Применение коник в теории квадратичных форм и центральных простых алгебр тема диссертации и автореферата по ВАК РФ 00.00.00, доктор наук Сивацкий Александр Станиславович

• Сивацкий Александр Станиславович
• доктор наук
• 2023

Оглавление диссертации доктор наук Сивацкий Александр Станиславович

Contents

Introduction

Chapter 1 Indecomposable algebras of exponent

1 Construction of certain quaternion algebra

2 Indecomposable algebras over Laurent series fields

3 Improvement of admissible triples

4 Cohomological invariants for central simple algebras of degree 8 and exponent

5 Application to torsion in the second Chow group

Chapter 2 The homology groups of the Brauer complex for a triquadratic field

extension

1 Standard elements for a triquadratic extension

2 Principal Lemma

3 The second divided power operation on the group 2 Br(k(/a,Vb, /c)/k)

4 The natural transformation h2 ® h\ ^ H3

5 The group h3 of the Brauer complex

Chapter 3 Nonexcellent field extensions of 2-primary degree

1 Special cup-product elements in the third cohomology group

2 Theorem on nonexcellent extensions

Chapter 4 Products of two conics and nonstandard triples of quaternion algebras

1 Nonexcellence of the function field of the product of two conics

2 Linked triples of quaternion algebras

3 A special example of a linked triple

4 Pairs of biquaternion algebras and standard triples

Chapter 5 Special quadratic forms and division algebras over the rational function fields

1 An application of the excellence property of conics

2 Behavior of quadratic forms under biquadratic extensions

Conclusion

Bibliography

Введение диссертации (часть автореферата) на тему «Применение коник в теории квадратичных форм и центральных простых алгебр»

Introduction

Relevance and the present state of the subject

In many problems of the algebraic theory of quadratic forms the function fields of quadrics are used. One of the earliest results, where this notion was applied, is the Arason-Pfister Hauptsatz, which was proved in 1971. This fundamental theorem claims that given a field F of characteristic different from 2, the dimension of any anisotropic quadratic form in the ideal In(F) is at least 2ra. Even today, 52 years later, there is no proof of this theorem on the level of the ground field F, i.e. without using transcendental extensions of F. Later in 1980 Andrey Suslin wrote a paper on the norm residue homomorphism, where the function field of a conic was a main ingredient. Recall that given a field F, the group K2 (F) is defined as the quotient group F* F*/H, where H is generated by all elements a ® (1 — a) with a = 0,1. There is a homomorphism hp : K2(F)/2 ^ 2Br(F), which is called the norm residue homomorphism, and which takes a symbol {a, b} := a ® b to the class of the quaternion algebra (a, b). In [65] Suslin proved that if a,b E F*, C is an affine conic over F determined by the equation ax2 + by2 = 1, and the maps hp and hFare bijective, then the map hp(c) is bijective as well. This result turned out to be very important for the proof of bijectivity of the norm residue map, which was produced by Alexander Merkurjev in 1981 ([34]). In particular, the last theorem claims that any central simple algebra A of exponent 2 over F is similar to a tensor product of quaternion algebras.

A few years later, applying conics, Merkurjev constructed a field with «-invariant 6, giving a negative anwer to the Kaplansky conjecture that the «-invariant of any field is a power of 2 or infinity ([35]). In 1989, Merkurjev generalized this counterexample, by constructing a field with «-invariant 2n for any prescribed positive integer n ([36]). Another result, where properties of conics are also applied, is a positive solution of the Pfister Factor Conjecture by K. Becher in 2007

([7]).

The proofs of the above problems demonstrate fruitfulness of this method in investigation of quadratic forms and central simple algebras over F. One can hope for existence other questions, still open, but where application of conics will be crucial for their solutions.

The purpose of the work

The problems presented in the thesis concern in particular various nonstandard objects such as indecomposable central simple algebras, nonexcellent field extensions, nonstandard triples of quatenion algebras, nonstandard pairs of biquaternion algebras, nontrivial elements in the homol-

ogy groups of the Brauer complex for a triquadratic extension. One of the purposes of this work is to construct these objects, establish a relationship between them and in some cases to measure "the level of nonstandardness". Another purpose is to determine if possible whether a given object is standard.

Scientific novelty of the work

All the results of the thesis are new, and their proofs are strict and detailed.

Practical and theoretical value

The work is purely theoretical. However, its methods and results can be used in further investigation of questions in the theory of quadratic forms and central simple algebras over fields. The variety of problems in the thesis solved by application of conics gives a hope that many others can be solved (completely or partially) applying similar ideas.

The methods of the research

The main feature of the thesis, which stress a relationship between the results, is a systematic application of conics. We use classical tools in the theory of quadratic forms and central simple algebras over fields. Also we apply Galois cohomology and Milnor's X-groups. Many results are based on deep theorems of Suslin, Merkurjev, Peyre and others, and can hardly be proven by "elementary" argument.

The results presented for the defence

1. Construction of indecomposable central simple algebras of exponent 2 and index 2n (n > 3) split by a given set of n square roots.

2. Construction of Brauer complexes for triquadratic field extensions with arbitrarily big homology groups hi and h3.

3. Description of strongly excellent 2-primary field extensions.

4. Necessary condition for standardness of a linked triple of quaternion algebras. Construction of nonlinked nonstandard triples of quaternion algebras.

5. Criterion for isotropicity of certain quadratic forms over the rational function field in one variable.

Approbation of the thesis and the publications

The main statements of this work were presented at the seminar "Algebraic groups" of the Chebi-shev Laboratory in May of 2022. All the results of the thesis were published in the author's papers [51]-[63] (without coauthors). All the corresponding journals are included in the list of ones recommended by the Higher Attestation Commission for competitors for the doctoral degree.

Notation and preliminaries

We start with certain preliminaries, which might be helpful while reading the text.

In the sequel all fields are supposed to be of characteristic different from 2. By a form we always mean a regular quadratic form over a field, which after choosing a basis of the underlying linear space, can be viewed as a homogeneous quadratic polynomial in a few variables. Any form p over a field F is isomorphic to a diagonal form with coefficients, say, a1,... ,am, with a,i E F *, which is denoted by p — (a1,..., am). If p — (a1,..., am) and ^ — (b1,... ,bn) , then the forms p ± ^ — (a1,..., am, b1... ,bn) and p ® ^ — (a1b1,..., a1bn, a2b1,..., a2bn,..., ambn) are called respectively the direct sum and (tensor) product of p and -0. It is easy to check that these operations are well defined. The element disc(^) := (—1) ~a1a2 ...am is called the discriminant of p.

The set of nonzero values of p is denoted by D(p). The form p is called isotropic if it has a nontrivial zero, or, equivalently, if it contains the hyperbolic plane H — (1, —1) as a direct summand. Otherwise p is called anisotropic. If p — nH for some n, then p is called hyperbolic. By definition the «-invariant of a field F is the maximal integer u(F) such that there exists a «(F)-dimensional anisotropic form over F (if such an integer does not exist, then set u(F) = ro). Up to isomorphism any form p decomposes uniquely as p — pan ± kH, where pan is anisotropic, and k > 0. The form pan is called the anisotropic part of p. Two forms p and ^ are called equivalent if the direct sum p ± is hyperbolic. By definition the Witt ring W(F) of F is the ring of classes of equivalent forms with respect to the operations of direct sum and tensor product. The ideal I(F) of W(F) consists of evendimensional classes of forms. Slightly abusing notation, we call elements of W(F) merely forms. For p E W(F) we denote by dim p the dimension of the anisotropic form corresponding to p.

If L/F is a field extension, then the form p can also be viewed over L. We denote it then as Pl, or res^/F(^). Put W(L/F) := ker(W(F) res > W(L)). In [64] Springer proved the following

Theorem. If L/F is an odd degree field extension, and p is an anisotropic form over F, then the form pl is anisotropic as well. In particular, W(L/F) = 0.

For a quadratic extension there is another classical result:

Theorem. If F(y/a)/F is a quadratic extension, and p is an anisotropic form over F, then p — (1, —a) ® r ± ^ for some forms r and where either is anisotropic, or ^ = 0.

This statement implies at once that W(F(y/a)/F) = (1, -a)W(F). A harder theorem is the equality

W(F(Va, Vb)/F) = (1, —a)W(F) + (1, —b)W(F).

for any a,b E F* ([11]). For a multiquadratic extension of degree 2ra > 8 the similar claim is false in general, which will be shown in chapter I.

Let L/F be a finite field extension, and let s : L ^ F be a nonzero F-linear map. Suppose V is a finite-dimensional linear space over L, and q : V ^ L is a regular quadratic form over L. Then the composition s o q is a form over F, and s induces a group homomorphism s* : W(L) ^ W(F), which is called Scharlau's transfer associated with the map s. From the definition of s* it easily follows that s* is a W(F)-module homomorphism, i.e. s*(pF(^ ® -0) = p ® s*(ip) if p E W(F), ^ E W(F(y/a)). Consider the following important example. Let a E F* be a nonsquare, L = F(y/a). Let further s be the F-linear map determined by s(1) = 0, s(y/a) = 1. Then s*((x + yy/a)) = (y, — y(x2 — ay2)), and s*((x)) = 0 if x, y E F, y = 0. The sequence

0 -► W(F) -— W(F(Va)) -W(F)

is exact ([47], Ch.2, §5).

There is an elegant description of the Witt group of the rational function field F(t) via the Witt groups of finite field extensions of F. More precisely the sequence of abelian groups

0 -► W(F) W(F(t)) —^ H W(Fp) -^ 0,

PeAi

is split exact (see, for example, [47], Ch.6, §3). We consider here a point p E A^ as a monic irreducible polynomial over F, Fp = F[t]/p is the corresponding residue field, and dp : W(F(t)) ^ W(Fp) is the residue homomorphism, which is well defined by the rule

i0 if vp(/) = 0

dP((/))= { _ P( J )

{(fp-1) if vp(f) = 1,

where vp is the discrete valuation associated with p. The splitting map W(F(t)) ^ W(F) takes a one-dimensional form ( f) to (I ( /)), where /( f) is the leading coefficient of the polynomial f E F [t].

If a1,... ,an E F*, then the n-fold Pfister form ((a1,... ,an)) is defined as the 2"-dimensional form (1, - a-]) ® ■ ■ ■ ® (1, -an). Obviously, the ideal In( F) C W (F) is generated by n-fold Pfister forms.

The following important statement is due to Pfister ([40]):

Theorem. Any isotropic Pfister form is hyperbolic.

In particular, if p is a subform of an n-fold isotropic Pfister form, and dimp > 2ra-1 + 1, then p is isotropic as well. It follows also that if n is a Pfister form, and s E D(tt), then n ~ sir.

Another crucial result in the theory of quadratic forms is the following

Theorem (Arason-Pfister Hauptsatz, [4]). If p is a nonzero form from In(F), then dim p > 2n. In particular, the intersection of all powers of I(F) is zero.

Quadratic forms have a close relationship with finite-dimensional central simple algebras and the 2-torsion of the Brauer group of the field. Recall that two finite-dimensional central simple algebras A and B over F are said to be Brauer-equivalent, or similar if their matrix algebras Mm(A) and Mn(B) are isomorphic for some m and n. The equivalence classes of algebras form a group Br(F) with respect to tensor product over F, which is called the Brauer group of F. By the Wedderburn theorem each finite-dimensional central simple algebra is isomorphic to an algebra MS(D), where D is a central division algebra, and D and s are uniquely determined. The dimension of D over F is a square, and the number ind(A) := Vdim D is called the index of A. The order exp(A) of A in Br(F) is called the exponent of A. It divides the index of A, and, moreover, the prime divisors of ind(A) and exp(A) are the same. There exists a field extension L/F of degree ind D such that AL := A L E 2Br(L) is zero. Moreover, if L/F is a finite field extension, and AL = 0 E 2Br(L), then the degree [L : F] is a multiple of ind D.

In this work we deal mostly with the two-torsion 2Br(F) of the Brauer group Br(F).

Similarly to forms we often identify a central simple algebra with its image in the Brauer group. We use the sign + for the group operation in the Brauer group. So, if A1,A2 are central simple algebras over F, then A1 + A2 is the element in the Brauer group associated with the algebra A1 ®F A2.

The simplest example of a central simple algebra is a quaternion algebra. For a,b E F* the quaternion algebra (a, b) is the F-algebra generated by i and j with relations i2 = a, j2 = b, ij = —ji. A biquaternion algebra is defined as the tensor product of two quaternion algebras.

If L/F is a field extension, then Br(L/F) := ker(Br(F) ^ Br(L)). It is easy to see that 2Br(L/F) = 0 for any odd degree extension, and A E Br(F(y/a)/F) iff A = (a, c) for some c E F*. Moreover, A E 2Br(F(^a, Vb)/F) iff A = (a,C1) + (b,C2) for some a,C2 E F* ([66, Prop. 2.4]). Similarly to the case of Witt rings the analogous statement for multiquadratic extensions is false in general.

A few times we will use a particular case of Tignol's result ([67, Prop. 2.4]).

Theorem. Let F be a field, D E Br(F), a E F*, x an indeterminate. Then ind(^ + (a,x)) = 2 ind DF .

In other words, the theorem claims that if D is a finite-dimensional central division algebra, then the algebra D ® (a, x) is division if and only if the algebra DF(^ is division.

For a form p there are notions of the Clifford algebra C(p) and its even part C0(p), which provides a relationship between quadratic forms and simple algebras. We don't need the invariant definition of these objects. However, recall that if p — (a1,... ,an), then C(p) is isomorphic to the algebra with generators ei and relations e2 = a,i, eiej = —ejei (1 < i, j < n, i = j). The

algebra CO« is the subalgebra of C«, generated by all e^j. If p E 1(F), then C« is a central simple algebra. If p E I(F) \ I2(F), then C0« is a central simple algebra over F(^disc p), and C0« = C(Of(vdisc f). If p E I(F), then the algebra C0« is central simple. Moreover, in the last case C0« = C(p), where p is the unique element of W(F) such that dim(p — p) = 1 and 1 E I2(F).

Denote by ind( p) the index of C« if dimp is even, and the index of C0« if dimp is odd.

For a field F we set Hn(F) := Hn(Gal(F),Z/2Z) the nth cohomology group of the Galois group of F with coefficients in Z/2Z. Denote by cd2 F the 2-cohomological dimension of the field F, i.e. the maximal number n such that Hn(L, Z/2Z) = 0 for some finite field extension L/F. The above cohomology groups have the following properties:

• Let a1,..., an E F *. The cohomological symbol (a1,..., an) E Hn (F) is defined as

(a1,..., an) = (a1) U ■ ■ ■ U (an), where (ai) E H 1(F) ~ F*/F*2 corresponds to ai. This symbol is multiplicative with respect to all arguments. There is a canonical isomorphism H2(F) ~ 2Br(F), where the symbol (a, b) corresponds to the quaternion algebra ( a, b).

• Let L be a field with a discrete valuation . Assume that the corresponding residue field Lv is of characteristic different from 2. There is a residue homomorphism dv : Hn(L) ^ Hn-1(LV) uniquely determined by the equalities dv(a 1,... ,an) = 0 if v(ai) = 0 for each i, and dv(a1,..., an) = (ai,... an-1) if v(ai) = 0 for 1 < i < n — 1, v(an) = 1 ([9], 99.11). If F is a subfield of L and v is trivial on F, then we denote the residue field Lv as Fv.

• For any finite field extension L/F there is a norm homomorphism NL/p : Hn(L) ^ Hn(F), which satisfies the following properties:

1) The norm map H 1(L) ^ H 1(F) coincides with the usual norm L*/L*2 ^ F*/F*2.

2) Nl/f (aL Up) = a U NL/F (p) for any a E Hm(F), p E Hn(L). In particular, (x2 -a, a) = Nf(V^)/f(x + Va, a) = NF(^)/f(0) = 0.

3) For any quadratic extension F(y/a)/F the sequence

Hn(F) -N—^ Hn(F(Va)) Hn(F)

is exact ([9], 99.13).

4) There is an exact sequence

0 ^Hn(F) Hn(F(t)) ®Hn-1(Fv) Hn-1 (F) ^ 0,

where v runs over all discrete valuations of F(t) over F ([9], 99.12). It follows that the sequence

0 ^ Hn(F) -NNU Hn(F(t)) ®Hn-1(Fv) ^ 0,

where v runs over all discrete F-valuations on F(t) except v^ is exact.

The statement that the composition

Hn(F (t)) —— ®Hn-1(Fv) —— Hn-1(F) is zero is called the reciprocity law.

Galois cohomology have a close relationship with Pfister forms. By very deep results of Voevod-sky ([69]) and Orlov, Vishik, Voevodsky ([38]) there is an isomorphism en : In/In+1(F) — Hn(F), which takes the class of the Pfister form ((a1,..., an)) to the symbol (a1,..., an) E Hn(F). In the case n = 2 the form p E 12(F) takes to C(p), and the fact that this map is surjective with the kernel 13(F) is equivalent to bijectivity of the norm residue homomorphism. This equivalence follows from the isomorphism K2(F)/2 — I2/13(F), which takes a symbol {a,b} to the class of the Pfister form ((a,b)).

A form <p is called an Albert form if dim <p = 6 and disc <p = 1. Jacobson proved that for any Albert forms <p1, <p2 the algebras C(p1) and C(p2) are isomorphic if and only if there exists A E F* such that p2 — \p1 ([23]). In view of the norm residue isomorphism this is equivalent to

— <P2 E 13(F).

For a form <p over F with dim <p > 2 denote by F(p) the function field of the projective quadric associated with p. Clearly, if p is isotropic, then F(p)/F is a purely transcendental extension. For forms p, ^ and r this implies that if pp(T) and ^f(^) are isotropic, then ^f(t) is isotropic as well. A similar statement holds if one replaces the form ^ by a division algebra D.

Obviously, the form <Pf(v) is isotropic, i.e. Pf(v) — ^ ^ kH for some form ^ over F(p) and k > 1. The number k is denoted as i1(p).

If p and ^ are anisotropic, and ^ E W(F(p)/F), then there exists a E F* such that ap C ^ ([47], Ch. IV, Th. 5.4). It follows that Br(F(p)/F) = 0 only if dim<p = 2, or dim<p = 3, or dim p = 4 and disc p = 1. In the last two cases Br(F(p)/F) = (Q), where Q is the quaternion algebra associated with p.

There is another important Merkurjev's theorem, which determines if the given finite-dimensional division algebra remains division over F(p):

Theorem. Let D be a finite-dimensional division algebra over F. Then the algebra Dp(v) is not division if and only if there exists an F-algebra homomorphism C0(p) ^ D.

If dim <p is odd, then C0(p) is a central simple algebra, hence the kernel of any homomorphism Cq(<p) ^ D is trivial. This means that in fact C0(p) is a subalgebra of D, hence D — C0(p) ®pD1 for some central simple algebra D1. In particular, if ind D = 2n and dim p > 2n + 3, then Dp(v) is a division algebra. Moreover, if n is a 3-fold Pfister form, then C0(p) is isomorphic to the matrix algebra M8(F) for any 7-dimensional subform p of n. Hence Dp(n) is a division algebra as well.

Now we introduce an important geometric ingredient. By a projective conic C over a field F we mean the projective variety determined by the equation ax2 + by2 = z2 in the projective space

PF. There is a bijective correspondence between projective conics, quaternion algebras and 2-fold Pfister forms. Namely, the projective conic above corresponds to the quaternion algebra (a, b) and the 2-fold Pfister form (( a, b)). It is rational if and only if the form (1, — a, —b) is isotropic, or, equivalently, ((a, b)) = 0. These conditions mean that the conic is isomorphic to the projective line PF.

By F(C) we denote the function field of C. By a point of C we always mean a closed point. Sometimes it is convenient to use an affine part of C, for instance, the affine conic in A2F determined by the equation ax2 + by2 = 1.

The points of C are in bijective correspondence with discrete F-valuations of F( C). The set of all points of C is the union of the points of the affine conic above and the infinity point determined by the equality = 0. The points of the affine conic are in bijective correspondence with the maximal ideals of the quotient ring F[x, y]/(ax2 + by2 — 1). The degree degp = [ Fp : F] of any point p E C is even ([9], 45.2).

By definition a divisor on C is a formal finite linear combination npp, where p is a point of

p

C, and np E Z. If D = npp, put deg D = J2np degp. This number is called the degree of D.

p p

Let f E F(C)*. Define the principal divisor div(/) as div(/) = Y1 vP(f)P, where vp is the

p

discrete valuation on F(C) corresponding to the point p. A divisor D is principal if and only if degD = 0 ([9], 45.2).

For the conic C one can define the Brauer group of C, which is a subgroup of Br F(C) and denoted by Br( C). We will need only its 2-torsion. There is an exact sequence

0 ^ 2Br(C) -► 2Br( F( C)) ——^ © H 1(Fp).

The quotient group 2Br( C)/2Br F is either trivial, or equals Z/2Z ([65]). The following important result on conics, which was obtained independently by Arason and Rost ([2], [43]), is crucial in the sequel. It is similar to the corresponding statement on quadratic extensions, but the proof is much more complicated.

Theorem. Let C be a conic over F. The extension F(C)/F is excellent, i.e. for any form p over F the anisotropic part of pF(C) is isomorphic to ^F(C) for some form ^ over F.

In chapters II and IV systems of quadratic forms (not necessarily regular) occur. There is a criterion, which permits to determine whether two given forms have a common zero [8].

Theorem. Let f and g be forms (not necessarily regular) in the same variables. They have a common zero if and only if the form f + tg over F(t) is isotropic.

Note that one implication is obvious, but the other is not. This statement and Springer's theorem imply at once that if L/F is an odd degree field extension, and the forms f and g have a common zero over L, then they have a common zero over F.

In chapters IV and V we use specialization maps for Witt groups and 2-torsion of the Brauer group. Let X be an irreducible algebraic variety over a field F, and let p E X be a rational nonsingular point of X. There are group homomorphisms J: F(X)* ^ F* and s : W(F(X)) ^ W(F) such that 's(f) = f (p) if the functions f, f-1 are regular at p, and s((f)) = (s(f)) for any f E F(X)*. In particular, dims(<p) < dimp for any p E W(F(X)), and dimp — dims(p) is even.

The maps s and J are called specialization homomorphisms. They are not defined canonically, but depend on the choice of the regular system of parameters in the regular local ring Op. Fix a regular system of parameters n1,... of the ring Op. Let Li be the quotient field of the ring Op/(n1,... ,-Ki) (0 < i < I). In particular, L° = F(X), Li = F. The field Li is a discrete valuation field with the residue field Li+1. Obviously, there are homomorphisms s1 : L* ^ L*+1 such that si(n'^+1u) = u for any unit u E Op/(-k1,. .. ) and k E Z. Put ? = s1-1 o ■ ■■ o : F (X )* ^ F *. Define s : W(F(X)) ^ W(F) by the rule s((f)) = (s(f)), f E F(X)*. Recall that for any field E the Witt group W(E) is the group with generators {a} subject to the relations {1} + { — 1} = 0, {ab2} = {a}, and {a} + {b} = {a + b} + {ab(a + b)} for a,b,a + b E E* [47, Chapter 2, Corollary 9.4]. It is straightforward to check that s respects these relations, hence s is a well defined group homomorphism. Note also that s is a homomorphism of rings and W(F)-modules.

Clearly, for any m > 0 the map 1 induces the specialization map Im(F(X)) ^ Im(F), and consequently, the map 2Br(F(X)) — I2/I3(F(X)) ^ I2/I3(F) — 2Br(F).

The structure of the contents

An overview of the author's results is given below in correspondence with the numeration of the chapters.

The thesis consists of five chapters. Its content is covered by author's papers [53], [54], and [51]-[63]. The main sources for the information are the books [9], [30] and [47], where all the necessary details can be found.

The first chapter concerns indecomposable central simple algebras of exponent 2. A well known result of Albert states that any central division algebra of index 4 and exponent 2 is a tensor product of two quaternion algebras ([3]). J.P.Tignol proved that any algebra of index 8 and exponent 2 is similar to a tensor product of four quaternion algebras ([66]). In view of the norm residue isomorphism, for any A E 2Br(F) there exists a multiquadratic extension L/F such that Al = 0. This observation makes natural to investigate elements from 2Br(F) with respect to their multiquadratic splitting fields.

v

If A E 2Br(F), ind A = 2n, and A = J2(ai,bi), then p > n, and if p = n, then A — (a1 ,b1) ®

i=1

■ ■ ■ ® (ap,bp). We call A decomposable if it is a tensor product of two nontrivial central simple algebras over F. Otherwise we say that A is indecomposable. Obviously, any indecomposable algebra is necessarily a division algebra. The first example of indecomposable algebra of index 8 and exponent 2 was given by Arason, Rowen and Tignol ([5]). Later Karpenko in [25] gave an example of an indecomposable algebra A of exponent 2m and index 2n for any m > 1, n > 3, n> m.

However, in Karpenko's examples in the case m =1 there is no upper bound for the minimal p such that A is a sum of p quaternion algebras. We construct an indecomposable algebra C of exponent 2 with a prescribed multiquadratic splitting field of degree at least 8 ([55], [53]). More precisely, let k be a field, n > 3. Suppose the elements a1,..., an E k*/k*2 are linearly independent over Z/2Z. Then there exists a field extension F/k, a quaternion algebra D over F and a division algebra C E 2Br F((t1))... ((tn)) of index 2n such that

1) The field F has no odd degree extensions, and u(F) = 4.

2) D E 2Br(L/F), where L = (F (/ai,..., /an).

3) For any tower F C F1 C F2 C L such that /a1 E F2*, [L : F1] = 8, [F2 : F1] = 4 we have

Df± E 2Br(Fi(/ar)/Fi) + 2Br(F2/Fi).

3) M2(C) ~ (a !,t 0 0e •••&E (an, tn) 0e D.

4) C is indecomposable over any odd degree extension of the iterated Laurent series field E = F(fa))... ((tn)).

In particular, cd2 E = n + 2, and u(E) = 2n+2. Moreover, in the case n =3, using a result of Barry and Merkurjev ([6]), we construct a field extension L/k and an indecomposable algebra D over L such that cd2 L = 3, expD = 2, and DL(^y^2,^3) = 0 ([61]). (It is easy to show that if cd2 K < 2, any central simple algebra D over K with exp D = 2 is isomorphic to a tensor product of quaternion algebras). Unfortunately, we do not know if there exist an indecomposable algebra of exponent 2 and index at least 16 over a field F with cd2 F = 3.

In chapter II we investigate the Brauer complex for an arbitrary triquadratic field extension, following [59].

Let L/F be a triquadratic field extension, i.e. L = F(y/a,\fb,y/c) for some a,b,c E F*, and [L : F] = 8. Let further G be the subgroup of F*/F*2 generated by the images of a,b, c. For

any I C {a ,b, c} put ai = n e (a0 = 1). Consider the abelian group complex (usually called the

eel

Brauer complex)

n K}/Kf —— G ®F*/F*2 —— 2BrF ——. 2BrL ——-^U 2Br E,

/=0 FcEcL, [L:E]=2

where Ki = F(y/aT) are all the quadratic extensions of F contained in L,

^({wi}) = J^a/ ® NKj/f(wi), ai 0 z) = (ar, z). 1

The homology group of this complex at the i + 1-th term from the left is denoted by hi(L/F) (1 < < 3) and in general is nontrivial. These groups were studied in many papers, for instance, by Arason, Rowen, Tignol, Elman, Shapiro, Wadsworth, Lam, Gille, and the author. In the present work we continue this investigation, in particular, we establish some kind of relation between the

groups h1 ( L/F) and h2(L/F).

Tignol proved that the similar homology groups for quadratic and biquadratic field extensions are trivial.

Note that by definition

h(LIF)- 2Br(L/F)

(a) U H 1(F) + (b) U H 1(F) + (c) U H 1(F)'

Since (e) U H 1(F) = Br(F(y/e)/F) for any e E F*, the group h2(L/F) measure the difference between the whole group 2Br(L/ F) and its subgroup of elements splitted over L by an obvious reason.

Given a field K and an element e E K* denote by Nk(e) the norm group Nk^)/kK(y/e)*. In [14] Gille proved that the group h1 (L/F) is isomorphic to the group

Nf (a) HNf (b) HNf (c)

F*2NF{va,vb,^)/FFVу/с)*.

Note that, obviously,

(a) U Nf (a) = (b) U Nf (b) = (c) U Nf (c) = 0,

and by the projection formula 2Br(L/ F) U Nl/fL* = 0. These observations and the above presentations of the groups h\(L/F) and h2(L/F) permit us to introduce a well defined group homomorphism fF : h2(L/F) h\(L/F) ^ H3(F) by the rule fF(x ® у) = x U y, where

* € 2Br(L/F) с H2(F), and у e Nf<"> ^ Nf<"> П NfM c h'(F).

F* 2

This pairing proves nontriviality of the given element у € h\(L/F) if there exists an element x € h2(L/F) such that f(x ® y) = 0.

Let к be a field, I = k(/a^Vb, /с). If к с F\ С F2, the maps fFl and fF2 are compatible. In other words, the system of all maps F, where с F, forms a natural transformation f : h2(lF/F) ® hi(lF/F) ^ H3(F). We call an element D e 2Br(L/F) standard with respect to the extension L/F, if

D e Br(F(/a)/F) + Br(F(/)/F) + Br(F(/)/F), i.e. if its image in the group h2(L/F) is zero. The groups

2Br(L/ F) and Br( F(/a)/F) + Br( F(Vb)/F) + Br( F(/)/ F)

do not coincide in general, so nonstandard elements exist (for instance, the quaternion algebra D in chapter I in the case of triquadratic extension is nonstandard). However, in any case, since

Df(^a) E 2Br(F(/a,y/b, /c)/F(/a)), we have

Df (ja) = (b ,x + y/a) + (c,u + v/a)

for some u,v,x,y E F, hence DF(^aVb) = (c,u + vy/a). We give a criterion, which determines whether D is standard with respect to the extension L/F, via this presentation for DF(^aVb) = (c,u + v/a) ([59]).

Certain examples of triquadratic extensions L/F with nontrivial group h1(L/F) were known earlier. However, in these examples the triples { a, , } are not arbitrary, and it is unclear how big the group h1(L/F) is. Using the natural transformation f : h2 0 h1 — H3, we show that given any a,b,c E k* such that [k(/a, y/b, /c) : k] = 8, there is an extension F/k with arbitrarily big h1(F(/a, yfb, /)/ F) ([59]).

We investigate also the group h3(L/F) in the case yj—1 E F*. We show that for any a,b,c E F* such that 0 = (a ,b, c) E H3(F) there exists a field extension F/F with arbitrarily big h3(LF/F)

([59]).

Recall that a field extension 10/k0 is called excellent (resp. n-excellent) if for any form p (resp. any form of dimension at most n) over k0 the anisotropic part of the form pi0 is defined over k0. In other words, the extension l0/k0 is excellent if for each anisotropic form p over k0 such that the form is isotropic there exists a form ^ over k0 such that dim ^ < dim p and

p — ^ E W(la/ka) = ker(W(ko) ——— W(lo)).

Otherwise, the extension 10/k0 is called nonexcellent. The extension 10/k0 is called strongly excellent if for any extension K/k0 linearly disjoint with l0/k0 the extension Kl0/K is excellent.

Obviously, by Springer's theorem any odd degree extension is strongly excellent. Also it is well known and can be easily checked that any quadratic extension is strongly excellent. The first examples of finite nonexcellent biquadratic extensions were constructed in [13].

It is easy to prove, using the results in chapter I, that no multiquadratic extension of degree bigger than 2 is strongly excellent ([51]). Moreover, the author constructed examples of nonexcellent extensions of arbitrary even degree, starting with 4 ([62]). These observations motivate the following

Conjecture. A finite even degree field extension I0/k0 is strongly excellent if and only if there is an intermediate field k0 C E C l0 ¡such that E/k0 is a quadratic extension, and l0/E is an odd degree extension.

Notice that the "if" part of the conjecture is trivial. "The only if" part seems to be very hard. One of the reasons is that if p, ^ are forms over K, dim< dimp, then trying to prove that ( 'pKi0)an ^ —klo , we should show that the element p — — in the Witt ring W(K) does not belong to the ideal W(Kl0/K) = ker(W(K) res ) W(Kl0)). However, W(Kl0/K) is computed only in a few very particular cases, and usually it is very hard to determine whether the given element of

W(K) belongs to this ideal or not.

In chapter III we prove the conjecture in the case when the degree [l0 : k0] is a power of 2 . In particular, the conjecture is valid if the field k0 has no proper odd degree extensions. The case of multiquadratic extension was covered in [51], and the general case of 2-primary degree in [60].

It is easy to prove that an extension L/K is not 4-excellent if there exists d E K* and a quaternion K-algebra Q such that = 0, but Q E 2Br(L/K) + Br(K(v^/K). This

statement is important in our proof of the 2-primary case of the conjecture, and it relates to the question if the obvious inclusion

2Br(L/K) + 2Br(K(Jd)/K) c 2Br(L(Vd)/K)

is equality. For example, it is easy to see that for any quadratic extension L/K

2Br(L/K) + 2Br(K (Jd)/K) = 2Br(L(Vd)/K).

On the other hand, if L/K is a biquadratic extension, then the last equality does not hold in general. In the case of cyclic Galois extension L/K of degree 4 the equality is false in general as well.

In chapter IV we deal with pairs and triples of quaternion algebras and pairs of biquaternion algebras. For a biquadratic extension there is a cohomological criterion for its excellency. Namely, let F be a field, a\, a2 E F*. Then the following conditions are equivalent:

a) The extension F(yfa^,, yfa2)/F is excellent.

b) For any a3 E F* the complex

F*/F*2 ®U — H2(F) H2(F(Ja1, /a2, Ja3i))

is exact, where U is the subgroup of F*/F*2 = H!(F) generated by a\, a2,^ ([13]).

Nonexactness in general of the above complex is the crucial point in the chapter I, where indecomposable algebras with index 8, exponent 2, and the splitting field F(^a!, ■sJa2, s/ay) have been constructed. However, for any field F and a\,a2 E F* the analogous sequence

F*/F*2 ®U — —> H2(F) H2(F(/F1, /F2)),

is exact, U being the subgroup of F*/F*2 generated by a\ and a2 ([66]).

Note also that if ai are quaternion algebras (1 < i < n) with associated conics Xi, then the complex

0 -► U -► H2(F) -—■ H2(F(X! x ■ ■ ■ x Xn)), (*)

is exact, this time U being the subgroup of H2(F) = 2Br(F) generated by the algebras ai.

With these important examples in mind it becomes quite natural to look at the complex

F*/F*2 0 U c— —— H3(F) H3(F(X1 *•••* Xn)), (**)

where the group U is as in complex (*). The last complex is always exact if n =1 ([1]), or n = 2

([39]).

We show that for n = 3 exactness of complex (**) does not hold in general, and give examples of nonexcellent extensions determined by the function field of the product of two conics ([52], [57]). More precisely, let k0 be a field. For a 2-fold Pfister form a over k0 denote by Xa the corresponding projective conic over k0 and by a the corresponding quaternion algebra in 2Br(k0). We prove that if a and 3 are 2-fold Pfister forms over k0 such that ind(a + 3) = 4, then there exists a field F over k0 such that the extension F(Xa x Xp)/F is not 6-excellent.

We do not know if one can replace in this result the condition ind(a + 3) = 4 by the condition ind(a + 3) = 2. However, we construct the following example: consider the quaternion algebra D over F, which is used in the construction of indecomposable algebra in chapter I for n = 3, and the corresponding elements a1, a2, a3. Let D ~ (u, v). The form

p ~ (—u, —v, uv, a3) ± —t((u, v))

over the Laurent series field F((t)) is anisotropic. Let Xi be the conic corresponding to the algebra (ai, t). It turns out that the form pF((t))(X!xx2) is isotropic, but its anisotropic part is not defined over F((¿)). In particular, the extension F((t))(X1 x X2)/F((t)) is not 8-excellent ([57]).

In this chapter we also introduce the notion of standard triple of quaternion algebras. As was shown by Peyre in [39], if X1, X2 are projective conics over a field F, then the torsion of the Chow group CH2(X1 x X2) is trivial. It is natural next to consider the case of three conics. The corresponding result was also obtained in [39]:

Theorem. Let F be a field, Q1,Q2,Q3 quaternion algebras over F, and X1,X2,X3 the corresponding projective conics. Denote by G the subgroup of 2Br(F) generated by all Qi. Then

1) The torsion of the group CH2(X1 x X2 x X3) is either 0, or Z/2Z.

2) Denote by d the least common multiple of all the numbers ind(a), where a runs over all the elements of G. The following two assertions are equivalent:

i) The algebras Qi have a common splitting field of degree dm, where m is odd.

ii) The torsion of the group CH2(X1 x X2 x X3) equals 0.

3) There is a monomorphism from the homology group of the complex

F*/f*2 0 g c— h3(F) —^ H3(F(X1 xX2 x X3))

to the torsion of CH2(X1 x X2 x X3).

We say that the triple {Q1,Q2,Q3} is standard (otherwise nonstandard), if the equivalent conditions from 2) hold. In particular, if the algebras Qi have a common slot, or, on the other

extreme, ind(Qi + Q2 + Q3) = 8, then the triple {Qi,Q2,Q3} is standard. Hence in these cases the complex

F*/f*2 0 G — h3(F) --U H3(F(Xi x X2 x X3))

is exact. Our result is as follows. For any field k0 and quaternion algebras Q1, Q2 such that

ind(Q1 + Q2) = 4 we construct the field extension F/k0 and a quaternion algebra Q3 over F such that the complex

F*/f*2 0 g — wod— H3(F) --U H3(F(Xi x X2 x X3))

is not exact. In particular, the triple {Q1,Q2,Q3} is nonstandard ([52]). Clearly, in this case d = 4. Moreover, it turns out that similarly to nonexactness of the cohomological complex above

W(F(Xi x X2 x X3)/F) = TiW(F)+ f2W(F)+ i3W(F),

where ni is the 2-fold Pfister form associated with Xi. This inequality is in contrast to the equalities W(K(Ci)/K) = nW(K), and W(K(Ci x C12)/K) = nW(K) + T2W(K) for any field K and conics Ci associated with 2-fold Pfister forms t, over K.

In the case d = 2, i.e. the case of quaternion algebras Qi, Q2, Q3 such that

ind(Qi + Q2) = ind(Qi + Q3) = ind(Q2 + Q3) = ind(Qi + Q2 + Q3) = 2

we say that the triple T = {Qi,Q2,Q3} is linked. We give a necessary condition for the linked triple T to be standard. Namely, put

c(T) = c(Qi) + c(Q2) + c(Q3) - c(Qi + Q2) - c(Qi + Q3) - c(Q2 + Q3) + c(Qi + Q2 + Q3) E W(F),

where c(Qi), c(Qi + Qj) and C(Qi + Q2 + Q3) are the corresponding 2-fold Pfister forms. It is easy to see that if Qi, have a common slot, then c(T) is a 4-fold Pfister form. Moreover, we show that if Qi, have a common splitting field of degree 2m, where m is odd, then Qi have a common slot. Thus, if c(T) is not a 4-fold Pfister form, then the triple {Qi,Q2,Q3} is nonstandard ([56]).

A natural question arises whether the condition c(T) e I4(F) implies that the algebras Qj, have a common slot, i.e. the triple {Qi,Q2,Q3} is standard. In general the answer is negative, and we give a corresponding counterexample.

The last problem in chapter IV concerns the property D(2) and common splitting fields of two biquaternion algebras. By definition, F has property D(2) if for any quadratic extension L/F and any binary quadratic forms qi, q2 over F the existence of a common value of the forms qiL, q2L implies the existence of a common value of the forms qiL, q2L, which lies in F. Examples of fields of characteristic 0, without this property, were given in [50]. As far as we know, such examples in positive characteristic have not been found. We are intereted in property D(2) by the following reason.

Let D1, D2 be biquaternion algebras over a field F. Assume that ind(D1 + D2) = 2. Risman proved that D1 and D2 have a common splitting field of degree 4 ([42]). A natural question arises whether there exists a common biquadratic splitting field for D1 and D2. We show that in general the answer is negative. More precisely, applying the same method as in construction of nonexcellent extension F(Xa x Xp)/F above, we prove the following

Theorem ([54]). For any field k with distinct modulo squares elements 1, b1, b2 such that Nk(b1) fl Nk(b2) = k*2 the field k(x) has not property D(2). There exist biquaternion algebras D]^, D2 over F = k(x)((t)) without common biquadratic splitting field such that ind(D1 + D2) = 2.

As a byproduct we obtain examples of standard triples of quaternion algebras with a common splitting field of degree 4, but without a common biquadratic splitting field.

In chapter V we consider some problems related to forms and algebras over rational function fields. The following theorem on algebras over the rational function field was proved in [46]:

Theorem. For every central simple k-algebra D and a,b E k* such that a E k*2 and ab E k*2 the following equality holds:

ind(D 0k{t) (a, t2 — b)) = 2 gcd{indDk(VE), indDk{^b)}.

In other words, the theorem claim that if D is a central division algebra, then D 0k (a, t2 — b) is a division one if and only if the algebras Dk(^ and Dk(^) are both division.

We give a new short proof of this theorem, and also obtain a similar theorem in the case of four square roots. Namely,we give a criterion for all the algebras Dk(^), Dk(^), Dk(^), to be division, provided that all the elements c, ac, be, abc E k* are nonsquares:

Theorem. Let a,b,c E k*, c,ca,cb,cab E k*2, a,[,z indeterminates. Let further

p(a, [, z) = z4 — 2(aa2 + bp2)z2 + (aa2 — bp2)2 E k(a, 3)[z]

be the minimal polynomial of a^a + [Vb over the field k(a,3). Suppose D is a central division algebra over k. The following two conditions are equivalent:

a) D 0k(a,p,z) (c,p(a,3, z)) is a division algebra over k(a,[3, z).

b) Dk(^-C), Dk{^c), and Dk(v^c) are division algebras.

Coming back to quadratic forms, let p1 and p2 be forms over a field k, and let p be a polynomial of odd degree over k. It is easy to see, that the form p1 ± pp2 is anisotropic if and only if both p1 and p2 are anisotropic. A natural question is to give a criterion for isotropicity of the form p1 ± (t2 — a) p2, where a E k*. Using the excellence property of conics, we get the following result

([58]):

Theorem. 1) Let k be a field, a E k, pi, p2 forms over k, m an odd positive integer. Then there exist forms r2 over k such that over k(t)

( pi ± (t2m - a)p2)an ^ Ti ± (t2m - a)T2.

2) Under notation above the following two conditions are equivalent:

a) The form ± (t2m - a)tp2 is isotropic.

b) There exist forms ^ and r2 over k such that - ^ = r2 - p2 E W(k(y/a)/k), and dim ^ + dim r2 < dim + dim p2.

By the same method we get the following result for 4-dimensional forms:

Theorem. Let k be afield, a,b E k*, p a 4-dimensional form overk. The following two conditions are equivalent.

1) The form is isotropic and its anisotropic part is defined over k.

2) The form <p((p(a,[, z))) is isotropic over k(a,[, z).

This problem has a relationship with indecomposable algebras. Suppose the form pk(^a,Vb) is isotropic, but its anisotropic part is not defined over k. If d = disc(p), and D is a quaternion algebra over k such that Dk^) ~ C0(p), then the algebra D ® (a, ti) ® (b, t2) ® (d, t3) over k((ti))((t2))((t3)) is similar to an indecomposable algebra.

Finally, we give examples of triples of conics {Xi, X2, X3} over a field K such that the corresponding triple of quaternion algebras is standard, but, on the other hand, the quotient group

W(K(Xi)/K)+W(KX^l)/K)lw(K(X3)/K) is nonzer°. Informally speaking, standard triples with respect to H3 are not necessarily standard with respect to the Witt ring.

Заключение диссертации по теме «Другие cпециальности», Сивацкий Александр Станиславович

ЗАКЛЮЧЕНИЕ

В заключение приведем открытые вопросы, непосредственно связанные с содержанием работы, которые представляются нам особенно интересными.

Глава I.

неразложимой над любым расширением нечетной степени основного поля ? Ответ неизвестен даже для случая индекса 8,

2, Существует ли алгебра экспоненты 2 и индекса 2п, те являющаяся суммой п +1 ква-тернионных алгебр в группе Брауэра ?

Глава II.

1. Может ли группа гомологий к2 в комплексе Брауэра для триквадратичного расширения быть сколь угодно большой ?

2. Можно ли спять условия Е к* и (а, Ь, с) = 0 в теореме 5.1 ?

Глава III.

1. Верпа ли гипотеза 2.5 о сильно превосходных расширениях ?

Глава IV.

1. Найти необходимые и достаточные условия для стандартности связанного триплета кватернионных алгебр.

2. Верно ли, что если кватернионные алгебры стандартного триплета имеют общее поле расщепления степени 4п, где п нечетно, то они имеют общее поле расщепления степени 4 ?

Глава V.

1. Существует ли стандартный связанный триплет кватернионных алгебр над полем Р такой, что Ш(Р(X1 х Х2 х Х3)/Р) = Ш(Р(Хф/Р) + Ш(Р(Х2)/Р) + Ш(Р(Х3)/Р), где Х^ -соответствующие коники ?

Список литературы диссертационного исследования доктор наук Сивацкий Александр Станиславович, 2023 год

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