Elementary matrix factorizations over Bézout domains
We study the homotopy category (and its -graded version ) of elementary factorizations, where is a Bézout domain which has prime elements and , where is a square-free element of and is a finite product of primes with order at least two. In this situation, we give criteria for detecting isomorphisms in and and formulas for the number of isomorphism classes of objects. We also study the full subcategory of the homotopy category of finite rank matrix factorizations of which is additively generated by elementary factorizations. We show that is Krull-Schmidt and we conjecture that it coincides with . Finally, we discuss a few classes of examples.
- 1 Matrix factorizations over an integral domain
- 2 Elementary matrix factorizations over a Bézout domain
- 3 The additive category for a Bézout domain and critically-finite
- 4 Counting elementary factorizations
- 5 Some examples
- A GCD domains
- B Bézout domains
The study of topological Landau-Ginzburg models lg1 (); lg2 (); lg1 (); lg2 () often leads to the problem of understanding the triangulated category of finite rank matrix factorizations of an element , where is a non-Noetherian commutative ring. For example, the category of B-type topological D-branes associated to a holomorphic Landau-Ginzburg pair with a non-compact Riemann surface and a non-constant holomorphic function has this form with , the non-Noetherian ring of holomorphic functions defined on . When is connected, the ring is a Bézout domain (in fact, an elementary divisor domain). In this situation, this problem can be reduced edd () to the study of the full subcategory whose objects are the elementary factorizations, defined as those matrix factorizations of for which the even and odd components of the underlying supermodule have rank one. In this paper, we study the category and the full category of which is additively generated by elementary matrix factorizations, for the case when is a Bézout domain. We say that is critically-finite if it is a product of a square-free element of with an element which can be written as a finite product of primes of multiplicities strictly greater than one. When is critically-finite, the results of this paper provide a detailed description of the categories and , reducing questions about them to the divisibility theory of .
The paper is organized as follows. In Section 1, we recall some basic facts about finite rank matrix factorizations over unital commutative rings and introduce notation and terminology which will be used later on. In Section 2, we study the category and its -graded completion when is any non-zero element of , describing these categories in terms of the lattice of divisors of and giving criteria for deciding when two objects are isomorphic. We also study the behavior of these categories under localization at a multiplicative set as well their subcategories of primary matrix factorizations. In Section 3, we show that the additive category is Krull-Schmidt when is a Bézout domain and is a critically-finite element of and propose a few conjectures about . In Section 4, we give a formula for the number of isomorphism classes in the categories and . Finally, Section 5 discusses a few classes of examples. Appendices A and B collect some information on greatest common denominator (GCD) domains and Bézout domains.
Notations and conventions
The symbols and denote the two elements of the field , where is the zero element. Unless otherwise specified, all rings considered are unital and commutative. Given a cancellative Abelian monoid , we say that an element divides if there exists such that . In this case, is uniquely determined by and and we denote it by or .
Let be a unital commutative ring. The set of non-zero elements of is denoted by , while the multiplicative group of units of is denoted by . The Abelian categories of all -modules is denoted , while the Abelian category of finitely-generated -modules is denoted . Let denote the category of -graded modules and outer (i.e. even) morphisms of such and denote the category of -graded modules and inner morphisms of such. By definition, an -linear category is a category enriched in the monoidal category while a -graded -linear category is a category enriched in the monoidal category . With this definition, a linear category is pre-additive, but it need not admit finite bi-products (direct sums). For any -graded -linear category , the even subcategory is the -linear category obtained from by keeping only the even morphisms.
For any unital integral domain , let denote the equivalence relation defined on by association in divisibility:
The set of equivalence classes of this relation coincides with the set of orbits for the obvious multiplicative action of . Since is a commutative domain, the quotient inherits a multiplicative structure of cancellative Abelian monoid. For any , let denote the equivalence class of under . Then for any , we have . The monoid can also be described as follows. Let be the set of non-zero principal ideals of . If are elements of , we have , so the product of principal ideals corresponds to the product of the multiplicative group and makes into a cancellative Abelian monoid with unit . Notice that coincides with the positive cone of the group of divisibility (see Subsection 5.2) of , when the latter is viewed as an Abelian group ordered by reverse inclusion. The monoids and can be identified as follows. For any , let denote the principal ideal generated by . Then depends only on and will also be denoted by . This gives a group morphism . For any non-zero principal ideal , the set of all generators of is a class in which we denote by ; this gives a group morphism . For all , we have and , which implies that and are mutually inverse group isomorphisms.
If is a GCD domain (see Appendix A) and are elements of such that , let be any greatest common divisor (gcd) of . Then is determined by up to association in divisibility and we denote its equivalence class by . The principal ideal does not depend on the choice of . The elements also have a least common multiple (lcm) , which is determined up to association in divisibility and whose equivalence class we denote by . For , we have:
If is a Bézout domain (see Appendix B), then we have , so the gcd operation transfers the operation given by taking the finite sum of principal ideals from to through the isomorphism of groups described above. In this case, we have . We also have and hence . Thus the lcm corresponds to the finite intersection of principal ideals.
1 Matrix factorizations over an integral domain
Let be an integral domain and be a non-zero element of .
1.1 Categories of matrix factorizations
We shall use the following notations:
denotes the -linear and -graded differential category of -valued matrix factorizations of of finite rank. The objects of this category are pairs , where is a free -graded -module of finite rank and is an odd endomorphism of such that . For any objects and of , the -graded -module of morphisms from to is given by the inner :
endowed with the differential determined uniquely by the condition:
denotes the -linear and -graded cocycle category of . This has the same objects as but morphism spaces given by:
denotes the -linear and -graded coboundary category of , which is an ideal in . This has the same objects as but morphism spaces given by:
denotes the -linear and -graded total cohomology category of . This has the same objects as but morphism spaces given by:
The subcategories of , , and obtained by restricting to morphisms of even degree are denoted respectively by , , and .
The categories , and admit double direct sums (and hence all finite direct sums of at least two elements) but do not have zero objects. On the other hand, the category is additive, the matrix factorization being a zero object. Finally, it is well-known that the category is triangulated (see Langfeldt () for a detailed treatment).
For later reference, recall that the biproduct (direct sum) of is defined as follows:
Given two matrix factorizations , of , their direct sum is the matrix factorization of , where and , with:
Given a third matrix factorization of and two morphisms in , their direct sum of is the ordinary direct sum of the -module morphisms and .
As a consequence, admits all finite but non-empty direct sums. The following result is elementary:
The following statements hold:
The subcategories and of are closed under finite direct sums (but need not have zero objects).
The direct sum induces a well-defined biproduct (which is again denoted by ) on the -linear categories and .
and are additive categories, a zero object in each being given by any of the elementary factorizations and , which are isomorphic to each other in . In particular, any finite direct sum of the elementary factorizations and is a zero object in and in .
1.2 Reduced rank and matrix description
Let be an object of , where . Taking the supertrace in the equation and using the fact that shows that . We call this natural number the reduced rank of and denote it by ; we have . Choosing a homogeneous basis of (i.e. a basis of and a basis of ) gives an isomorphism of -supermodules , where and denotes the -supermodule with -homogeneous components . This isomorphism allows us to identify with a square matrix of size which has block off-diagonal form:
where and are square matrices of size with entries in . The condition amounts to the relations:
where denotes the identity matrix of size . Since , these conditions imply that the matrices and have maximal rank111To see this, it suffices to consider equations (1) in the field of fractions of .:
Matrix factorizations for which form a dg subcategory of which is essential in the sense that it is dg-equivalent with . Below, we often tacitly identify with this essential subcategory and use similar identifications for , and .
Given two matrix factorizations and of , write , with . Then:
An even morphism has the matrix form:
with and we have:
An odd morphism has the matrix form:
with and we have:
The cocycle condition satisfied by an even morphism amounts to the system:
which in turn amounts to any of the following equivalent conditions:
Similarly, the cocycle condition defining an odd morphism amounts to the system:
which in turn amounts to any of the following equivalent conditions:
1.3 Strong isomorphism
Recall that denotes the even subcategory of . This category admits non-empty finite direct sums but does not have a zero object.
Two matrix factorizations and of over are called strongly isomorphic if they are isomorphic in the category .
It is clear that two strongly isomorphic factorizations are also isomorphic in , but the converse need not hold.
Let and be two matrix factorizations of over , where . Then the following statements are equivalent:
and are strongly isomorphic.
(a quantity which we denote by ) and there exist invertible matrices such that one (and hence both) of the following equivalent conditions is satisfied:
and are strongly isomorphic iff there exists which is an isomorphism in . Since is an even morphism in the cocycle category, we have:
The condition that be even allows us to identify it with a matrix of the form , while invertibility of in amounts to invertibility of the matrix , which in turn means that and are square matrices (thus ) belonging to . Thus relation (2) reduces to either of conditions 1. or 2., which are equivalent since and . ∎
1.4 Critical divisors and the critical locus of
A divisor of which is not a unit is called critical if .
be the set of all critical divisors of . The ideal:
is called the critical ideal of . Notice that consists of those elements of which are divisible by all critical divisors of . In particular, we have and hence there exists a unital ring epimorphism .
A critical prime divisor of is a prime element such that . The critical locus of is the subset of consisting of the principal prime ideals of generated by the critical prime divisors of :
1.5 Critically-finite elements
Let be a Bézout domain. Then is a GCD domain, hence irreducible elements of are prime. This implies that any factorizable element222I.e. an element of which has a finite factorization into irreducibles. of has a unique prime factorization up to association in divisibility.
A non-zero non-unit of is called:
non-critical, if has no critical divisors;
critically-finite if it has a factorization of the form:
where , are critical prime divisors of (with for ) and is non-critical and coprime with .
Notice that the elements , and in the factorization (4) are determined by up to association, while the integers are uniquely determined by . The factors and are called respectively the non-critical and critical parts of . The integers are called the orders of the critical prime divisors .
For a critically-finite element with decomposition (4), we have:
where333The notation indicates the integral part of a real number .:
is called the reduction of . Notice that is determined up to association in divisibility.
1.6 Two-step factorizations of
Recall that a two-step factorization (or two-step multiplicative partition) of is an ordered pair such that . In this case, the divisors and are called -conjugate. The transpose of is the ordered pair (which is again a two-step factorization of ), while the opposite transpose is the ordered pair . This defines an involution of the set of two-step factorizations of . The two-step factorizations and are called similar (and we write ) if there exists such that and . We have .
The support of a two-step factorization of is the principal ideal .
Let be a gcd of and . Since (where , ), it is clear that is a critical divisor of . Notice that the opposite transpose of the two step factorization has the same support as .
1.7 Elementary matrix factorizations
A matrix factorization of over is called elementary if it has unit reduced rank, i.e. if .
Any elementary factorization is strongly isomorphic to one of the form , where is a divisor of and , with . Let denote the full subcategory of whose objects are the elementary factorizations of over . Let and denote respectively the cocycle and total cohomology categories of . We also use the notations and . Notice that an elementary factorization is indecomposable in , but it need not be indecomposable in the triangulated category .
The map which sends to the ordered pair is a bijection. The suspension of is given by , since:
In particular, corresponds to the opposite transpose and we have:
Hence preserves the subcategory of and the subcategories and of and . This implies that is equivalent with the graded completion . We thus have natural isomorphisms:
for any divisors of , where and .
The support of an elementary matrix factorization is the ideal of defined through:
Notice that this ideal is generated by any gcd of and and that is a critical divisor of .
We will see later that an elementary factorization is trivial iff its support equals .
Two elementary matrix factorizations and of are called similar if or equivalently . This amounts to existence of a unit such that and .
Two elementary factorizations and are strongly isomorphic iff they are similar. In particular, strong isomorphism classes of elementary factorization are in bijection with the set of those principal ideals of which contain .
Suppose that and are strongly isomorphic. By Proposition 1.4, there exist units such that and , where . Setting gives and , hence and are similar. Conversely, suppose that . Then there exists a unit such that and . Setting and gives and , which shows that and are strongly isomorphic upon using Proposition 1.4. The map which sends the strong isomorphism class of to the principal ideal gives the bijection stated. ∎
It is clear that and are similar iff the corresponding two-step factorizations and of are similar. Since any strong isomorphism induces an isomorphism in , it follows that similar elementary factorizations are isomorphic in .
1.8 The categories and
Let denote the smallest full -linear subcategory of which contains all objects of and is closed under finite direct sums. It is clear that is a full dg subcategory of . Let denote the total cohomology category of . Let denote the subcategory obtained from by keeping only the even morphisms. Notice that coincides with the smallest full subcategory of which contains all elementary factorizations of .
2 Elementary matrix factorizations over a Bézout domain
Throughout this section, let be a Bézout domain and be a non-zero element of .
2.1 The subcategory of elementary factorizations
Let be divisors of and , be the corresponding elementary matrix factorizations of . Let , . Let be a gcd of and . Define: