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\section*{Demazure, lectures on p-divisible groups}

[[!redirects lectures on p-divisible groups]] This entry is about the text

\begin{itemize}%
\item Michel Demazure, lectures on [[p-divisible group|p-divisible groups]] \href{http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups}{web}

\end{itemize}
\hypertarget{chapter_i_schemes_and_formal_schemes}{}\subsection*{{chapter I: schemes and formal schemes}}\label{chapter_i_schemes_and_formal_schemes}

\hypertarget{1_functors}{}\subsubsection*{{1. $k$-functors}}\label{1_functors}

[[I.1, k-functors]]

\hypertarget{2_affine_schemes}{}\subsubsection*{{2. affine $k$-schemes}}\label{2_affine_schemes}

[[I.1, k-functors|I.2, affine k-schemes]]

\hypertarget{3_closed_and_open_subfunctors_schemes}{}\subsubsection*{{3. closed and open subfunctors; schemes}}\label{3_closed_and_open_subfunctors_schemes}

[[I.3, open- and closed subfunctors; schemes]]

\hypertarget{4_the_geometric_point_of_view}{}\subsubsection*{{4. the geometric point of view}}\label{4_the_geometric_point_of_view}

[[I.4, the geometric point of view]]

\hypertarget{5_finiteness_conditions}{}\subsubsection*{{5. finiteness conditions}}\label{5_finiteness_conditions}

[[I.5, finiteness conditions]]

\hypertarget{6_the_four_definitions_of_formal_schemes}{}\subsubsection*{{6. the four definitions of formal schemes}}\label{6_the_four_definitions_of_formal_schemes}

[[I.6, the four definitions of formal schemes]]

\hypertarget{7_operations_on_formal_schemes}{}\subsubsection*{{7. operations on formal schemes}}\label{7_operations_on_formal_schemes}

[[I.7. operations on formal schemes]]

\hypertarget{8_constant_and_tale_schemes}{}\subsubsection*{{8. constant- and \'e{}tale schemes}}\label{8_constant_and_tale_schemes}

[[I.8, constant- and étale schemes]], [[the fundamental theorem of Galois theory]], [[Grothendieck's Galois theory]]

\hypertarget{9_the_frobenius_morphism}{}\subsubsection*{{9. the Frobenius morphism}}\label{9_the_frobenius_morphism}

[[I.9, the Frobenius morphism]]

\hypertarget{10_frobenius_morphism_and_symmetric_products}{}\subsubsection*{{10. Frobenius morphism and symmetric products}}\label{10_frobenius_morphism_and_symmetric_products}

[[I.10, Frobenius morphism and symmetric products]]

\hypertarget{chapter_ii_groupschemes_and_formal_groupschemes}{}\subsection*{{chapter II: group-schemes and formal group-schemes}}\label{chapter_ii_groupschemes_and_formal_groupschemes}

\hypertarget{1_groupfunctors}{}\subsubsection*{{1. group-functors}}\label{1_groupfunctors}

[[II.1, group -functors]], [[k-group-functor]], [[k-group]] (=[[k-group scheme]])

\hypertarget{2_constant_and_tale_kgroups}{}\subsubsection*{{2. constant and \'e{}tale k-groups}}\label{2_constant_and_tale_kgroups}

[[II.2, constant and étale k-groups]]

\hypertarget{3_affine_kgroups}{}\subsubsection*{{3. affine k-groups}}\label{3_affine_kgroups}

[[II.3, affine k-groups]]

\hypertarget{4_kformal_groups_cartier_duality}{}\subsubsection*{{4. k-formal groups, Cartier duality}}\label{4_kformal_groups_cartier_duality}

[[II.4, k-formal groups, Cartier duality]], [[Cartier duality]]

\hypertarget{5_the_frobenius_and_the_verschiebung_morphism}{}\subsubsection*{{5. the Frobenius and the Verschiebung morphism}}\label{5_the_frobenius_and_the_verschiebung_morphism}

[[II.5, the Frobenius and the Verschiebung morphism]]

\hypertarget{6_the_category_of_affine_kgroups}{}\subsubsection*{{6. the category of affine k-groups}}\label{6_the_category_of_affine_kgroups}

[[II.6, the category of affine k-groups]]

From now on `'$k$-group'` will mean by default `'commutative $k$-group'` and the field $k$ will be of characteristic $p\gt 0$. The case $p=0$ is rather trivial.

\hypertarget{7_tale_and_connected_formal_kgroups}{}\subsubsection*{{7. \'e{}tale and connected formal k-groups}}\label{7_tale_and_connected_formal_kgroups}

[[II.7, étale and connected formal k-groups]]

\hypertarget{8_multiplicative_affine_groups}{}\subsubsection*{{8. multiplicative affine groups}}\label{8_multiplicative_affine_groups}

[[II.8, multiplicative affine groups]], [[diagonalizable group scheme]]

\hypertarget{9_unipotent_affine_groups_decomposition_of_affine_groups}{}\subsubsection*{{9. unipotent affine groups, decomposition of affine groups}}\label{9_unipotent_affine_groups_decomposition_of_affine_groups}

[[II.9, unipotent affine groups, decomposition of affine groups]]

\hypertarget{10_smooth_formal_groups}{}\subsubsection*{{10. smooth formal groups}}\label{10_smooth_formal_groups}

[[II.10, smooth formal groups]]

\hypertarget{11_pdivisible_formal_groups}{}\subsubsection*{{11. p-divisible formal groups}}\label{11_pdivisible_formal_groups}

[[II.11, p-divisible formal groups]]

\hypertarget{12_appendix}{}\subsubsection*{{12 appendix}}\label{12_appendix}

[[II.12, appendix]]

\hypertarget{chapter_iii_witt_groups_and_dieudonn_modules}{}\subsection*{{chapter III: Witt groups and Dieudonn\'e{} modules}}\label{chapter_iii_witt_groups_and_dieudonn_modules}

\hypertarget{1_the_artinhasse_exponential_series}{}\subsubsection*{{1. the Artin-Hasse exponential series}}\label{1_the_artinhasse_exponential_series}

[[III.1 the Artin-Hasse exponential series]]

\hypertarget{2_the_witt_rings_over_}{}\subsubsection*{{2. the Witt rings over $\mathbb{Z}$}}\label{2_the_witt_rings_over_}

[[III.2, the Witt rings over Z]]

\hypertarget{3_the_witt_rings_over_}{}\subsubsection*{{3. the Witt rings over $k$}}\label{3_the_witt_rings_over_}

[[III.3, the Witt rings over k]]

\hypertarget{4_duality_of_finite_witt_groups}{}\subsubsection*{{4. duality of finite Witt groups}}\label{4_duality_of_finite_witt_groups}

[[III.4, duality of finite Witt groups]]

\hypertarget{5_dieudonn_modules_affine_unipotent_groups}{}\subsubsection*{{5. Dieudonn\'e{} modules (affine unipotent groups)}}\label{5_dieudonn_modules_affine_unipotent_groups}

[[III.5, Dieudonné modules (affine unipotent groups)]]

\hypertarget{6_dieudonn_modules_torsion_finite_groups}{}\subsubsection*{{6. Dieudonn\'e{} modules ($p$-torsion finite $k$-groups)}}\label{6_dieudonn_modules_torsion_finite_groups}

[[III.6, Dieudonné modules (p-torsion finite k-groups)]]

\hypertarget{8_dieudonne_modules_pdivisible_groups}{}\subsubsection*{{8. Dieudonne modules (p-divisible groups)}}\label{8_dieudonne_modules_pdivisible_groups}

[[III.8, Dieudonné modules (p-divisible groups)]]

\hypertarget{9_dieudonn_modules_connected_formal_groups_of_finite_type}{}\subsubsection*{{9. Dieudonn\'e{} modules (connected formal groups of finite type)}}\label{9_dieudonn_modules_connected_formal_groups_of_finite_type}

[[III.9, Dieudonné modules (connected formal groups of finite type)]]

\hypertarget{chapter_iv_classification_of__divisible_groups}{}\subsection*{{chapter IV: classification of $p$ divisible groups}}\label{chapter_iv_classification_of__divisible_groups}

Unless otherwise stated let $k$ be a perfect field of prime characteristic.

We denote write $B(K):=Quot(W(k))$ for the [[quotient field]] of the [[Witt ring]] $W(k)$.

We extend the [[Frobenius morphism]] $x\mapsto x^{(p)}$ to an automorphism of $B(k)$. The set of fixed points of $x\mapsto x^{(p)}$ in $W(k)$ is $W(F_p)=\mathbb{Z}_p$. The set of fixed points of $x\mapsto x^{(p)}$ in $B(k)$ is $B(F_p)=\mathbb{Q}_p$.

\hypertarget{1_isogenies}{}\subsubsection*{{1. isogenies}}\label{1_isogenies}

[[Demazure, lectures on p-divisible groups, IV.1, isogenies]]

\hypertarget{2_the_category_of_spaces}{}\subsubsection*{{2. the category of $F$-spaces}}\label{2_the_category_of_spaces}

[[Demazure, lectures on p-divisible groups, IV.2, the category of F-spaces]]

\hypertarget{3_the_spaces__}{}\subsubsection*{{3. the spaces $E^\lambda$, $\lambda \ge 0$}}\label{3_the_spaces__}

[[Demazure, lectures on p-divisible groups, IV.3, the spaces E{\tt \symbol{94}}lambda, lambda $\backslash$ge 0]]

\hypertarget{4_classificaton_of_spaces_over_an_algebraically_closed_field}{}\subsubsection*{{4. classificaton of $F$-spaces over an algebraically closed field}}\label{4_classificaton_of_spaces_over_an_algebraically_closed_field}

[[Demazure, lectures on p-divisible groups, IV.4, classificaton of F-spaces over an algebraically closed field]]

\hypertarget{5_slopes}{}\subsubsection*{{5. slopes}}\label{5_slopes}

[[Demazure, lectures on p-divisible groups, IV.5, slopes]]

\hypertarget{6_the_characteristic_class_of_an_endomorphism}{}\subsubsection*{{6. the characteristic class of an endomorphism}}\label{6_the_characteristic_class_of_an_endomorphism}

[[Demazure, lectures on p-divisible groups, IV.6, the characteristic class of an endomorphism]]

\hypertarget{7_specialization_of_divisible_groups}{}\subsubsection*{{7. specialization of $p$-divisible groups}}\label{7_specialization_of_divisible_groups}

[[Demazure, lectures on p-divisible groups, IV.7, specialization of p-divisible groups]]

\hypertarget{8_some_particular_cases}{}\subsubsection*{{8. some particular cases}}\label{8_some_particular_cases}

[[Demazure, lectures on p-divisible groups, IV.8, some particular cases]]

\hypertarget{chapter_v_adic_cohomology_of_abelian_varieties}{}\subsection*{{chapter V: $p$-adic cohomology of abelian varieties}}\label{chapter_v_adic_cohomology_of_abelian_varieties}

\hypertarget{1_abelian_varieties_known_facts}{}\subsubsection*{{1. abelian varieties, known facts}}\label{1_abelian_varieties_known_facts}

[[Demazure, lectures on p-divisible groups, V.1, abelian varieties, known facts]]

\hypertarget{2_points_of_finite_order_and_endomorphisms}{}\subsubsection*{{2. points of finite order and endomorphisms}}\label{2_points_of_finite_order_and_endomorphisms}

[[Demazure, lectures on p-divisible groups, V.2, points of finite order and endomorphisms]]

\hypertarget{3_structure_of_the_divisible_group_}{}\subsubsection*{{3. structure of the $p$-divisible group $A(p)$}}\label{3_structure_of_the_divisible_group_}

[[Demazure, lectures on p-divisible groups, V.3, structure of the p-divisible group A(p)]]

\hypertarget{related_nlab_entries}{}\subsection*{{Related nlab entries}}\label{related_nlab_entries}

[[relations of certain classes of group schemes]]



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