nLab Demazure, lectures on p-divisible groups

This entry is about the text

• Michel Demazure, lectures on p-divisible groups web

chapter I: schemes and formal schemes

1. $k$-functors

I.1, k-functors

2. affine $k$-schemes

I.2, affine k-schemes

3. closed and open subfunctors; schemes

I.3, open- and closed subfunctors; schemes

4. the geometric point of view

I.4, the geometric point of view?

5. finiteness conditions

I.5, finiteness conditions

6. the four definitions of formal schemes

I.6, the four definitions of formal schemes

7. operations on formal schemes

I.7. operations on formal schemes

8. constant- and étale schemes

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

9. the Frobenius morphism

I.9, the Frobenius morphism

10. Frobenius morphism and symmetric products

I.10, Frobenius morphism and symmetric products

chapter II: group-schemes and formal group-schemes

1. group-functors

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

2. constant and étale k-groups

II.2, constant and étale k-groups

3. affine k-groups

II.3, affine k-groups

4. k-formal groups, Cartier duality

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

5. the Frobenius and the Verschiebung morphism

II.5, the Frobenius and the Verschiebung morphism

6. the category of affine k-groups

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.

7. étale and connected formal k-groups

II.7, étale and connected formal k-groups

8. multiplicative affine groups

II.8, multiplicative affine groups, diagonalizable group scheme

9. unipotent affine groups, decomposition of affine groups

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

10. smooth formal groups

II.10, smooth formal groups

11. p-divisible formal groups

II.11, p-divisible formal groups

12 appendix

II.12, appendix?

chapter III: Witt groups and Dieudonné modules

1. the Artin-Hasse exponential series

III.1 the Artin-Hasse exponential series

2. the Witt rings over $\mathbb{Z}$

III.2, the Witt rings over Z

3. the Witt rings over $k$

III.3, the Witt rings over k

4. duality of finite Witt groups

III.4, duality of finite Witt groups

5. Dieudonné modules (affine unipotent groups)

III.5, Dieudonné modules (affine unipotent groups)

6. Dieudonné modules ($p$-torsion finite $k$-groups)

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

8. Dieudonne modules (p-divisible groups)

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

9. Dieudonné modules (connected formal groups of finite type)

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

chapter IV: classification of $p$ 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 $B(F_p)=\mathbb{Q}_p$.

1. isogenies

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

2. the category of $F$-spaces

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

3. the spaces $E^\lambda$, $\lambda \ge 0$

Demazure, lectures on p-divisible groups, IV.3, the spaces E^lambda, lambda \ge 0?

4. classificaton of $F$-spaces over an algebraically closed field

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

5. slopes

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

6. the characteristic class of an endomorphism

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

7. specialization of $p$-divisible groups

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

8. some particular cases

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

chapter V: $p$-adic cohomology of abelian varieties

1. abelian varieties, known facts

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

2. points of finite order and endomorphisms

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

3. structure of the $p$-divisible group $A(p)$

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

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