        ]
        =
        
          D
          
            1
          
        
        ∘
        
          D
          
            2
          
        
        −
        
          D
          
            2
          
        
        ∘
        
          D
          
            1
          
        
        .
      
    
    {\displaystyle [D_{1},D_{2}]=D_{1}\circ D_{2}-D_{2}\circ D_{1}.}

since it is readily verified that the commutator of two derivations is again a derivation.
<ul><li>There is an A-module ΩA/K (called the <a href="/facts/K%C3%A4hler_differentials/1aPfygbf">Kähler differentials</a>) with a K-derivation d: A → ΩA/K through which any derivation D: A → M factors. That is, for any derivation D there is a A-module map φ with</li></ul>

D
        :
        A
        
          
            
              
                ⟶
              
              
                d
              
            
          
        
        
          Ω
          
            A
            
              /
            
            K
          
        
        
          
            
              
                ⟶
              
              
                φ
              
            
          
        
        M
      
    
    {\displaystyle D:A{\stackrel {d}{\longrightarrow }}\Omega _{A/K}{\stackrel {\varphi }{\longrightarrow }}M}

The correspondence 
 
 
 
 D
 ↔
 φ
 
 
 {\displaystyle D\leftrightarrow \varphi }
 
 is an isomorphism of A-modules:

Der
          
            K
          
        
        ⁡
        (
        A
        ,
        M
        )
        ≃
        
          Hom
          
            A
          
        
        ⁡
        (
        
          Ω
          
            A
            
              /
            
            K
          
        
        ,
        M
        )
      
    
    {\displaystyle \operatorname {Der} _{K}(A,M)\simeq \operatorname {Hom} _{A}(\Omega _{A/K},M)}

<ul><li>If k ⊂ K is a <a href="/facts/Subring/M0perybt">subring</a>, then A inherits a k-algebra structure, so there is an inclusion</li></ul>

Der
          
            K
          
        
        ⁡
        (
        A
        ,
        M
        )
        ⊂
        
          Der
          
            k
          
        
        ⁡
        (
        A
        ,
        M
        )
        ,
      
    
    {\displaystyle \operatorname {Der} _{K}(A,M)\subset \operatorname {Der} _{k}(A,M),}

since any K-derivation is a fortiori a k-derivation.
<h2 id="graded-derivations">Graded derivations</h2>
Given a <a href="/facts/Graded_algebra/AIUjkSaP">graded algebra</a> A and a homogeneous linear map D of grade |D| on A, D is a homogeneous derivation if

D
          (
          a
          b
          )
          =
          D
          (
          a
          )
          b
          +
          
            ε
            
              
                |
              
              a
              
                |
              
              
                |
              
              D
              
                |
              
            
          
          a
          D
          (
          b
          )
        
      
    
    {\displaystyle {D(ab)=D(a)b+\varepsilon ^{|a||D|}aD(b)}}

for every homogeneous element a and every element b of A for a commutator factor ε = ±1. A graded derivation is sum of homogeneous derivations with the same ε.
If ε = 1, this definition reduces to the usual case. If ε = −1, however, then

D
          (
          a
          b
          )
          =
          D
          (
          a
          )
          b
          +
          (
          −
          1
          
            )
            
              
                |
              
              a
              
                |
              
              
                |
              
              D
              
                |
              
            
          
          a
          D
          (
          b
          )
        
      
    
    {\displaystyle {D(ab)=D(a)b+(-1)^{|a||D|}aD(b)}}

for odd |D|, and D is called an anti-derivation.
Examples of anti-derivations include the <a href="/facts/Exterior_derivative/p7HA0wze">exterior derivative</a> and the <a href="/facts/Interior_product/DjfNykHd">interior product</a> acting on <a href="/facts/Differential_form/T9gyHtMv">differential forms</a>.
Graded derivations of <a href="/facts/Superalgebra/96vSHP12">superalgebras</a> (i.e. Z2-graded algebras) are often called superderivations.

<h2 id="related-notions">Related notions</h2>
<a href="/facts/Hasse%E2%80%93Schmidt_derivation/A3xoT2AJ">Hasse–Schmidt derivations</a> are K-algebra homomorphisms

A
        →
        A
        [
        [
        t
        ]
        ]
        .
      
    
    {\displaystyle A\to A[[t]].}

Composing further with the map that sends a <a href="/facts/Formal_power_series/CNmaG0Vk">formal power series</a> 
 
 
 
 ∑
 
 a
 
 n
 
 
 
 t
 
 n
 
 
 
 
 {\displaystyle \sum a_{n}t^{n}}
 
 to the coefficient 
 
 
 
 
 a
 
 1
 
 
 
 
 {\displaystyle a_{1}}
 
 gives a derivation.

<h2 id="see-also">See also</h2>
<ul><li>In <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a> derivations are <a href="/facts/Tangent_space/crvpbSeG">tangent vectors</a></li>
<li><a href="/facts/K%C3%A4hler_differential/1aPfygbf">Kähler differential</a></li>
<li><a href="/facts/Hasse_derivative/B6mPTDeJ">Hasse derivative</a></li>
<li><a href="/facts/P-derivation/g5f5cHJg">p-derivation</a></li>
<li><a href="/facts/Wirtinger_derivatives/n2n6EJ8p">Wirtinger derivatives</a></li>
<li><a href="/facts/Derivative_of_the_exponential_map/1gkglfwg">Derivative of the exponential map</a></li></ul>

<ul><li><a href="/facts/Nicolas_Bourbaki/qRJhs9nr">Bourbaki, Nicolas</a> (1989), Algebra I, Elements of mathematics, Springer-Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-64243-9.</li>
<li><a href="/facts/David_Eisenbud/hw2yEnnx">Eisenbud, David</a> (1999), Commutative algebra with a view toward algebraic geometry (3rd. ed.), Springer-Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-94269-8.</li>
<li>Matsumura, Hideyuki (1970), Commutative algebra, Mathematics lecture note series, W. A. Benjamin, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8053-7025-6.</li>
<li>Kolař, Ivan; Slovák, Jan; Michor, Peter W. (1993), <a href="http://www.emis.de/monographs/KSM/index.html">Natural operations in differential geometry</a>, Springer-Verlag.</li></ul>

Derivation (differential algebra) open-in-new

Derivation (differential algebra)