Witt’s Extension Theorem for Quadratic Spaces over Semiperfect Rings
We prove that every isometry of between (not-necessarily orthogonal) summands of a unimodular quadratic space over a semiperfect ring can be extended an isometry of the whole quadratic space. The same result was proved by Reiter for the broader class of semilocal rings, but with certain restrictions on the base modules, which cannot be removed in general.
Our result implies that unimodular quadratic spaces over semiperfect rings cancel from orthogonal sums. This improves a cancellation result of Quebbemann, Scharlau and Schulte, which applies to quadratic spaces over hermitian categories. Combining this with other known results yields further cancellation theorems. For instance, we prove cancellation of (1) systems of sesquilinear forms over henselian local rings, and (2) non-unimodular hermitian forms over (arbitrary) valuation rings.
Finally, we determine the group generated by the reflections of a unimodular quadratic space over a semiperfect ring.
Key words and phrases:quadratic form, Witt’s Theorem, reflection, isometry group, semiperfect ring, semilocal ring, Dickson’s invariant, hermitian category, sesquilinear form.
Let be a field of characteristic not and let be a nondegenerate quadratic space over . The following theorem, known as Witt’s Theorem or Witt’s Extension Theorem, is fundamental in the theory of quadratic forms.
Theorem 1.1 (Witt).
Let be subspaces of and let be an isometry. Then extends to an isometry of . Furthermore, is a product of reflections.
Among the theorem’s consequences are cancellation of nondegenerate quadratic spaces and the fact that , the isometry group of , acts transitively on maximal totally isotropic subspaces of .
The works of Bak , Wall  and others have led to defining a notion of quadratic forms over arbitrary (non-commutative) rings, and also to an appropriate definition of reflections (see , , ). In this context, Witt’s Extension Theorem was generalized by Reiter  to semilocal rings, but with certain restrictions on the quadratic spaces (see also  and  for earlier results). Cancellation of unimodular quadratic spaces was likewise generalized to various families of semilocal rings including the cases where is commutative (,  or [11, §3.4.3]), local ([11, Rm. 3.4.2], for instance), or ([19, §3.4]; this case includes all one-sided artinian rings). However, despite the previous evidence, Keller  has demonstrated that cancellation fails over arbitrary semilocal rings, implying that the restrictions in Reiter’s Theorem cannot be removed in general. (Most of the results mentioned here can also be found in [15, Ch. VI].)
In this paper, we restrict our attention to a family of semilocal rings called semiperfect rings, and study to what extent Witt’s Extension Theorem holds in this setting. Recall that a ring is called semiperfect if it is semilocal and its Jacobson radical, , is idempotent lifting. For example, local rings and semilocal rings satisfying are semiperfect. See §2.5 below for further examples and details.
Let be a unimodular quadratic space over a semiperfect ring (the definition is recalled below) and let be summands of . Our main results are:
We determine the subgroup of generated by reflections (Theorem 5.8). Apart from an obvious exception in which there are no reflections, this subgroup is always of finite index in .
The proofs are based on Reiter’s ideas with certain improvements. In particular, we generalize Reiter’s -reflections (see ). We also stress that (1)–(3) hold without assuming is invertible.
Our results imply that unimodular quadratic spaces over semiperfect rings cancel from orthogonal sums. (Note that the base ring in Keller’s counterexample [11, §2] is semilocal but not semiperfect.) This in turn leads to other cancellation theorems as follows: In ,  and , it was shown that systems of (not-necessarily unimodular) sesquilinear forms can be treated as (single) unimodular hermitian forms over a different base ring. Thus, cancellation holds when this base ring is semiperfect. Using this, we show that cancellation holds for
arbitrary (i.e. not-necessarily unimodular) hermitian forms over involutary valuation rings (Corollary 4.18),
systems of sesquilinear forms over involutary henselian valuation rings (Corollary 4.17).
We also strengthen a cancellation theorem of Quebbeman, Scharlau and Schulte [19, §3.4] which applies to quadratic spaces over hermitian categories (Corollary 4.14). Specifically, the cancellation of [19, §3.4] assumes that the underlying hermitian category satisfies: (i) all idempotents split, (ii) every object is the direct sum of objects with local endomorphism ring, and (iii) if is the endomorphism ring of an object, then . We show that cancellation holds even without assuming condition (iii).
The paper is organized as follows: In section 2, we recall the definitions of quadratic forms over rings and several results to be used throughout. Section 3 introduces quasi-reflections and reflections. In section 4, we prove our version of Witt’s Extension Theorem and discuss its applications. Finally, in section 5, we describe the group spanned by the reflections of a unimodular quadratic space (over a semiperfect ring).
This section collects several preliminary topics that will be used throughout the paper: We recall quadratic forms over unitary rings, several facts concerning them, a notion of orthogonality for unitary rings, and several facts about semiperfect rings.
§2.1. Quadratic Forms
Let be a ring. An anti-structure on consists of a pair such that is an anti-automorphism (written exponentially) and satisfies and for all .
Denote by the category of finitely generated projective right -modules. As usual, a sesquilinear space is a pair such that and is a biadditive map satisfying for all , . In this case, we call a sesquilinear form. The form is called -hermitian if it also satisfies .
We say that is unimodular if the map given by sending to is an isomorphism. Note that can be made into a right -module by setting for all , , . This makes is a homomorphism of -modules. There is a natural isomorphism give by for all , .
Next, set and . A form parameter (for ) consists of an additive group such that
In this case, the quartet is called a unitary ring.
For , let denote the abelian group of sesquilinear forms on , and let denote the subgroup consisting of sesquilinear forms satisfying and for all . The image of in is denoted .
A quadratic space (over ) is a pair with and . Associated with are the -hermitian form
and the quadratic map given by
Both and are determined by the class , and conversely, is determined by and . We also have
We say that is unimodular if is a unimodular -hermitian form, i.e. if is an isomorphism ( itself may be non-unimodular).
Isometries between quadratic (resp. sesquilinear, -hermitian) spaces are defined in the standard way (cf. [15, §I.2.2, §I.5.2]). We let denote the group of isometries of . The category of unimodular quadratic spaces over (with isometries as morphisms) is denoted by .
When , we have , so there is only one form parameter . Furthermore, in this case, is isomorphic to the the category of unimodular -hermitian forms over . Indeed, can be recovered from via .
§2.2. Conjugation and Transfer
We now introduce two well-known procedures that we refer to as conjugation and -transfer. They allow one to alter a unitary ring while maintaining data about isometries between quadratic forms (isometry groups in particular). We shall use these manipulations several times in the sequel.
Proposition 2.4 (“Conjugation”).
Let . Define by
We call the conjugation of by . Then is a unitary ring and .
This proposition is essentially [20, Lm. 1.6]; everything follows by straightforward computation. The categorical equivalence is constructed as follows: For any and , define by
Then is s sesquilinear form over , and the assignment defines the required categorical isomorphism (isometries are mapped to themselves). ∎
Proposition 2.5 (“-transfer”).
Let be an idempotent satisfying and . For every sesquilinear form , denote by the restriction of to . Then:
is a unitary ring.
is a quadratic space over . It is unimodular if and only if is unimodular.
The map is an abelian group isomorphism. In particular, .
The assignment , called -transfer, gives rise to an equivalence of categories .
Part (i) is straightforward.
For parts (ii), (iii) and (iv), view and as hermitian categories with a form parameter as in Remark 2.3. Let be the functor given by . By Morita Theory (see [18, §18D] for instance), is an equivalence of categories. In addition, there is a natural isomorphism from to given by (check this for , the general case follows by additivity). It is routine to check that is strictly duality preserving functor from to (see [19, §2]). This means parts (ii) and (iii) hold tautologically, and part (iv) follows from [19, Lm. 2.1] (for instance). ∎
§2.3. Simple Unitary Rings
A unitary ring is called simple if the only ideals of which are invariant under are and . It is not hard to show that in this case, is either simple, or , where is a simple ring, and is given by for some .
Assume is simple and is artinian. Then the Artin-Wedderburn Theorem implies that where is a division ring or a product of a division ring and its opposite. Identifying with , we say that such is standard or in standard form if:
is of the form for some involution . (In particular, is an involution.)
When with a division ring, is the exchange involution .
In this case, we have (because when , we have ).
It is not true that any simple artinian unitary ring is isomorphic to a unitary ring in standard form. However, this is true after applying a suitable conjugation in the sense of Proposition 2.4, and conjugation does not essentially change the category of quadratic spaces.
After a suitable conjugation (cf. Proposition 2.4), any simple artinian unitary ring is isomorphic to a unitary ring in standard form.
Apart from a small difference in condition (3), this proposition is [20, Pr. 2.1]. We have included here a full proof of the sake of completeness.
Let be as above. By [8, Th. 7.8], is conjugate to some of the form where is an anti-automorphism, so assume is in this form. This implies that commutes with the standard matrix units (because they satisfy ), hence we may view as an element of (embedded diagonally in ) which satisfies for all . Now, it is enough to show that can be made standard by conjugation.
Assume that there exists with . Then , so by Proposition 2.4 (or by computation), is an involution. Observe that is an involutary additive map, hence always satisfies . If with a division ring, take to get . Otherwise, any with will do. If such does not exist, then for all . Taking implies and hence . Thus, either can be conjugated to with an involution, or and (in which case is a field). This implies (1) and (3).
It remains to check (2). Indeed, when is not a division ring, there is an isomorphism and under that isomorphism is given by for some . Since is an involution, it must be the exchange involution. ∎
Proposition 2.6 is the reason why many authors require to be an involution in the definition of unitary rings. The author does not know if there exists a similar result for semilocal rings (i.e. a statement guaranteeing that can always be conjugated into an involution). See [9, Rm. 7.7] for further discussion.
Let be a unitary ring such that is a semisimple (artinian) ring. Then factors into a product
with each simple artinian.
See [20, p. 486], for instance. ∎
We now define a notion of orthogonality for simple artinian unitary rings which will be used later in the text (compare with the orthogonality defined in [3, Ch. 4, §2] in the commutative case). This notion is used implicitly and repeatedly in .
A simple artinian unitary ring is called orthogonal if:
is simple and of finite dimension over its center, denoted ,
is a -vector space and where .
If in addition (i.e. is split as a central simple -algebra), then we say that is split-orthogonal.
We use the term “orthogonal” because isometry groups of unimodular quadratic forms over an orthogonal unitary ring are forms of the the orthogonal group , when viewed as algebraic groups over . This follows from the discussion in §5.1 below. (A symplectic unitary ring can likewise be defined by replacing with in condition (3).)
If is simple artinian and in standard from (see §2.3), then it is split-orthogonal if and only if for a field , is the matrix transposition, , and .
Generalizing the example, let be a unitary ring such that is an involution. If satisfies conditions (1) and (2), then is a central simple algebra over its center (see [16, §1]) and is an involution of the first kind (i.e. it fixes ). This easily implies . By [16, Pr. 2.6], when , there is such that
where . When (resp. ) is called orthogonal (resp. symplectic). Furthermore, when , we always have
Thus, when is an involution, condition (3) is equivalent to having one of the following:
, is orthogonal and ,
, is symplectic and ,
See [16, §2] for further details about orthogonal and symplectic involutions.
We further recall that the index of a central simple -algebra admitting an involution of the first kind is a power of ([16, Cr. 2.8]). Thus, if is odd, then is split (i.e. ).
Orthogonality (resp. split-orthogonality) of simple artinian unitary rings is preserved under conjugation (see Proposition 2.4). Furthermore, if is an idempotent satisfying , then is orthogonal (resp. split-orthogonal) if and only if is orthogonal (resp. split-orthogonal).
That orthogonality (resp. split-orthogonality) is invariant under conjugation is clear from the definitions, so we turn to prove the second statement. Note that since is simple and , we have (because ). Morita Theory (see [18, §18D], for instance) now implies that is simple if and only if is simple, and . Writing , it follows that is a (split) central simple -algebra if and only if is. Furthermore, in this case, it is easy to see that is of the first kind if and only if is. Therefore, we may assume is a central simple -algebra and is of the first kind.
We claim that is a -vector space if and only if is a -vector space. (In fact, this is clear when because and is of the first kind.) One direction is evident so we turn to show the other. Assume is a -vector space and let and . Write for . Then where . Observe that since , . Since is a -vector space, for all , hence . Now, , as required.
Assume is a -vector space. It is left to show that if and only if , where and . Observe that by the above discussion, when is an involution, we always have and . Thus, by Proposition 2.6, the same holds for arbitrary . Let . There is nothing to prove if . Otherwise, , hence . It is easy to check that , and this implies if and only if . ∎
§2.5. Semiperfect Rings
We finish this section with recalling several facts about semiperfect rings. Proofs and additional details can be found in [21, §2.7–§2.9].
A ring is called semiperfect if it satisfies the following equivalent conditions:
is is semilocal and is idempotent lifting.
All finitely generated right (or left) -modules have a projective cover (see [21, Df. 2.8.31]).
There exists orthogonal idempotents with and such that is local for all .
In this case, any system of orthogonal idempotents in can be lifted to a system of orthogonal idempotents in . Furthermore, is semiperfect for any idempotent .
Examples of semiperfect rings include all one-sided artinian rings, and more gerenally, all semilocal rings with . Further examples that will be used later can be obtained from the following proposition.
Let be a henselian local (commutative) ring, and let be an -algebra. Then is semiperfect if one of the following holds:
is noetherian and is finitely generated as an -module.
is a valuation ring and is -torsion-free and of finite rank over .111 For an integral domain , the rank an -module is , where is the fraction field of .
When (1) holds, this follows from [1, Th. 22] or [25, Lm. 12]. When (2) holds, is semilocal by [27, Th. 5.4]. Let be an idempotent such that has no idempotents other than and . The proof of [25, Lm. 14] then implies that is local. Replacing with and repeating this procedure yields a (finite) system of orthogonal idempotents with and such that local for all , so is semiperfect. ∎
Let be a semiperfect ring. Then, up to isomorphism, there exist finitely many indecomposable projective -modules, , and every can be written as with uniquely determined. If , then are the simple -modules, up to isomorphism.
The modules can be constructed as follows: Write as a product of simple artinian rings , let be a primitive idempotent in , and let be a lifting of to . Then are the indecomposable projective right -modules, up to isomorphism.
3. Reflections and Quasi-Reflections
In this section we introduce and study quasi-reflections, which slightly extend a notion of reflections used by Reiter . Throughout, is a unitary ring, and be a quadratic space over .
Let be idempotents. An element is called -invertible if there exists such that and . It is easy to see that is unique and has as its -inverse. We hence write , or just when are understood from the context. Notice that there exists an -invertible element if and only , in which case we write . Indeed, left multiplication by an -invertible element gives an isomorphism from to , and any isomorphism is easily seen to be of this form.
Let be idempotents, and set for all . Then is -invertible if and only if is -invertible.
We only show the non-trivial direction. Assume has an -inverse with . Then , hence . (This follows from the easy fact that .) Likewise, . Let be the inverse of in , and let be the inverse of in . Then , , and . Thus, is an -inverse of . ∎
Let be an idempotent, let , and let be -invertible. We define by
Observe that is completely determined by the class . Following , we call an -reflection. We will also use the name quasi-reflection, which does not restrict us to a particular idempotent . A reflection of is a -reflection. When we want to stress the quadratic form , we shall write instead of .
Reiter’s definition of -reflections ([20, Df. 1.2]) is essentially the same, except that he assumes (in which case is just the inverse of in ). The generalization defined here will play a crucial role later in the text.
In the previous setting, is an isometry of . Its inverse is .
This is similar to the proof of [20, Pr. 1.3]; replace the usual inverses with -inverses. ∎
Let be idempotents.
If , then -reflections and -reflections coincide.
If , then the composition of an -reflection and an -reflection is an -reflection. Specifically, .
(i) Let be an -invertible element. It is a straightforward computation to verify that , which proves the claim.
(ii) Throughout the proof, we shall make repeated implicit usage of the fact that and , which easily follows from .
Observe first that
and that is -invertible with inverse . Thus, is an -reflection. Now, for all , we have
as required. ∎
Let be an idempotent and let .
If is -invertible, then .
If there exist and -invertible such that is -invertible for , then .
This is essentially the same as the proofs of Lemma 1.4 and Lemma 1.5 in ; replace the usual inverses with -inverses. ∎
(i) When applying conjugation (see Proposition 2.4) with respect to , -reflections remain -reflections. Indeed, it is straightforward to check that (note that if , then ). (In Reiter’s setting, which assumes , -reflections are preserved only when commutes with , for otherwise, is not invariant under the conjugation of by .)
(ii) Let be idempotents with and . Then -transfer (see Proposition 2.5) sends -reflections of to -reflections of . Indeed, .
4. Witt’s Extension Theorem
Using methods of Reiter  and the notion of quasi-reflections above, we now show that every isometry between subspaces of a unimodular quadratic space over a semiperfect ring can be extended to an isometry of the whole quadratic space. Furthermore, with small exception, the resulting isometry is a product of quasi-reflections. We compare our results with those of Reiter in Remark 4.12.
§4.1. General Setting
We set some general notation that will be used throughout. Let be a semiperfect unitary ring. For , set . In particular, . We shall occasionally view as a right -module. The image of in will be denoted by . Note that are isomorphic if and only if , because is a projective cover of and projective covers are unique up to isomorphism.
Let and let be the map induced by on . Then is a semisimple unitary ring, hence, by Proposition 2.8, it factors into a product
with each a simple artinian unitary ring (see §2.3). We write with a division ring, or a product of a division ring and its opposite.
By Proposition 2.6, for every , there exists such that the conjugation (see Proposition 2.4) of by is standard (see §2.3). Choose whose image in is . Then, by conjugating with , we may assume that is in standard form for all . Note that conjugation preserves quasi-reflections by Remark 3.6(i), so this is allowed if our goal is to prove that certain isometries extend to a product of quasi-reflections.
Now, let denote the standard matrix unit in . Then are orthogonal -invariant idempotents. Since is idempotent lifting, we can lift to orthogonal idempotents . The idempotents may not be invariant under , but we have as right -modules (because ), and hence .
Next, we set
Note that by Proposition 2.12, is split-orthogonal if and only if is split-orthogonal, and in this case, is a field, , and . Also note that .
Let be a quadratic space over . Then gives rise to a quadratic space over ; the map is defined by
Since factors into a product of unitary rings, the datum of is equivalent to the datum of quadratic spaces over . Specifically, if we write with a right -module, then is just the restriction of to the copy of in . We further set