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''module of Kähler derivations''
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An algebraic analogue of the concept of the differential of a function. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d0312501.png" /> be a commutative ring regarded as an algebra over a subring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d0312502.png" /> of it. The module of derivations of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d0312503.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d0312504.png" /> is defined as the quotient module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d0312505.png" /> of the free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d0312506.png" />-module with basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d0312507.png" /> by the submodule generated by the elements of the type
+
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d0312508.png" /></td> </tr></table>
+
''module of Kähler derivations''
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d0312509.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125010.png" />. The canonical homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125011.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125012.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125013.png" />-derivation in the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125014.png" /> (cf. [[Derivation in a ring|Derivation in a ring]]) with values in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125015.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125016.png" /> having the following universality property: For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125017.png" />-derivation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125018.png" /> with values in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125019.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125020.png" /> there exists a uniquely defined homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125021.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125022.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125023.png" />. The correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125024.png" /> defines an isomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125025.png" />-modules:
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125026.png" /></td> </tr></table>
 
  
In particular, the module of derivations of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125027.png" /> into itself is isomorphic to the dual <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125028.png" />-module to the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125029.png" />.
+
An algebraic analogue of the concept of the differential of a function. Let  $  A $
 +
be a commutative ring regarded as an algebra over a subring  $  B $
 +
of it. The module of derivations of the  $  B $-
 +
algebra  $  A $
 +
is defined as the quotient module  $  \Omega _ {A/B}  ^ {1} $
 +
of the free  $  A $-
 +
module with basis  $  ( dx ) _ {x \in A }  $
 +
by the submodule generated by the elements of the type
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125030.png" /> is regarded as an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125031.png" />-algebra with respect to the homomorphism
+
$$
 +
d ( x + y ) - dx - dy ,\  d ( xy ) - x  dy - y  dx,\  db ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125032.png" /></td> </tr></table>
+
where  $  x, y \in A $,
 +
$  b \in B $.
 +
The canonical homomorphism of  $  A $-
 +
modules  $  d: A \rightarrow \Omega _ {A/B}  ^ {1} $
 +
is a  $  B $-
 +
derivation in the ring  $  A $(
 +
cf. [[Derivation in a ring|Derivation in a ring]]) with values in the  $  A $-
 +
module  $  \Omega _ {A/B}  ^ {1} $
 +
having the following universality property: For any  $  B $-
 +
derivation  $  \partial  : A \rightarrow M $
 +
with values in an  $  A $-
 +
module  $  M $
 +
there exists a uniquely defined homomorphism of  $  A $-
 +
modules  $  \overline \partial \; : \Omega _ {A/B}  ^ {1} \rightarrow M $
 +
such that  $  \overline \partial \; \circ d = \partial  $.  
 +
The correspondence  $  \partial  \rightarrow \overline \partial \; $
 +
defines an isomorphism of  $  A $-
 +
modules:
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125033.png" /> is the ideal generated by the elements of the type
+
$$
 +
\mathop{\rm Der} _ {B} ( A , M)  \simeq  \mathop{\rm Hom} _ {A} ( \Omega _ {A/B}  ^ {1} , M).
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125034.png" /></td> </tr></table>
+
In particular, the module of derivations of a ring  $  A $
 +
into itself is isomorphic to the dual  $  A $-
 +
module to the module  $  \Omega _ {A/B}  ^ {1} $.
  
then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125035.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125036.png" /> is isomorphic to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125037.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125038.png" />.
+
If  $  A \otimes _ {B} A $
 +
is regarded as an  $  A $-
 +
algebra with respect to the homomorphism
  
The module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125039.png" /> of derivations has the following properties:
+
$$
 +
A  \rightarrow  A \otimes _ {B} A \  ( a \rightarrow a \otimes 1 )
 +
$$
  
1) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125040.png" /> is a multiplicatively closed set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125042.png" />, then there is a canonical localization isomorphism:
+
and $  I $
 +
is the ideal generated by the elements of the type
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125043.png" /></td> </tr></table>
+
$$
 +
a \otimes 1 - 1 \otimes a ,
 +
$$
  
2) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125044.png" /> is a homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125045.png" />-algebras, then there is a canonical exact sequence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125046.png" />-modules:
+
then the  $  A $-
 +
module  $  \Omega _ {A/B}  ^ {1} $
 +
is isomorphic to the  $  A $-
 +
module  $  I / I  ^ {2} $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125047.png" /></td> </tr></table>
+
The module  $  \Omega  ^ {1} $
 +
of derivations has the following properties:
  
3) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125048.png" /> is an ideal of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125050.png" />, then there is an exact canonical sequence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125051.png" />-modules:
+
1) If $  S $
 +
is a multiplicatively closed set in  $  A $
 +
and $  T = S \cap B $,  
 +
then there is a canonical localization isomorphism:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125052.png" /></td> </tr></table>
+
$$
 +
( \Omega _ {A/B}  ^ {1} ) _ {S}  \simeq  \Omega _ {A _ {S}  / B _ {T} }  ^ {1} .
 +
$$
  
where the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125053.png" /> is induced by the derivation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125054.png" />.
+
2) If  $  \phi :  A \rightarrow A _ {1} $
 +
is a homomorphism of  $  B $-
 +
algebras, then there is a canonical exact sequence of  $  A _ {1} $-
 +
modules:
  
4) A field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125055.png" /> is a separable extension of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125056.png" /> of finite transcendence degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125057.png" /> if and only if there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125058.png" />-space isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125059.png" />.
+
$$
 +
\Omega _ {A/B}  ^ {1} \otimes _ { A } A _ {1}  \mathop \rightarrow \limits ^  \alpha    \Omega _ {A _ {1}  / B }  ^ {1}  \rightarrow  \Omega _ {A _ {1}  / A }  \rightarrow  0 .
 +
$$
  
5) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125060.png" /> is an algebra of polynomials, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125061.png" /> is a free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125062.png" />-module with as basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125063.png" />.
+
3) If $  I $
 +
is an ideal of the ring  $  A $
 +
and  $  A _ {1} = A/I $,  
 +
then there is an exact canonical sequence of  $  A _ {1} $-
 +
modules:
  
6) An algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125064.png" /> of finite type over a perfect field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125065.png" /> is a regular ring if and only if the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125066.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125067.png" /> is projective.
+
$$
 +
I / I  ^ {2}  \rightarrow ^ { {d _ 1} } \Omega _ {A/B}  ^ {1} \otimes _ { A } A _ {1}  \rightarrow \
 +
\Omega _ {A _ {1}  / B }  ^ {1}  \rightarrow  0 ,
 +
$$
  
7) Concerning 2) above, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125068.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125069.png" /> of finite type is smooth over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125070.png" /> if and only if the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125071.png" /> is injective while the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125072.png" /> of derivations is projective and its rank is equal to the relative dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125073.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125074.png" />.
+
where the homomorphism d _ {1} $
 +
is induced by the derivation  $  d: A \rightarrow \Omega _ {A/B}  ^ {1} $.
  
The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125075.png" />-th exterior power <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125076.png" /> of the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125077.png" /> of derivations is said to be the module of (differential) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125081.png" />-forms of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125082.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125083.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125084.png" />.
+
4) A field  $  K $
 +
is a separable extension of a field  $  k $
 +
of finite transcendence degree  $  n $
 +
if and only if there is a  $  K $-
 +
space isomorphism  $  \Omega _ {K/k}  ^ {1} \simeq K  ^ {n} $.
  
By virtue of 1) it is possible to define, for any morphism of schemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125085.png" />, the sheaf of relative (or Kähler) derivations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125086.png" /> and its exterior powers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031250/d03125087.png" />.
+
5) If  $  A = B [ T _ {1} \dots T _ {n} ] $
 
+
is an algebra of polynomials, then  $  \Omega _ {A/B} ^ {1} $
====References====
+
is a free  $  A $-
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Grothendieck (ed.) et al. (ed.) , ''Revêtements étales et groupe fondamental. SGA 1'' , ''Lect. notes in math.'' , '''224''' , Springer (1971) {{MR|0354651}} {{ZBL|1039.14001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Grothendieck, "Eléments de géométrie algébrique IV. Etude locale des schémes et des morphismes de schémes" ''Publ. Math. IHES'' , '''20''' (1964) {{MR|0173675}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> E. Kähler, "Algebra und Differentialrechnung" , Deutsch. Verlag Wissenschaft. (1958) {{MR|0094593}} {{ZBL|0079.05701}} </TD></TR></table>
+
module with as basis  $  dT _ {1} \dots dT _ {n} $.
  
 +
6) An algebra  $  A $
 +
of finite type over a perfect field  $  k $
 +
is a regular ring if and only if the  $  A $-
 +
module  $  \Omega _ {A/k}  ^ {1} $
 +
is projective.
  
 +
7) Concerning 2) above, the  $  A $-
 +
algebra  $  A _ {1} $
 +
of finite type is smooth over  $  A $
 +
if and only if the homomorphism  $  \alpha $
 +
is injective while the module  $  \Omega _ {A _ {1}  / A }  ^ {1} $
 +
of derivations is projective and its rank is equal to the relative dimension of  $  A _ {1} $
 +
over  $  A $.
  
====Comments====
+
The  $  i $-
 +
th exterior power  $  \wedge  ^ {i} \Omega _ {A/B}  ^ {1} $
 +
of the module  $  \Omega _ {A/B}  ^ {1} $
 +
of derivations is said to be the module of (differential)  $  i $-
 +
forms of the  $  B $-
 +
algebra  $  A $
 +
and is denoted by  $  \Omega _ {A/B}  ^ {i} $.
  
 +
By virtue of 1) it is possible to define, for any morphism of schemes  $  X \rightarrow Y $,
 +
the sheaf of relative (or Kähler) derivations  $  \Omega _ {X/Y}  ^ {1} $
 +
and its exterior powers  $  \Omega _ {X/Y}  ^ {i} $.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top"> A. Grothendieck (ed.) et al. (ed.) , ''Revêtements étales et groupe fondamental. SGA 1'' , ''Lect. notes in math.'' , '''224''' , Springer (1971) {{MR|0354651}} {{ZBL|1039.14001}} </TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top"> A. Grothendieck, "Eléments de géométrie algébrique IV. Etude locale des schémas et des morphismes de schémas" ''Publ. Math. IHES'' , '''20''' (1964) {{MR|0173675}} {{ZBL|}} </TD></TR>
 +
<TR><TD valign="top">[4]</TD> <TD valign="top"> E. Kähler, "Algebra und Differentialrechnung" , Deutsch. Verlag Wissenschaft. (1958) {{MR|0094593}} {{ZBL|0079.05701}} </TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR>
 +
</table>

Latest revision as of 06:08, 16 July 2024


module of Kähler derivations

An algebraic analogue of the concept of the differential of a function. Let $ A $ be a commutative ring regarded as an algebra over a subring $ B $ of it. The module of derivations of the $ B $- algebra $ A $ is defined as the quotient module $ \Omega _ {A/B} ^ {1} $ of the free $ A $- module with basis $ ( dx ) _ {x \in A } $ by the submodule generated by the elements of the type

$$ d ( x + y ) - dx - dy ,\ d ( xy ) - x dy - y dx,\ db , $$

where $ x, y \in A $, $ b \in B $. The canonical homomorphism of $ A $- modules $ d: A \rightarrow \Omega _ {A/B} ^ {1} $ is a $ B $- derivation in the ring $ A $( cf. Derivation in a ring) with values in the $ A $- module $ \Omega _ {A/B} ^ {1} $ having the following universality property: For any $ B $- derivation $ \partial : A \rightarrow M $ with values in an $ A $- module $ M $ there exists a uniquely defined homomorphism of $ A $- modules $ \overline \partial \; : \Omega _ {A/B} ^ {1} \rightarrow M $ such that $ \overline \partial \; \circ d = \partial $. The correspondence $ \partial \rightarrow \overline \partial \; $ defines an isomorphism of $ A $- modules:

$$ \mathop{\rm Der} _ {B} ( A , M) \simeq \mathop{\rm Hom} _ {A} ( \Omega _ {A/B} ^ {1} , M). $$

In particular, the module of derivations of a ring $ A $ into itself is isomorphic to the dual $ A $- module to the module $ \Omega _ {A/B} ^ {1} $.

If $ A \otimes _ {B} A $ is regarded as an $ A $- algebra with respect to the homomorphism

$$ A \rightarrow A \otimes _ {B} A \ ( a \rightarrow a \otimes 1 ) $$

and $ I $ is the ideal generated by the elements of the type

$$ a \otimes 1 - 1 \otimes a , $$

then the $ A $- module $ \Omega _ {A/B} ^ {1} $ is isomorphic to the $ A $- module $ I / I ^ {2} $.

The module $ \Omega ^ {1} $ of derivations has the following properties:

1) If $ S $ is a multiplicatively closed set in $ A $ and $ T = S \cap B $, then there is a canonical localization isomorphism:

$$ ( \Omega _ {A/B} ^ {1} ) _ {S} \simeq \Omega _ {A _ {S} / B _ {T} } ^ {1} . $$

2) If $ \phi : A \rightarrow A _ {1} $ is a homomorphism of $ B $- algebras, then there is a canonical exact sequence of $ A _ {1} $- modules:

$$ \Omega _ {A/B} ^ {1} \otimes _ { A } A _ {1} \mathop \rightarrow \limits ^ \alpha \Omega _ {A _ {1} / B } ^ {1} \rightarrow \Omega _ {A _ {1} / A } \rightarrow 0 . $$

3) If $ I $ is an ideal of the ring $ A $ and $ A _ {1} = A/I $, then there is an exact canonical sequence of $ A _ {1} $- modules:

$$ I / I ^ {2} \rightarrow ^ { {d _ 1} } \Omega _ {A/B} ^ {1} \otimes _ { A } A _ {1} \rightarrow \ \Omega _ {A _ {1} / B } ^ {1} \rightarrow 0 , $$

where the homomorphism $ d _ {1} $ is induced by the derivation $ d: A \rightarrow \Omega _ {A/B} ^ {1} $.

4) A field $ K $ is a separable extension of a field $ k $ of finite transcendence degree $ n $ if and only if there is a $ K $- space isomorphism $ \Omega _ {K/k} ^ {1} \simeq K ^ {n} $.

5) If $ A = B [ T _ {1} \dots T _ {n} ] $ is an algebra of polynomials, then $ \Omega _ {A/B} ^ {1} $ is a free $ A $- module with as basis $ dT _ {1} \dots dT _ {n} $.

6) An algebra $ A $ of finite type over a perfect field $ k $ is a regular ring if and only if the $ A $- module $ \Omega _ {A/k} ^ {1} $ is projective.

7) Concerning 2) above, the $ A $- algebra $ A _ {1} $ of finite type is smooth over $ A $ if and only if the homomorphism $ \alpha $ is injective while the module $ \Omega _ {A _ {1} / A } ^ {1} $ of derivations is projective and its rank is equal to the relative dimension of $ A _ {1} $ over $ A $.

The $ i $- th exterior power $ \wedge ^ {i} \Omega _ {A/B} ^ {1} $ of the module $ \Omega _ {A/B} ^ {1} $ of derivations is said to be the module of (differential) $ i $- forms of the $ B $- algebra $ A $ and is denoted by $ \Omega _ {A/B} ^ {i} $.

By virtue of 1) it is possible to define, for any morphism of schemes $ X \rightarrow Y $, the sheaf of relative (or Kähler) derivations $ \Omega _ {X/Y} ^ {1} $ and its exterior powers $ \Omega _ {X/Y} ^ {i} $.

References

[1] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
[2] A. Grothendieck (ed.) et al. (ed.) , Revêtements étales et groupe fondamental. SGA 1 , Lect. notes in math. , 224 , Springer (1971) MR0354651 Zbl 1039.14001
[3] A. Grothendieck, "Eléments de géométrie algébrique IV. Etude locale des schémas et des morphismes de schémas" Publ. Math. IHES , 20 (1964) MR0173675
[4] E. Kähler, "Algebra und Differentialrechnung" , Deutsch. Verlag Wissenschaft. (1958) MR0094593 Zbl 0079.05701
[a1] R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001
How to Cite This Entry:
Derivations, module of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Derivations,_module_of&oldid=23806
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article