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Journal of Generalized Lie Theory and Applications
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Generalized Matric Massey Products for Graded Modules

Arvid Siqveland

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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Abstract

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.

1 Introduction

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 equation be a set of elements in equation. For equation and equation we have the ordinary cup-products equation given by

equation

For example,

equation

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

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 equation of M and consider the degree zero part equation of the Yoneda complex. Then, equation is a differential graded k-algebra. Let

equation

be a k-basis. Then, the relation algebra equation is isomorphic to the prorepresenting hull equation of the (graded) deformation functor DefM, that is equation. 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 equation for equation being a coherent equation-module. Assume that

equation

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

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

2 Classical graded theory

2.1 Notation

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

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

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

If V is a vector space, equation 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:

equation

where equation has the additional property equation.

2.3 Construction of graded S-modules

For the graded R-modules M and N, equation does certainly not inherit a total grading by equationequation. 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 equation is a homogenous homomorphism (of degree 0), then ker(f) and im(f) are both graded submodules.

Lemma 2. Let equation be a graded R-module, M any R-module and equation a surjective R-module homomorphism. Then, equation 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 equation for some f. In fact, this is equivalent with M being graded.

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

equation

sending ei of degree pi to mi (also of degree pi).

We easily see that equation, making equation into a graded module. As the kernel is also generated by a finite number of homogenous elements, say by equation, where equation with equation 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

equation

Conversely, given homogenous elements equation of degrees equation, respectively, then equationequation maps surjectively onto the graded module equation 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 equation of equation-modules. In the same way, an ideal equation 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 equation for short, and let X be a scheme/k. Then, a sheaf of graded equation-modules is an equation-module equation such that equation is a graded equation-module for every open equation. We define the contravariant functorequation by

equation

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

equation

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

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

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

equation

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

equation

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

equation

given by

equation

For the rest of this section, we assume that equation. We will use the notations DefM andequation when no confusion is possible.

By definition, the tangent spaces of the moduli spaces are

equation

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

equation

For a surjective small morphism equation in equation, given a diagram

equation

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

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

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

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

3 Deformation theory

3.1 Generalized Massey products

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

Let equation be a set of d elements in equation, let equation, and put equationequation.The first-order Massey products are then the ordinary cup-products in A. That is

equation

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

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

equation

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

equation

where equation are chosen linear coefficients for each pair equation such that equation.

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

equation

Definition 8. The first-order Massey products are the ordinary cup-products in A. That is, for equation,

equation

Choose a k-basis equation for equation, and put

equation

Let equation, and choose equation such that equation is a monomial basis for

equation

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

equation

and for each equation we have

equation

Choose for each equation an equation such that equation. Put

equation

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

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

Assume that the k-algebras

equation

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

equation

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

equation

Put equation. For each equation we have a unique relation in equation:

equation

Definition 10. The Nth-order Massey products are

equation

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 OS-algebra.

Put

equation

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

equation

For each equation, choose equation such that equation.

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

Choose a monomial basis equation for

equation

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

Definition 12. Let (A , d) be a differential graded OS k-algebra, and let equation be a set of elements in equation. Let equation be a k-basis for equation, and for equation, let

equation

Then, one defines

equation

and calls it the relation algebra of equation.

3.2 Obstruction theory

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

equation

with equation. Consider a small surjective morphism equation in equation, and let equation. Then, an element equation such that equation is called a lifting of MV to U.

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

equation

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

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

equation

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 equation be a lifting of equation, which obviously exists. By assumption, equation. Thus, equation, that is equation. The commutativity of the first rectangle then follows from the fact that equation, that is, equationis a small morphism.

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

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

equation

This gives the desired lifting, and we may continue this way with equation. 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 equation in the top row, we get a lifting as in the above diagram. By the induction hypothesis, the bottom row is exact with equation 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, equation is flat over U because equation as k-vector space implies that MU is U-free and thus flat.

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

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

equation

and where the differential

equation

is given by

equation

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

Lemma 14. equation.

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

equation

given in terms of the 2-cocycle equation, such that equation if and only if MV may be lifted to U. Moreover, if equation, then the set of liftings of MV to U is a principal homogenous space over equation.

Proof. Since Li is free for each i, we can choose a lifting equation making the following diagram commutative for each i:

equation

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

equation

Also, o is a cocycle, and

equation

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

If equation then there is an element equation such that equation. Put equationequation. and one finds that equation. Thus, it follows from Lemma 13 that MV can be lifted to U.

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

equation

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

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

Pick a basis

equation

and a basis

equation

Denote by equation and equation the corresponding dual bases. Put

equation

We set equation and equation. Let equation and equation denote the tangent spaces of equation and DefM, respectively. A deformation equation corresponding to an isomorphism equation is represented by the lifting equationequation of {L, d} where

equation

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

equation

with

equation

This is to say equation for each equation. Translating, we get

equation

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

equation

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

equation

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

equation

For equation, letting

equation

as in Section 3.1 gives

equation

Dividing out by the obstructions, that is letting equation, makes the obstruction 0, that is equation for each equation, 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 equationequation be a k-vector space basis. Then, the relation algebra of equation is isomorphic to the prorepresenting hull equation of DefM, that is equation.

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

Proof of Proposition 5. For each small morphism equation,if MV is unobstructed, so is equation. Thus, if there are no relations in HM, there are none in equation either.

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

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

4 An example of an obstructed determinantal variety in the postulation Hilbert scheme

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 equation and consider the two R-matrices

equation

We let I and J be the ideals generated by the minors of GI and GJ , respectively. Then the graded modules MI = R/I and MJ = R/J belong to the same component in GradAlg. (This is because the irreducible variety of fixed degree polynomials maps to an irreducible subset of GradAlg, contained in the same component.) Thus if the dimension of the tangent space of the two modules differs, the one with the highest dimension necessarily has to be obstructed (meaning that it corresponds to a singular point). Computing with Singular [1], we find that equation. We then know that the first is an example of an obstructed module.

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

equation

Then equation and equation are given by the minimal resolution

equation

with

equation

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

equation

We then notice that equation and so equation.

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

equation

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

equation

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

equation

Letting

equation

we may conclude from Proposition 16 the following.

Proposition 18. The determinantal scheme given by the minors of the matrix GI has first-order relations given by its second-order local formal moduli

equation

From [3] it follows that the obstruction space for M = R/I is H2(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 GI 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; equation, 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 equation, it is always sufficient to have the two leading morphisms equationequation. Also, it is known that finding these by the methods below, they can always be extended to the full complex; see [7].

We find

equation

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

equation

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

equation

We put

equation

and the restricted local formal moduli to the second order is

equation

We follow the algorithm given in Section 3.1 further. We choose a basis B2 for

equation

for example,

equation

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

equation

Similarly,

equation

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

equation

Again, following the algorithm given in Section 3.1, we choose a monomial basis equationequation for equation, 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):

equation

We now put

equation

and so

equation

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

equation

We choose

equation

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

equation

Now, put equation 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 equation with at least one of equation. We see that equation for all equation with equation and so all fifth-order Massey products are zero. Noting also that equation we are ready to conclude the following proposition.

Proposition 20. Let equation. 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 GI such that its intersection with the hyperplane equation is isomorphic to

equation

with the versal family

equation

for equation with

equation

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 equation 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.

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