Splitting algebras, factorization algebras, and residues

21 Pages·2009·0.207 MB·English

by  Dan Laksov

Splitting algebras, factorization algebras, and residues
Dan Laksov
Department of Mathematics, KTH
26 Januar 2009
Abstract.
Wedefineandconstructsplittingandfactorizationalgebras. Inordertostudythese
algebrasweintroduceresiduesthatgeneralizeclassicalSchurpolynomials. Inparticular
we show how residues induceGysin maps betweensplittingand factorization algebras.
We base our presentation upon well known and classical results on alternating and
symmetric polynomials, that we prove. Throughout we focus on results that are used
to describe the cohomology theories for flag and Grassmann manifolds, both in the
classical, quantum, equivariant, and quantum-equivariant sense. As a consequence we
obtain, for example, an interpretation by factorization algebras of the the astonishing
description by L. Gatto of the cohomology of grassmannians via exterior powers.
Contents
Introduction
1. Formal Laurent series and formal power series
2. Residues
3. Symmetric and alternating polynomials
4. Residues and symmetric polynomials
5. Splitting and factorization algebras
6. Applications to symmetric polynomials
7. Gysin maps and module bases for factorization algebras
8. Exterior products and factorization algebras
1991 Mathematics Subject Classification. 14N15, 14M15, 05E05, 13F20.
Key words and phrases. alternating polynomials, determinantal formulas, exterior products,
factorization algebras, formal Laurent series, formal power series, Jacobi-Trudi Lemma, residues,
Schubert cycles, splitting algebras, symmetric groups, symmetric polynomials,.
Typeset by AMS-TEX
1
Introduction
Splitting and factorization algebras appear in several branches of mathematics. A
well known and illustrative example is polynomial rings considered as algebras over
the symmetric polynomials. This is the generic example of splitting algebras (see [B],
[EL], [LT1], and [PZ], and further references there).
Another well known example we obtain by adjoining a root to a monic polynomial
in one variable with coefficients in a ring. This useful, and frequently used, construc-
tion splits off a linear term from the polynomial. Successively adjoining roots we
obtain splitting algebras of higher order of the polynomial.
An important class of examples of splitting and factorization algebras, that is
our main motivation for writing these notes, is the cohomology rings of flag and
Grassmann manifolds (see [G], [GS1], [LT1], [LT2] and [LT3], and further references
there). The cohomology ring of the manifold of complete flags in a vector space V of
dimension n is, for example, the splitting algebra for Tn. Moreover, the cohomology
ring of the grassmannian of d-dimensional subspaces of V is the factorization algebra
of Tn in factors of degrees d and n − d. For families of flags or grassmannians
we get the corresponding splitting, respectively factorization, algebra of the Chern
polynomial of the locally free sheaf that defines the family. Quantum, equivariant,
and quantum-equivariant cohomology of grassmannians give more examples. These
are obtained from the factorization algebras by changing bases or by varying the
polynomial we factor (see [GS2], [L1] and [L2], and further references there).
A more unusual example comes from Galois theory, where splitting algebras can
be used to give a presentation of the theory that lies close to the original point of
view (see [EL], [K], and [T]).
In this article we give a self-contained presentation of those parts of the theory
of splitting and factorization algebras that are related to the above examples. In
particular we give bases for factorization algebras that correspond to bases consisting
of classical Schur polynomials for symmetric polynomials and to Schubert cycles in
cohomology. We also construct Gysin maps from splitting algebras to factorization
algebras that correspond to the Gysin maps from cohomology rings of flag manifolds
to cohomology rings of grassmannians in geometry. Our treatment is different from
that in the works mentioned above in that it is based upon the theory of symmetric
polynomials, and thus illustrates the connections between splitting and factorization
algebras and the theory of symmetric polynomials.
It is worth noticing that as a consequence of our presentation we immediately
obtain an interpretation by factorization algebras of the astonishing description of
the cohomology of grassmannians via exterior products given by Letterio Gatto (see
[G], [GS1], [LT1] and [LT2]). In particular we obtain the fundamental determinantal
formulas that correspond to the Giambelli formula and the determinantal formula in
Schubert calculus.
We also indicate how the connection between splitting algebras and polynomial
algebras, mentioned above, gives somewhat exotic proofs of the different parts of the
Main Theorem of Symmetric Polynomials.
The prerequisites for reading this article are knowledge of the definition of rings
and ideals, and of the residue ring of a ring by an ideal. In addition, some knowledge
is needed of basic results on polynomial rings and determinants.
2
1. Formal Laurent series and formal power series
In this section we remind the reader of formal Laurent series, and power series
over commutative rings with unit. We also recall some basic results on inversion of
power series, and in particular, the usual relation between elementary and complete
symmetric polynomials.
1.1 Algebras. By a ring A we always mean a commutative ring with unity. An
A-algebra ϕ : A → B is a homomorphism of rings. We shall throughout denote by
A[T] the A-algebra of polynomials in the variable T with coefficients in A.
1.2 Formal Laurent series. A formal Laurent series in the variable 1 is a formal
T
expression
a a
g(T) = ···+a T2 +a T +a + 1 + 2 +··· ,
−2 −1 0
T T2
where each a lies in a ring A. For every A-algebra ϕ : A → B we write
i
ϕ(a ) ϕ(a )
ϕg(T) = ···+ϕ(a )T2 +ϕ(a )T +ϕ(a )+ 1 + 2 +··· .
−2 −1 0 T T2
1.3 Formal power series. When 0 = a = a = ··· we say that the formal Laurent
1 2
series g(T) is a formal power series, and we change the indexing to
g(T) = b +b T +b T2 +··· .
0 1 2
1.4 Inverting formal power series. Let
p(T) = Tn −c Tn−1 +···+(−1)nc
1 n
be a polynomial in the variable T with coefficients in the ring A. An easy calculation
shows that the equation
1 = (1−c T +···+(−1)nc Tn)(1+s T +s T2 +···)
1 n 1 2
of formal power series has an unique solution with s ,s ,... in A, and that each
1 2
element s can be expressed as a polynomial in c ,...,c with integer coefficients.
i 1 n
Conversely, an equally easy calculation shows that the elements s ,s ,... determine
1 2
c ,...,c uniquely and that each element c can be expressed as a polynomial in
1 n i
s ,...,s with integer coefficients.
1 n
1.5 Example. For every natural number h we have
Th 1 s s
= Th−n = Th−n 1+ 1 + 2 +···
p(T) 1− c1 +···+(−1)n cn T T2
T Tn
(cid:16) (cid:17)
s s s
h−n+1 h−n+2 h
= ···+ + +···+ +··· . (1.5.1)
T T2 Tn
3
1.6 Formal power series and symmetric polynomials. Let T ,...,T be alge-
1 n
braically independent elements over the ring A and write
P(T) = (T −T )···(T −T ) = Tn −C Tn−1 +···+(−1)nC .
1 n 1 n
Then
C = T ···T
j i1 ij
0≤i1<X···<ij≤n
is the j’th elementary symmetric polynomial in the variables T ,...,T . We obtain
1 n
1 1
=
1−C T +···+(−1)nC Tn (1−T T)···(1−T T)
1 n 1 n
n
= (1+T T +T2T2 +···) = 1+S T +S T2 +··· ,
i i 1 2
i=1
Y
where
S = T ···T
j i1 ij
0≤i1X,...,ij≤n
is the j’th complete symmetric polynomial in the variables T ,...,T .
1 n
2. Residues
Residues will play an important role in the remaining part of this article. We here
define residues and give their main properties.
2.1 Definition. Let
a a
g (T) = ···+a T2 +a T +a + i1 + i2 +···
i i−2 i−1 i0 T T2
for i = 1,...,n be formal Laurent series in the variable 1. We write
T
a11 ... a1n
. . .
Res(g1,...,gn) = det .. .. .. .
!
an1 ... ann
The main properties of residues are summarized in the following result:
2.2 Lemma. Let A be a ring.
(1) Res is A-linear in g ,...,g . That is, for every index i, for every formal
1 n
Laurent series g′, and for every pair of elements a,a′ in A, we have
i
Res(g ,...,ag +a′g′,...,g ) = aRes(g ,...,g ,...,g )+a′Res(g ,...,g′,...,g ).
1 i i n 1 i n 1 i n
(2) Res is alternating in g ,...,g . That is, if g = g for some i 6= j we have
1 n i j
Res(g ,...,g ) = 0.
1 n
(3) Res is zero on polynomials. That is, if at least one g is a polynomial in T
i
we have
Res(g ,...,g ) = 0.
1 n
(4) Res is functorial. That is, when ϕ : A → B is an A-algebra and g ,...,g
1 n
have coefficients in A, then
ϕ(Res(g ,...,g )) = Res(ϕg ,...,ϕg ).
1 n 1 n
4
Proof. All the properties of Res follow directly from Definition 2.1, and the corre-
sponding properties of determinants.
The following result shows, in particular, that the residue generalizes the classical
Schur polynomials (see e.g. [La2], [M2] and [Ma]). It will be used repeatedly in the
following.
2.3 Proposition. Let p(T) = Tn −c Tn−1 +···+(−1)nc be in the algebra A[T]
1 n
of polynomials in T with coefficients in A. Moreover, let s = 0 for i = 1,2,..., let
−i
s = 1, and let s ,s ,... be determined by the equation
0 1 2
1 = (1−c T +···+(−1)nc Tn)(1+s T +s T2 +···),
1 n 1 2
of formal power series. For all natural numbers h ,...,h , we have
1 d
Th1 Thd sh1−.n+1 .... sh1−.n+d
Res ,..., = det .. .. .. .
p p  
(cid:18) (cid:19) shd−n+1 ... shd−n+d
 
In particular, when 0 ≤ h ≤ n−i for i = 1,...,d, we have
i
Th1 Thd 1 when hi = n−i for i = 1,...,d
Res ,..., =
p p 0 when h < n−j for some j.
(cid:18) (cid:19) (cid:26) j
Proof. The first part of the proposition follows immediately from Definition 2.1 and
equation (1.5.1).
From the first part of the proposition it follows that when 0 ≤ h ≤ n − i for
i
i = 1,...,d the d×d-matrix (s ) is upper triangular. Moreover, it follows that
hi−n+j
there are ones on the diagonal when h = n −i for i = 1,...,d, and a zero on the
i
diagonal in position (j,j) when h < n − j. Thus the last part of the proposition
j
holds.
3. Symmetric and alternating polynomials
Here we recall some terminology concerning symmetric and alternating polynomi-
als.
3.1 Notation, terminology and elementary properties. Let Abearingandlet
T ,...,T be algebraically independent elements over A. We denote by A[T ,...,T ]
1 n 1 n
the A-algebra of polynomials in these variables with coefficients in A.
The symmetric group S , that is, the group of permutations of 1,2,...,n, operates
n
on A[T ,...,T ] by
1 n
(σf)(T ,...,T ) = f(T ,...,T )
1 n σ(1) σ(n)
for all f ∈ A[T ,...,T ] and σ ∈ S . A polynomial f ∈ A[T ,...,T ] is symmetric
1 n n 1 n
when
(σf)(T ,...,T ) = f(T ,...,T ) for all σ ∈ S .
1 n 1 n n
The symmetric polynomials in A[T ,...,T ] form an A-algebra A[T ,...,T ]sym.
1 n 1 n
5
For every element f ∈ A[T ,...,T ] we write
1 n
alt(f) = sign(σ)(σf),
σ∈S
Xn
where sign(σ) is the sign of the permutation σ. A polynomial of the form alt(f)
for f ∈ A[T ,...,T ] is called alternating. The alternating polynomials form an
1 n
A-submodule A[T ,...,T ]alt of A[T ,...,T ].
1 n 1 n
For all polynomials f ,...,f in A[T] we write
1 n
f1(T1) ... f1(Tn)
. . .
(f (T )) = . . . .
i j . . .
 
fn(T1) ... fn(Tn)
 
Then
alt(f (T )···f (T )) = sign(σ)f (T )···f (T ) = det(f (T )).
1 1 n n 1 σ(1) n σ(n) i j
σ∈S
Xn
3.2 Remark. The determinant det(Tn−i) = alt(Tn−1···T0) is an alternating poly-
j 1 n
nomial and is called the Vandermonde determinant. An easy calculation shows that
it is equal to (T − T ). We want however to make the point that this is
0≤i<j≤n i j
not needed in the following. All that we need is that the Vandermonde determinant
Q
is not a zero divisor in A[T ,...,T ], or equivalently, that it is not zero. This follows,
1 n
for example, from the expansion of the determinant det(Tn−i) that contains a single
j
monomial of the form Tn−1···T0.
1 n
3.3 Lemma. Let T ,...,T be algebraically independent elements over A, and let
1 n
P(T) = (T −T )···(T −T ) = Tn −C Tn−1 +···+(−1)nC .
1 n 1 n
For all f ,...,f in A[C ,...,C ] we have that Res(f1,..., fn) is in A[C ,...,C ].
1 n 1 n P P 1 n
Proof. Itfollowsfrom 1.4and 1.6that fi iscontainedinA[C ,...,C ]. Theassertion
P 1 n
of the lemma thus follows from Definition 2.1.
4. Residues and symmetric polynomials
In this section we first show how we can use residues to give a natural proof of
a general version of the Jacobi-Trudi Lemma. This Lemma is then used to give the
well known bases for the alternating polynomials as an A-module, and as a module
over the symmetric polynomials. As a consequence we obtain the well known bases
of the symmetric polynomials in terms of Schur polynomials. We also observe that
the residue can be used to define a Gysin type map from a polynomial algebra to
the corresponding algebra of symmetric polynomials. The results of this section will
later be generalized to splitting algebras.
6
4.1 The Jacobi-Trudi Lemma. Let T ,...,T be algebraically independent ele-
1 n
ments over the ring A and let
P(T) = (T −T )···(T −T ).
1 n
For all polynomials f ,...,f in A[T] we obtain
1 n
f f
det(f (T )) = Res 1,..., n det(Tn−i).
i j P P j
(cid:18) (cid:19)
Proof. We use the division algorithm to the polynomial f (T) modulo P(T) over the
i
algebra A[T ,...,T ] and obtain
1 n
f (T) = q (T)P(T)+r (T)
i i i
where q (T) and r (T) have coefficients in A[T ,...,T ], and r (T) is of degree less
i i 1 n i
that n in T. Since Res is linear in f ,...,f and zero on polynomials by Lemma 2.2
1 n
we obtain
f f r r r r
1 n 1 n 1 n
Res ,..., = Res q + ,...,q + = Res ,..., .
1 n
P P P P P P
(cid:18) (cid:19)
(cid:16) (cid:17) (cid:16) (cid:17)
Moreover, since P(T ) = 0 for j = 1,...,n, we obtain the equations det(f (T )) =
j i j
det(q (T )P(T )+r(T )) = det(r (T )). Since bothResanddetarelinearinr ,...,r
i j j j i j 1 n
it suffices to prove the lemma when r = Thi with 0 ≤ h < n for i = 1,...,n.
i i
Moreover, since Res is alternating in r ,...,r by Lemma 2.2, and the same is true
1 n
for det, we can assume that n > h > ··· > h ≥ 0, that is, we can assume that
1 n
h = n − i for i = 1,...,n. However, then the lemma follows from the equality
i
Res(Tn−1,..., T0) = 1 of Proposition 2.3.
P P
The next result gives the bases of A[T ,...,T ]alt alluded to above. As a corollary
1 n
we obtain a basis for A[T ,...,T ]sym as an A-module in terms of Schur polynomials.
1 n
4.2 Theorem. Let T ,...,T be algebraically independent elements over the ring A.
1 n
(1) The A-module A[T ,...,T ]alt is an A[T ,...,T ]sym-submodule of the alge-
1 n 1 n
bra A[T ,...,T ], and the homomorphism A[T ,...,T ] → A[T ,...,T ]alt
1 n 1 n 1 n
that maps f to alt(f) is A[T ,...,T ]sym-linear.
1 n
(2) As an A[T ,...,T ]sym-module A[T ,...,T ]alt is free of rank 1 with basis the
1 n 1 n
Vandermonde determinant det(Tn−i).
j
(3) As an A-module A[T ,...,T ]alt is free with basis det(Thi) for h > ··· >
1 n j 1
h ≥ 0.
n
Proof. (1) For g ∈ A[T ,...,T ]sym and f ∈ A[T ,...,T ]alt we have
1 n 1 n
alt(gf) = sign(σ)σ(gf)= sign(σ)gσ(f)= galt(f),
σ∈S σ∈S
Xn Xn
that proves assertion (1).
7
(2) It follows from assertion (1) that we have an inclusion of A[T ,...,T ]sym-
1 n
modules A[T ,...,T ]symdet(Tn−i) ⊆ A[T ,...,T ]alt. Since det(Tn−i) is not a zero
1 n j 1 n j
divisor in A[T ,...,T ], as observed in 3.2, it suffices to show the converse inclusion.
1 n
From the definition of A[T ,...,T ]alt it is clear that this A-module is generated
1 n
by the elements alt(f (T )···f (T )) = det(f (T )) for all f ,...f i A[T]. Thus
1 1 n n i j 1 n
it follows from the Jacobi-Trudi Lemma 4.1 that the A-module A[T ,...,T ]alt is
1 n
generated by the elements Res(f1,..., fn)det(Tn−i), where P(T) = (T−T )···(T−
P P j 1
T ). SinceRes(f1,..., fn)isinA[T ,...,T ]sym forallf ,...,f inA[T],aswenoted
n P P 1 n 1 n
in Lemma 3.3, it follows that A[T ,...,T ]alt ⊆ A[T ,...,T ]symdet(Tn−i), as we
1 n 1 n j
wanted to show.
(3) As we just saw the A-module A[T ,...,T ]alt is generated by the elements
1 n
Res(f1,..., fn)det(Tn−i) for all f ,...,f in A[T]. Since Res is linear in f ,...,f
P P j 1 n 1 n
by Lemma 2.2 it is generated by the elements with f = Thi for i = 1,...,n, and
i
since Res is alternating by the same lemma we can assume that h > ··· > h ≥
1 n
0. However, the elements det(Thi) = Res(Th1,..., Thn)det(Tn−i) for h > ··· >
j P P j 1
h ≥ 0 are linearly independent over A because det(Thi) is the only one of these
n j
polynomials in T ,...,T that contains the monomial Th1 ···Thn.
1 n 1 n
4.3 Corollary. Let P(T) = (T −T )···(T −T ) = Tn −C Tn−1 +···+(−1)nC .
1 n 1 n
(1) The homomorphism
∂(P) : A[T ,...,T ] → A[T ,...,T ]sym
1 n 1 n
defined by ∂(P)(f (T )···f (T )) = Res(f1,..., fn) is linear as a homomor-
1 1 n n P P
phism of A[T ,...,T ]sym-modules.
1 n
(2) The A-module A[T ,...,T ]sym is free with basis Res(Th1,..., Thn) for h >
1 n P P 1
··· > h ≥ 0.
n
(3) We have A[T ,...,T ]sym = A[C ,...,C ].
1 n 1 n
Proof. It follows from assertion (2) of the theorem that multiplication by det(Tn−i)
j
gives an isomorphism µ : A[T ,...,T ]sym → A[T ,...,T ]alt of A[T ,...,T ]sym-
1 n 1 n 1 n
modules. Moreover, it follows from the Jacobi-Trudi Lemma 4.1 that µ∂(P) = alt.
Thus assertions (1) and (2) follow from assertions (1) and (3) of the theorem.
(3) It is clear that we have an inclusion A[C ,...,C ] ⊆ A[T ,...,T ]sym. The
1 n 1 n
converse inclusion follows from assertion (2) since Lemma 3.3 shows that the element
Res(Th1,..., Thn) lies in A[C ,...,C ].
P P 1 n
5. Splitting and factorization algebras
Here we first define splitting and factorization algebras and give the two most
common constructions of splitting algebras. We obtain a basis for splitting algebras
as a module, and also a proof of the existence of factorization algebras.
Both splitting and factorization algebras can be constructed in many alternative
ways(see[B],[EL],[LT1]and[PZ]).Eachconstructionprovidesadifferentperspective
of the field. Properties that are obvious in one construction may be complicated in
another. The connections between the different constructions are thus of separate
interest.
8
5.1 Definition. Let
p(T) = T −c Tn−1 +···+(−1)nc
n 1 n
be in the algebra A[T] of polynomials in the variable T over the ring A. Moreover,
let m ,...,m be positive integers such that
1 r
m +···+m = n.
1 r
A factorization of p(T) in factors of degrees m ,...,m over an A-algebra ψ : A → B
1 r
is an ordered set of polynomials q ,...,q of degrees m ,...,m in B[T] such that
1 r 1 r
ψp(T) = Tn −ψ(c )Tn−1 +···+(−1)nψ(c ) = q (T)···q (T).
1 n 1 r
We say that an A-algebra ϕ : A → Factm1,...,mr(p) is a factorization algebra for p(T)
A
over A in factors of degrees m ,...,m when we have a factorization
1 r
ϕp(T) = p (T)···p (T)
1 r
over Factm1,...,mr(p) in factors of degrees m ,...,m , and when ϕ satisfies the fol-
A 1 r
lowing universal property:
For every A-algebra ψ : A → B such that we have a factorization
ψp(T) = q (T)···q (T)
1 r
of p(T) over B in factors of degrees m ,...,m , there is a unique A-algebra homo-
1 r
morphism
χ : Factm1,...,mr(p) → B
A
such that
χϕp (T) = q (T) for i = 1,...,r.
i i
We call ϕp(T) = p (T)···p (T) the universal factorization of p(T) over A.
1 r
When r = d+1 and 1 = m = ··· = m we write
1 d
Fact1,...,1,mr(p) = Splitd(p)
A A
and call Splitd(p) the d’th splitting algebra for p(T) over A. The universal factoriza-
A
tion
ϕp(T) = (T −ξ )···(T −ξ )p (T)
1 d d+1
we call the universal splitting and we call ξ ,...,ξ the universal roots. Moreover,
1 d
we let
Split (p) = Splitn(p)
A A
and call Split (p) the splitting algebra for p(T) over A. For every integer d such that
A
0 ≤ d ≤ n we write
Factd,n−d(P) = Factd(p).
A A
9
5.2 Example. As mentioned in the introduction we obtain a first splitting algebra
for a monic polynomial p of degree n in A[T] by adjunction of a root of p(T) to A.
More precisely, we have that A[T]/(p) is a first splitting algebra Split1 (p) of p(T)
A
over A, or a factorization algebra Fact1 (p) of p(T) over A in factors of degrees 1 and
A
n − 1. The universal splitting is ϕp(T) = (T − ξ)p (T), where ξ is the class of T
2
modulo p(T). Similarly A[T]/(p) is a factorization algebra Factn−1(p) of p(T) over
A
A in factors of degrees n−1 and 1 with universal factorization p (T)(T −ξ).
1
5.3 Remark and convention. From Example 5.2 and the well known properties
of the A-algebra A[T]/(p) it follows that Split1 (p) is a free A-module with a basis
A
1,ξ,...,ξn−1. Since 1 is part of this basis the map A → Split1 (p) is injective. When
A
it can cause no confusion we shall identify A with its image in Split1 (p) via this map.
A
5.4 Remark. Two factorizationalgebrasB ,B forp(T) over Ain factorsof degrees
1 2
m ,...,m are canonically isomorphic as A-algebras. This is because it follows from
1 r
theuniversal propertiesofB andB that we haveunique A-algebrahomomorphisms
1 2
ψ : B → B , respectively ψ : B → B , and, again by the uniqueness, we have
2 2 1 1 1 2
that ψ ψ and ψ ψ are the identity maps of B , respectively of B .
2 1 1 2 1 2
We now prove the existence of splitting algebras.
5.5 Theorem (Construction 1). Let p(T) = Tn −c Tn−1 +···+(−1)nc be in
1 n
A[T] and let
P(T) = (T −T )···(T −T ) = Tn −C Tn−1 +···+(−1)nC ,
1 n 1 n
where T ,...,T are algebraically independent elements over A. Then the residue
1 n
algebra
A[T ,...,T ]/(C −c ,...,C −c )
1 n 1 1 n n
of the polynomial algebra A[T ,...,T ] modulo the ideal generated by the elements
1 n
C − c ,...,C − c is a splitting algebra for p(T) over A. The residue classes
1 1 n n
ξ ,...,ξ of T ,...,T are the universal roots.
1 n 1 n
Proof. Let
ϕ : A → A[ξ ,...,ξ ]
1 n
be the A-algebra. The residue class of C by the canonical map
i
χ : A[T ,...,T ] → A[ξ ,...,ξ ]
1 n 1 n
is then ϕ(c ), and we have
i
ϕp(T) = Tn −ϕ(c )Tn−1 +···+(−1)nϕ(c )
1 n
= Tn −χ(C )Tn−1 +···+(−1)nχ(C ) = χP(T) = (T −ξ )···(T −ξ ).
1 n 1 n
Thus p(T) splits completely over A[ξ ,...,ξ ].
1 n
Let ψ : A → B be an A-algebra such that
ψp(T) = (T −b )···(T −b )
1 n
10