1. such that
∂p−1,q ◦ ∂p,q = ∂p,q−1 ◦ ∂p,q
P = ∂p,q−1 ◦ ∂p,q + ∂p−1,q ◦ ∂p,q = 0 .
We define the associated chain complex (Xn , ∂)
p-adic numbers The completion of the ratio- by setting
nal field Q with respect to the p-adic valuation
Xn = Xp,q , ∂n = ∂p,q + ∂p,q .
|·|p . See p-adic valuation. See also completion.
p+q=n p+q=n
p-adic valuation For a fixed prime integer p,
We call ∂ the total boundary operator, and ∂ ,
the valuation |·|p , defined on the field of rational
∂ the partial boundary operators.
numbers as follows. Write a rational number in
the form pr m/n where r is an integer, and m, n partial derived functor Suppose F is a func-
are non-zero integers, not divisible by p. Then tor of n variables. If S is a subset of {1, . . . , n},
|p r m/n|p = 1/p r . See valuation. we consider the variables whose indices are
in S as active and those whose indices are in
parabolic subalgebra A subalgebra of a Lie {1, . . . n}S as passive. By fixing all the passive
algebra g that contains a maximal solvable sub- variables, we obtain a functor FS in the active
algebra of g. variables. The partial derived functors are then
defined as the derived functors R k FS . See also
parabolic subgroup A subgroup of a Lie
functor, derived functor.
group G that contains a maximal connected
solvable Lie subgroup of G. An example is the partial differential The rate of change of a
subgroup of invertible upper triangular matrices function of more than one variable with respect
in the group GLn (C) of invertible n×n matrices to one of the variables while holding all of the
with complex entries. other variables constant.
parabolic transformation A transformation partial fraction An algebraic expression of
of the Riemann sphere whose fixed points are ∞ the form
and another point.
nj
aj m
paraholic subgroup A subgroup of a Lie m .
group containing a Borel subgroup. j m=1
z − αj
parametric equations The name given to
equations which specify a curve or surface by partially ordered space Let X be a set. A
expressing the coordinates of a point in terms of relation on X that satisfies the conditions:
a third variable (the parameter), in contrast with (i.) x ≤ x for all x ∈ X
a relation connecting x, y, and z, the cartesian (ii.) x ≤ y and y ≤ x implies x = y
coordinates. (iii.) x ≤ y and y ≤ z implies x ≤ z
is called a partial ordering.
partial boundary operator We call (Xp,q ,
∂ , ∂ ) over A a double chain complex if it is partial pivoting An iterative strategy, using
a family of left A-modules Xp,q for p, q ∈ Z pivots, for solving the equation Ax = b, where
together with A-automorphisms A is an n × n matrix and b is an n × 1 matrix.
In the method of partial pivoting, to obtain the
∂p,q : Xp,q → Xp−1,q matrix Ak (where A0 = A), the pivot is chosen
to be the entry in the kth column of Ak−1 at
and or below the diagonal with the largest absolute
∂p,q : Xp,q → Xp,q−1 value.
c 2001 by CRC Press LLC

2. partial product Let {αn }∞ be a given se-
n=1 1 − e and e are orthogonal idempotent elements,
quence of numbers (or functions defined on a and
common domain in Rn or Cn ) with terms R = eR + (1 − e)R
αn = 0 for all n ∈ N. The formal infinite prod- is the direct sum of left ideals. This is called
uct α1 · α2 · · · is denoted by ∞ αj . We call
j =1 Peirce’s right decomposition.
n
Pell’s equation The Diophantine equations
Pn = αj
x 2 − ay 2 = ±4 and ±1, where a is a positive
j =1
integer, not a perfect square, are called Pell’s
its nth partial product. equations. The solutions of such equations can
be found by continued fractions and are used in
peak point See peak set. the determination of the units of rings such as
√
Z[ a]. This equation was studied extensively
peak set Let A be an algebra of functions by Gauss. It can be regarded as a starting point
on a domain ⊂ Cn . We call p ∈ a peak of modern algebraic number theory.
point for A if there is a function f ∈ A such that When a < 0, then Pell’s equation has only
f (p) = 1 and |f (z)| < 1 for all z ∈ {p}. finitely many solutions. If a > 0, then all solu-
The set P(A) of all peak points for the algebra tions xn , yn of Pell’s equation are given by
A is called the peak set of A. √ n √
x1 + ay1 xn + ayn
± = ,
Peirce decomposition Let A be a semisimple 2 2
Jordan algebra over a field F of characteristic 0 provided that the pair x1 , y1 is a solution with
√
and let e be an idempotent of A. For λ ∈ F , let the smallest x1 + ay1 > 1. Using continued
Ae (λ) = {a ∈ A : ea = λa}. Then fractions, we can determine x1 , y1 explicitly.
A = Ae (1) ⊕ Ae (1/2) ⊕ Ae (0) . penalty method of solving non-linear pro-
gramming problem A method to modify a
This is called the Peirce decomposition of A, constrained problem to an unconstrained prob-
relative to E. If 1 is the sum of idempotents ej , lem. In order to minimize (or maximize) a func-
let Aj,k = Aej (1) when j = k and Aej ∩ Aek tion φ(x) on a set which has constraints (such
when j = k. These are called Peirce spaces, as f1 (x) ≥ 0, f2 (x) ≥ 0, . . . fm (x) ≥ 0), a
and A = ⊕j ≤k Aj,k . See also Peirce space. penalty or penalty function, ψ(x, a), is intro-
duced (where a is a number), where ψ(x, a) = 0
Peirce’s left decomposition Let e be an idem- if x ∈ X or ψ(x, a) > 0 if x ∈ X and ψ in-
/
potent element of a ring R with identity 1. Then volves f1 (x) ≥ 0, f2 (x) ≥ 0, . . . fm (x) ≥ 0.
Then, one minimizes (or maximizes) φa (x) =
R = Re ⊕ R(1 − e)
φ(x) + ψ(x, a) without the constraints.
expresses R as a direct sum of left ideals. This
is called Peirce’s left decomposition. percent Percent means hundredths. The
symbol % stands for 100 . We may write a per-
Peirce space Suppose that the unity element cent as a fraction with denominator 100. For ex-
1 ∈ K can be represented as a sum of the mutu- ample, 31% = 100 , 55% = 100 , . . . etc. Simi-
31 55
ally orthogonal idempotents ej . Then, putting larly, we may write a fraction with denominator
100 as a percent.
Aj,j = Aej (1), Aj,k = Aej (1/2)∩Aek (1/2) ,
perfect field A field such that every algebraic
we have A = j ≤k ⊕Aj,k . Then Aj,k are
extension is separable. Equivalently, a field F is
called Peirce spaces. perfect if each irreducible polynomial with co-
efficients in F has no multiple roots (in an alge-
Peirce’s right decomposition Let e be an braic closure of F ). Every field of characteristic
idempotent element of a unitary ring R, then 0 is perfect and so is every finite field.
c 2001 by CRC Press LLC

3. perfect power An integer or polynomial corresponding to the sign of the permutation is
which can be written as the nth power of another missing from each summand.
integer or polynomial, where n is a positive in-
teger. For example, 8 is a perfect cube, because permutation group Let A be a finite set with
8 = 23 , and x 2 + 4x + 4 is a perfect square, #(A) = n. The permutation group on n ele-
because x 2 + 4x + 4 = (x + 2)2 . ments is the set Sn consisting of all one-to-one
functions from A onto A under the group law:
period matrix Let R be a compact Riemann
surface of genus g. Let ω1 , . . . , ωg be a ba- f ·g =f ◦g
sis for the complex vector space of holomorphic
differentials on R and let α1 , . . . , α2g be a ba- for f, g ∈ Sn . Here ◦ denotes the composition
sis for the 1-dimensional integral homology of of functions.
R. The period matrix M is the g × 2g matrix
whose (i, j )-th entry is the integral of ωj over permutation matrix An n × n matrix P ,
αi . The group generated by the 2g columns of obtained from the identity matrix In by permu-
M is a lattice in Cg and the quotient yields a g- tations of the rows (or columns). It follows that a
dimensional complex torus called the Jacobian permutation matrix has exactly one nonzero en-
variety of R. try (equal to 1) in each row and column. There
are n! permutation matrices of size n × n. They
period of a periodic function Let f be a are orthogonal matrices, namely, P T P = P P T
function defined on a vector space V satisfying = In (i.e., P T = P −1 ). Multiplication from
the relation the left (resp., right) by a permutation matrix
permutes the rows (resp., columns) of a matrix,
f (x + ω) = f (x) corresponding to the original permutation.
for all x ∈ V and for some ω ∈ V . The number permutation representation A permutation
ω is called a period of f (x), and f (x) with a representation of a group G is a homomorphism
period ω = 0 is call a periodic function. from G to the group SX of all permutations of a
set X. The most common example is when X =
period relation Conditions on an n×n matrix G and the permutation of G obtained from g ∈
which help determine when a complex torus is G is given by x → gx (or x → xg, depending
an Abelian manifold. In Cn , let be generated on whether a product of permutations is read
by (1, 0, . . . , 0), (0, 1, 0, . . . , 0), . . . , (0, 0, . . . , right-to-left or left-to-right).
0, 1), (a11 , a12 , . . . , a1n ), (a21 , a22 , . . . , a2n ),
. . . (an1 , an2 , . . . , ann ). Then Cn / is an Peron-Frobenius Theorem See Frobenius
Abelian manifold if there are integers d1 , d2 , . . . , Theorem on Non-Negative Matrices.
dn = 0 such that, if A = (aij ) and D = (δij di ),
then (i.) AD is symmetric; and (ii.) (AD) is Perron’s Theorem of Positive Matrices If
positive symmetric. Conditions (i.) and (ii.) are A is a positive n × n matrix, A has a positive
the period relations. real eigenvalue λ with the following properties:
(i.) λ is a simple root of the characteristic equa-
permanent Given an m×n matrix A = (aij ) tion.
with m ≤ n, the permanent of A is defined by (ii.) λ has a positive eigenvector u.
(iii.) If µ is any other eigenvalue of A, then
permA = a1i1 a2i2 . . . amim , |µ| < λ.
where the summation is taken over all m- Peter-Weyl theory Let G be a compact Lie
permutations (i1 , i2 , . . . , im ) of the set {1, 2, group and let C(G) be the commutative asso-
. . . , n}. When A is a square matrix, the per- ciative algebra of all complex valued continuous
manent therefore has an expansion similar to functions defined on G. The multiplicative law
that of the determinant, except that the factor defined on C(G) is just the usual composition
c 2001 by CRC Press LLC

4. of functions. Denote (1856–1941). The first Picard theorem was
  proved in 1879: An entire function which is not a
  polynomial takes every value, with one possible
s(G) = f ∈ C(G) : dim CLg f < ∞
  exception, an infinity of times.
g∈G
The second Picard theorem was proved in
where Lg f = f (g·). The Peter-Weyl theory 1880: In a neighborhood of an isolated essen-
tells us that the subalgebra s(G) is everywhere tial singularity, a single-valued, holomorphic
dense in C(G) with respect to the uniform norm function takes every value, with one possible ex-
f ∞ = maxg∈G |f (g)|. ception, an infinity of times. In other words, if
f (z) is holomorphic for 0 < |z − z0 | < r, and
Pfaffian differential form The name given there are two unequal numbers a, b, such that
to the expression f (z) = a, f (z) = b, for |z − z0 | < r, then z0
is not an essential singularity.
n
dW = Xi dxi . Picard variety Let V be a complete normal
i=1 variety. The factor group of the divisors on V ,
algebraically equivalent to 0 modulo the group
of divisors linearly equivalent to 0, has a natural
p-group A group G such that the order of
canonical structure of an Abelian variety, called
G is p n , where p is a prime number and n is a
the Picard variety.
non-negative integer.
Picard-Vessiot theory One of two main the-
Picard-Lefschetz transform Let W be a lo-
ories of differential rings and fields. See Galois
cal system attached to the monodromy repre-
theory of differential fields. The Picard-Vessiot
sentation ϕp : π1 (U, 0) → GL(H p (W, Q)).
theory deals with linear homogeneous differen-
For each point tj there corresponds a cycle δj of
tial equations.
H n−1 (W, Q) called a vanishing cycle such that
if γj is a loop based at 0 going once around tj ,
pi-group Let π be a set of prime numbers and
we have for each x ∈ H n−1 (W, Q),
let π be the set of prime numbers not in π . A π -
ϕp γj (x) = x ± x, δj δj . group is a finite group whose order is a product
of primes in π . A finite group is π -solvable if
every Jordan-Hölder factor is either a π -group
Picard number Let V be a complete nor- or a solvable π -group. For a π -solvable group
mal variety and let D(V ), Da (V ) be the group G, define a series of subgroups
of divisors and group of divisors algebraically
equivalent to zero, respectively. The rank of 1 = P0 ⊆ N0 ⊂ P1 ⊂ N1 · · · ⊂ Pn ⊆ Nn = G
the quotient group N S(V ) = D(V )/Da (V ) is
called the Picard number of V . such that Pj /Nj −1 is a maximal normal π -
subgroup of G/Pj . This is called the π -series
∗
Picard scheme Let OV be the sheaf of mul- of G and n is called the π -length of G.
tiplicative group of the invertible elements in
OV . The group of linear equivalence classes pi-length See pi-group.
of Cartier divisors can be identified with
H 1 (V , OV ). From this point of view, we can pi-series See pi-group.
generalize the theory of the Picard variety to the
case of schemes. The theory thus obtained is pi-solvable group A finite group G such that
called the theory of Picard schemes. the order of each composition factor of G is ei-
ther an element of a collection, π , of prime num-
Picard’s Theorem There are two important bers or mutually prime to any element of π .
theorems in one complex variable proved by
the French mathematician Charles Émile Picard pivot See Gaussian elimination.
c 2001 by CRC Press LLC

5. pivoting See Gaussian elimination. bers Pi = (iK), (i = 2, 3, . . . ). The plurigen-
era Pi , (i = 2, 3, . . . ) are the same for any two
place A mapping φ : K → {F, ∞}, where K birationally equivalent nonsingular surfaces.
and F are fields, such that, if φ(a) and φ(b) are
defined, then φ(a +b) = φ(a)+φ(b), φ(ab) = plus sign The symbol “+” indicating the al-
φ(a)φ(b) and φ(1) = 1. gebraic operation of addition, as in a + b.
place value The value given to a digit, de- Poincaré Let R be a commutative ring with
pending on that digit’s position in relation to the unit. Let U be an orientation over R of a com-
units place. For example, in 239.71, 9 repre- pact n-manifold X with boundary. Then for all
sents 9 units, 3 represents 30 units, 2 represents indices q and R-modules G there is an isomor-
7
200 units, 7 represents 10 units and 1 represents phism
1
100 units.
γU : Hq (X; G) ≈ H n−q (X; G) .
Plancherel formula Let G be a unimodular
ˆ
locally compact group and G be its quasidual. This is called Poincaré-Lefschetz duality. The
Let U be a unitary representation of G and U ∗ analogous result for a manifold X without bound-
be its adjoint. For any f , g ∈ L1 (G) ∩ L2 (G), ary is called Poincaré duality.
the Plancherel formula
Poincaré-Birkhoff-Witt Theorem Let G be
f (x)g(x) dx = ∗
t (Ug (ξ )Uf (ξ )) dµ(ξ ) a Lie algebra over a number field K. Let X1 , . . . ,
G ˆ
G Xn be a basis of G, and let R = K[Y1 , . . . , Yn ]
be a polynomial ring on K in n indeterminates
holds, where Uf (ξ ) = G f (x)Ux (ξ )dx. The
ˆ Y1 , . . . , Yn . Then there exists a unique alge-
measure µ is called the Plancherel measure. bra homomorphism ψ : R → G such that
ψ(1) = 1 and ω(Yj ) = Xj , j = 1, . . . , n.
plane trigonometry Plane trigonometry is
Moreover, ψ is bijective, and the j th homoge-
related to the study of triangles, which were
neous component Rj is mapped by ψ onto G j .
studied long ago by the Babylonians and an- k k k
Thus, the set of monomials {X11 X22 . . . Xnn },
cient Greeks. The word trigonometry is derived
k1 , . . . , kn ≥ 0, forms a basis of U (G) over
from the Greek word for “the measurement of
K. This is the so-called Poincaré-Birkhoff-Witt
triangles.” Today trigonometry and trigonomet-
Theorem. Here U (G) = T (G)/J is the quotient
ric functions are indispensable tools not only in
associative algebra of G where J is the two-sided
mathematics, but also in many practical appli-
ideal of T (G) generated by all elements of the
cations, especially those involving oscillations
form X ⊗ Y − Y ⊗ X − [X, Y ] and T (G) is the
and rotations.
tensor algebra over G.
Plücker formulas Let m be the class, n the
degree, and δ, χ, i, and τ be the number of Poincaré differential invariant Let w =
nodes, cups, inflections, tangents, and bitan- α(z − z◦ )/(1 − z◦ z) with |α| = 1 and |z◦ | <
gents. Then 1, be a conformal mapping of |z| < 1 onto
|w| < 1. Then the quantity |dw|/(1 − |w|2 ) =
n(n − 1) = m + 2δ + 3χ |dz|/(1 − |z|2 ) is called Poincaré’s differen-
tial invariant. The disk {|z| < 1} becomes
m(m − 1) = n + 2τ + 3i
a non-Euclidean space using any metric with
3n(n − 2) = i + 6δ + 8χ ds = |dz|/(1 − |z|2 ).
3m(m − 2) = χ + 6τ + 8i
3(m − n) = i − χ . Poincaré duality Any theorem general-
izing the following: Let M be a com-
pact n-dimensional manifold without bound-
plurigenera For an algebraic surface S with a ary. Then, for each p, there is an isomor-
canonical divisor K of S, the collection of num- phism H p (M; Z2 ) ∼ Hn−p (M; Z2 ). If, in
=
c 2001 by CRC Press LLC

6. addition, M is assumed to be orientable, then be simple if W has no nonzero proper subco-
H p (M) ∼ Hn−p (M).
= algebra. The co-algebra V is called a pointed
co-algebra if all of its simple subco-algebras are
Poincaré-Lefschetz duality Let R be a com- one-dimensional. See coalgebra.
mutative ring with unit. Let U be an orientation
over R of a compact n-manifold X with bound- pointed set Denoted by (X, p), a set X where
ary. Then for all indices q and R-modules G p is a member of X.
there is an isomorphism
polar decomposition Every n × n matrix A
γU : Hq (X; G) ≈ H n−q (X; G) . with complex entries can be written as A = P U ,
where P is a positive semidefinite matrix and U
This is called Poincaré-Lefschetz duality. The is a unitary matrix. This factorization of A is
analogous result for a manifold X without bound- called the polar decomposition of the polar form
ary is called Poincaré duality. of A.
Poincaré metric The hermitian metric polar form of a complex number Let z =
2 x + iy be a complex number. This number has
ds 2 = dz ∧ dz
(1 − |z|2 )2 the polar representation
is called the Poincaré metric for the unit disc in z = x + iy = r(cos θ + i sin θ )
y
the complex plane. where r = x 2 + y 2 and θ = tan−1 x .
Poincaré’s Complete Reducibility Theorem polarization Let A be an Abelian variety and
A theorem which says that, given an Abelian let X be a divisor on A. Let X be a divisor on
variety A and an Abelian subvariety X of A, A such that m1 X ≡ m2 X for some positive in-
there is an Abelian subvariety Y of A such that tegers m1 and m2 . Let X be the class of all such
A is isogenous to X × Y . divisors X . When X contains positive nonde-
generate divisors, we say that X determines a
point at infinity The point in the extended polarization on A.
complex plane, not in the complex plane itself.
More precisely, let us consider the unit sphere polarized Abelian variety Suppose that V
in R3 : is an Abelian variety. Let X be a divisor on
V and let D(X) denote the class of all divisors
S = (x1 , x2 , x3 ) ∈ R3 : x1 + x2 + x3 = 1 ,
2 2 2
Y on V such that mX ≡ nY , for some inte-
gers m, n > 0. Further, suppose that D(X) de-
which we define as the extended complex num-
termines a polarization of V . Then the couple
bers. Let N = (0, 0, 1); that is, N is the north
(V , D(X)) is called a polarized Abelian variety.
pole on S. We regard C as the plane {(x1 , x2 , 0)
See also Abelian variety, divisor, polarization.
∈ R3 : x1 , x2 ∈ R} so that C cuts S along the
equator. Now for each point z ∈ C consider the pole Let z = a be an isolated singularity of
straight line in R3 through z and N . This in- a complex-valued function f . We call a a pole
tersects the sphere in exactly one point Z = N . of f if
By identifying Z ∈ S with z ∈ C, we have S lim |f (z)| = ∞ .
identified with C ∪ {N }. If |z| > 1 then Z is in z→a
the upper hemisphere and if |z| < 1 then z is in That is, for any M > 0 there is a number ε >
the lower hemisphere; also, for |z| = 1, Z = z. 0, such that |f (z)| ≥ M whenever 0 < |z −
Clearly Z approaches N when |z| approaches a| < ε. Usually, the function f is assumed to
∞. Therefore, we may identify N and the point be holomorphic, in a punctured neighborhood
∞ in the extended complex plane. 0 < |z − a| < .
pointed co-algebra Let V be a co-algebra. pole divisor Suppose X is a smooth affine va-
A nonzero subco-algebra W of V is said to riety of dimension r and suppose Y ⊂ X is a sub-
c 2001 by CRC Press LLC

7. variety of dimension r − 1. Given f ∈ C(X) the usual addition and multiplication of polyno-
(0), let ordY f < 0 denote the order of vanish- mials. The ring R[X] is called the polynomial
ing of f on Y . Then (f ) = Y (ordY f ) · Y is ring of X over R.
called a pole divisor of f in Y . See also smooth
affine variety, subvariety, order of vanishing. polynomial ring in m variables Let R be
a ring and let X1 , X2 , . . . , Xm be indetermi-
polynomial If a0 , a1 , . . . , an are elements nates. The set R[X1 , X2 , . . . , Xm ] of all poly-
of a ring R, and x does not belong to R, then nomials in X1 , X2 , . . . , Xm with coefficients in
R is a ring with respect to the usual addition and
a 0 + a1 x + · · · + an x n multiplication of polynomials and is called the
polynomial ring in m variables X1 , X2 , . . . , Xm
is a polynomial. over R.
polynomial convexity Let ⊆ Cn be a do-
main (a connected open set). If E ⊆ is a Pontrjagin class Let F be a complex PL
subset, then define sheaf over a PL manifold M. The total Pontrja-
gin class p([F]) ∈ H 4∗ (M; R) of a coset [F]
E = {z ∈ : |p(z)| ≤ sup |p(w)| of real PL sheaves via complexification of [F]
w∈E satisfies these axioms:
(i.) If [F] is a coset of real PL sheaves of rank
for all p a polynomial} . m on a PL manifold M, then the total Pontrja-
gin class p([F]) is an element 1 + p1 ([F]) +
The set E is called the polynomially convex hull · · · + p[m/2] ([F]) of H ∗ (M; R) with pi ([F]) ∈
of E in . If the implication E ⊂⊂ implies H 4i (M; R);
E ⊂⊂ always holds, then is said to be (ii.) p( ! [F]) = ∗ p([F]) ∈ H 4∗ (N ; R) for
polynomially convex. any PL map : N → M;
(iii.) p([F] ⊕ [G]) = p([G]) for any cosets [F]
polynomial equation An equation P = 0 and [G] over M;
where P is a polynomial function of one or more (iv.) If [F] contains a bona fide real vector bun-
variables. dle ξ over M, then p([F]) is the classical total
Pontrjagin class p(ξ ) ∈ H 4∗ (M; R).
polynomial function A function which is a
finite sum of terms of the form an x n , where n is
a nonnegative integer and an is a real or complex Pontryagin multiplication A multiplication
number.
h∗ : H∗ (X) ⊗ H∗ (X) → H∗ (X) .
polynomial identity An equation P (X1 , X2 ,
. . . , Xn ) = 0 where P is a polynomial in n
(H∗ (X) are homology groups of the topological
variables with coefficients in a field K such that
space X.)
P (a1 , a2 , . . . , an ) = 0 for all ai in an algebra A
over K.
Pontryagin product The result of Pontrya-
polynomial in m variables A function which gin multiplication. See Pontryagin multiplica-
n n n tion.
is a finite sum of terms ax1 1 x2 2 . . . xmm , where
n1 , n2 , . . . , nm are nonnegative integers and a is
a real or complex number. For example, 5x 2 y 3 + positive angle Given a vector v = 0 in Rn ,
3x 4 z − 2z + 3xyz is a polynomial in three vari- then its direction is described completely by the
ables. angle α between v and i = (1, 0, . . . , 0), the unit
vector in the direction of the positive x1 -axis. If
polynomial ring Let R be a ring. The set we measure the angle α counterclockwise, we
R[X] of all polynomials in an indeterminate X say α is a positive angle. Otherwise, α is a neg-
with coefficients in R is a ring with respect to ative angle.
c 2001 by CRC Press LLC

8. positive chain complex A chain complex X positive Weyl chamber The set of λ ∈ V ∗
such that the only possible non-zero terms Xn such that (β, λ) > 0 for all positive roots β,
are those Xn for which n ≥ 0. where V is a vector space over a subfield R of
the real numbers.
positive cycle An r-cycle A = ni Ai such
that ni ≥ 0 for all i, where Ai is not in the power Let a1 , . . . , an be a finite sequence of
singular locus of an irreducible variety V for all elements of a monoid M. We define the “prod-
i. uct” of a1 , . . . , an by the following: we define
1
j =1 aj = a1 , and
positive definite function A complex val-  
ued function f on a locally compact topological k+1 k
group G such that aj =  aj  ak+1 .
j =1 j =1
f (s − t)φ(s)φ(t)dsdt ≥ 0
G Then
k m k+m
for every φ, continuous and compactly supported aj ak+ = aj .
on G. j =1 =1 j =1
If all the aj = a, we denote a1 · a2 . . . an as a n
positive definite matrix An n × n matrix A, and call this the nth power of a.
such that, for all u ∈ Rn , we have
power associative algebra A distributive al-
(A(u), u) ≥ 0 , gebra A such that every element of A generates
an associative subalgebra.
with equality only when u = 0.
power method of computing eigenvalues
positive divisor A divisor that has only pos- An iterative method for determining the eigen-
itive coefficients. value of maximum absolute value of an n ×
n matrix A. Let λ1 , λ2 , . . . , λn be eigenval-
positive element An element g ∈ G, where
ues of A such that |λ1 | > |λ2 | ≥ · · · ≥ |λn |
G is an ordered group, such that g ≥ e.
and let y1 be an eigenvector such that (λ1 I −
A)y1 = 0. Begin with a vector x (0) such that
positive exponent For an expression a b , the (0)
exponent b if b > 0. (y1 , x (0) ) = 0 and for some i0 , xi0 = 1. De-
termine θ (0) , θ (1) , . . . , θ (m) , . . . and x (1) , x (2) ,
positive matrix An n × n matrix A with real . . . , x (m+1) , . . . by Ax (j ) = θ (j ) x (j +1) . Then
entries such that aj k > 0 for each j and k. See limj →∞ θ (j ) = λ1 and limj →∞ x (j ) is the
also positive definite matrix. eigenvector corresponding to λ1 .
positive number A real number greater than power of a complex number Let z = x +
zero. iy = r(cos θ + i sin θ ) be a complex number
y
with r = x 2 + y 2 and θ = tan−1 x . Let n be
positive root Let S be a basis of a root system a positive number. The nth power of z will be
φ in a vector space V such that each root β can be the complex number r n (cos nθ + i sin nθ ).
written as β = a∈S ma a, where the integers
ma have the same sign. Then β is a positive root power-residue symbol Let n be a positive
if all ma ≥ 0. integer and let K be an algebraic number field
containing the nth roots of unity. Let α ∈ K ×
positive semidefinite matrix An n×n matrix and let ℘ be a prime ideal of the ring such that
A such that, for all u ∈ Rn , we have ℘ is relatively prime to n and α. The nth power
is a positive integer and let K be an algebraic
(A(u), u) ≥ 0 . number field containing the nth roots of unity.
c 2001 by CRC Press LLC

9. Let α ∈ K × and let ℘ be a prime ideal of the groups (or rings, modules, etc.). There is a stan-
ring of integers of K such that ℘ is relatively dard procedure for constructing a sheaf from a
prime to n and α. The nth power residue symbol presheaf.
α
℘ is the unique nth root of unity that is
n primary Abelian group An Abelian group
congruent to α (N℘−1)/n mod ℘. When n = 2 in which the order of every element is a power
and K = Q, this symbol is the usual quadratic of a fixed prime number.
residue symbol.
primary component Let R be a commuta-
predual Let X and Y be Banach spaces such tive ring with identity 1 and let J be an ideal
that X is the dual of Y , X = Y ∗ . Then Y is of R. Assume J = I1 ∩ · · · ∩ In with each Ii
called the predual of X. primary and with n minimal among all such rep-
resentations. Then each Ii is called a primary
preordered set A structure space for a non- component of J .
empty set R is a nonempty collection X of non-
empty proper subsets of R given the hull-kernel primary ideal Let R be a ring with identity
topology. If there exists a binary operation ∗ on 1. An ideal I of R is called primary if I = R
R such that (R, ∗) is a commutative semigroup and all zero divisors of R/I are nilpotent.
and the structure space X consists of prime semi-
group ideals, then it is said that R has an X - primary linear programming problem A
compatible operation. For p ∈ R, let Xp = linear programming problem in which the goal
{A ∈ X : p ∈ A}. A preorder (reflexive and
/ is to maximize the linear function z = cx with
transitive relation) ≤ is defined on R by the rule the linear conditions n=1 aij xj = bi (i =
j
that a ≤ b if and only if Xa ⊆ Xb . Then R is 1, 2, . . . , m) and x ≥ 0, where x = (x1 , x2 , . . . ,
called a preordered set. xn ) is the unknown vector, c is an n × 1 vec-
tor of real numbers, bi (i = 1, 2, . . . , n) and
preordered set A structure space for a non- aij (i = 1, 2, . . . , n, j = 1, 2, . . . , n) are real
empty set R is a nonempty collection X of non- numbers.
empty proper subsets of R given the hull-kernel
topology. If there exists a binary operation ∗ on primary ring Let R be a ring and let N be
R such that (R, ∗) is a commutative semigroup the largest ideal of R containing only nilpotent
and the structure space X consists of prime semi- elements. If R/N is nonzero and has no nonzero
group ideals, then it is said that R has an X - proper ideals, R is called primary.
compatible operation. For p ∈ R, let Xp =
{A ∈ X : p ∈ A}. A preorder (reflexive and
/ primary submodule Let R be a commuta-
transitive relation) ≤ is defined on R by the rule tive ring with identity 1. Let M be an R-module.
that a ≤ b if and only if Xa ⊆ Xb . Then R is A submodule N of M is called primary if when-
called a preordered set. ever r ∈ R is such that there exists m ∈ M/N
with m = 0 but rm = 0, then r n (M/N ) =
presheaf Let X be a topological space. Sup- 0 for some integer n.
pose that, for each open subset U of X, there is
an Abelian group (or ring, module, etc.) F(U ). prime A positive integer greater than 1 with
Assume F(φ) = 0. In addition, suppose that the property that its only divisors are 1 and itself.
whenever U ⊆ V there is a homomorphism The numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 27 are
the first ten primes. There are infinitely many
ρU V : F(V ) → F(U ) prime numbers.
such that ρU U = identity and such that ρU W = prime divisor For an integer n, a prime di-
ρU V ρV W whenever U ⊆ V ⊆ W . The col- visor is a prime that occurs in the prime factor-
lection of Abelian groups along with the homo- ization of n. For an algebraic number field or
morphisms ρU V is called a presheaf of Abelian for an algebraic function field of one variable (a
c 2001 by CRC Press LLC

10. field that is finitely generated and of transcen- C-characters of G, where C denotes the field of
dence degree 1 over a field K), a prime divi- complex numbers. Then χ ∈ Irr(G) is called a
sor is an equivalence class of nontrivial valua- primitive character if χ = ϕ G for any character
tions (over K in the latter case). In the number ϕ of a proper subgroup of G. See also character
field case, the prime divisors correspond to the of group, irreducible character.
nonzero prime ideals of the ring of integers and
the archimedean valuations of the field. primitive element Let E be an extension
field of the field F (E is a field containing F
prime element In a commutative ring R with as subfield). If u is an element of E and x is an
identity 1, a prime element p is a nonunit such indeterminate, then we have the homomorphism
that if p divides a product ab with a, b ∈ R, then g(x) → g(u) of the polynomial ring F [x] into
p divides at least one of a, b. When R = Z, the E, which is the identity on F and send x → u. If
prime elements are of the form ±p for prime the kernel is 0, then F [u] ∼ F [x]. Otherwise,
=
numbers p. we have a monic polynomial f (x) of positive
degree such that the kernel is the principal ideal
prime factor A prime factor of an integer n (f (x)), and then F [u] ∼ F [x]/(f (x)). Then
=
is a prime number p such that n is a multiple of we say E = F (u) is a simple extension of F
p. and u a primitive element (= field generator of
E/F ).
prime field The rational numbers and the
fields Z/pZ for prime numbers p are called primitive equation An equation f (X) = 0
prime fields. Every field contains a unique sub- such that a permutation of roots of f (X) = 0 is
field isomorphic to exactly one of these prime primitive, where f (X) ∈ K[X] is a polynomial,
fields. and K is a field.
prime ideal Let R be a commutative ring primitive hypercubic set A finite subgroup
with identity 1 and let I = R be an ideal of R. K of the orthogonal group O(V ) is called fully
Then I is prime if whenever a, b ∈ R are such transitive if there is a set S = {e1 , . . . , es } that
that ab ∈ I , then at least one of a and b is in I . spans V on which K acts transitively and K has
no invariant subspace in V . In this case, one can
prime number A positive integer p is said choose S as either
to be prime if (i.) the primitive hypercubic type:
(i.) p > 1,
(ii.) p has no positive divisors except 1 and p. S = {e1 , . . . , en } , ei , ej = δij ;
The first few prime numbers are 2, 3, 5, 7, 11,
13, 17. or
(ii.) the primitive hyperbolic type:
prime rational divisor A divisor p =
ni Pi on X over k satisfying the following S = {f1 , . . . , fn+1 } ,
three conditions: (i.) p is invariant under any au- 1, i=1,...n+1,i=j
¯
tomorphism σ of k/k; (ii.) for any j , there exists (fi , fj ) = 1 .
σ −n, i,j =1,...,n+1,i=j
¯
an automorphism σj of k/k such that Pj = P1 j ;
(iii.) n1 = · · · = nt = [k(P1 ) : k]i , where X is primitive ideal Let R be a Banach algebra.
a nonsingular irreducible complete curve, k is a A two-sided ideal I of R is primitive if there is a
subfield of the universal domain K such that X regular maximal left ideal J such that I is the set
is defined over k. Prime rational divisors gen- of elements r ∈ R with rR ⊆ I . The regularity
erate a subgroup of the group of divisors G(X), of J means that there is an element u ∈ R such
which is called a group of k-rational divisors. that r − ru ∈ J for all r ∈ R.
primitive character Let G be a finite group primitive idempotent element An idempo-
and let Irr(G) denote the set of all irreducible tent element that cannot be expressed as a sum
c 2001 by CRC Press LLC

11. a + b with a and b nonzero idempotents satis- over V , and I (Q) is the two-sided ideal of T (V )
fying ab = ba = 0. generated by elements x ⊗ x − Q(x) · 1 for
x ∈ V . Compare with principal automorphism,
primitive permutation representation Let i.e., the unique automorphism α of C(Q) such
G be a group acting as a group of permutations that α(x) = −x, for all x ∈ V .
of a set X. This is called a permutation represen-
tation of G. This representation is called primi- principal automorphism Let A be a com-
tive if the only equivalence relations R(x, y) on mutative ring and let M be a module over A. Let
X such that R(x, y) implies R(gx, gy) for all a ∈ A. The homomorphism
x, y ∈ X and all g ∈ G are equality and the
M x → ax
trivial relation R(x, y) for all x, y ∈ X.
is called the principal homomorphism associ-
primitive polynomial Let f (x) be a poly- ated with a, and is denoted aM . When aM is
nomial with coefficients in a commutative ring one-to-one and onto, then we call aM a princi-
R. When R is a unique factorization domain, pal automorphism of the module M.
f (x) is called primitive if the greatest common
divisor of the coefficients of f (x) is 1. For an principal divisor of functions The formal
arbitrary ring, a slightly different definition is sum
sometimes used: f (x) is primitive if the ideal
generated by the coefficients of f (x) is R. (φ) = m1 p1 + · · · + mj pj + n1 q1 + · · · + nk qk
where p1 , . . . , pj are the zeros and q1 , . . . , qk
primitive ring A ring R is called left prim-
are the poles of a meromorphic function φ, mi
itive if there exists an irreducible, faithful left
is the order of pi and ni is the order of qi .
R-module, and R is called right primitive if
there exists an irreducible, faithful right R- principal genus An ideal group of K formed
module. See also irreducible R-module, faithful by the set of all ideals U of K relatively prime to
R-module. m such that NK/k (U) belongs to H (m), where
k is an algebraic number field, m is an integral
primitive root of unity Let m be a positive divisor of k, T (m) is the multiplicative group
integer and let R be a ring with identity 1. An of all fractional ideals of k which are relatively
element ζ ∈ R is called a primitive mth root prime to m, S(m) is the ray modulo m, H (m)
of unity if ζ m = 1 but ζ k = 1 for all positive is an ideal group modulo m (i.e., a subgroup of
integers k < m. T (m) containing S(m)), and K/k is a Galois
extension.
primitive transitive permutation group Let
G be a transitive group of permutations of a set principal H -series An H -series which is
X. If the stabilizer of each element of X is a strictly decreasing and such that there exists no
maximal subgroup of G, then G is called prim- normal series distinct from , finer than , and
itive. strictly decreasing. See also H -series, normal
series, finer.
principal adele Let K be an algebraic num-
ber field and let AK be the adeles of K. The principal ideal Let R be a commutative ring
image of the diagonal injection of K into AK is with identity 1. A principal ideal is an ideal of
the set of principal adeles. the form aR = {ar|r ∈ R} for some a ∈ R.
principal antiautmorphism A unique an- principal ideal domain An integral domain
tiautomorphism β of a Clifford algebra C(Q) in which every ideal is principal. See principal
such that β(x) = x, for all x ∈ V , where ideal.
C(Q) = T (V )/I (Q), V is an n-dimensional
linear space over a field K, and Q is a qua- principal ideal ring A ring in which every
dratic form on V , T (V ) is the tensor algebra ideal is principal. See principal ideal.
c 2001 by CRC Press LLC

12. Principal Ideal Theorem There are at least cipal submatrix of A. Its determinant is called
two results having this name: a principal minor of A. For example, let
(1) Let K be an algebraic number field and  
let H be the Hilbert class field of K. Every ideal a11 a12 a13
of the ring of integers of K becomes principal A = a21 a22 a23  .
when lifted to an ideal of the ring of integers of a31 a32 a33
H . This was proved by Furtwängler in 1930.
Then, by deleting row 2 and column 2 we obtain
(2) Let R be a commutative Noetherian ring
the principal submatrix of A
with 1. If x ∈ R and P is minimal among the
prime ideals of R containing x, then the codi- a11 a13
mension of P is at most 1 (that is, there is no .
a31 a33
chain of prime ideals P ⊃ P1 ⊃ P2 (strict in-
clusions) in R). This was proved by Krull in Notice that the diagonal entries and A itself are
1928. principal submatrices of A.
principal idele Let K be an algebraic number principal value (1) The principal values of
field. The multiplicative group K × injects di- arcsin, arccos, and arctan are the inverse func-
agonally into the group IK of ideles. The image tions of the functions sin x, cos x, and tan x, re-
is called the set of principal ideles. stricted to the domains − π ≤ x ≤ π , 0 ≤ x ≤
2 2
π , and − π < x < π , respectively. See arc sine,
2 2
principal matrix Suppose A = [Aij ] is an arc cosine, arc tangent.
n × n matrix. The principal matrices associated (2) Let f (x) have a singularity at x = c,
with A are A(k) = [Aij ], 1 ≤ i, j ≤ k ≤ n. with a ≤ c ≤ b. The Cauchy principal value of
b
a f (x) dx is
principal minor See principal submatrix.
c− b
principal order Let K be a finite extension lim f (x) dx + f (x) dx .
→0 a c+
of the rational field Q. The ring of all algebraic
integers in K is called the principal order of K. The Cauchy principal value of an improper
∞ c
integral −∞ f (x)dx is limc→∞ −c f (x)dx.
principal root A root with largest real part (if
this root is unique) of the characteristic equation principle of counting constants Let X and
of a differential-difference equation. Y be algebraic varieties and let C be an ir-
reducible subvariety of X × Y . Let pX and
principal series For a semisimple Lie group, pY denote the projection maps onto the fac-
those unitary representations induced from finite tors of X × Y . Let a1 = dim(pX (C)) and
dimensional unitary representations of a mini- a2 = dim(pY (C)). There exist a nonempty
mal parabolic subgroup. open subset U1 of pX (C), contained in pX (C),
and a nonempty open subset U2 of pY (C)), con-
principal solution A solution F (x) of the tained in pY (C), such that all irreducible com-
equation F (x)/ x = g(x), where F (x) = ponents of C(x) = {y ∈ Y : (x, y ∈ C} have
F (x + x) − F (x). Such a solution F (x) can the same dimension b2 for all x ∈ U1 and such
be obtained by a formula in terms of integral, that all irreducible components of C −1 (y) =
series, and limits. {x ∈ X : (x, y) ∈ C} have the same dimension
b2 for all y ∈ U2 . These dimensions satisfy
principal submatrix A submatrix of an m×n a1 + b2 = a2 + b1 .
matrix A is an (m − k) × (n − ) matrix obtained
from A by deleting certain k rows (k < m) and principle of reflection Two complex num-
columns ( < n) of A. If m = n and if the set bers z1 and z2 are said to be symmetric with
of deleted rows coincides with the set of deleted respect to a circle of radius r and center z0 if
columns, we call the submatrix obtained a prin- (z1 − z0 )(z2 − z0 ) = r 2 . The principle of
c 2001 by CRC Press LLC

13. reflection states that if the image of the circle the following way:
under a linear fractional transformation w =
(az + b)/(cz + d) is again a circle (this hap- Zp,q = Xp × Yq
pens unless the image is a line), then the images ∂p,q = ∂p × 1
w1 and w2 of z1 and z2 are symmetric with re-
∂ p,q = (−1)p 1 × ∂q .
spect to this new circle. See also linear fractional
function.
product formula (1) Let K be a finite exten-
The Schwarz Reflection Principle of complex
sion of the rational numbers Q. Then v |x|v =
analysis deals with the analytic continuation of
1 for all x ∈ K, x = 0, where the product is over
an analytic function defined in an appropriate
all the normalized absolute values (both p-adic
set S, to the set of reflections of the points of S.
and archimedean) of K.
(2) Let K be an algebraic number field con-
product A term which includes many phe- taining the nth roots of unity, and let a and b
nomena. The most common are the following: be nonzero elements of K. For a place v of K
(as in (1) above), let ( a,b )n be the nth norm-
v
(1) The product of a set of numbers is the re-
residue symbol. Then v ( a,b )n = 1. See also
v
sult obtained by multiplying them together. For
norm-residue symbol.
an infinite product, this requires considerations
of convergence.
profinite group Any group G can be made
(2) If A1 , . . . , An are sets, then the product into a topological group by defining the collec-
A1 × · · · × An is the set of ordered n-tuples tion of all subgroups of finite index to be a neigh-
(a1 , . . . , an ) with ai ∈ Ai for all i. This defini- borhood base of the identity. A group with this
tion can easily be extended to infinite products. topology is called a profinite group.
(3) Let A1 and A2 be objects in a category C. projection matrix A square matrix M such
A triple (P , π1 , π2 ) is called the product of A1 that M 2 = M.
and A2 if P is an object of C, πi : P → Ai is
a morphism for i = 1, 2, and if whenever X is projective algebraic variety Let K be a
another object with morphisms fi : X → Ai , ¯ ¯
field, let K be its algebraic closure, and let Pn (K)
for i = 1, 2, then there is a unique morphism ¯
be n-dimensional projective space over K. Let S
f : X → P such that πi f = fi for i = 1, 2. be a set of homogeneous polynomials in X0 , . . . ,
Xn . The set of common zeros Z of S in Pn (K) ¯
(4) See also bracket product, cap product, is called a projective algebraic variety. Some-
crossed product, cup product, direct product, times, the definition also requires the set Z to be
Euler product, free product, Kronecker prod- irreducible, in the sense that it is not the union
uct, matrix multiplication, partial product, ten- of two proper subvarieties.
sor product, torsion product, wedge product.
projective class Let A be a category. A pro-
product complex Let C1 be a complex of jective class is a class P of objects in A such
right modules over a ring R and let C2 be a that for each A ∈ A there is a P ∈ P and a
complex of left R-modules. The tensor product P-epimorphism f : P → A.
C1 ⊗R C2 gives a complex of Abelian groups,
projective class group Consider left mod-
called the product complex.
ules over a ring R with 1. Two finitely gener-
ated projective modules P1 and P2 are said to
product double chain complex The dou- be equivalent if there are finitely generated free
ble chain complex (Zp,q , ∂ , ∂ ) obtained from modules F1 and F2 such that P1 ⊕F1 P2 ⊕F2 .
a chain complex X of right A-modules with The set of equivalence classes, with the opera-
boundary operator ∂p and a chain complex Y tion induced from direct sums, forms a group
of left A-modules with boundary operator ∂q in called the projective class group of R.
c 2001 by CRC Press LLC

14. projective cover An object C in a category there is a surjection f : A → M, there is a
C is the projective cover of an object A if it sat- homomorphism g : M → A such that f g is the
isfies the following three properties: (i.) C is identity map of M.
a projective object. (ii.) There is an epimor-
phism e : C → A. (iii.) There is no projective projective morphism A morphism f : X →
object properly between A and C. In a gen- Y of algebraic varieties over an algebraically
eral category, this means that if g : C → A closed field K which factors into a closed im-
and f : C → C are epimorphisms and C is mersion X → Pn (K) × Y , followed by the pro-
projective, then f is actually an isomorphism. jection to Y . This concept can be generalized to
Thus, projective covers are simply injective en- morphisms of schemes.
velopes “with the arrows turned around.” See
also epimorphism, injective envelope. projective object An object P in a category C
In most familiar categories, objects are sets satisfying the following mapping property: If e :
with structure (for example, groups, topologi- C → B is an epimorphism in the category, and
cal spaces, etc.) and morphisms are particu- f : P → B is a morphism in the category, then
lar kinds of functions (for example, group ho- there exists a (usually not unique) morphism g :
momorphisms, continuous functions, etc.), and P → C in the category such that e ◦ g = f .
epimorphisms are onto functions (surjections) This is summarized in the following “universal
of a particular kind. Here is an example of a mapping diagram”:
projective cover in a specific category: In the P
category of compact Hausdorff spaces and con- f ∃g
tinuous maps, the projective cover of a space X e
always exists, and is called the Gleason cover B ←− C
of the space. It may be constructed as the Stone Projectivity is simply injectivity “with the ar-
space (space of maximal lattice ideals) of the rows turned around.” See also epimorphism,
Boolean algebra of regular open subsets of X. injective object, projective module.
A subset is regular open if it is equal to the inte- In most familiar categories, objects are sets
rior of its closure. See also Gleason cover, Stone with structure (for example, groups, topologi-
space. cal spaces, etc.), and morphisms are particular
kinds of functions (for example, group homo-
projective dimension Let R be a ring with morphisms, continuous maps, etc.), so epimor-
1 and let M be an R-module. The projective phisms are onto functions (surjections) of partic-
dimension of M is the length of the smallest ular kinds. Here are two examples of projective
projective resolution of M; that is, the projective objects in specific categories: (i.) In the cate-
dimension is n if there is an exact sequence 0 → gory of Abelian groups and group homomor-
Pn → · · · → P0 → M → 0, where each phisms, free groups are projective. (An Abelian
Pi is projective and n is minimal. If no such group G is free if it is the direct sum of copies
finite resolution exists, the projective dimension of the integers Z.) (ii.) In the category of com-
is infinite. pact Hausdorff spaces and continuous maps, the
projective objects are exactly the extremely dis-
projective general linear group The quo- connected compact Hausdorff spaces. (A com-
tient group defined as the group of invertible pact Hausdorff space is extremely disconnected
matrices (of a fixed size) modulo the subgroup if the closure of every open set is again open.)
of scalar matrices. See also compact topological space, Hausdorff
space.
projective limit The inverse limit. See in-
verse limit. projective representation A homomorphism
from a group to a projective general linear group.
projective module A module M for which
there exists a module N such that M ⊕ N is projective resolution Let B be a left R mod-
free. Equivalently, M is projective if, whenever ule, where R is a ring with unit. A projective
c 2001 by CRC Press LLC

15. resolution of B is an exact sequence, projective symplectic group The quotient
group defined as the group of symplectic ma-
φ2 φ1 φ0
· · · −→ E1 −→ E0 −→ B −→ 0 , trices (of a given size) modulo the subgroup
{I, −I }, where I is the identity matrix. See sym-
where every Ei is a projective left R module. plectic group.
(We shall define exact sequence shortly.) There
is a companion notion for right R modules. Pro- projective unitary group The quotient group
jective resolutions are extremely important in defined as the group of unitary matrices modulo
homological algebra and enter into the dimen- the subgroup of unitary scalar matrices. See uni-
sion theory of rings and modules. See also flat tary matrix.
resolution, injective resolution, projective mod-
ule, projective dimension. proper component Let U and V be irre-
An exact sequence is a sequence of left R ducible subvarieties of an irreducible algebraic
modules, such as the one above, where every variety X. A simple irreducible component of
φi is a left R module homomorphism (the φi U ∩ V is called proper if it has dimension equal
are called “connecting homomorphisms”), such to dim U + dim V - dim X.
that Im(φi+1 ) = Ker(φi ). Here Im(φi+1 ) is the
image of φi+1 , and Ker(φi ) is the kernel of φi . In proper equivalence An equivalence relation
the particular case above, because the sequence R on a topological space X such that R[K] =
ends with 0, it is understood that the image of φ0 {x ∈ X : xRk for some k ∈ K} is compact for
is B, that is, φ0 is onto. There is a companion all compact sets K ⊆ X.
notion for right R modules.
proper factor Let a and b be elements of a
projective scheme A projective scheme over commutative ring R. Then a is a proper factor
a scheme S is a closed subscheme of projective of b if a divides b, but a is not a unit and there
space over S. is no unit u with a = bu.
projective space Let K be a field and con- proper fraction A positive rational number
sider the set of (n + 1)-tuples (x0 , . . . , xn ) in such that the numerator is less than the denom-
K n+1 with at least one coordinate nonzero. Two inator. See also improper fraction.
tuples (. . . , xi , . . . ) and (. . . , xi , . . . ) are equiv-
alent if there exists λ ∈ K × such that xi = proper intersection Let Y and Z be subva-
λxi for all i. The set Pn (K) of all equivalence rieties of an algebraic variety X. If every irre-
classes is called n-dimensional projective space ducible component of Y ∩ Z has codimension
over K. It can be identified with the set of equal to codimY +codimZ, then Y and Z are said
lines through the origin in K n+1 . The equiv- to intersect properly.
alence class of (x0 , x1 , . . . , xn ) is often denoted
(x0 : x1 : · · · : xn ). More generally, let R be a proper Lorentz group The group formed
commutative ring with 1. The scheme Pn (R) is by the Lorentz transformations whose matrices
given as the set of homogeneous prime ideals of have determinants greater than zero.
R[X0 , . . . , Xn ] other than (X0 , . . . , Xn ), with a
structure sheaf defined in terms of homogeneous proper morphism of schemes A morphism
rational functions of degree 0. It is also possible of schemes f : X → Y such that f is sepa-
to define projective space Pn (S) for a scheme rated and of finite type and such that for every
S by patching together the projective spaces for morphism T → Y of schemes, the induced mor-
appropriate rings. phism X ×T Y → T takes closed sets to closed
sets.
projective special linear group The quotient
group defined as the group of matrices (of a fixed proper orthogonal matrix An orthogonal
size) of determinant 1 modulo the subgroup of matrix with determinant +1. See orthogonal
scalar matrices of determinant 1. matrix.
c 2001 by CRC Press LLC

16. proper product Let R be an integral domain ness condition for all prime ideals P is called
with field of quotients K and let A be an algebra pseudogeometric.
over K. Let M and N be two finitely generated
R-submodules of A such that KM = KN = A. pseudovaluation A map v from a ring R into
If {a ∈ A : Ma ⊆ M} = {a ∈ A : aN ⊆ N }, the nonnegative real numbers such that (i.) v(r)
and this is a maximal order of A, then the product = 0 if and only if r = 0, (ii.) v(rs) ≤ v(r)v(s),
MN is called a proper product. (iii.) v(r + s) ≤ v(r) + v(s), and (iv.) v(−r) =
v(r) for all r, s ∈ R.
proper transform Let T : V → W be a
rational mapping between irreducible varieties p-subgroup A finite group whose order is
and let V be an irreducible subvariety of V . A a power of p is called a p-group. A p-group
proper transform of V by T is the union of all that is a subgroup of a larger group is called a
irreducible subvarieties W of W for which there p-subgroup.
is an irreducible subvariety of T such that V and
W correspond. p-subgroup For a finite group G and a prime
integer p, a subgroup S of G such that the order
proportion A statement equating two ratios, of S is a power of p.
a c
b = d , sometimes denoted by a : b = c : d.
The terms a and d are the extreme terms and the pure imaginary number An imaginary num-
terms b and c are the mean terms. ber. See imaginary number.
pure integer programming problem A
proportional A term in the proportion a =
b
c problem similar to the primary linear program-
d . Given numbers a, b, and c, a number x which ming problem in which the solution vector x =
satisfies a = x is a fourth proportional to a, b,
b
c
(x1 , x2 , . . . , xn ) is a vector of integers: the prob-
and c. Given numbers a and b, a number x
lem is to minimize z = cx with the conditions
which satisfies a = x is a third proportional to
b
b
Ax = b, x ≥ 0, and xj (j = 1, 2, . . . , n) is an
a
a and b, and a number x which satisfies x = x b integer, where c is an n × 1 vector of real num-
is a mean proportional to a and b.
bers, A is an m × n matrix of real numbers, and
b is an m × 1 matrix of real numbers.
proportionality The state of being in pro-
portion. See proportion.
purely infinite von Neumann algebra A
von Neumann algebra A which has no semifinite
Prüfer domain An integral domain R such normal traces on A.
that every nonzero ideal of R is invertible. Equiv-
alently, the localization RM is a valuation ring purely inseparable element Let L/K be an
for every maximal ideal M. Another equivalent extension of fields of characteristic p > 0. If
condition is that an R-module is flat if and only n
α ∈ L satisfies α p ∈ K for some n, then α is
if it is torsion-free. called a purely inseparable element over K.
Prüfer ring An integral domain R such that purely inseparable extension An extension
all finitely generated ideals in R are inversible L/K of fields such that every element of L is
in the field of quotients of R. See also integral purely inseparable over K. See purely insepa-
domain. rable element.
pseudogeometric ring Let A be a Noethe- purely inseparable scheme Given an ir-
rian integral domain with field of fractions K. If reducible polynomial f (X) over a field k, if
the integral closure of A in every finite extension the formal derivative df/dX = 0, then f (X)
of K is finitely generated over A, then A is said is inseparable; otherwise, f is separable. If
to satisfy the finiteness condition. A Noethe- char(k) = 0, every irreducible polynomial
rian ring R such that R/P satisfies the finite- f (X)(= 0) is separable. If char(k) = p > 0,
c 2001 by CRC Press LLC

17. an irreducible polynomial f (X) is inseparable Pythagorean field
√ A field F that contains
if and only if f (X) = g(X p ). An algebraic a 2 + b2 for all a, b ∈ F .
element α over k is called separable or insep-
arable over k if the minimal polynomial of α Pythagorean identities The following ba-
over k is separable or inseparable. An alge- sic identities involving trigonometric functions,
braic extension of k is called separable if all resulting from the Pythagorean Theorem:
elements of K are separable over k; otherwise,
K is called inseparable. If α is inseparable, then sin2 (x) + cos2 (x) = 1
k has nozero characteristic p and the minimal
polynomial f (X) of α can be decomposed as tan2 (x) + 1 = sec2 (x)
r r r
f (X) = (X − α1 )p (X − α2 )p . . . (X − αm )p , 1 + cot 2 (x) = csc2 (x) .
r ≥ 1, where α1 , . . . , αm are distinct roots of
r
f (X) in its splitting field. If α p ∈ k for some
Pythagorean numbers Any combination of
r, we call α purely inseparable over k. An al-
three positive integers a, b, and c such that a 2 +
gebraic extension K of k is called purely in-
b2 = c2 .
separable if all elements of the field are purely
inseparable over k. Let V ⊂ k n be a reduced
Pythagorean ordered field An ordered field
irreducible affine algebraic variety. V is called P such that the square root of any positive ele-
purely inseparable if the function field k(V ) is ment of P is in P .
purely inseparable over k. Therefore, we can de-
fine the same notion for reduced irreducible al- Pythagorean Theorem Consider a right tri-
gebraic varieties. Since there is a natural equiva- angle with legs of length a and b and hypotenuse
lence between the category of algebraic varieties of length c. Then a 2 + b2 = c2 .
over k and the category of reduced, separated, al-
gebraic k-schemes, by identifying these two cat- Pythagorean triple A solution in positive
egories, we define purely inseparable reduced, integers x, y, z to the equation x 2 + y 2 = z2 .
irreducible, algebraic k-schemes. Some examples are (3, 4, 5), (5, 12, 13), and
(20, 21, 29). If x, y, z have no common divi-
purely transcendental extension An exten- sor greater than 1, then there are integers a, b
sion of fields L/K such that there exists a set such that x = a 2 −b2 , y = 2ab, and z = a 2 +b2
of elements {xi }i∈I , algebraically independent (or the same equations with the roles of x and
over K, with L = K({xi }). See algebraic inde- y interchanged). Also called Pythagorean num-
pendence. bers.
pure quadratic A quadratic equation of the
form ax 2 + c = 0, that is, a quadratic equation
with the first degree term bx missing.
c 2001 by CRC Press LLC