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ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Phase Spaces and Deformation Theory

Olav Arnfinn Laudal*

 

Institute of Mathematics, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway

*Corresponding Author:
Olav Arnfinn Laudal
Institute of Mathematics University of Oslo
P.O. Box 105Blindern 0316 Oslo, Norway
E-mail:
[email protected]

Received date 7 October 2009; Accepted date 10 December 2010

 

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Abstract

We have previously introduced the notion of non-commutative phase space (algebra) associated to any associative algebra, defined over a field. The purpose of the present paper is to prove that this construction is useful in non-commutative deformation theory for the construction of the versal family of finite families of modules. In particular, we obtain a much better understanding of the obstruction calculus, that is, of the Massey products.

1 Introduction

In [8], we sketched a physical “toy model,” where the space-time of classical physics became a section of a universal fiber space image , defined on the moduli space image of the physical systems we chose to consider (in this case, the systems composed of an observer and an observed, both sitting in a Euclidean 3-space). This moduli space was called the time-space. Time, in this mathematical model, was defined to be a metric ρ on the time-space, measuring all possible infinitesimal changes of the state of the objects in the family we are studying. This gave us a model of relativity theory, in which the set of all (relative) velocities turned out to be a projective space. Dynamics was introduced into this picture, via the general construction, for any associative algebra A, of a phase space Ph(A). This is a universal pair of a homomorphism of algebras,image and a derivation,image such that for any homomorphism of A into a k-algebra R, the derivations of A in R are induced by unique homomorphisms image composed with d. Iterating this Ph(−)-construction, we obtained a limit morphism imageimage with imageimage, and a universal derivationimagethe Dirac-derivation. A general dynamical structure of order n is now a two-sided δ-ideal σ in image inducing a surjective homomorphismimage

In [8] and later in [10], we have shown that, associated to any such time space H with a fixed dynamical structure H(σ), there is a kind of “Quantum field theory”. In particular, we have stressed the point that, if H is the affine ring of a moduli space of the objects we want to study, the ring image is the complete ring of observables, containing the parameters not only of the iso-classes of the objects in question, but also of all dynamical parameters. The choice made by fixing the dynamical structure σ, and reducing to the k-algebra H(σ), would classically correspond to the introduction of a parsimony principle (e.g. to the choice of some Lagrangian).

The purpose of this paper is to study this phase-space construction in greater detail. There is a natural descending filtration of two-sided ideals,image.The corresponding quotientsimage are finite dimensional vector spaces, and considered as affine varieties; these are our non-commutative Jet-spaces.

We will first see, in Section 2, that we may extend the usual prolongation-projection procedure of Elie Cartan to this non-commutative setting, and obtain a framework for the study of general systems of (non-commutative) PDEs; see also [9].

In Section 3, we present a short introduction to non-commutative deformations of modules, and the generalized Massey products, as exposed in [4,5].

Then, in Section 4, the main part of the paper follows: the construction for finitely generated associative algebras A of the versal family of the non-commutative deformation functor of any finite family of finitely dimensional A-modules, based on the phase-space of a resolution of the k-algebra A.

Notice that our image is a non-commutative analogue of the notion of higher differentials treated in many texts (see [1] and the more recent paper [2]).

2 Phase spaces and the Dirac derivation

Given a k-algebra A, denote by A/k − alg the category where the objects are homomorphisms of k-algebras κ : A → R, and the morphisms image are commutative diagrams:

image

and consider the functor

image

defined by image It is representable by a k-algebra-morphism, image ith a universal family given by a universal derivation image It is easy to construct Ph(A). In fact, let imagebe a surjective homomorphism of algebras, with image freely generated by theimage and put I = kerπ. Let,

image

where image is a formal variable. Clearly there is a homomorphismimage and a derivationimage defined by puttingimage the equivalence class ofimageSince image both kill the ideal I, they define a homomorphism image and a derivationimage To see that i0 and d have the universal property,image be an object of the categoryimageAny derivation image defines a derivationimage mappingimage be the homomorphism defined by

image

where image sends I and dI to zero, and so defines a homomorphismimage such that the composition with image isimage The unicity is a consequence of the fact that the images of i0 and d generate Ph(A) as k-algebra.

Clearly Ph(−) is a covariant functor on k − alg, and we have the identities,

image

with the last one associating d to the identity endomorphism of Ph(A). In particular, we see that i0 has a cosection,image corresponding to the trivial (zero) derivation of A.

Let now V be a right A-module, with the structure morphismimage).We obtain a universal derivation:

image

defined by image Using the long exact sequence

image

we obtain the non-commutative Kodaira-Spencer class

image

inducing the Kodaira-Spencer morphism

image

via the identity image then the exact sequence above proves that there exist aimage such thatimage This is just another way of proving that c(V ) is the obstruction for the existence of a connection,

image

It is well known, I think, that in the commutative case, the Kodaira-Spencer class gives rise to a Chern character by putting

image

and that if c(V ) = 0, the curvature image of the connection ∇ induces a curvature class in a generalized Lie-algebra cohomology:

image

Any Ph(A)-module W, given by its structure map,

image

corresponds bijectively to an induced A-module structureimagetogether with a derivation imageimage defining an elementimage Fixing this last element, we find that the set of Ph(A)-module structures on the A-module W is in one-to-one correspondence withimage Conversely, starting with an A-module V and an elementimage we obtain a Ph(A)-moduleimage It is then easy to see that the kernel of the natural map

image

induced by the linear map

image

is the quotient

image

and the image is a subspace image which is rather easy to compute; see examples below.

Remark 1. Defining time as a metric on the moduli space, Simp(A), of simple A-modules, in line with the philosophy of [8], noticing thatimage is the tangent space of Simp(A) at the point corresponding to V , we see that the non-commutative space Ph(A) also parametrizes the set of generalized momenta, that is, the set of pairs of a point image and a tangent vector at that point.

Example 2. (i) Let image then obviously,image and d is given by d(t) = dt, such that forimage we findimage with the notations of [7], that is, the non-commutative derivation of f with respect to t. One should also compare this with the non-commutative Taylor formula of loc.cit. If imageis an A-module, defined is an A-module, defined by the matriximage andimage is defined in terms of the matriximage then the Ph(A)-module image is the image-module defined by the action of the two matricesimage and we find

image

We have the following inequalities:

image

(ii) Let image then we find

image

In particular, we have a surjective homomorphism

image

with the right-hand side algebra being theWeyl algebra. This homomorphism exists in all dimensions.We also have a surjective homomorphism,

image

that is, onto the affine algebra of the classical phase-space.

The phase-space construction may, of course, be iterated. Given the k-algebra A, we may form the sequence image defined inductively by

image

Let image be the canonical imbedding, and letimage be the corresponding derivation. Since the composition of image and the derivation dn+1 is a derivationimage there exists by universality a homomorphism imagesuch that

image

Notice that we compose functions and functors from left to right. Clearly, we may continue this process constructing new homomorphisms

image

with the property

image

Notice also that we have the “bi-gone”image and the “hexagone”

image

and, in general,

image

which is all easily proved by composing with image Thus, the Ph* (A) is a semi-cosimplicial algebra with a cosection onto A. Therefore, for any object

image

the semi-cosimplicial algebra above induces a semi-simplicial k-vector space image and one should be interested in its homology.

The system of k-algebras and homomorphisms of k-algebrasimage has an inductive (direct) limit, image together with homomorphismsimage satisfying

image

Moreover, the family of derivations image define a unique derivationimage such that imageimagePut

image

The k-algebra image has a descending filtration of two-sided ideals, withimage given inductively by

image image

such that the derivation δ induces derivations image Using the canonical homomorphismimageimagewe pull the filtration image back toimagenot bothering to change the notation.

Definition 3. Let image be the completion ofimage in the topology given by the filtration image The k-algebra imagewill be referred to as the k-algebra of higher differentials, and D(A) will be called the k-algebra of formalized higher differentials. Put

image

Clearly, δ defines a derivation on D(A), and an isomorphism of k-algebras

image

and, in particular, an algebra homomorphism

image

inducing the algebra homomorphisms

image

which, by killing, in the right-hand side algebra, the image of the maximal ideal, image of A corresponding to a pointimage induces a homomorphism of k-algebras

image and an injective homomorphism

 

image

see [8]. More generally, let A be a finitely generated k-algebra and letimage be an n-dimensional representation (e.g. a point of Simpn(A)) corresponding to a two-sided ideal m = ker ρ of A. Then image induces a homomorphism

image

and we will be interested in the image; see Section 4.

The k-algebras image are our generalized jet spaces. In fact, any homomorphism of A-algebras

image

composed with

image

is a usual differential operator of order ≤ n on A. Notice also the commutative diagram

image

Here the upper vertical morphisms are injective, with the lower line being the sequence of symbols.

It is easy to see that the differential operators form an associative k-algebra, Diff(A). In fact, assume two differential operators

image

and consider the functorially defined diagram

image

then the product is defined by the composition

image

Let now V be, as above, a right A-module, with structure morphism image Consider the linear map

image

Assume that the non-commutative Kodaira-Spencer class, defined above,

image

vanishes. Then, as we know, there exist a connection, that is, a linear map

image

such that image It is also easy to see that this connection induces higher-order connections, that is, k-linear maps,

defined by

image

In fact, we just have to prove that ∇(n) is well defined, that is, we have to prove that

image

Noticing that

image

where we have put image, we find

image

These higher-order connections will induce a diagram

image

where the lower line is the sequence of symbols. Notice that

image

as given above, by definition has the property that for all a ∈ A and all v ∈ V we have

image

Assume, in particular, that V and the A-moduleW are free of ranks p and q, respectively. Let image be a family of A-homomorphismsimage defining a generalized differential operator

image

The solution space of image is by definitionimage There are natural generalizations of this set-up, which we will, hopefully, return to in a later paper, extending the classical prolongation-projection method of Elie Cartan to this non-commutative setting. See Example 4 for the commutative analogue.

In [8], we introduced the notion of a dynamical structure for a k-algebra A, as a two-sided δ-stable ideal σ ⊂ Ph (A), or equivalently as the corresponding quotient A(σ) of the δ-algebra image Any such A(σ)will be given in terms of a sequence of ideals, image with the property thatimage The solution space of such a system, should be considered as the non-commutative scheme parametrized by A(σ), that is, as the geometric system of all simple representations of A(σ); see [6].

This is, in a sense, dual to the classical theory of PDEs, as we will show by considering the following example, leaving the general situation to the hypothetical paper referred to above.

Example 4 (see [9]). (i) Let image and consider the situation corresponding to a free particle (see [8]) that is, where we have obtained A(σ) by killing image for everyimage then the commutativizationimage is a free A-module generated by the basis

image

Put image The dual basisimage may be identified with a basisimage of the A-module of all (classical higher-order) differential operators of order less or equal to k. In fact, consider the composition

image

then, for f ∈ A we have

image

where we assume

image

and where image is μpth-order derivation with respect to image we let image to be the identity operator on A..

Now, consider the commutativization of A(σ), as a k-linear space, and for every k ≥ 1,

image

as a family of affine spaces fibered over Simp1(A),

image

This family is defined by the homomorphism of k-algebras

image

Let image then the system of equations

image

is a system of partial differential equations (an SPDE, for short). Suppose there is a solution, that is, an f ∈ A, such that

image

then, for every j, we must have

image

which amounts to extending the SPDE by, including together with imagethe polynomials

image

where it should be clear how to interpret the indices. Let us denote by P the extended family of polynomials,

image

and let image be the ideal, generated by the polynomials in P, contained in image Denote by Sm := image the corresponding subvariety. Clearly, the canonical mapimage induced by the trivial derivation of image has a canonical restriction image Denote also byimage the restriction of the morphism image defined above, to Sk. Classically, the system is called regular if all πk are fiber bundles, so smooth, for allimage Now, for any closed point of Spec(A), that is, for any point image consider the sequence of fibers overimage and the corresponding sequence of mapsimage An element image corresponds exactly to an elementimagefor which

image

that is, to a formal solution of the SPDE. Thus, the projective limit of schemes image is the space of formal solutions of the SPDE at image

A fundamental problem in the classical theory of PDE is then the following.

Find necessary and sufficient conditions on the SPDE image to be non-empty, and find, based on image its structure. In particular, compute its dimension image

We will not, here, venture into this vast theory, but just add one remark. The solution space is in fact a family, with parameter-space Simp1(A). Given any point image the (formal) scheme, image of formal solutions may have deformations.We might want to compute the formal moduli image and relate the given family to the corresponding mini-versal family.

The tangent space of image is given as

image

see [3]. A tangent at the point image of Simp1(A) is the value at t of a linear combination of the fundamental vector fields, the derivations image of A. The map between the tangent space of the given family and the tangent space of H is then easily seen to be the following:

image

where image is the class of the map, associating aimage to the class atimage The image of the tangent at image ofimage corresponding to Dj , in the tangent space of H, is zero if this map is a derivation. Now, this is exactly what we have arranged, together with any P ∈ p, and also including

image

in the ideal p. Thus, the map η is trivial, and the given pro-family is formally constant, as one probably should have suspected! Moreover, it is easy to see that if image has a local section, thenimage is formally constant at image The basic problem is to find computable conditions under which the constancy of πk implies the surjectivity of pl1, and thereby the non-triviality of S(P)(t).

We will, hopefully, come back to these questions in a later paper.

image

and it is easy to see that image and, of course, image In particular, there is a homomorphism onto

image

(iii) Let now image so that there are no natural surjective homomorphisms image is, however, injective. The difference between examples (i) and (ii) is, of course, due to the fact that in the first case A is graded, and in the second it is not; see Section 4.

3 Non-commutative deformations of families of modules

In [5,6,7], we introduced non-commutative deformations of families of modules of non-commutative k-algebras, and the notion of swarm of right modules (or more generally of objects in a k-linear abelian category). Let image denote the category of r-pointed not necessarily commutative k-algebras R. The objects are the diagrams of k-algebras

image

such that the composition of ι and π is the identity. Any such r-pointed k-algebra R is isomorphic to a k-algebra of image The radical of R is the bilateral ideal Rad image The dual k-vector space of Rad(R)/ Rad(R)2 is called the tangent space of R.

For r = 1, there is an obvious inclusion of categories image where l, as usual, denotes the category of commutative local Artinian k-algebras with residue field k.

Fix a (not necessarily commutative) associative k-algebra A and consider a right A-module M. The ordinary deformation functor image is then defined. Assumingimage has a finite k-dimension for i = 1, 2, it is well known (see [12] or [5]) that DefM has a pro-representing hull H, the formal moduli of M. Moreover, the tangent space of H is isomorphic to image and H can be computed in terms of image and their matric Massey products; see [5].

In the general case, consider a finite family image of right A-modules. Assume that dimkimage ∞. Any such family of A-modules will be called a swarm. We will define a deformation functor image generalizing the functor DefM above. Given an objectimage consider the k-vector space and the left R-module image It is easy to see that

image

Clearly, π defines a k-linear and left R-linear map

image

inducing a homomorphism of R-endomorphism rings,

image

The right A-module structure on the Vi s is defined by a homomorphism of k-algebras:

image

Notice that this homomorphism also provides each image with an A-bimodule structure. Let imageSets be the set of isoclasses of homomorphisms of k-algebras

image

such that image where the equivalence relation is defined by inner automorphisms in the k-algebra imageimage inducing the identity onimage One easily proves that DefV has the same properties as the ordinary deformation functor and we may prove the following theorem (see [5]).

Theorem 5. The functorDefV has a pro-representable hull, that is, an object H of the category of pro-objectsimage ofimage together with a versal family

image

where m = Rad(H), such that the corresponding morphism of functors on image

image

defined for image is smooth and an isomorphism on the tangent level. Moreover, H is uniquely determined by a set of matric Massey products defined on subspaces

image

with values in image

The right action of imagedefines a homomorphism of k-algebras,

image

and the k-algebra O(V) acts on the family of A-modules image extending the action of A. If image for all i = 1, . . . , r, the operation of associating image turns out to be a closure operation.

Moreover, we prove the crucial result.

Theorem 6 (a generalized Burnside theorem). Let A be a finite dimensional k-algebra, with k being an algebraically closed field. Consider the familyimage of all simple A-modules, then

image

is an isomorphism.

We also prove that there exists, in the non-commutative deformation theory, an obvious analogy to the notion of pro-representing (modular) substratum H0 of the formal moduli H; see [3]. The tangent space of H0 is determined by a family of subspaces

image

the elements of which should be called the almost split extensions (sequences) relative to the family V, and by a subspace

image

which is the tangent space of the deformation functor of the full subcategory of the category of A-modules generated by the family image is the set of all indecomposables of some Artinian k-algebra A, we show that the above notion of almost split sequence coincides with that of Auslander; see [11].

Using this we consider, in [5,7], the general problem of classification of iterated extensions of a family of modules image and the corresponding classification of filtered modules with graded components in the family V, and extension type given by a directed representation graph Γ. The main result is the following; see [7].

Proposition 7. Let A be any k-algebra and image any swarm of A-modules, such that

image

(i) Consider an iterated extension E of V, with representation graph Γ. Then there exists a morphism of k-algebras

image

(ii) The set of equivalence classes of iterated extensions of V with representation graph Γ is a quotient of the set of closed points of the affine algebraic varietyimage

(iii) There is a versal family image of A-modules defined onimage containing as fibers all the isomorphism classes of iterated extensions of V with representation graph Γ.

To any, not necessarily finite, swarm image of right-A-modules, we have associated two associative k-algebras (see [6,7]):

image

and a sub-quotient imagetogether with natural k-algebra homomorphisms

image

and image with the property that the A-module structure on c is extended to an O-module structure in an optimal way. We then defined an affine non-commutative scheme of right A-modules to be a swarm image of right A-modules, such that image is an isomorphism. In particular, we considered, for finitely generated k-algebras, the swarm image consisting of the finite dimensional simple A-modules, and the generic point A, together with all morphisms between them. The fact that this is a swarm, that is for all objects image we haveimage is easily proved. We have in [7] proved the following result (see [7, Proposition 4.1] for the definition of the notion of geometric k-algebra)

Proposition 8. Let A be a geometric k-algebra, then the natural homomorphism

image

is an isomorphism, that is, image is a scheme for A.

In particular, image is a scheme forimage To analyze the local structure ofimage we need the following lemma (see [7, Lemma 3.23]).

Lemma 9. Let image be a finite subset ofimage then the morphism of k-algebras,

image

is topologically surjective.

Proof. Since the simple modules image are distinct, there is an obvious surjection

image

Put image and consider for m ≥ 2 the finite dimensional k-algebra, image Clearly, Simp(B) = V so that by the generalized Burnside theorem (see [5, Theorem 3.4]) we find

image

Consider the commutative diagram

.

where all morphisms are natural. In particular α exists since image maps intoimage and therefore induces the morphism α commuting with the rest of the morphisms. Consequently, α has to be surjective, and we have proved the contention.

Example 10. As an example of what may occur in rank infinity, we will consider the invariant problem image Here we are talking about the algebraimage crossed product of C[x] with the group image the product in A is given byimage There are two “points” (i.e. orbits) modeled by the obvious origin image We may also choose the two pointsimage in line with the definitions of [6]. Obviously, C[x] corresponds to the closure of the orbit image This choice is the best if we want to make visible the adjacencies in the quotient, and we will therefore treat both cases.

We need to compute

image

Now,

image

and since x acts as zero on image acts as identity on V1 and as a homogenous multiplication on V0, we find

image

Any image is determined by its valuesimage Moreover, since in A we have image image

The left-hand side of the last equation isimage and the right-hand side is δ(x), and since this must hold for all image we must have δ(x) = 0. Moreover, since image it is clear that the continuity of δ implies that δ must be equal to α ln image for some α ∈ C. (To simplify the writing, we will put log := ln(| · |).)Therefore,

image

The cup-product of this class, log ∪ log, sits in image and is given by the 2-cocycle

image

This is seen to be a boundary, that is, there exists a map image such that for all image we have

image

Just put image Therefore, the cup product is zero, and if we, in general, put

image

where n is the number of 1s in the first index, then computing the Massey products of the element log ∈image we find the nth Massey product

image

and this is easily seen to be the boundary of the 1-cochain

image

Therefore, all Massey products are zero. Of course, we have not yet proved that they could be different from zero, that is, we have not computed the obstruction group image and found it non-trivial! Now this is unnecessary. Now, assume firstimage image

is determined by the values of image Sinceimage we may find a trivial derivation such that subtracting from δ we may assume δ(x) = 0. But then the formula

image

implies

image

from which it follows that

image

Now, sinceimage we find

image

which should hold for any pair of image and any p. This obviously implies δ = 0.

This argument shows not only that

image

when image but also whenimage Finally, we find that the formula above,

image

shows that for

image

we have image for all p. Therefore,

image

when image However, whenimage we find that δ with image for p ≥ 1, survives. These will, as above, give rise to a logarithm of the real part of C ∗. Therefore, in this case image The miniversal families look like

image

when image image

when V1 = C[x].

4 The infinite phase space construction and Massey products

Let, as above, image be a family of A-modules. To compute the relevant cohomology for the deformation theory, that is, the image we may use the Leray spectral sequence of [3], together with the formulas

image

where W is any A-bimodule. Choose a surjective morphism image of a free k-algebra F onto A, and put I = ker μ, then we find that

image

where Der is the restriction of the derivations , image Moreover, consider a commutative diagram of homomorphisms of algebras, in which imageis not yet included

image

and where J := ker π has square 0. The composition map image induces an elementimage independent upon the choice of ρ′ . If this (obstruction) element vanishes, then O is the restriction to I of a derivation image Subtracting this from ρ′,we may assume that ρ′ (I) = 0, so there exists a liftingimage If there exists a lifting image then we may obviously assume that O = 0.

Now, let . represent a basis ofimage and letimageimage denote the dual basis. Consider, the free matrix k-algebra (quiver)image generated in slot (i, j) by the (formal) elements of Ei,j . There is a unique homomorphism

image

Denote by the same letter the completion of T1 with respect to the powers of the radical Rad(T1) := kerπ. Then image Consider the k-algebra and the π-induced homomorphism

image

Clearly, π1 splits, and it is easy to see that.

image

defined by

image

is a derivation, image therefore inducing a unique homomorphism, image makes the following diagram commute

image

Now, we would have liked to extend this diagram, completing it with commuting homomorphisms,

image

where

image

However, as will be clear in the next construction, the obvious continuation of this procedure does not work. In fact, the formalized higher differentials D(A) is not really the natural phase-space to work with for all purposes. In an obvious sense it is too homogenous. We are therefore led to the construction of a kind of projective resolution of A. Consider as above a surjective homomorphism,image withimage a free k-algebra, and image Obviouslyimage are also free, andimage is a freeimage algebra. Let exp(δ) : F →D(F) be defined as in Section 2 by

image

and denote by image the induced homomorphism. Define

image

Clearly, image there are only natural surjective homomorphisms,image By functoriality, the diagram above induces another commutative diagram, which may be completed to the commutative diagram image image

where we, in expectation of later constructions, put

image

Now the map

image

is zero, and the resulting map image is, as deformation of the family V, the universal family at the tangent level. Since image is a free algebra over Ph(n)(F), there is lifting image We want an induced ρ2. Consider the composition

image

lifting image The restriction to I vanishes on I2 and induces a map

image

It is easily seen to be F-linear, both from left and right, and so it induces the obstruction

image

independent upon the choice of extension imageNow

image

may be identified with

image

which is a subspace of

image

Denote by T2 the free matrix algebra (quiver), in image generated by imagejust like the construction of T1 above, such that

image

We may now state and prove the main result of this paper.

Theorem 11. (i) For any finite family of (finite dimensional) A-modules,image there is a homomorphism image, making the following diagram commutative

image

such that the versal family image

(ii) Moreover, image may be constructed recursively, as a quotient ofimage by annihilating a series of obstructions, on, defining a morphism in image

Proof. We have above constructed an obstruction for lifting ρ1 to a ρ2. It is a unique element;

image

Obviously, the image

image

generates an ideal of T1, contained in image and put

image

Then, there is a commutative diagram

image

In fact, since we have divided out with the obstruction, we know that the morphism

image

is the restriction of a derivation

image

Now change the morphism image It is easily seen that for this new morphism, image is zero, restricted to I, proving the existence of image Recall that D1(A) = H1.

Now image definesimage be the two-sided ideal in T1 generated by

image

and let us put

image

The diagram above induces a commutative diagram, image constructed as above, but where image is the problem,

image

Consider now the map image ending up in

image

which clearly is killed by image and therefore really is a matrix of vector spaces, as an image As above, this map is easily seen to be a left and right linear map as F-modules, F acting on O(3) via image Moreover, the induced element

image

is independent on the choice of imageNow, we define image where σ3 is defined by the image of o3, and define image as above. Since, by functoriality, the morphism

image

must induce the zero element in the corresponding

image

it must be the restriction of a derivation image Now changeimage by sendingimage leaving the other values of the parameters unchanged. Then, a little calculation shows that the new image maps eachimage to zero, inducing a morphism We now have a new situation, given by a commutative diagram, not yet including image image

and it is clear how to proceed. This proves (i), and the rest is a consequence of the general theorem [3, Theorem 4.2.4].

We cannot replace H by D. This follows from the trivial Example 4(iii) above. However, if we are in a graded situation, things are nicer.

Corollary 12. Assume that A is a finitely generated, graded, in degree 1, k-algebra, and assume that V is a family of graded A-modules. Then there is a corresponding graded formal moduli image and there is a commutative diagram,

image

such that the graded versal family image

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