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Buskerud University College, Department of Technology, P.O. Box 251, N-3601 Kongsberg, Norway **Email:** [email protected]

**Received date:** 1 October 2009; **Accepted date:** 10 December 2010

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The theory of generalized matric Massey products has been applied for some time to A-modules M, A being a k-algebra. The main application is to compute the local formal moduli ˆHM, isomorphic to the local ring of the moduli of A-modules. This theory is also generalized to OX-modulesM, X being a k-scheme. In these notes, we consider the definition of generalized Massey products and the relation algebra in any obstruction situation (a differential graded k-algebra with certain properties), and prove that this theory applies to the case of graded Rmodules, R being a graded k-algebra and k algebraically closed. When the relation algebra is algebraizable, that is, the relations are polynomials rather than power series, this gives a combinatorial way to compute open (´etale) subsets of the moduli of graded R-modules. This also gives a sufficient condition for the corresponding point in the moduli of O Proj(R)-modules to be singular. The computations are straightforwardly algorithmic, and an example on the postulation Hilbert scheme is given.

The theory of generalized matric Massey products (GMMP) for *A*-modules, *A* being a k-algebra, is given by Laudal
in [4], and applied to the theory of moduli of global and local modules in [6,7]. This theory can obviously be
applied also to the study of various Hilbert schemes, leading to GMMP for graded *R*-modules *M*, *R* being a graded
*k*-algebra. Let (*A*^{•} , *d*_{•}) be a differential graded *k*-algebra, and let be a set of elements in . For and we have the ordinary cup-products** ** given by

For example,

By inductively adding elements due to some relations, we define the higherorder
generalized matric Massey products , for some of higher-order, provided *A*^{•}
satisfies certain properties. The inductive definition of GMMP is controlled at each step by the relations between the
monomials in an algebra constructed in parallel. We call this algebra the relation algebra of . It is interesting
in its own right to study the GMMP structure and the relation algebra of various sets of .

Deformation theory is introduced as a tool for studying local properties of various moduli spaces. It is well
known that the prorepresenting hull of the deformation functor of a point *M* in moduli is the completion of
the local ring in that point [5]. Consider a graded *R*-module *M, R* being a graded k-algebra. Choose a minimal
resolution of *M* and consider the degree zero part of the Yoneda complex. Then, is a differential graded *k*-algebra. Let

be a *k*-basis. Then, the relation algebra is isomorphic to the prorepresenting hull of the (graded) deformation
functor Def_{M}, that is . In addition to the definition of the graded GMMP, this is the main result of
the paper, implying that general results about the GMMP give local information about moduli. In addition, we get
the following result, telling us how GMMP on *R* can be used to study the singular locus of sheaves on Proj(*R*).

**Proposition 1.** *Let for being a coherent -module. Assume that*

*Then, the hulls of the two deformation functors and are isomorphic, that is *.

We conclude the paper with an explicit application to the postulation Hilbert scheme, suggested by Professor Jan Kleppe.

2.1 Notation

We let be a graded *k*-algebra, *k* being algebraically closed of characteristic 0 and *R* finitely generated
in degree 1. We let be a graded R-module, and let denote the twisted module of *M* with
grading .

We will reserve the name *S* for the free polynomial *k*-algebra , so that *R* is a quotient of some
*S* by some homogenous ideal *I*, that is .

By we mean the category of local Artinian *k*-algebras with residue field *k*. If , we will use the
notation for the maximal ideal in *U*. A surjective morphism in is called small if ker . The ring of dual numbers is denoted by , that is .

If *V* is a vector space, denotes its dual.

2.2 Homomorphisms

Classically, homomorphisms of graded *k*-algebras *R* are *homogenous of degree* 0. This is also the case with morphisms
of graded *R*-modules. We might extend this definition by giving a grading to the homomorphisms:

where has the additional property .

2.3 Construction of graded *S*-modules

For the graded *R*-modules *M* and *N*, does certainly not inherit a total grading by . Thus, the sentence “a free graded *R*-module” does simply not make any sense. In this
section, we clarify how the grading is given.

Recall that if *M, N* are graded *R*-modules and is a homogenous homomorphism (of degree 0), then
ker(*f*) and im(*f*) are both graded submodules.

**Lemma 2.** *Let be a graded R-module, M any R-module and a surjective R-module homomorphism. Then, has a natural structure of graded R-module.*

**Remark 3.** The proof of the above lemma is immediate, but it is not always the case that for some
*f*. In fact, this is equivalent with *M* being graded.

For the sake of simplicity, assume that is a finitely generated graded module, generated by a finite number of homogenous elements of degrees , respectively. Then, we have a surjective homomorphism

sending *e _{i}* of degree

We easily see that , making into a graded module. As the kernel is also generated by a finite number of homogenous elements, say by , where with being homogenous in *R*, we have the following proposition.

**Proposition 4.** *Every finitely generated, graded R-module M has a minimal resolution of the form*

*Conversely, given homogenous elements of degrees , respectively, then maps surjectively onto the graded module and so it is a graded R-module.*

2.4 Families of graded modules

A finitely generated graded *R*-module *M* defines a coherent sheaf of -modules. In the same way, an
ideal defines a sheaf of ideals on Proj(*R*), and this gives a subscheme of Proj(*R*). Thus, the study of various
moduli spaces is influenced by the study of graded *R*-modules.

Let us denote for short, and let *X* be a scheme/*k*. Then, a sheaf of graded -modules is an -module such that is a graded -module for every open . We define the
contravariant functor by

The moduli spaces that we want to give results about are the schemes representing various restrictions of the functor given by

The restrictions can be that is an ideal sheaf with fixed Hilbert polynomial . Then, the above functor in the
case where is the Hilbert functor . If is locally free of rank *r* with fixed chern classes,
we get and so forth.

The two functors and are usually not equivalent, that is, there exist graded modules *M* such that . However, , and this, as we will see, is sufficient for given applications.

In general, for a contravariant functor , and an element , we define the fiber
functor from the category of pointed schemes over *k* to the category of sets by

If is represented by a scheme , and if is a geometric point, then the tangent space in this point is

The fiber functors define covariant functors and . In our situation, we obtain the two deformation functors

given by

For the rest of this section, we assume that . We will use the notations Def_{M} and when no confusion
is possible.

By definition, the tangent spaces of the moduli spaces are

respectively will be defined below). For every , the morphism is surjective with section .

For a surjective small morphism in , given a diagram

with mapping to mapping to , then it follows by functoriality of that is a lifting of .

This has obvious consequences, and we will eventually prove the following.

**Proposition 5.** *Let for . Assume that . Then
the hulls of the two deformation functors and are isomorphic, that is *.

**Remark 6.** This proof says that if , where , are the liftings to corresponding to the
canonical morphism , then . This can be used correspondingly in the higher-order liftings,
but it is very hard to check.

3.1 Generalized Massey products

In this subsection, we consider a differential graded *k*-algebra (*A*^{•} , *d*_{•}) with certain properties. We will assume that , and for , we will use the notation . For , we
will define some generalized Massey products , where . Notice
that the Massey products may not be defined for all (if any) . The overall idea is the following.

Let be a set of *d* elements in , let , and put .The first-order Massey products are then the ordinary cup-products in *A*^{•}. That is

**Definition 7.** One will say that the Massey product is *identically zero* if .

The higher-order Massey products are defined inductively. Assume that the Massey products are defined for , . For each , assume that there exists a fixed linear relation

and choose an such that . The set , is called a * defining
system* for the

where are chosen linear coefficients for each pair such that .

One way to construct Massey products, that is to construct the relations *b* given above, is as follows. Let be a set of representatives of *d* elements in . Let

**Definition 8.** The first-order Massey products are the ordinary cup-products in *A*^{•}. That is, for ,

Choose a *k*-basis for , and put

Let , and choose such that is a monomial basis for

Put . For each with , we have a unique relation in :

and for each we have

Choose for each an such that . Put

Choose a monomial basis for such that for each for some and some .

**Definition 9.** The set is called a defining system for the Massey products .

Assume that the *k*-algebras

and the sets have been constructed for according to the above, in particular

and we assume (by induction) that is a basis for

Put . For each we have a unique relation in :

**Definition 10.** The *N*th-order Massey products are

For this to be well defined, we need both that *y*(*n*) is a coboundary, and that its class is independent of the
choices of *α*. We will only consider algebras *A*^{•} that obey this, and we call *A*^{•} * an obstruction situation algebra*, or
concisely an

Put

pick a monomial basis for such that , and put . For each , we have a unique relation in , and for each ,

For each , choose such that .

**Definition 11.** The set is called a defining system for the Massey products .

Choose a monomial basis for

such that for each , we have for some and . The construction then continues by induction.

Definition 12. Let (*A*^{•} , *d*_{•}) be a differential graded OS *k*-algebra, and let be a set of elements in . Let be a *k*-basis for , and for , let

Then, one defines

and calls it the **relation algebra** of .

3.2 Obstruction theory

In this section, fix once and for all a minimal (graded) resolution of the graded *R*-module *M*:

with . Consider a small surjective morphism in , and let . Then, an element such that is called a lifting of *M _{V}* to

**Lemma 13.** *Giving a lifting of the graded R-module M is equivalent to giving a lifting of
complexes:*

*One also has that for all , and that the top row is exact*.

*Proof.* Because *U* is Artinian, its maximal ideal is nilpotent. We will prove the lemma by induction on *n* such
that , the case *n* = 1 being obvious. Assuming the result true for *n*, then assume and put . Thus, the sequence of *U*-modules with is exact with being a
small morphism, and such that . Notice that is *V* -flat and that such that . Also notice that . Thus, tensorizing over *U* with *M _{U}* we get the exactness of the first vertical sequence in the diagram

The exactness of the horizontal top row follows from exactness of *I* over *k*, the bottom row is exact by
assumption, and the middle row is constructed as follows: let be a lifting of , which obviously exists. By
assumption, . Thus, , that is . The commutativity of the first rectangle then follows from the fact that , that is, is a small morphism.

Now, choose a lifting of . As above, by the induction hypothesis, and therefore it commutes with .

For each generator , choose an such that , and put . Then

This gives the desired lifting, and we may continue this way with . We have proved that the middle sequence is a complex.

Conversely, given a lifting of complexes as in the lemma, then taking the tensor product over *U* with in the top row, we get a lifting as in the above diagram. By the induction hypothesis, the bottom row is exact with being a lifting of M.

Writing up the long exact sequence of the short exact sequence of complexes, we have that the sequence in the
middle is also exact, is flat over *U* because as *k*-vector space implies that *M _{U}* is

As the category of graded *R*-modules is the (abelian) category of representations of the graded *k*-algebra *R*, a
homomorphism between the two graded *R*-modules *M* and *N* is by definition homogenous of degree 0. This implies that the derived functors of Hom_{R} in the category of graded *R*-modules are the derived functors of , where denotes *R*-linear homomorphisms of degree 0. Thus, we use the notation .

We have fixed the minimal graded resolution of *M*, and we define the graded Yoneda complex by , where

and where the differential

is given by

where the composition is given by . It is straightforward to prove the following.

**Lemma 14. **.

**Proposition 15.** *Let be a small morphism in with kernel I. Let correspond to the
lifting of the complex . Then, there is a uniquely defined obstruction*

*given in terms of the 2-cocycle , such that if and only if M _{V} may be
lifted to U. Moreover, if , then the set of liftings of M_{V} to U is a principal homogenous space over *.

*Proof.* Since *L _{i}* is free for each

As is small, the composition is induced by a unique morphism , and so

Also, *o* is a cocycle, and

Another choice leads to an differing by the image of an element in *I* such that is independent of the choice of liftings. This also proves the “only if” part.

If then there is an element such that . Put . and one finds that . Thus, it follows from Lemma 13 that *M _{V}* can be lifted to

For the last statement, given two liftings and corresponding to and . Then
their difference induces morphisms , and (for each choice of basis element in *I*) is a cocycle and thus defining . This gives the claimed surjection

Notice that this proves that is an OS-algebra as well.

We now combine the theory of Massey products and the theory of obstructions. We let denote the
prorepresenting hull (the local formal moduli) of Def_{M}. All the way we will use the notations and constructions in
Section 3.1.

Pick a basis

and a basis

Denote by and the corresponding dual bases. Put

We set and . Let and denote the tangent spaces of and Def_{M}, respectively.
A deformation corresponding to an isomorphism is represented by the lifting of {*L*^{•}, *d*_{•}} where

Now, put , choose as in Section 3.1 and put . Then

with

This is to say for each . Translating, we get

Following the construction in Section 3.1, for each , we pick a 1-cochain such that

Then the family is a defining system for the Massey products . Define by

Then , and so, by Lemma 13, corresponds to a lifting . We continue by induction: given a defining system for the Massey products , assume that is defined by Then it follows that

For , letting

as in Section 3.1 gives

Dividing out by the obstructions, that is letting , makes the obstruction 0, that is for each , such that the next order defining system can be chosen.

Thus we have proved the following.

**Proposition 16.** Let R be a graded k-algebra and M a graded R-module. Let be a k-vector space basis. Then, the relation algebra of is isomorphic
to the prorepresenting hull of Def_{M}, that is .

*Proof.* This follows directly from Schlessinger’s article [5].

*Proof of Proposition 5.* For each small morphism ,if *M _{V}* is unobstructed, so is . Thus, if there are
no relations in

**Proposition 17.** *Let R be a graded k-algebra and a point in the moduli space of -modules corresponding
to . If all cup-products of are identically zero and , then is nonsingular in the point *.

*Proof.* We can choose all defining systems for *M* equal to zero so that there are no relations in *H _{M}*. The result then
follows from Proposition 5.

The * postulation Hilbert scheme* is the scheme GradAlg parameterizing graded algebras with fixed Hilbert function.
The following example is given to me by Jan Kleppe. The theory is treated in [2,3].

Let and consider the two *R*-matrices

We let *I* and *J* be the ideals generated by the minors of *G _{I}* and

Notice that a computer program (a library in Singular [1]) can be made for these computations. This will be clear in this example. However, for large tangent space dimensions, it seems that the common computers of today are too small.

In this section, we will cut out the tangent space by a hyperplane where the variety in question is obstructed. This will give readable information about the relations in the point corresponding to the variety, and the example will be possible to read.

We put

Then and are given by the minimal resolution

with

To compute a basis for , we apply the functor , resulting in the sequence

We then notice that and so .

Programming in Singular’s work [1], a basis for is given by the columns in the following 6×24- matrix:

Following the algorithm and notation given in Section 3.1, we compute cup-products. Of the 300 computed, 79 are identically zero in the meaning that

Of the remaining 221, 205 are zero in cohomology, giving in total 16 nonzero cup-products. With respect to a basis for , these products can be expressed by

Letting

we may conclude from Proposition 16 the following.

**Proposition 18.** *The determinantal scheme given by the minors of the matrix G _{I} has first-order relations given by
its second-order local formal moduli*

From [3] it follows that the obstruction space for *M* = *R/I* is *H*^{2}(*M,M,R*), where *M* is considered as a graded
*R*-algebra. This *k*-vector space has dimension 3, and so we may conclude the following.

**Corollary 19.** *The determinantal scheme given by the minors of the matrix G _{I} is maximally obstructed.*

We are now going to put most (21) of the variables above to zero. That is, we choose the most interesting of
the 24 variables above; , all others are put to zero. We follow the algorithm given in
Section 3.1 and we work in the Yoneda complex Hom^{•} (*L*_{•}, *L*_{•}), where *L*_{•} denotes the *R*-free resolution of *M* given
above.

Notice that for , it is always sufficient to have the two leading morphisms . Also, it is known that finding these by the methods below, they can always be extended to the full complex; see [7].

We find

This given, it is an easy match to compute the cup products (or the first order generalized Massey products). These are

As classes in cohomology, we find (as we already knew)

We put

and the *restricted* local formal moduli to the second order is

We follow the algorithm given in Section 3.1 further. We choose a basis *B _{2}* for

for example,

and choose a third-order defining system. Notice that by we mean a *representative* of the cohomology class

Similarly,

Writing up why these are cocycles, we find what to choose for :

Again, following the algorithm given in Section 3.1, we choose a monomial basis for , and can compute the Massey products (notice again that the next to last expression is the representative in the Yoneda complex of its cohomology class):

We now put

and so

We put , and the next order defining system is easy to find, only one of the representations of the elements is different from zero:

We choose

The rest of the elements in the defining system are chosen identically zero, and we put and compute the fourth-order Massey products:

Now, put Because these then are homogenous of degree two, the next order defining systems involves only fourth-order Massey products, and these can all be chosen identically zero. Then, the fifthorder Massey products involve with at least one of . We see that for all with and so all fifth-order Massey products are zero. Noting also that we are ready to conclude the following proposition.

**Proposition 20.** *Let . Then, there exists an open subset of the component
of GradAlg, the moduli scheme of graded R-algebras, containing the determinantal scheme corresponding to the
matrix G _{I} such that its intersection with the hyperplane is isomorphic to*

*with the versal family*

*for with*

**Remark 21.** When the local formal moduli with its formal family is algebraizable in this way, we get an open subset
of the moduli (at least ´etale). Thus, we get a lot more than just the local formal information. The conditions for when is algebraizable is an interesting question.

**Acknowledgment** The author would like to thank Jan Kleppe for introducing him to the theory of postulation Hilbert schemes.

- Greuel GM, Pfister G, Schonemann H, Singular 3.0. (2005) A Computer Algebra System for Polynomial Computations. Centre for Computer Algebra, University of Kaiserslautern.
- Kleppe JO (2006) Maximal families of Gorenstein algebras. Trans Amer Math Soc 358: 3133–3167.
- Kleppe JO, Miro-Roig RM (2005) Dimension of families of determinantal schemes. Trans Amer Math. Soc 357: 2871–2907.
- Laudal OA (1986) Matric Massey products and formal moduli I. Algebra, Algebraic Topology and Their Interactions (Stockholm, 1983 Lecture Notes in Math, Springer, Berlin 1183: 218–240.
- Schlessinger M (1968) Functors of Artin rings. Trans Amer Math Soc 130: 208–222.
- Siqveland (2001) Global matric Massey products and the compactified Jacobian of the E6-singularity. J Algebra 241: 259–291.
- Siqveland (2001) The method of computing formal moduli. J Algebra 241: 292–327.

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