Timeline for Picard functor of an algebraic group
Current License: CC BY-SA 3.0
2 events
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| Dec 6, 2014 at 11:07 | comment | added | user74230 | This lacks motivation (beyond the proper case). Consider $P={\rm{Pic}}_{Y/K}$ defined by etale sheafification for smooth $Y$ over $K$. This satisfies the functorial criterion to be locally of finite presentation and $P(R)={\rm{Pic}}(Y_R)$ for strictly henselian local $R$ over $K$, so the tangent space at 1 is $\ker(P(K[\varepsilon])\rightarrow P(K))={\rm{H}}^1(Y,O_Y)$. For affine $Y$ this is 0, so if representable for such $Y$ it is the constant $K$-group associated to Pic($Y$), and hence ${\rm{Pic}}(Y_R)={\rm{Pic}}(Y)$ for strictly henselian local $R$ over $K$. False for $Y$ an affine line. | |
| Dec 6, 2014 at 3:53 | history | asked | Question Mark | CC BY-SA 3.0 |