Branched covers and matrix factorizations

Let (S,n)$(S,\mathfrak {n})$ be a regular local ring and f$f$ a non‐zero element of n2$\mathfrak {n}^2$ . A theorem due to Knörrer states that there are finitely many isomorphism classes of maximal Cohen–Macaulay (CM) R=S/(f)$R=S/(f)$ ‐modules if and only if the same is true for the double branched cover of R$R$ , that is, the hypersurface ring which is defined by f+z2$f+z^2$ in S⟦z⟧$S\llbracket z \rrbracket$ . We consider an analogue of this statement in the case of the hypersurface ring defined instead by f+zd$f+z^d$ for d⩾2$d\geqslant 2$ . In particular, we show that this hypersurface, which we refer to as the d$d$ ‐fold branched cover of R$R$ , has finite CM representation type if and only if, up to isomorphism, there are only finitely many indecomposable matrix factorizations of f$f$ with d$d$ factors. As a result, we give a complete list of polynomials f$f$ with this property in characteristic zero. Furthermore, we show that reduced d$d$ ‐fold matrix factorizations of f$f$ correspond to Ulrich modules over the d$d$ ‐fold branched cover of R$R$ .


Introduction
Let (S, n, k) be a complete regular local ring and let d ≥ 2 be an integer.Assume that k is algebraically closed and that the characteristic of k does not divide d.Fix a non-zero element f ∈ n 2 and let R = S/(f ) be the hypersurface ring defined by f .A finitely generated module M over a local ring A is called maximal Cohen-Macaulay (MCM) if depth A (M ) = dim(A), the Krull dimension of A. We will consider MCM modules over R and over the d-fold branched cover of R, that is, the hypersurface ring R ♯ = S z /(f + z d ).The main objective of this paper is to further understand the connection, established for d = 2 by Knörrer [Knö87] and extended for d > 2 in [Tri21], between MCM R ♯ -modules and matrix factorizations of f with d factors which we define below.
Definition 1.1.A matrix factorization of f with d factors is a d-tuple of homomorphisms between finitely generated free S-modules of the same rank such that ϕ 1 ϕ 2 • • • ϕ d = f • 1 F 1 .We denote the category of matrix factorizations of f with d factors by MF d S (f ).If rank S (F i ) = n for all i, then we say X is of size n.
A local ring A is said to have finite Cohen-Macaulay (CM) type if there are only finitely many isomorphism classes of indecomposable objects in the category MCM(A) of MCM A-modules.We adopt the following analogous terminology for the representation type of MF d S (f ).
Definition 1.2.We say that f has finite d-MF type if the category MF d S (f ) has, up to isomorphism, only finitely many indecomposable objects.
In [Knö87], Knörrer proved that R = S/(f ) has finite CM type if and only if R ♯ = S z /(f + z 2 ) has finite CM type.The correspondence, given by Eisenbud [Eis80,Corollary 6.3], between matrix factorizations and MCM R-modules implies that the number of isomorphism classes of indecomposable objects in MCM(R) and MF 2 S (f ) differ by only one.Since R ♯ is also a hypersurface ring, the same is true for MCM(R ♯ ) and MF 2 S z (f + z 2 ).With this in mind, we can state the following version of Knörrer's theorem.
In Section 2 we investigate which of the analogous implications for d-fold factorizations hold when d ≥ 2. Our main result in this direction is the following.As a consequence of Theorem A, we give a complete list of polynomials which have only finitely many indecomposable d-fold matrix factorizations up to isomorphism.
Theorem B. Let k be an algebraically closed field of characteristic zero and S = k y, x 2 , . . ., x r .Assume 0 = f ∈ (y, x 2 , . . ., x r ) 2 and d > 2. Then f has finite d-MF type if and only if, after a possible change of variables, f and d are one of the following: The main tool in Section 2 is a pair of functors (−) ♭ : MCM(R ♯ ) → MF d S (f ) and (−) ♯ : MF d S (f ) → MCM(R ♯ ).The functors we define play a similar role as the ones in [LW12, Chapter 8, §2] given by the same notation.Following the results in [LW12, Chapter 8, §3] and [Knö87, Proposition 2.7], we investigate the decomposability of (−) ♭ and (−) ♯ in Section 4.
A matrix factorization X = (ϕ 1 , . . ., ϕ d ) ∈ MF d S (f ) is called reduced if the map ϕ k is minimal for each k ∈ Z d .Equivalently, X is reduced if, after choosing bases, all entries of ϕ k lie in the maximal ideal of S for all k ∈ Z d .In the original setting given by Eisenbud (d = 2), matrix factorizations which are reduced are the only interesting ones.In particular, the only indecomposable non-reduced matrix factorizations with 2 factors are (1, f ) and (f, 1).For d > 2, there are often non-reduced dfold matrix factorizations of f which are also indecomposable and therefore play a role in Theorem A and Theorem B.
In Section 5, we focus only on reduced matrix factorizations.We show that, as long as d is not too large, reduced matrix factorizations of f with d factors correspond to Ulrich modules over R ♯ , that is, modules which are MCM and maximally generated in the sense of [BHU87].We let ord(f ) denote the maximal integer e such that f ∈ n e .
Theorem C. Assume that d ≤ ord(f ) and that f + z d is irreducible.Then f has finite reduced d-MF type if and only if there are, up to isomorphism, only finitely many indecomposable Ulrich R ♯ -modules.
Notation.All indices are taken modulo d unless otherwise specified.

Finite matrix factorization type
The key lemma in Knörrer's proof of Theorem 1.3 states that for MCM modules M over R and N over the double branched cover R # = S z /(f + z 2 ), we have isomorphisms . In this section we extend Knörrer's lemma to arbitrary d-fold branched covers and use it to prove Theorem A.
As in the Introduction, we fix d ≥ 2 and let (S, n, k) be a complete regular local ring with k algebraically closed of characteristic not dividing d.We also fix a non-zero element f ∈ n 2 and let R = S/(f ).We recall a few facts about matrix factorizations of f with two or more factors.
decomposes into a finite direct sum of indecomposable objects such that the decomposition is unique up to isomorphism and permutation of the summands.
The d-fold branched cover of R is the hypersurface ring R ♯ = S z /(f +z d ).As an S-module, R ♯ is finitely generated and free with basis given by {1, z, z 2 , . . ., z d−1 }.Consequently, a finitely generated R ♯ -module N is MCM over R ♯ if and only if it is free over S [Yos90, Proposition 1.9].Furthermore, multiplication by z on N defines an S-linear map ϕ : N → N which satisfies ϕ d = −f • 1 N .Conversely, given a free S-module F and a homomorphism ϕ : F → F satisfying ϕ d = −f • 1 F , the pair (F, ϕ) defines an MCM R ♯ -module whose z-action is given by the map ϕ.We will use these perspectives interchangeably throughout.
Since S is complete and char k ∤ d, [LW12, A.31] implies that the polynomials x d + 1 and x d − 1 in S[x] each have d distinct roots in S. We let ω ∈ S be a primitive d th root of 1 in the sense that ω d = 1 and ω s = 1 for any 1 ≤ s < d.Let µ ∈ S be any d th root of −1.We start with a pair of functors between the categories MCM(R ♯ ) and MF d S (f ).Definition 2.2.
(i) Let N ∈ MCM(R ♯ ) and set ϕ : N → N to be the S-linear homomorphism representing multiplication by z on N .Since (µϕ as an S-module with z-action given by: z Remark 2.3.The role of µ in the definition of N ♭ is to obtain a d-fold factorization of f (instead of −f ) and to do so in a symmetric way.It is important to note that the isomorphism class of N ♭ ∈ MF d S (f ) is independent of choice of µ.To see this, observe that if µ ′ is another root of x d + 1, then µ ′ = ω j µ for some j ∈ Z d .Using this fact, one can construct an isomorphism (µϕ N , µϕ N , . . ., µϕ N ) ∼ = (µ ′ ϕ N , µ ′ ϕ N , . . ., µ ′ ϕ N ) in MF d S (f ).Similarly, µ −1 in the definition of X ♯ ensures that we obtain a module over R ♯ .The isomorphism class of X ♯ in MCM(R ♯ ) is also independent of the choice of µ.
Consider the automorphism σ : R ♯ → R ♯ which fixes S pointwise and maps z to ωz.This automorphism acts on the category of MCM R ♯ -modules in the following sense: For each N ∈ MCM(R ♯ ), let (σ k ) * N denote the MCM R ♯ -module obtained by restricting scalars along σ k .Since σ d = 1 R ♯ , the mapping N → σ * N forms an autoequivalence of the category MCM(R ♯ ).
Notice that the first half of the proof is valid in any characteristic as long as there exists an element µ ∈ S satisfying µ d = −1.For instance, if d is odd, then µ = −1 is a valid choice.However, the second half of the proof explicitly makes use of the fact that char k does not divide d.
In order to show the second isomorphism, let N ∈ MCM(R ♯ ) and set ϕ : N → N to be the S-linear map representing multiplication by z on N .Then N ♭ = (µϕ : N → N, . . ., µϕ : N → N ) ∈ MF d S (f ).Thus, as an S-module, N ♭♯ = N ⊕ N ⊕ • • • ⊕ N, the direct sum of d copies of the free S-module N .The z-action on N ♭♯ is given by z for any n ∈ N ♭♯ .Note that for m ∈ (σ k ) * N , z • m = ω k zm by definition.Therefore, for n = (n d , . . ., n 1 ) ∈ N ♭♯ , we have that In other words, g k is an R ♯ -homomorphism.Putting these maps together we have an R ♯ -homomorphism In the other direction, we have R ♯ -homomorphisms Therefore, setting s = s 0 s 1 • • • s d−1 , we have that gs is the identity on d−1 k=0 (σ k ) * N and so g is a split surjection.However, since both the target and source of g have the same rank as free S-modules, we can conclude that g is an isomorphism of R ♯ -modules.
(i) For each X ∈ MF d S (f ), there exists We can now prove Theorem A. The proof is lifted directly from the d = 2 case.Once again, the characteristic assumption on char k is only needed in half of the proof as long as there exists µ ∈ S satisfying µ d = −1.
Proof of Theorem A. Let X 1 , X 2 , . . ., X t be a representative list of the isomorphism classes of indecomposable d-fold matrix factorizations of f and let N ∈ MCM(R ♯ ) be indecomposable.Since for some 1 ≤ i ≤ t.Hence, every indecomposable MCM R ♯ -module is isomorphic to one appearing in the finite list consisting of all summands of all X ♯ j , 1 ≤ j ≤ t.The converse follows similarly from Proposition 2.4 and KRS in MF d S (f ).
As above, σ : R ♯ → R ♯ is the automorphism of R ♯ which pointwise fixes S and and maps z to ωz.In [Tri21, Section 4], it was shown the the category MF d S (f ) is equivalent to the category of finitely generated modules over the skew group algebra R ♯ [σ] which are MCM as R ♯ -modules (equivalently, free as S-modules).We denote this category by MCM σ (R ♯ ).The equivalence is given by a pair of inverse functors and therefore multiplication by z defines an S-linear map defines a matrix factorization which we denote as A(N ) ∈ MF d S (f ).To finish this section, we make note of the connection between the functors A and B and the functors (−) ♯ and (−) ♭ .
Lemma 2.6.Let H : MCM σ (R ♯ ) → MCM(R ♯ ) be the functor which forgets the action of σ and Proof.The first statement follows directly from the definition of (−) ♯ and B. For the second, consider the idempotents These idempotents have three important properties:

Hypersurfaces of finite matrix factorization type
Let (A, m) be a regular local ring and g ∈ m 2 be non-zero.Then the hypersurface ring A/(g) is called a simple hypersurface singularity if there are only finitely many proper ideals I ⊂ A such that g ∈ I 2 .In the case that A is a power series ring over an algebraically closed field of characteristic zero, the pair of papers [BGS87] and [Knö87] prove the following theorem.
Essential to their conclusion is the classification of simple hypersurface singularities, due to Arnol'd [Arn73], which gives explicit normal forms for all polynomials defining such a singularity.These are often referred to as the ADE singularities.The culmination of these results is a complete list of polynomials which define hypersurface rings of finite CM type in all dimensions (see [Yos90,Theorem 8.8] or [LW12, Theorem 9.8]).Equivalently, the polynomials in this list are precisely the ones with only finitely many indecomposable 2-fold matrix factorizations up to isomorphism.
Using Theorem A and the classification described above, we are able to compile a list of all f with finite d-MF type for d > 2.
Theorem 3.2.Let k be an algebraically closed field of characteristic zero and S = k y, x 2 , . . ., x r .Assume 0 = f ∈ (y, x 2 , . . ., x r ) 2 and d > 2. Then f has finite d-MF type if and only if, after a possible change of variables, f and d are one of the following: Proof.Let f and d be a pair in the list given.Then f + z d ∈ S z defines a simple hypersurface singularity and therefore R ♯ has finite CM type by Theorem 3.1.By Theorem A, f has finite d-MF type.
Conversely, let 0 = f ∈ (y, x 2 , . . ., x r ) 2 , d > 2, and assume f has finite d-MF type.Then R ♯ = S z /(f + z d ) has dimension r and is of finite CM type by Theorem A. We consider two cases.
First, assume dim R ♯ = 1, that is, assume S = k y .Then f = uy k for some unit u ∈ S and k ≥ 2. Since S is complete and char k = 0, there exists a kth root v of u −1 in S [LW12, A.31].Therefore, after replacing y with vy, we may assume that f = y k .Since dim R ♯ = 1, [Yos90, 8.2.1] implies that ord(y k + z d ) ≤ 3. Hence, either k ≤ 3 or d ≤ 3.If k = 2, there are no restrictions on d since y 2 + z d defines a simple (A d−1 ) singularity for all d > 2. If k = 3, then the fact that y 3 + z d defines a 1-dimensional simple hypersurface singularity implies that d = 3, 4, or 5. Similarly, if d = 3, then k = 2, 3, 4, or 5.
Next, assume dim R ♯ ≥ 2. In this case, [Yos90, 8.2.2] implies that ord(f + z d ) ≤ 2. Since d > 2 and f ∈ (y, x 2 , . . ., x r ) 2 , we have that ord(f ) = 2.By the Weierstrass Preparation Theorem [LW12, Corollary 9.6], there exists a unit u ∈ S and g ∈ k y, x 2 , . . ., x r−1 such that f = (g + x 2 r )u.As above, we may neglect the unit and assume that f = g + x 2 r for some g ∈ k y, x 2 , . . ., x r−1 .Since f + z d = g + x 2 r + z d has finite 2-MF type, Knörrer's theorem (Theorem 1.3) implies that g + z d has finite 2-MF type as well.Thus, g has finite d-MF type by Theorem A. We repeat this argument until f = g ′ + x 2 2 + • • • + x 2 r for some g ′ ∈ k y with finite d-MF type.Finally, we apply the first case to g ′ to finish the proof.
Corollary 3.3.Let k be an algebraically closed field of characteristic zero, S = k y, x 2 , . . ., x r , and f ∈ (y, x 2 , . . ., x r ) 2 be non-zero.If f has finite d-MF type for some d ≥ 2, then R = S/(f ) is an isolated singularity, that is, R p is a regular local ring for all non-maximal prime ideals p.
Proof.The polynomials listed in Theorem 3.2 are a subset of the ones in [Yos90, Theorem 8.8] (or [LW12, Theorem 9.8]), all of which define isolated singularities.
Suppose we have a pair f and d from the list in Theorem 3.2 such that R ♯ has dimension 1.Then [Yos90, Chapter 9] gives matrix factorizations for every indecomposable MCM R ♯ -module.By computing multiplication by z on each of these R ♯ -modules, we can compile a representative list of all isomorphism classes of indecomposable d-fold factorizations of f .We give one such computation in the following example.
Example 3.4.Let k be algebraically closed of characteristic zero.Let S = k y , f = y 4 ∈ S, and R = S/(f ).The hypersurface ring R ♯ = k x, y /(y 4 + x 3 ) is a simple curve singularity of type E 6 and has finite CM type.Here we are viewing R ♯ as the 3-fold branched cover of R. By Theorem A, the category MF 3 S (y 4 ) has only finitely many non-isomorphic indecomposable objects.We give a complete list below.
A complete list of non-isomorphic indecomposable MCM R ♯ -modules is given in [Yos90, 9.13].By Corollary 2.5, we may compute multiplication by x on each of these modules to obtain a representative from each isomorphism class of indecomposable matrix factorizations of y 4 with 3 factors.By Remark 2.3, we may choose µ = −1.
Following the notation of [Yos90, 9.13], we let ϕ 1 = x y y 3 −x 2 and M 1 = cok ϕ 1 .Let e 1 and e 2 in M 1 denote the images of the standard basis on S x 2 .Then e 1 and e 2 satisfy xe 1 = −y 3 e 2 and x 2 e 2 = ye 1 .As an S-module, M 1 is free with basis {e 1 , e 2 , xe 2 }.Multiplication by x on M 1 is therefore given by Hence, M ♭ 1 = (−ϕ, −ϕ, −ϕ) ∈ MF 3 S (y 4 ).Furthermore, we have a commutative diagram Thus, M ♭ 1 is isomorphic to the direct sum of the indecomposable factorization X ϕ 1 := (y 3 , y, 1) and its corresponding shifts, that is, Similarly, multiplication by x can be computed for each of the indecomposable MCM R ♯ modules listed in [Yos90,9.13].From this computation, we obtain a list of 3-fold matrix factorizations of y 4 given in the table below.
The factorizations P 1 , X ϕ 1 , X ψ 1 , X ϕ 2 , and X β are each indecomposable since they are of size one.By [Tri21,Corollary 5.15], the syzygy (and cosyzygy) of an indecomposable reduced matrix factorization is again indecomposable.Here a reduced matrix factorization means all the entries of all the matrices lie in the maximal ideal of S (see Section 5).Using [Tri21, 2.11], we have that .By considering cokernels, both cases lead to contradictions and so X ξ must be indecomposable.
By Proposition 2.4, these seven factorizations, and each of their corresponding shifts, give the complete list of non-isomorphic indecomposable objects in MF 3 S (y 4 ) (21 in total).To end this section, we discuss the relationship between the 2-MF type of f and the d-MF type of f for d > 2. In one direction, we have the following consequence of Theorem B. In general, the converse of Lemma 3.5 does not hold.The example below gives a polynomial of finite 2-MF type but of infinite 3-MF type.
Example 3.6.Let S = k x, y for an algebraically closed field with characteristic char k = 2, 3 and let f = x 3 + y 3 ∈ S. The hypersurface ring R = S/(f ) is a simple singularity of type D 4 and therefore has finite CM type.
Consider R ♯ = k x, y, z /(x 3 + y 3 + z 3 ), the 3-fold branched cover of R. Following [BP15], to each point (a, b, c) ∈ k 3 satisfying a 3 + b 3 + c 3 = 0 and abc = 0, we associate the Moore matrix The R ♯ -module N abc := cok(X abc ) is MCM and is given by the matrix factorization (X abc , 1 abc adjX abc ) ∈ MF 2 S z (x 3 + y 3 + z 3 ), where adjX abc is the classical adjoint of X abc .Furthermore, N abc is indecomposable since det X abc = abc(x 3 + y 3 + z 3 ) and x 3 + y 3 + z 3 is irreducible.Buchweitz and Pavlov give precise conditions for X abc to be matrix equivalent to X a ′ b ′ c ′ (see [BP15, Proposition 2.13]).In particular, their results imply that the collection {N abc }, as (a, b, c) varies over the curve x 3 + y 3 + z 3 , gives an infinite collection of non-isomorphic indecomposable MCM R ♯ -modules.It now follows from Theorem A that x 3 + y 3 has infinite 3-MF type.
With respect to the images of the standard basis on S z 3 , multiplication by z on N abc is given by the S-matrix Therefore, we have that N ♭ abc = (µϕ abc , µϕ abc , µϕ abc ) ∈ MF 3 S (x 3 + y 3 ), where µ 3 = −1.Hence, the collection of non-isomorphic indecomposable summands of N ♭ abc , for all (a, b, c) as above, forms an infinite collection of indecomposable objects in MF d S (f ).Furthermore, the entries of ϕ abc lie in the maximal ideal of S so x 3 + y 3 has infinite reduced 3-MF type as well (see Section 5).

Decomposability of N ♭ and X ♯
Let d ≥ 2 and (S, n, k) be a complete regular local ring.We maintain the same assumptions on k as in Section 2, that is, we assume that k is algebraically closed of characteristic not dividing d.Let f ∈ n2 be non-zero, R = S/(f ), and R ♯ = S z /(f + z d ).
Proposition 2.4 showed that both N ♭♯ and X ♯♭ decompose into a sum of d objects.In this section we investigate the decomposability of N ♭ and X ♯ .
Recall that the shift functor In particular, for any X ∈ MF d S (f ), there exists a smallest integer k ∈ {1, 2, . . ., d − 1, d} such that T k X ∼ = X.We call k the order of X.
Lemma 4.1.For any X ∈ MF d S (f ), the order of X is a divisor of d.Proof.For a given X ∈ MF d S (f ), the cyclic group of order d generated by T acts on the set of equivalence classes {[T i X] : i ∈ Z d }.In particular, the stabilizer of [X] is generated by T k for some k | d which can be taken to be the smallest possible in {1, 2, . . ., d}.It follows that the order of X is k.
The next result builds on an idea of Knörrer [Knö87, Lemma 1.3] and Gabriel [Gab81,p. 95].The proof is based on [LW12, Lemma 8.25] which states that a matrix factorization (ϕ, ψ) ∈ MF 2 S (f ) satisfying (ϕ, ψ) ∼ = (ψ, ϕ) is isomorphic to a factorization of the form (ϕ 0 , ϕ 0 ) 2 .For d > 2, the situation is similar, but the divisors of d play a role.Specifically, if X has order k, then X is isomorphic to the concatenation of k matrices, repeated d/k times.Proposition 4.2.Let X ∈ MF d S (f ) be indecomposable of size n and assume X has order k < d.Then there exist S-homomorphisms ϕ By assumption, there is an isomorphism α = (α 1 , . . ., α d ) : X → T k X.By applying T k (−) repeatedly, we obtain an automorphism α of X defined by the composition In particular, α is the d-tuple (α Since k is algebraically closed, it cannot have any non-trivial finite extensions which are division rings.Hence the division ring Λ/ rad Λ must be isomorphic to k.This allows us to write α = c • 1 X + ρ for some c ∈ k × and ρ ∈ rad Λ.Since char k | d, we may scale α by c − 1 r and assume α = 1 Represent the function g(x) = (1 + x) −1/r by its Maclaurin series and define, for each i ∈ Z d , By repeatedly applying α i g(ρ i ) = g(ρ i+k )α i , we have that Hence, β i is an isomorphism for each i ∈ Z d and therefore the morphism β is an isomorphism of matrix factorizations.
Corollary 4.3.Let X ∈ MF d S (f ) be indecomposable of size n and assume that X ∼ = T X.Then there exists a homomorphism ϕ : , and assume both X and N are indecomposable objects. (i Proof.If X ∼ = T X, then Corollary 4.3 implies that there exists a free S-module F and an endomorphism ϕ : F → F such that ϕ d = f • 1 F and X ∼ = (ϕ, ϕ, . . ., ϕ) ∈ MF d S (f ).The pair (F, µ −1 ϕ) defines an MCM(R ♯ ) module M as follows: As an S-module, M = F , and the z-action on M is given by z • m = µ −1 ϕ(m) for all m ∈ M , where µ ∈ S satisfies M is naturally an R ♯ -module.Since M = F is free over S, it is MCM over R ♯ .By applying (−) ♭ , we have that M ♭ = (ϕ, ϕ, . . ., ϕ) ∼ = X.
Assume N ∼ = σ * N .Using a similar technique to the proof of Proposition 4.2, we obtain an isomorphism of R ♯ -modules θ : This contradicts KRS since the left side has precisely d indecomposable summands while the right hand side has at least 2d indecomposable summands.Hence, M is indecomposable.• Suppose that σ * M ∼ = M .Then, since M is indecomposable, the arguments above imply that M ♭ decomposes into a sum of at least d indecomposable summands.Since T (M ♭ ) ∼ = M ♭ , we have Since X is indecomposable, the left hand side has precisely d indecomposable summands while the right hand side has at least d 2 indecomposable summands.Once again, we have a contradiction and so M ∼ = σ * M .This completes the proof of (i).We omit the remaining assertions from (iii) as they follow similarly.
In order to prove (ii), suppose The left hand side has precisely d indecomposable summands and therefore t ≤ d.
If X ♯ decomposes into exactly d indecomposables, that is, if t = d, then (4.1) implies that M ♭ i is indecomposable for each i and that X ∼ = M ♭ j for some 1 The proof of (iv) is similar, observing that σ * (X ♯ ) ∼ = X ♯ for any X ∈ MF d S (f ).

Reduced matrix factorizations and Ulrich modules
Let (S, n, k) be a complete regular local ring, 0 = f ∈ n 2 , and let d ≥ 2 be an integer.Assume k is algebraically closed of characteristic not dividing d.In this section, we will consider the following special class of matrix factorizations in MF d S (f ).
Equivalently, after choosing bases, X is reduced if and only if the entries of ϕ k lie in n for all k ∈ Z d .We say that f has finite reduced d-MF type if there are, up to isomorphism, only finitely many indecomposable reduced matrix factorizations X ∈ MF d S (f ).In the case d = 2, any indecomposable non-reduced matrix factorization is isomorphic to either (1, f ) or (f, 1) in MF 2 S (f ) [Yos90, Remark 7.5].In particular, this implies that finite 2-MF type is equivalent to finite reduced 2-MF type.
For d > 2, the situation is quite different.There at least as many non-reduced indecomposable d-fold factorizations of f as there are reduced ones [Tri21,  In the case d = 2, Theorem 3.1 implies that reduced 2-MF type is determined by the cardinality of the set c 2 (f ).One implication of Theorem 3.1 is proven explicitly in [BGS87].The authors show that the association X → I(X) forms a surjection from the set of isomorphism classes of reduced 2-fold matrix factorizations of f onto the set c 2 (f ).Hence, if there are only finitely many indecomposable reduced 2-fold matrix factorizations of f up to isomorphism, then the set c 2 (f ) is finite.
The following result of Herzog, Ulrich, and Backelin shows that the association X → I(X) remains surjective in the case d > 2.
Since X k is reduced and the MCM R-module cok x y k 0 −x is indecomposable, [Tri21,Lemma 6.6] implies that X k is indecomposable.Thus, x 2 y has infinite reduced 3-MF type.On the other hand, it's not hard to see that c 3 (x 2 y) contains only the maximal ideal.So c 3 (x 2 y) is a finite set but x 2 y has infinite reduced 3-MF type.
Let N be an MCM R ♯ -module and let µ R ♯ (N ) denote the minimal number of generators of N .Recall that N is finitely generated and free over S. We will see below that there is an inequality (5.1) µ R ♯ (N ) ≤ rank S (N ).In the following, we consider MCM R ♯ -modules N where the equality µ R ♯ (N ) = rank S (N ) is attained.
As we saw in Example 3.4, a matrix factorization of the form N ♭ , obtained by computing multiplication by z on an MCM R ♯ -module N , can be non-reduced.We will show below that the matrix representing multiplication by z on N contains unit entries precisely when µ R ♯ (N ) < rank S (N ).In other words, the restriction of the functor (−) ♭ : MCM(R ♯ ) → MF d S (f ) to the subcategory of MCM R ♯ -modules satisfying µ R ♯ (N ) = rank S (N ) produces only reduced matrix factorizations of f with d factors.Conversely, the image of the functor (−) ♯ : MF d S (f ) → MCM(R ♯ ), restricted to the subcategory of reduced matrix factorizations of f , consists exactly of the MCM R ♯ -modules N satisfying µ R ♯ (N ) = rank S (N ).
Lemma 5.6.Let N be an MCM R ♯ -module and assume that f + z d is irreducible.Then N is a finitely generated free S-module satisfying Proof.Let (Φ : S z n → S z n , Ψ : S z n → S z n ) ∈ MF 2 S z (f + z d ) be a matrix factorization of f + z d such that Φ is minimal and cok Φ = N .Since Φ is minimal, n = µ R ♯ (N ).Then det Φ = u(f + z d ) k for some 1 ≤ k ≤ n and some unit u ∈ S z .Recall that k = rank R ♯ (N ) by [Eis80,Propoistion 5.6].By tensoring with S = S z /(z), we find that det Φ = v • f k , where Φ = Φ ⊗ S z 1 S and v ∈ S is a unit.Moreover, Φ is injective since Φ Ψ = f • 1 S n = Ψ Φ, and we have a minimal presentation of N/zN over S: Φ On the other hand, since N is MCM over R ♯ , it is finitely generated and free as an S-module.Let r = rank S (N ) and consider the map ϕ : S r → S r representing multiplication by z on N .This map also gives a presentation of N/zN over S, though the presentation may not be minimal (see Example 3.4).Thus, there exists a commutative diagram with vertical isomorphisms This implies that µ R (N/zN ) ≤ r, where R = S/(f ) as usual.The desired inequality now follows from the fact that µ Proof.Let N ∈ MCM(R ♯ ) and set r = rank S (N ).Let ϕ : S r → S r be the S-linear map representing multiplication by z on N .Then the presentation of N/zN given by ϕ in (5.2) is minimal if and only if r = rank S (N ) = µ R (N/zN ), where R = S/(f ).Since µ R (N/zN ) = µ R ♯ (N ), we have that ϕ is minimal if and only if rank S (N ) = µ R ♯ (N ).This proves the first statement since N ♭ = (µϕ, µϕ, • • • , µϕ) ∈ MF d S (f ).By Proposition 2.4, X ♯♭ ∼ = k∈Z d T k X which is reduced if and only if X is reduced.The second statement now follows from the first by taking N = X ♯ ∈ MCM(R ♯ ).Lemma 5.7 gives us a specialization of Corollary 2.5 and Theorem A.
In particular, f has finite reduced d-MF type if and only if there are, up to isomorphism, only finitely many indecomposable MCM R ♯ -modules N satisfying rank S (N ) = µ R ♯ (N ).
Proof.Both (i) and (ii) follow from Lemma 5.7 and Proposition 2.4.The final statement follows as in the proof of Theorem A by noticing that a matrix factorization Y ∈ MF d S (f ) is reduced if and only if every summand of Y is reduced and that an MCM R ♯ -module N satisfies µ R ♯ (N ) = rank S (N ) if and only if every summand of N satisfies the same equality.In the case of the d-fold branched cover of R, we have the following connection between reduced d-fold matrix factorizations of f and Ulrich modules over R ♯ .
Corollary 5.9.Assume d ≤ ord(f ) and that f Remark 5.10.Let d = 2 so that R ♯ = S z /(f + z 2 ) is the double branched cover.The condition rank S (N ) = µ R ♯ (N ) is redundant in this case.An MCM R ♯ -module N satisfies rank S (N ) = µ R ♯ (N ) if and only if N has no summands isomorphic to R ♯ (this follows from the proof of [LW12, Lemma 8.17 (iii)]).In other words, the conclusion of Proposition 5.8 is simply a restatement of Knörrer's Theorem (Theorem 1.3 above) when d = 2. Furthermore, Corollary 5.9 implies that any MCM R ♯ -module with no free summands is an Ulrich module.This is a known result of Herzog-Kühl [HK87, Corollary 1.4] since the multiplicity of R ♯ is two.
Example 5.11.Let k be an algebraically closed field of characteristic zero and consider the onedimensional hypersurface ring R a,i = k x, y /(x a + y a+i ), a ≥ 2, i ≥ 0. If i = 1 or i = 2, then, by [HUB91, Theorem A.3], R a,i has only finitely many isomorphism classes of indecomposable Ulrich modules.Set S = k y and consider R a,i as the a-fold branched cover of R = k y /(y a+i ).Since e(R a,i ) = a, Corollary 5.9 implies that y a+i , for i ∈ {1, 2}, has only finitely many isomorphism classes of reduced indecomposable a-fold matrix factorizations.In other words, y a+i has finite reduced a-MF type for i = 1, 2 and any a ≥ 2.
The methods in [HUB91,Theorem A.3] can be used to compute the isomorphism classes of indecomposable reduced matrix factorizations of y a+i .For instance, let a ≥ 2 and i = 1.Then R a,1 ∼ = k t a , t a+1 and t a is a minimal reduction of the maximal ideal m of R a,1 .Hence, R ′ a,1 = R a,1 [{ r t a : r ∈ m}] = k t is the first quadratic transform of R a,1 .By [HUB91, Corollary A.1], an R a,1 -module M is Ulrich over R a,1 if and only if it is MCM over R ′ a,1 .Since R ′ a,1 = k t is a regular local ring, the only indecomposable MCM R ′ a,1 -module is R ′ a,1 itself.As an S ∼ = k t a -module, R ′ a,1 = k t is free with basis given by {1, t, t 2 , . . ., t a−1 }.Thus, multiplication by x = t a+1 on the basis {1, t, . . ., t a−1 } is given by the mapping t k → t a+1+k = t a t k+1 for 0 ≤ k ≤ a − 1.Since y = t a , it follows that multiplication by x on the MCM R a,1 -module R ′ It follows that (R ′ a,1 ) ♭ ∼ = i∈Za T i ((y 2 , y, y, . . ., y)) ∈ MF a k y (y a+1 ).By Proposition 5.8 and Corollary 5.9, the matrix factorization (y 2 , y, y, . . ., y) ∈ MF a k y (y a+1 ), and its corresponding shifts, are the only indecomposable reduced matrix factorizations of y a+1 with a factors.
Notice that for a ≥ 4, the polynomial y a+1 does not appear on the list given in Theorem B for any d > 2. Thus, the conclusions of this example imply that, for a ≥ 4, the polynomial y a+1 has infinitely many isomorphism classes of indecomposable matrix factorizations with a factors but only finitely many which are reduced.
The last example shows the necessity of the assumption d ≤ ord(f ) in Corollary 5.9.
Example 5.12.Let k be algebraically closed of characteristic zero.Set S = k x , f = x 3 , and R = S/(f ).The hypersurface ring R ♯ = k x, y /(x 3 + y 4 ) is the same ring given in Example 3.4, however, here we are viewing R ♯ as the 4-fold branched cover of R = k x /(x 3 ).Again using the notation of [Yos90, 9.13], we take B = cok β where The MCM R ♯ -module B is, in this case, free of rank 4 over S = k x .In particular, if e 1 , e 2 , e 3 ∈ B are the images of the standard basis on S y 3 , then an S-basis for B is {e 1 , e 2 , e 3 , ye 2 }.Multiplication by y on B is therefore given by the S-matrix ϕ =     0 0 x 0 −x 0 0 0 0 0 0 x 0 1 0 0 Notice that B is an Ulrich R ♯ -module but multiplication by y on B is given by a non-reduced matrix.In other words, the condition d ≤ ord(f ) in Proposition 5.8 is needed to produce reduced matrix factorizations of f .

Theorem A .
Let d ≥ 2. Then f has finite d-MF type if and only if the d-fold branched cover R ♯ = S z /(f + z d ) has finite CM type.

Remark 2. 7 .
In the case of an Artin algebra Λ, the relationship between Λ and the skew group algebra Λ[G] for a finite group G was studied by Reiten and Riedtmann in [RR85].They show that many representation theoretic properties hold simultaneously for Λ and Λ[G].In particular, Λ has finite representation type if and only if the same is true of Λ[G].The equivalence of categories MF d S (f ) ≈ MCM σ (R ♯ ) [Tri21, Theorem 4.4] and Theorem A give an analogous relationship between R ♯ and the skew group algebra R ♯ [σ].
size 2, a non-trivial decomposition would be of the form (y, b, c) ⊕ (y 3 , b ′ , c ′ ) for some b, c, b ′ , c ′ ∈ S. Since det 0 y y 1 = −y 2 , the possibilities for b and b ′ are, up to units, b = y 2 and b ′ = 1 or b = y = b ′ Corollary 3.5.Let S be a regular local ring, f a non-zero non-unit in S, and d ≥ 2. If f has finite d-MF type, then f has finite k-MF type for all 2 ≤ k ≤ d.In particular, if f has finite d-MF type for some d ≥ 2, then R = S/(f ) has finite CM type.
Corollary 5.15].Moreover, finite d-MF type clearly implies finite reduced d-MF type but the converse does not hold for d > 2 as we will show in Example 5.11.Definition 5.2.(i) Let X = (ϕ 1 , . . ., ϕ d ) ∈ MF d S (f ) and pick bases to consider ϕ k , k ∈ Z d , as a square matrix with entries in S. Following [BGS87], we define I(ϕ k ) to be the ideal generated by the entries of ϕ k and set I(X) = k∈Z d I(ϕ k ).Note that the ideal I(X) does not depend on the choice of bases.(ii) Let c d (f ) denote the collection of proper ideals I of S such that f ∈ I d .
Theorem 5.3 ([HUB91], Theorem 1.2).Let I be a proper ideal of S and d ≥ 2. If f ∈ I d , then there exists a reduced matrix factorization X ∈ MF d S (f ) such that I(X) = I.Corollary 5.4.Suppose f has finite reduced d-MF type.Then c d (f ) is a finite collection of ideals of S. Corollary 5.4 extends one direction of Theorem 3.1; however, the converse does not hold for d > 2 as shown by the next example.Example 5.5.Let S = k x, y with char k = 2 and f = x 2 y ∈ S. Then the one-dimensional D ∞ singularity R = S/(f ) has countably infinite CM type by [BGS87, Proposition 4.2].For each k ≥ 1, we have a reduced matrix factorization of x 2 y with 3 factors: For a module M over a local ring A, we let e(M ) denote the multiplicity of M .If M is an MCM A-module, there is a well known inequality µ A (M ) ≤ e(M ).The class of MCM modules satisfying µ A (M ) = e(M ) are called Ulrich modules.For background on Ulrich modules we refer the reader to [Bea18], [BHU87], [HK87], and [HUB91].If A is a domain, then we may compute the multiplicity of M as e(M ) = e(A) • rank A (M ) (see [Yos90, Proposistion 1.6]).
f ) is a reduced matrix factorization.In particular, f has finite reduced d-MF type if and only if there are, up to isomorphism, only finitely many indecomposable Ulrich R ♯ -modules.Proof.Since d ≤ ord(f ), the multiplicity of R ♯ = S z /(f + z d ) is d.Hence, an MCM R ♯ -module N is Ulrich if and only if µ R ♯ (N ) = d • rank R ♯ (N ).By Lemma 5.6, the quantity d • rank R ♯ (N ) is equal to the rank of N as a free S-module.Thus, N is Ulrich if and only if µ R ♯ (N ) = rank S (N ).Both statements now follow from Proposition 5.8.

a, 1
is given by the a × a matrix with entries in k y