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
its subgroup scheme of
• 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
– 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
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
• 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