NOTATION AND TERMINOLOGY 9

Notation and terminology

• Numerical labeling of text items and displayed expressions.

– We use “x.y.z”, “x.y.z.w”, etc. for text items (sub-subsections, results,

remarks, definitions, etc.), arranged lexicographically without repetition.

– Any labeling of displayed expressions (equations, commutative diagrams,

etc.) is indicated with parentheses, so “see (x.y.z)” means that one should

look at the zth displayed expression in subsection x.y. This convention

avoids confusion with the use of “x.y.z” to label a text item.

– Any label for a text item is uniquely assigned, so even though “see x.y.z”

does not indicate if it is a sub-subsection or theorem (or lemma, etc.),

there is no ambiguity for finding it in this book.

• Convention on notation.

– p denotes a prime number.

– CM fields are usually denoted by L.

– K often stands for an arbitrary field, κ is usually used to denote either a

residue field or a finite field of characteristic p.

– V ∨ denotes the dual of a finite-dimensional vector space V over a field.

– k denotes a perfect field, often of characteristic p 0. In 4.2–4.6, k is an

algebraically closed field of characteristic p.

– K0 is the fraction field of W (k), where k is a perfect field of characteristic

p 0 and W (k) is the ring of p-adic Witt vectors with entries in k.

– Abelian varieties are usually written as A, B, or C, and p-divisible groups

are often denoted as G or as X or Y .

– The p-divisible group attached to an abelian variety or an abelian scheme

A is denoted by

A[p∞];

its subgroup scheme of

pn-torsion

points is

A[pn].

• Fields and their extensions.

– For a field K, we write K to denote an algebraic closure and Ks to denote

a separable closure.

– An extension of fields K /K is primary if K is separably algebraically

closed in K (i.e., the algebraic closure of K in K is purely inseparable

over K).

– For a number field L we write OL to denote its ring of integers. Similar

notation is used for non-archimedean local fields.

– If q is a power of a prime p, Fq

denotes a finite field with size q (sometimes

understood to be the unique subfield of order q in a fixed algebraically

closed field of characteristic p). If κ and κ are abstract finite fields with

respective sizes q = pn and q = pn for integers n, n 1 then κ ∩ κ

denotes the unique subfield of either κ or κ with size

pgcd(n,n );

the

context will always make clear if this is being considered as a subfield of

either κ or κ . Likewise, κκ denotes κ ⊗κ∩κ κ , a common extension of

κ and κ with size

plcm(n,n ).

• Base change.

– If T → S is a map of schemes and S is an S-scheme, then TS denotes

the S -scheme T ×S S if S is understood from context.

– When S = Spec(R) and S = Spec(R ) are aﬃne, we may write TR

to denote T ⊗R R := T ×Spec(R) Spec(R ) when R is understood from

context.