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A natural generalization of the concept of the [[Picard variety|Picard variety]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p0726701.png" /> for a smooth algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p0726702.png" /> within the framework of the theory of schemes. To define the Picard scheme for an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p0726703.png" />-scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p0726704.png" /> one considers the relative Picard functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p0726705.png" /> in the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p0726706.png" /> of schemes over the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p0726707.png" />. The value of this functor on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p0726708.png" />-scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p0726709.png" /> is the group
+
{{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/p/p072/p072670/p07267010.png" /></td> </tr></table>
+
A natural generalization of the concept of the [[Picard variety|Picard variety]]  $  \mathfrak P (X) $
 +
for a smooth algebraic variety  $  X $
 +
within the framework of the theory of schemes. To define the Picard scheme for an arbitrary  $  S $-
 +
scheme  $  X $
 +
one considers the relative Picard functor  $  \mathop{\rm Pic}\nolimits _{X/S} $
 +
in the category  $  \mathop{\rm Sch}\nolimits /S $
 +
of schemes over the scheme  $  S $.  
 +
The value of this functor on an  $  S $-
 +
scheme  $  S ^ \prime  $
 +
is the group $$
 +
H ^{0} (S ^ \prime  ,\  R _{fpqc} ^{1} f _{*} ^{ {\ } \prime} (G _{ {m,\ } X ^ \prime } )),
 +
$$
 +
where  $  f ^{ {\ } \prime} : \  X \times _{S} S ^ \prime  \rightarrow S ^ \prime  $
 +
is the base-change morphism and  $  R _{fpqc} ^{1} f _{*} ^{ {\ } \prime} (G _{ {m,\ } X ^ \prime } ) $
 +
is the sheaf in the Grothendieck topology  $  S _{fpqc} ^ \prime  $
 +
of strictly-flat quasi-compact morphisms associated with the pre-sheaf $$
 +
T  \rightarrow  H ^{1} (T _{fpqc} ,\  G _{m} )  =  H ^{1} (T _{ \textrm  et} ,\  G _{m} ),
 +
$$
 +
and  $  G _{m} $
 +
denotes the standard multiplicative group sheaf. If the Picard functor  $  \mathop{\rm Pic}\nolimits _{X/S} $
 +
is representable on  $  \mathop{\rm Sch}\nolimits /S $,
 +
then the  $  S $-
 +
scheme representing it is called the relative Picard scheme for the  $  S $-
 +
scheme  $  X $
 +
and is denoted by  $  \mathop{\rm Pic}\nolimits under _{X/S} $.
 +
If  $  X $
 +
is an algebraic scheme over a certain field  $  k $
 +
having a rational  $  k $-
 +
point, then $$
 +
\mathop{\rm Pic}\nolimits _{X/k} (S ^ \prime  )  =    \mathop{\rm Pic}\nolimits (X \times _{k} S ^ \prime  )/
 +
\mathop{\rm Pic}\nolimits (S ^ \prime  )
 +
$$
 +
for any  $  k $-
 +
scheme  $  S ^ \prime  $[[#
 +
References|[3]]]. In particular,  $  \mathop{\rm Pic}\nolimits _{X/k} (k) =  \mathop{\rm Pic}\nolimits (X) $
 +
can be identified with the group of  $  k $-
 +
rational points  $  \mathop{\rm Pic}\nolimits _{X/k} (k) $
 +
of  $  \mathop{\rm Pic}\nolimits _{X/k} $
 +
if such exists.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267011.png" /> is the base-change morphism and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267012.png" /> is the sheaf in the Grothendieck topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267013.png" /> of strictly-flat quasi-compact morphisms associated with the pre-sheaf
+
If  $  f: \  X \rightarrow S $
 +
is a projective morphism with geometrically-integral fibres, then the scheme  $  \mathop{\rm Pic}\nolimits under _{X/S} $
 +
exists and is a locally finitely representable separable group  $  S $-
 +
scheme. If  $  S =  \mathop{\rm Spec}\nolimits (k) $,
 +
then the connected component of the unit,  $  \mathop{\rm Pic}\nolimits under _{X/k} ^{0} $,
 +
of  $  \mathop{\rm Pic}\nolimits under _{X/k} $
 +
is an algebraic  $  k $-
 +
scheme, and the corresponding reduced  $  k $-
 +
scheme  $  (  \mathop{\rm Pic}\nolimits _{X/k} ^{0} ) _{ {fnnme} red} $
 +
is precisely the Picard variety  $  \mathfrak P _{c} (X) $[[#
 +
References|[4]]]. The nilpotent elements in the local rings of the scheme  $  \mathop{\rm Pic}\nolimits under _{X/k} ^{0} $
 +
give much additional information on the Picard scheme and enable one to explain various "pathologies" in algebraic geometry over a field of characteristic  $  p > 0 $.  
 +
On the other hand, over a field of characteristic 0 the scheme  $  \mathop{\rm Pic}\nolimits under _{K/k} ^{0} $
 +
is always reduced [[#References|[6]]]. It is also known that  $  \mathop{\rm Pic}\nolimits _{F/k} $
 +
is a reduced scheme if  $  F $
 +
is a smooth algebraic surface and  $  H ^{2} (F,\  {\mathcal O} _{F} ) = 0 $[[#
 +
References|[5]]].
  
<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/p/p072/p072670/p07267014.png" /></td> </tr></table>
+
For any proper flat morphism $  f: \  X \rightarrow S $(
 
+
finitely representable if the base $  S $
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267015.png" /> denotes the standard multiplicative group sheaf. If the Picard functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267016.png" /> is representable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267017.png" />, then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267018.png" />-scheme representing it is called the relative Picard scheme for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267019.png" />-scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267020.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267021.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267022.png" /> is an algebraic scheme over a certain field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267023.png" /> having a rational <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267024.png" />-point, then
+
is Noetherian) for which $  f _{*} ^{ {\ } \prime} ( {\mathcal O} _{ {X} ^ \prime } ) = {\mathcal O} _{ {S} ^ \prime } $,  
 
+
the functor $  \mathop{\rm Pic}\nolimits _{X/S} $
<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/p/p072/p072670/p07267025.png" /></td> </tr></table>
+
is an algebraic space over $  S $
 
+
for any base-change morphism $  f ^{ {\ } \prime} : \  X ^ \prime  = X \times _{S} S ^ \prime  \rightarrow S $[[#
for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267026.png" />-scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267027.png" /> [[#References|[3]]]. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267028.png" /> can be identified with the group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267029.png" />-rational points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267030.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267031.png" /> if such exists.
+
References|[1]]]. In particular, the functor $  \mathop{\rm Pic}\nolimits _{X/S} $
 
+
is representable if the ground scheme $  S $
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267032.png" /> is a projective morphism with geometrically-integral fibres, then the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267033.png" /> exists and is a locally finitely representable separable group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267034.png" />-scheme. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267035.png" />, then the connected component of the unit, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267036.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267037.png" /> is an algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267038.png" />-scheme, and the corresponding reduced <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267039.png" />-scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267040.png" /> is precisely the Picard variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267041.png" /> [[#References|[4]]]. The nilpotent elements in the local rings of the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267042.png" /> give much additional information on the Picard scheme and enable one to explain various "pathologies" in algebraic geometry over a field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267043.png" />. On the other hand, over a field of characteristic 0 the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267044.png" /> is always reduced [[#References|[6]]]. It is also known that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267045.png" /> is a reduced scheme if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267046.png" /> is a smooth algebraic surface and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267047.png" /> [[#References|[5]]].
+
is the spectrum of a local Artinian ring.
 
 
For any proper flat morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267048.png" /> (finitely representable if the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267049.png" /> is Noetherian) for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267050.png" />, the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267051.png" /> is an algebraic space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267052.png" /> for any base-change morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267053.png" /> [[#References|[1]]]. In particular, the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267054.png" /> is representable if the ground scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267055.png" /> is the spectrum of a local Artinian ring.
 
  
 
====References====
 
====References====
Line 23: Line 75:
  
 
====Comments====
 
====Comments====
The standard multiplicative sheaf over a scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267056.png" /> assigns to an affine open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267057.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267058.png" /> the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267059.png" /> of units of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267060.png" />.
+
The standard multiplicative sheaf over a scheme $  X $
 +
assigns to an affine open set $  U $
 +
in $  X $
 +
the group $  \Gamma ( U,\  {\mathcal O} _{X} ) ^{*} $
 +
of units of $  \Gamma (U ,\  {\mathcal O} _{X} ) $.
 +
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Grothendieck, "Fondements de la géométrie algébrique" , Secr. Math. Univ. Paris (1961/62) (Extracts Sem. Bourbaki 1957–1962) {{MR|1611235}} {{MR|1086880}} {{MR|0146040}} {{ZBL|0239.14002}} {{ZBL|0239.14001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A. Altman, S. Kleiman, "Compactification of the Picard scheme I" ''Adv. in Math.'' , '''35''' (1980) pp. 50–112 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A. Altman, S. Kleiman, "Compactification of the Picard scheme II" ''Amer. J. Math.'' , '''101''' (1979) pp. 10–41 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> J.P. Murre, "On contravariant functors from the category of preschemes over a field into the category of abelian groups (with an application to the Picard functor)" ''Publ. Math. IHES'' , '''23''' (1964) pp. 581–619 {{MR|206011}} {{ZBL|0142.18402}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> F. Oort, "Sur le schéma de Picard" ''Bull. Soc. Math. France'' , '''90''' (1962) pp. 1–14 {{MR|0138627}} {{ZBL|0123.13901}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Grothendieck, "Fondements de la géométrie algébrique" , Secr. Math. Univ. Paris (1961/62) (Extracts Sem. Bourbaki 1957–1962) {{MR|1611235}} {{MR|1086880}} {{MR|0146040}} {{ZBL|0239.14002}} {{ZBL|0239.14001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A. Altman, S. Kleiman, "Compactification of the Picard scheme I" ''Adv. in Math.'' , '''35''' (1980) pp. 50–112 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A. Altman, S. Kleiman, "Compactification of the Picard scheme II" ''Amer. J. Math.'' , '''101''' (1979) pp. 10–41 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> J.P. Murre, "On contravariant functors from the category of preschemes over a field into the category of abelian groups (with an application to the Picard functor)" ''Publ. Math. IHES'' , '''23''' (1964) pp. 581–619 {{MR|206011}} {{ZBL|0142.18402}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> F. Oort, "Sur le schéma de Picard" ''Bull. Soc. Math. France'' , '''90''' (1962) pp. 1–14 {{MR|0138627}} {{ZBL|0123.13901}} </TD></TR></table>

Latest revision as of 10:54, 20 December 2019


A natural generalization of the concept of the Picard variety $ \mathfrak P (X) $ for a smooth algebraic variety $ X $ within the framework of the theory of schemes. To define the Picard scheme for an arbitrary $ S $- scheme $ X $ one considers the relative Picard functor $ \mathop{\rm Pic}\nolimits _{X/S} $ in the category $ \mathop{\rm Sch}\nolimits /S $ of schemes over the scheme $ S $. The value of this functor on an $ S $- scheme $ S ^ \prime $ is the group $$ H ^{0} (S ^ \prime ,\ R _{fpqc} ^{1} f _{*} ^{ {\ } \prime} (G _{ {m,\ } X ^ \prime } )), $$ where $ f ^{ {\ } \prime} : \ X \times _{S} S ^ \prime \rightarrow S ^ \prime $ is the base-change morphism and $ R _{fpqc} ^{1} f _{*} ^{ {\ } \prime} (G _{ {m,\ } X ^ \prime } ) $ is the sheaf in the Grothendieck topology $ S _{fpqc} ^ \prime $ of strictly-flat quasi-compact morphisms associated with the pre-sheaf $$ T \rightarrow H ^{1} (T _{fpqc} ,\ G _{m} ) = H ^{1} (T _{ \textrm et} ,\ G _{m} ), $$ and $ G _{m} $ denotes the standard multiplicative group sheaf. If the Picard functor $ \mathop{\rm Pic}\nolimits _{X/S} $ is representable on $ \mathop{\rm Sch}\nolimits /S $, then the $ S $- scheme representing it is called the relative Picard scheme for the $ S $- scheme $ X $ and is denoted by $ \mathop{\rm Pic}\nolimits under _{X/S} $. If $ X $ is an algebraic scheme over a certain field $ k $ having a rational $ k $- point, then $$ \mathop{\rm Pic}\nolimits _{X/k} (S ^ \prime ) = \mathop{\rm Pic}\nolimits (X \times _{k} S ^ \prime )/ \mathop{\rm Pic}\nolimits (S ^ \prime ) $$ for any $ k $- scheme $ S ^ \prime $[[# References|[3]]]. In particular, $ \mathop{\rm Pic}\nolimits _{X/k} (k) = \mathop{\rm Pic}\nolimits (X) $ can be identified with the group of $ k $- rational points $ \mathop{\rm Pic}\nolimits _{X/k} (k) $ of $ \mathop{\rm Pic}\nolimits _{X/k} $ if such exists.

If $ f: \ X \rightarrow S $ is a projective morphism with geometrically-integral fibres, then the scheme $ \mathop{\rm Pic}\nolimits under _{X/S} $ exists and is a locally finitely representable separable group $ S $- scheme. If $ S = \mathop{\rm Spec}\nolimits (k) $, then the connected component of the unit, $ \mathop{\rm Pic}\nolimits under _{X/k} ^{0} $, of $ \mathop{\rm Pic}\nolimits under _{X/k} $ is an algebraic $ k $- scheme, and the corresponding reduced $ k $- scheme $ ( \mathop{\rm Pic}\nolimits _{X/k} ^{0} ) _{ {fnnme} red} $ is precisely the Picard variety $ \mathfrak P _{c} (X) $[[# References|[4]]]. The nilpotent elements in the local rings of the scheme $ \mathop{\rm Pic}\nolimits under _{X/k} ^{0} $ give much additional information on the Picard scheme and enable one to explain various "pathologies" in algebraic geometry over a field of characteristic $ p > 0 $. On the other hand, over a field of characteristic 0 the scheme $ \mathop{\rm Pic}\nolimits under _{K/k} ^{0} $ is always reduced [6]. It is also known that $ \mathop{\rm Pic}\nolimits _{F/k} $ is a reduced scheme if $ F $ is a smooth algebraic surface and $ H ^{2} (F,\ {\mathcal O} _{F} ) = 0 $[[# References|[5]]].

For any proper flat morphism $ f: \ X \rightarrow S $( finitely representable if the base $ S $ is Noetherian) for which $ f _{*} ^{ {\ } \prime} ( {\mathcal O} _{ {X} ^ \prime } ) = {\mathcal O} _{ {S} ^ \prime } $, the functor $ \mathop{\rm Pic}\nolimits _{X/S} $ is an algebraic space over $ S $ for any base-change morphism $ f ^{ {\ } \prime} : \ X ^ \prime = X \times _{S} S ^ \prime \rightarrow S $[[# References|[1]]]. In particular, the functor $ \mathop{\rm Pic}\nolimits _{X/S} $ is representable if the ground scheme $ S $ is the spectrum of a local Artinian ring.

References

[1] M. Artin, "Algebraization of formal moduli I" D.C. Spencer (ed.) S. Iyanaga (ed.) , Global analysis (papers in honor of K. Kodaira) , Univ. Tokyo Press (1969) pp. 21–72 MR0260746 Zbl 0205.50402
[2] C. Chevalley, "Sur la théorie de la variété de Picard" Amer. J. Math. , 82 (1960) pp. 435–490 MR0118723 Zbl 0127.37701
[3] A. Grothendieck, "Technique de déscente et théorèmes d'existence en géometrie algébrique. V. Les schémas de Picard. Théorèmes d'existence" Sém. Bourbaki , 14 (1962) pp. 232/01–232/19 MR1611170
[4] A. Grothendieck, "Eléments de géomètrie algébrique. I Le langage des schémas" Publ. Math. IHES : 4 (1960) pp. 1–228 MR0217083 MR0163908 Zbl 0118.36206
[5] D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) MR0209285 Zbl 0187.42701
[6] F. Oort, "Algebraic group schemes in character zero are reduced" Invent. Math. , 2 : 1 (1966) pp. 79–80 MR206005
[7] I.V Dolgachev, "Abstract algebraic geometry" J. Soviet Math. , 2 : 3 (1974) pp. 264–303 Itogi Nauk. i Tekhn. Algebra. Topol. Geom. , 10 (1972) pp. 47–112 Zbl 1068.14059


Comments

The standard multiplicative sheaf over a scheme $ X $ assigns to an affine open set $ U $ in $ X $ the group $ \Gamma ( U,\ {\mathcal O} _{X} ) ^{*} $ of units of $ \Gamma (U ,\ {\mathcal O} _{X} ) $.


References

[a1] A. Grothendieck, "Fondements de la géométrie algébrique" , Secr. Math. Univ. Paris (1961/62) (Extracts Sem. Bourbaki 1957–1962) MR1611235 MR1086880 MR0146040 Zbl 0239.14002 Zbl 0239.14001
[a2] A. Altman, S. Kleiman, "Compactification of the Picard scheme I" Adv. in Math. , 35 (1980) pp. 50–112
[a3] A. Altman, S. Kleiman, "Compactification of the Picard scheme II" Amer. J. Math. , 101 (1979) pp. 10–41
[a4] J.P. Murre, "On contravariant functors from the category of preschemes over a field into the category of abelian groups (with an application to the Picard functor)" Publ. Math. IHES , 23 (1964) pp. 581–619 MR206011 Zbl 0142.18402
[a5] F. Oort, "Sur le schéma de Picard" Bull. Soc. Math. France , 90 (1962) pp. 1–14 MR0138627 Zbl 0123.13901
How to Cite This Entry:
Picard scheme. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Picard_scheme&oldid=44309
This article was adapted from an original article by V.V. Shokurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article