# nLab Demazure, lectures on p-divisible groups

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## chapter I: schemes and formal schemes

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

### 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

### 2. constant and étale k-groups

II.2, constant and étale k-groups

### 3. affine k-groups

II.3, affine k-groups

### 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

### 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

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)?

relations of certain classes of group schemes

Last revised on February 4, 2018 at 19:09:37. See the history of this page for a list of all contributions to it.