Knörrer Periodicity and Bott Periodicity

Michael K. Brown

Received: July 19, 2016

Communicated by Henning Krause

Abstract. The goal of this article is to explain a precise sense in which Knörrer periodicity in commutative algebra and Bott periodicity in topological K-theory are compatible phenomena. Along the way, we prove an 8-periodic version of Knörrer periodicity for real isolated hypersurface singularities, and we construct a homomorphism from the Grothendieck group of the homotopy category of matrix factorizations of a complex (real) polynomial f into the topological K-theory of its Milnor fiber (positive or negative Milnor fiber).

2010 Mathematics Subject Classification: 13D15, 18D20, 18E30, 32S55, 55N15 Keywords and Phrases: Atiyah-Bott-Shapiro construction, Bott periodicity, Knörrer periodicity, matrix factorizations, Milnor fibration

Contents

1 Introduction 1460

2 Knörrer periodicity over ℝ 1462

2.1 Matrix factorization categories . . . . . . . . . . . . . . . . . . 1462 2.1.1 Definitions and some properties . . . . . . . . . . . . . . 1462 2.1.2 Triangulated structure . . . . . . . . . . . . . . . . . . . 1466 2.1.3 Stabilization . . . . . . . . . . . . . . . . . . . . . . . . 1467

2.2 The tensor product of matrix factorizations . . . . . . . . . . . 1468 2.3 Matrix factorizations of quadratics . . . . . . . . . . . . . . . . 1472 2.4 Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1473

3 Matrix factorizations and the topological K-theory of the Milnor fiber 1476 3.1 The real and complex Milnor fibers . . . . . . . . . . . . . . . . 1476

3.1.1 Construction of the Milnor fibration and some properties

of the Milnor fiber . . . . . . . . . . . . . . . . . . . . . 1477 3.1.2 The Sebastiani-Thom homotopy equivalence . . . . . . . 1478 3.1.3 An analogue of the Milnor fibration over ℝ . . . . . . . 1480

3.2 Relative topological K-theory . . . . . . . . . . . . . . . . . . . 1481 3.3 A generalized Atiyah-Bott-Shapiro construction applied to ma-

trix factorizations . . . . . . . . . . . . . . . . . . . . . . . . . . 1486 3.3.1 The Atiyah-Bott-Shapiro construction . . . . . . . . . . 1486 3.3.2 A more general construction . . . . . . . . . . . . . . . . 1487 3.3.3 The kernel and image of ϕ_{f}^{ℂ} . . . . . . . . . . . . . . . . 1492

3.4 Knörrer periodicity and Bott periodicity . . . . . . . . . . . . . 1494

1 Introduction

Let k be a field. In this article, we study hypersurface rings of the form k[x_{1}, …, x_{n}]/(f) from both an algebraic and topological point of view. An important algebraic invariant of such a ring is its homotopy category of matrix factorizations, which we denote by [MF(k[x_{1}, …, x_{n}], f)] (we recall the definition of this category in Section 2.1.1). Matrix factorizations were introduced by Eisenbud in [Eis80] as a tool for studying the homological behavior of modules over a hypersurface ring. More recently, matrix factorizations have begun appearing in a wide variety of contexts, for instance homological mirror symmetry (e.g. [KKP08], by Katzarkov-Kontsevich-Pantev) and knot theory (e.g. [KR08], by Khovanov-Rozansky). In the present work, we continue the study of an interplay between matrix factorizations and topological K-theory that was begun in the inspiring paper [BvS12] of Buchweitz-van Straten. A fundamental result in the theory of matrix factorizations is Knörrer’s periodicity theorem:

Theorem 1.1 ([Knö87] Theorem 3.1). Suppose k is algebraically closed and char(k) ≠ 2. If f ∈ (x_{1}, …, x_{n}) ⊆ k[[x_{1}, …, x_{n}]], there is an equivalence of categories

[MF(k[[x_{1}, …, x_{n}]], f)] ⟶^{≅} [MF(k[[x_{1}, …, x_{n}, u, v]], f + u^{2} + v^{2})].

This result plays an important role in the classification of local hypersurface rings of finite maximal Cohen-Macaulay type; we refer the reader to Chapter 9 of Leuschke-Wiegand’s text [LW12] for details. Knörrer’s periodicity theorem also demonstrates that one cannot recover f from its homotopy category of matrix factorizations. The main goal of this article is explain a precise sense in which Knörrer periodicity is a manifestation of Bott periodicity in topological K-theory. In Section 2, we motivate this project with a proof of an 8-periodic version of Knörrer periodicity for isolated hypersurface singularities over the real numbers:

Theorem 1.2. Let f ∈ (x_{1}, …, x_{n}) ⊆ ℝ[x_{1}, …, x_{n}], and suppose ℝ[x_{1}, …, x_{n}]/(f) has an isolated singularity at the origin (i.e. dim_{ℝ} _{\frac{ℝ[[x_{1},...,x_{n}]]}{( \frac{∂f}{∂x_{1}} ,..., \frac{∂f}{∂xn})}} < ∞). Then there exists an equivalence of triangulated

categories

[MF(ℝ[[x_{1}, …, x_{n}]], f)] ⟶^{≅} [MF(ℝ[[x_{1}, …, x_{n}, u_{1}, …, u_{8}]], f − u_{1}^{2} − ⋯ − u_{8}^{2})].

We point out that the “period” here is exactly 8; that is, for 1 ⩽ l < 8, it can happen that

[MF(ℝ[[x_{1}, …, x_{n}]], f)] ≇ [MF(ℝ[[x_{1}, …, x_{n}, u_{1}, …, u_{l}]], f − u_{1}^{2} − ⋯ − u_{l}^{2})].

Our proof relies heavily on machinery developed by Dyckerhoff and Toën in [Dyc11] and [Toë07]. This result draws a distinction between the maximal Cohen-Macaulay representation theory of hypersurface rings with ground field ℝ and those whose ground field is algebraically closed and has characteristic not equal to 2, since the latter exhibit 2-periodic Knörrer periodicity. The maximal Cohen-Macaulay representation theory of hypersurface rings with ground field ℝ does not seem to be well-studied, and we hope this work motivates further investigation in this direction. The presence of 2- and 8-periodic versions of Knörrer periodicity over ℂ and ℝ, respectively, suggests the possibility of a compatibility between Knörrer periodicity and Bott periodicity. Such a compatibility statement is formulated and proved in Section 3. We state here the version of this result over ℂ; a version over ℝ is also proven in Section 3 (Theorems 3.32 and 3.33).

Theorem 1.3. Suppose f ∈ (x_{1}, …, x_{n}) ⊆ Q := ℂ[x_{1}, …, x_{n}] and either Q/(f) has an isolated singularity at the origin (i.e. dim_{ℂ} _{\frac{ℂ[[x_{1},...,x_{n}]]}{( \frac{∂f}{∂x_{1}} ,..., \frac{∂f}{∂xn})}} < ∞) or

f is quasi-homogeneous. Then there exists a commutative diagram

\xymatrix{K_{0}[MF(Q, f)] \ar[r]^{ϕ_{f}^{ℂ}} \ar[dd]^{K} & KU^{0}(B_{ϵ}, F_{f}) \ar[d]^{β} \\  & KU^{0}(B_{ϵ}, F_{f}) ⊗ KU^{0}(B_{ϵ′}, F_{u^{2}+v^{2}}) \ar[d]^{_{ST_{KU}}} \\ K_{0}[MF(Q[u, v], f + u^{2} + v^{2})] \ar[r]^{ϕ_{f+u^{2}+v^{2}}^{ℂ}} & KU^{0}(B_{ϵ′′}, F_{f+u^{2}+v^{2}})}

where F_{f}, F_{u^{2}+v^{2}}, and F_{f+u^{2}+v^{2}} denote the Milnor fibers of f, u^{2} + v^{2}, and f + u^{2} + v^{2}; ϵ, ϵ′, ϵ′′ > 0; B_{ϵ}, B_{ϵ′}, and B_{ϵ′′} are closed balls of radius ϵ, ϵ′, and ϵ′′ in ℂ^{n}, ℂ^{2}, and ℂ^{n+2}, respectively; K is induced by the Knörrer functor; β is the Bott periodicity isomorphism; and ST_{KU} is given by the product in relative K-theory followed by the inverse of the map induced by pullback along the Sebastiani-Thom homotopy equivalence.

The Sebastiani-Thom homotopy equivalence to which we refer in Theorem 1.3 is discussed in Section 3.1.2. The key construction in this section yields the horizontal maps above; specifically, given a polynomial f over the complex (real) numbers, we build a map Φ_{f}^{ℂ} (Φ_{f}^{ℝ}) that assigns to a matrix factorization of a complex (real) polynomial f a class in the topological K-theory of the Milnor fiber (positive or negative Milnor fiber) of f; this map first appeared in [BvS12] in the setting of complex isolated hypersurface singularities. We prove that this construction induces a map ϕ_{f}^{ℂ} (ϕ_{f}^{ℝ}) on the Grothendieck group of the homotopy category of matrix factorizations of f, and we show that it recovers the Atiyah-Bott-Shapiro construction when f is a non-degenerate quadratic over ℝ or ℂ. The Atiyah-Bott-Shapiro construction, introduced in Part III of [ABS64], provides the classical link between ℤ/2ℤ-graded modules over Clifford algebras and vector bundles over spheres; the maps ϕ_{f}^{ℂ} and ϕ_{f}^{ℝ} we discuss in Section 3 can be thought of as providing a more general link between algebra and topology. Acknowledgements. This work is adapted from my Ph.D. thesis at the University of Nebraska-Lincoln. I must first of all thank my thesis advisor, Mark E. Walker, for his support during my time as a graduate student at Nebraska. I thank Luchezar Avramov and Brian Harbourne for their comments on preliminary versions of this paper, and also Ragnar Buchweitz, Jesse Burke, Michael Hopkins, and Claudia Miller for valuable conversations with regard to this work. I owe special thanks to Hai Long Dao for pointing out to me that one may use Proposition 3.3 in [Dao13] to prove Proposition 3.26 below. I would also like to gratefully acknowledge support from NSF Award DMS-0966600 and the University of Nebraska-Lincoln MCTP grant (NSF Award DMS-0838463). Finally, I thank the anonymous referee for his or her helpful suggestions.

2 Knörrer periodicity over ℝ

In this section, we recall some foundational material concerning matrix factorizations in commutative algebra, and we exhibit an 8-periodic version of Knörrer periodicity for matrix factorization categories associated to isolated hypersurface singularities over the real numbers.

2.1 Matrix factorization categories

We provide some background on matrix factorization categories. Fix a commutative algebra Q over a field k and an element f of Q. Henceforth, when we use the term “dg category”, we mean “k-linear differential ℤ/2ℤ-graded category”. We cite results on differential ℤ-graded categories from [Toë11] several times throughout this section; we refer the reader to Section 5.1 of [Dyc11] for a discussion as to how one may reformulate the results in [Toë11] so that they apply to the ℤ/2ℤ-graded setting.

2.1.1 Definitions and some properties

Definition 2.1. The dg category MF(Q, f) of matrix factorizations of f over Q is given by the following: Objects in MF(Q, f) are pairs (P, d), where P is a finitely generated projective ℤ/2ℤ-graded Q-module, and d is an odd-degree endomorphism of P such that d^{2} = f · id_{P}. Henceforth, we will often denote an object (P, d) in MF(Q, f) by just P.

The morphism complex of a pair of matrix factorizations P, P′, which we will denote by Hom_{MF}(P, P′), is the ℤ/2ℤ-graded module of Q-linear maps from P to P′ equipped with the differential ∂ given by

∂(α) = d′ ∘ α − (−1)^{|α|}α ∘ d

for homogeneous maps α : P → P′.

We will often express an object P in MF(Q, f) with the notation

P_{1} \underleftarrow{−}=\overrightarrow{−}_{d_{0}}^{d_{1}} P_{0},

where P_{1}, P_{0} are the odd and even degree summands of P, and d_{1}, d_{0} are the restrictions of d to P_{1} and P_{0}, respectively. A degree 0 morphism α in MF(Q, f) can be represented by a diagram of the following form:

P_{1} ⟶^{d_{1}} P_{0} ⟶^{d_{0}} P_{1}
α_{1}↓  ^{α_{0}}↓  ↓^{α_{1}}
P′_{1} ⟶^{d′_{1}} P′_{0} ⟶^{d′_{0}} P′_{1}

It is straightforward to check that α is a cycle if and only if this diagram commutes. In fact, if f ∈ Q is a non-zero-divisor, it is easy to see that the left square commutes if and only if the right square commutes.

Remark 2.2. If P_{1} and P_{0} are free and f is non-zero-divisor, P_{1} and P_{0} must have the same rank.

Define Z^{0}MF(Q, f) to be the category with the same objects as MF(Q, f) and with morphisms given by the degree 0 cycles in MF(Q, f). When Q is regular with finite Krull dimension and f is a regular element of Q (i.e. f is a non-unit, non-zero-divisor), Z^{0}MF(Q, f) is an exact category with the evident family of exact sequences ([Orl03] Section 3.1). The homotopy category, [MF(Q, f)], of the dg category MF(Q, f) is defined to be the quotient of Z^{0}MF(Q, f) by morphisms that are boundaries in MF(Q, f). That is, objects in [MF(Q, f)] are the same as those of MF(Q, f), and the morphisms in [MF(Q, f)] between objects P, P′ are classes in H^{0}Hom_{MF}(P, P′).

Definition 2.3. We call a matrix factorization trivial if it is a direct sum of matrix factorizations that are isomorphic in Z^{0}MF(Q, f) to either

E \underleftarrow{−}=\overrightarrow{−}_{id_{E}}^{f·id_{E}} E

or

E \underleftarrow{−}=\overrightarrow{−}_{f·id_{E}}^{id_{E}} E

for some finitely generated projective Q-module E.

The following result gives an alternative characterization for when a morphism in Z^{0}MF(Q, f) is a boundary in MF(Q, f); the straightforward proof is omitted.

Proposition 2.4. A morphism α : P → P′ in Z^{0}MF(Q, f) is a boundary in MF(Q, f) if and only if it factors through a trivial matrix factorization in Z^{0}MF(Q, f).

We conclude this section with a technical result that will be used in the proof of Proposition 3.19:

Proposition 2.5. Let P = (P_{1} \underleftarrow{−}=\overrightarrow{−}_{d_{0}}^{d_{1}} P_{0}) be a matrix factorization of f over

Q. Assume f is a non-zero-divisor. Then the following are equivalent:

(1) coker(d_{1}) is isomorphic to L/fL for some projective Q-module L.

(2) There exists a trivial matrix factorization E and a matrix factorization E′ that is isomorphic in Z^{0}MF(Q, f) to one of the form

F \underleftarrow{−}=\overrightarrow{−}_{f}^{id_{F}} F

such that P ⊕ E′ is isomorphic to E in Z^{0}MF(Q, f).

We will use the following general fact about idempotent complete categories. We suspect that this result is well-known to experts; we omit the purely formal proof.

Lemma 2.6. Let C be an idempotent complete additive category, and let E be a collection of objects in C that is

• closed under isomorphisms,

• closed under finite coproducts, and

• closed under taking summands; that is, whenever X is an object in C such that id_{X} factors through an object in E, X is an object in E.

Denote by L the quotient of C by those morphisms that factor through an object in E. If X and Y are objects in C, their images in L are isomorphic if and only if there exist objects E_{X}, E_{Y} in E such that

X ⊕ E_{X} ≅ Y ⊕ E_{Y} .

We now prove Proposition 2.5:

Proof. (2) ⇒ (1): Since the cokernel of d_{1} is isomorphic to the cokernel of

d_{1} ⊕ id_{F} : P_{1} ⊕ F → P_{0} ⊕ F,

we may assume P is trivial. In this case, the result is obvious.

Knörrer Periodicity and Bott Periodicity 1465

(1) ⇒ (2): Choose a projective Q-module L such that there exists an isomorphism

coker(d_{1}) ⟶^{≅} L/fL.

We have Q-projective resolutions

0 → P_{1} ⟶^{d_{1}} P_{0} → coker(d_{1}) → 0

0 → L ⟶^{f} L → L/fL → 0

Thus, there exist maps

β_{i} : P_{i} → L, γ_{i} : L → P_{i}

for i = 0, 1 making the following diagrams commute:

0 ⟶ P_{1} ⟶^{d_{1}} P_{0} ⟶ coker(d_{1}) ⟶ 0
β_{1}↓ ^{β_{0}}↓ ↓≅
0 ⟶ L ⟶^{f} L ⟶ L/fL ⟶ 0
0 ⟶ L ⟶^{f} L ⟶ L/fL ⟶ 0
γ_{1}↓ ^{γ_{0}}↓ ↓≅
0 ⟶ P_{1} ⟶^{d_{1}} P_{0} ⟶ coker(d_{1}) ⟶ 0

Hence, we have maps

h_{P} : P_{0} → P_{1}, h_{L} : L → L

such that

γ_{1} ∘ β_{1} − id_{P_{1}} = h_{P} ∘ d_{1}, γ_{0} ∘ β_{0} − id_{P_{0}} = d_{1} ∘ h_{P} .
β_{1} ∘ γ_{1} − id_{L} = fh_{L}, β_{0} ∘ γ_{0} − id_{L} = fh_{L}.

We have commutative diagrams

\xymatrix{P_{1} \ar[r]^{d_{1}} \ar[d]^{h_{P} ∘d_{1}} & P_{0} \ar[d]^{h_{P}} & L \ar[r]^{f·id_{L}} \ar[d]^{f·h_{L}} & L \ar[d]^{h_{L}} \\ P_{1} \ar[r]^{_{id_{P_{1}}}} \ar[d]^{_{id_{P_{1}}}} & P_{1} \ar[d]^{d_{1}} & L \ar[r]^{_{id_{L}}} \ar[d]^{_{id_{L}}} & L \ar[d]^{f·id_{L}} \\ P_{1} \ar[r]^{d_{1}} & P_{0} & L \ar[r]^{f·id_{L}} & L}

Denote by E the collection of matrix factorizations of f over Q isomorphic in Z^{0}MF(Q, f) to a matrix factorization of the form

E \underleftarrow{−}=\overrightarrow{−}_{f}^{id_{E}} E.

Notice that Z^{0}MF(Q, f) is an idempotent complete additive category, and E is closed under direct sums and direct summands in Z^{0}MF(Q, f). Letting L denote the quotient of Z^{0}MF(Q, f) by those morphisms that factor through an object in E, we have that

(P_{1} \underleftarrow{−}=\overrightarrow{−}_{d_{0}}^{d_{1}} P_{0}) ≅ (L \underleftarrow{−}=\overrightarrow{−}_{id_{L}}^{f} L)

in L. The result now follows from Lemma 2.6.

2.1.2 Triangulated structure

Suppose Q is regular with finite Krull dimension and f is a regular element of Q. A feature of the homotopy category [MF(Q, f)] is that it may be equipped with a triangulated structure in the following way ([Orl03] Section 3.1): The shift functor maps the object

P = (P_{1} \underleftarrow{−}=\overrightarrow{−}_{d_{0}}^{d_{1}} P_{0})

to the object

P [1] = (P_{0} \underleftarrow{−}=\overrightarrow{−}_{−d_{1}}^{−d_{0}} P_{1}).

Given a morphism α : (P_{1} \underleftarrow{−}=\overrightarrow{−}_{d_{0}}^{d_{1}} P_{0}) → (P′_{1} \underleftarrow{−}=\overrightarrow{−}_{d′_{0}}^{d′_{1}} P′_{0}) in Z^{0}MF(Q, f), the

mapping cone of α is defined as follows:

cone(α) = (P′_{0} ⊕ P_{1} \begin{matrix}( & \begin{matrix}d′_{0} & α_{1} \\ 0 & −d_{1} \end{matrix} & ) \\ \underleftarrow{−} & = & = & = & = & = & = & = & = & = & = & = & \overrightarrow{−} \\ ( & \begin{matrix}d′_{1} & α_{0} \\ 0 & −d_{0} \end{matrix} & ) \end{matrix} P′_{1} ⊕ P_{0})

There are canonical morphisms P′ → cone(α) and cone(α) → P [1] in Z^{0}MF(Q, f). Taking the distinguished triangles in [MF(Q, f)] to be the triangles isomorphic in [MF(Q, f)] to those of the form

P ⟶^{α} P′ → cone(α) → P [1],

[MF(Q, f)] may be equipped with the structure of a triangulated category.

The Grothendieck group, K_{0}[MF(Q, f)], of the triangulated category [MF(Q, f)] is defined to be the free abelian group generated by isomorphism classes of [MF(Q, f)] modulo elements of the form [P_{1}] − [P_{2}] + [P_{3}], where P_{1}, P_{2}, and P_{3} fit into a distinguished triangle in the following way:

P_{1} → P_{2} → P_{3} → P_{1}[1].

Remark 2.7. The category MF(Q, f) is not always triangulated in the dg sense (see Section 4.4 of [Toë11] for the definition of a triangulated dg category). When MF(Q, f) is triangulated in the dg sense, the induced triangulated structure on [MF(Q, f)] agrees with the triangulated structure just described.

Remark 2.8. When Q is a regular local ring and f is a regular element of Q, one has an equivalence of triangulated categories

[MF(Q, f)] ⟶^{≅} \underline{MCM}(Q/(f)),

where \underline{MCM}(Q/(f)) denotes the stable category of maximal Cohen-Macaulay (MCM) modules over the ring Q/(f). The stable category of MCM modules is obtained by taking the quotient of the category of MCM modules over Q/(f) by those morphisms that factor through a projective Q/(f)-module. The above equivalence is given, on objects, by

(P_{1} \underleftarrow{−}=\overrightarrow{−}_{d_{0}}^{d_{1}} P_{0}) ↦ coker(d_{1}).

Matrix factorizations were first defined by Eisenbud in [Eis80]; this interplay between matrix factorizations and MCM modules over hypersurface rings provided the original motivation for the study of matrix factorization categories.

2.1.3 Stabilization

Assume now that Q is a regular local ring of Krull dimension n, and suppose f is a regular element of Q. Denote by D^{b}(Q/(f)) the bounded derived category of Q/(f), and set Perf(Q/(f)) to be the full subcategory of D^{b}(Q/(f)) given by perfect complexes. Perf(Q/(f)) is a thick subcategory of D^{b}(Q/(f)); define \underline{D^{b}}(Q/(f)) to be the Verdier quotient of D^{b}(Q/(f)) by Perf(Q/(f)). In [Buc86], Buchweitz calls this quotient the stabilized derived category of Q/(f). By [Buc86], the functor

\underline{MCM}(Q/(f)) → \underline{D^{b}}(Q/(f))

that sends an MCM module M to the complex with M concentrated in degree 0 is a triangulated equivalence. Hence, composing with the equivalence in Remark 2.8, one has an equivalence

[MF(Q, f)] → \underline{D^{b}}(Q/(f))

Following [Dyc11], given an object C in \underline{D^{b}}(Q/(f)), we denote by C^{stab} the isomorphism class in [MF(Q, f)] corresponding to C under the above equivalence (“stab” stands for “stabilization”). In particular, thinking of the residue field k of Q/(f) as a complex concentrated in degree 0, we may associate to k an isomorphism class k^{stab} in [MF(Q, f)]. We now construct an object E_{f} in MF(Q, f) that represents k^{stab}; this construction appears in [Dyc11]. Choose a regular system of parameters x_{1}, …, x_{n} for Q, and consider the Koszul complex

(⨁_{i=0}^{n}∧^{i}Q^{n}, s_{0})

as a ℤ/2ℤ-graded complex of free Q-modules with even (odd) degree piece given by the direct sum of the even (odd) exterior powers of Q^{n}. Here, s_{0} denotes the ℤ/2ℤ-folding of the Koszul differential associated to x_{1}, …, x_{n}. Choose an expression of f ∈ Q of the form

f = g_{1}x_{1} + ⋯ + g_{n}x_{n}.

Fix a basis e_{1}, …, e_{n} of Q^{n}, and set s_{1} to be the odd-degree endomorphism of ⨁_{i=0}^{n} ∧^{i}Q^{n} given by exterior multiplication on the left by g_{1}e_{1} + ⋯ + g_{n}e_{n}. Define

E_{f} := (⨁_{i=0}^{n}∧^{i}Q^{n}, s_{0} + s_{1}).

It is easy to check that E_{f} is a matrix factorization of f. By Corollary 2.7 in [Dyc11], E_{f} represents k^{stab} in [MF(Q, f)]. In particular, E_{f} does not depend on the choice of regular system of parameters x_{1}, …, x_{n} or coefficients g_{1}, …, g_{n} up to homotopy equivalence. Henceforth, we shall denote the dg algebra End_{MF}(E_{f}) by A_{(Q,f)}.

2.2 The tensor product of matrix factorizations

Let k be a field. We begin this section with a technical definition:

Definition 2.9. Suppose Q is a commutative algebra over k and f ∈ Q. If the pair (Q, f) satisfies

• Q is essentially of finite type over k

• Q is equidimensional of dimension n

• The module Ω_{Q/k}^{1} of Kähler differentials is locally free of rank n

• The zero locus of df ∈ Ω_{Q/k}^{1} is a 0-dimensional scheme supported on a unique closed point 𝔪 of Spec(Q) with residue field k and f ∈ 𝔪

we shall call the pair Q/(f) an isolated hypersurface singularity, or IHS.

Remark 2.10. Our IHS condition above is precisely condition (B) in Section 3.2 of [Dyc11]. As noted in loc. cit., if Q/(f) and Q′/(f′) are IHS, Q ⊗_{k} Q′/(f ⊗ 1 + 1 ⊗ f′) is as well. Suppose Q and Q′ are commutative algebras over k, f ∈ Q, and f′ ∈ Q′. Given objects P and P′ in MF(Q, f), MF(Q′, f′), one can form their tensor product over k:

((P_{1} ⊗_{k} P′_{0})⊕(P_{0} ⊗_{k} P′_{1}) \begin{matrix}( & d_{1} ⊗ id_{P′_{0}} id_{P_{0}} ⊗ d′_{1} −id_{P_{1}} ⊗ d′_{0} d_{0} ⊗ id_{P′_{1}} & ) \\ \underleftarrow{−} & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & \overrightarrow{−} \\ ( & d_{0} ⊗ id_{P′_{0}} −id_{P_{1}} ⊗ d′_{1} id_{P_{0}} ⊗ d′_{0} d_{1} ⊗ id_{P′_{1}} & ) \end{matrix} (P_{0} ⊗_{k} P′_{0})⊕(P_{1} ⊗_{k} P′_{1})).

We will denote the tensor product by P ⊗_{MF} P′. This construction first appeared in [Yos98]; it can be thought of as a ℤ/2ℤ-graded analogue of the tensor product of complexes. It is straightforward to check that P ⊗_{MF} P′ is an object in MF(Q⊗_{k} Q′, f ⊗1+1⊗f′). In fact, setting f ⊕f′ := f ⊗1+1⊗f′ ∈ Q⊗_{k} Q′, and noting that there is a canonical map of complexes

Hom_{MF}(P, L) ⊗_{k} Hom_{MF}(P′, L′) → Hom_{MF}(P ⊗_{MF} P′, L ⊗_{MF} L′),

we have the following:

Proposition 2.11. There is a dg functor

ST_{MF} : MF(Q, f) ⊗_{k} MF(Q′, f′) → MF(Q ⊗_{k} Q′, f ⊕ f′)

that sends an object (P, P′) to P ⊗_{MF} P′.

Remark 2.12. It is straightforward to verify that ST_{MF} induces a pairing

K_{0}[MF(Q, f)] ⊗ K_{0}[MF(Q′, f′)] → K_{0}[MF(Q ⊗_{k} Q′, f ⊕ f′)].

Remark 2.13. The “ST” in the name ST_{MF} stands for “Sebastiani-Thom”, since this tensor product operation is related to the Sebastiani-Thom homotopy equivalence discussed in Section 3.1.2. A precise sense in which the tensor product of matrix factorizations is related to the Sebastiani-Thom homotopy equivalence is illustrated by the proof of Proposition 3.29 below; see Remark 3.31 for further details. Now, suppose Q/(f) and Q′/(f′) are IHS. Set Q′′ := Q ⊗_{k} Q′. We will denote by Q̂ the 𝔪-adic completion of Q_{𝔪}, where 𝔪 is as in the definition of IHS. Define Q̂′ and Q̂′′ similarly, and let

ϕ : Q̂ ⊗_{k} Q̂′ → Q̂′′

denote the canonical ring homomorphism. ϕ induces a dg functor

MF(ϕ) : MF(Q̂ ⊗_{k} Q̂′, f ⊕ f′) → MF(Q̂′′, f ⊕ f′).

Set ŜT_{MF} to be the composition of MF(ϕ) with the tensor product functor

MF(Q̂, f) ⊗_{k} MF(Q̂′, f′) → MF(Q̂ ⊗_{k} Q̂′, f ⊕ f′).

Proposition 2.14. If Q/(f) and Q′/(f′) are IHS,

ŜT_{MF} : MF(Q̂, f) ⊗_{k} MF(Q̂′, f′) → MF(Q̂′′, f ⊕ f′)

is a Morita equivalence of dg categories.

Remark 2.15. We emphasize that Proposition 2.14 is really a straightforward application of several results in [Dyc11]; we include a proof for completeness. We refer the reader to Section 4.4 of [Toë11] for the definition of a Morita equivalence of dg categories.

Proof. Let 𝔪 and 𝔪′ be the maximal ideals of Q and Q′ arising in Definition 2.9. Suppose Q_{𝔪} and Q′_{𝔪′} have Krull dimensions n and m, respectively. Q_{𝔪} and Q′_{𝔪′} are regular local rings; choose regular systems of parameters x_{1}, …, x_{n} and y_{1}, …, y_{m} in Q_{𝔪} and Q′_{𝔪′}, and choose expressions

f = g_{1}x_{1} + ⋯ + g_{n}x_{n}
f′ = h_{1}y_{1} + ⋯ + h_{m}y_{m}

of f and f′. Use these expressions to construct the dga’s A_{(Q_{𝔪},f)} and A_{(Q′_{𝔪′} ,f′)}, as in Section 2.1.3. Note that x_{1}, …, x_{n} and y_{1}, …, y_{m} form regular systems of parameters in Q̂ and Q̂′ as well, so we may use these expressions to construct A_{(Q̂,f)} and A_{(Q̂′,f′)}. Also, x_{1} ⊗ 1, …, x_{n} ⊗ 1, 1 ⊗ y_{1}, …, 1 ⊗ y_{m} is a regular system of parameters in Q′′_{𝔪′′}, where 𝔪′′ := 𝔪 ⊗ 1 + 1 ⊗ 𝔪′, so we may use the expression

f ⊕ f′ = (g_{1}x_{1} ⊗ 1) + ⋯ + (g_{n}x_{n} ⊗ 1) + (1 ⊗ h_{1}y_{1}) + ⋯ + (1 ⊗ h_{m}y_{m})

to construct A_{(Q′′_{𝔪′′} ,f⊕f′)} and A_{(Q̂′′,f⊕f′)}. By Section 6.1 of [Dyc11], we have a quasi-isomorphism

F : A_{(Q_{𝔪},f)} ⊗_{k} A_{(Q′_{𝔪′} ,f′)} ⟶^{≅} A_{(Q′′_{𝔪′′} ,f⊕f′)}.

We also have a canonical map

G : A_{(Q̂,f)} ⊗_{k} A_{(Q̂′,f′)} → A_{(Q̂′′,f⊕f′)}.

By the proof of Theorem 5.7 in [Dyc11], the inclusions

A_{(Q_{𝔪},f)} ↪ A_{(Q̂,f)}
A_{(Q′_{𝔪′} ,f′)} ↪ A_{(Q̂′,f′)}
A_{(Q′′_{𝔪′′} ,f⊕f′)} ↪ A_{(Q̂′′,f⊕f′)}

are all quasi-isomorphisms. Since a tensor product of Morita equivalences is again a Morita equivalence ([Toë11] Section 4.4), it follows that the map

A_{(Q_{𝔪},f)} ⊗_{k} A_{(Q′_{𝔪′} ,f′)} → A_{(Q̂,f)} ⊗_{k} A_{(Q̂′,f′)} is a Morita equivalence. We have the following commutative square:

\xymatrix{A_{(Q_{𝔪},f)} ⊗_{k} A_{(Q′_{𝔪′} ,f′)} \ar[r] \ar[d] & A_{(Q̂,f)} ⊗_{k} A_{(Q̂′,f′)} \ar[d]^{G} \\ A_{(Q′′_{𝔪′′} ,f⊕f′)} \ar[r] & A_{(Q̂′′,f⊕f′)}}

It follows that G is a Morita equivalence. One may think of a dga as a dg category with a single object. Adopting this point of view, we have inclusion functors

i : A_{(Q̂,f)} ↪ MF(Q̂, f)

j : A_{(Q̂′,f′)} ↪ MF(Q̂′, f′)
l : A_{(Q̂′′,f⊕f′)} ↪ MF(Q̂′′, f ⊕ f′)

Combining Theorem 5.2 and Lemma 5.6 in [Dyc11], we conclude that i, j, and l are Morita equivalences. In particular, we have that

i ⊗ j : A_{(Q̂,f)} ⊗_{k} A_{(Q̂′,f′)} → MF(Q̂, f) ⊗_{k} MF(Q̂′, f′)

is a Morita equivalence. Finally, consider the following commutative diagram:

\xymatrix{A_{(Q̂,f)} ⊗_{k} A_{(Q̂′,f′)} \ar[r]^{i⊗j} \ar[d]^{G} & MF(Q̂, f) ⊗_{k} MF(Q̂′, f′) \ar[d]^{_{ŜT_{MF}}} \\ A_{(Q̂′′,f⊕f′)} \ar[r]^{l} & MF(Q̂′′, f ⊕ f′)}

Since the left-most vertical map and both horizontal maps are Morita equivalences, ŜT_{MF} is as well.

Remark 2.16. Using Theorem 4.11 of [Dyc11] along with a similar argument to the one above, one may show that, under the assumptions of Proposition 2.14, the functor

MF(Q, f) ⊗_{k} MF(Q′, f′) → MF(Q′′, f ⊕ f′)

given by tensor product of matrix factorizations is also a Morita equivalence.

2.3 Matrix factorizations of quadratics

Fix a field k such that char(k) ≠ 2 and a finite-dimensional vector space V over k. Let q : V → k be a quadratic form, and let Cliff_{k}(q) denote the Clifford algebra associated to q. Cliff_{k}(q) is a ℤ/2ℤ-graded k-algebra; let mod_{ℤ/2ℤ}(Cliff_{k}(q)) denote the category of finitely generated ℤ/2ℤ-graded left modules over Cliff_{k}(q). Henceforth, when we refer to a module over a Clifford algebra, we will always mean it to be a left module. Assume q is non-degenerate, and choose a basis {e_{1}, …, e_{n}} of V with respect to which q is diagonal; that is,

q = a_{1}x_{1}^{2} + ⋯ + a_{n}x_{n}^{2} ∈ S^{2}(V^{∗})

where the x_{i} comprise the dual basis corresponding to the e_{i}, and the a_{i} are nonzero elements of k. Denote by Q the localization of S(V^{∗}) at the ideal (x_{1}, …, x_{n}). The following theorem, due to Buchweitz-Eisenbud-Herzog, yields a relationship between Clifford modules and matrix factorizations of non-degenerate quadratic forms:

Theorem 2.17 ([BEH87]). There is an equivalence of k-linear categories

mod_{ℤ/2ℤ}(Cliff_{k}(q)) ⟶^{≅} [MF(Q̂, q)].

Denote by Θ the explicit construction of this equivalence described in the proof of Theorem 14.7 in [Yos90]. Remark 2.18. The inclusion

k[x_{1}, …, x_{n}] ↪ Q̂

induces an equivalence

[MF(k[x_{1}, …, x_{n}], q_{n})] ⟶^{≅} [MF(Q̂, q_{n})].

To see this, we first recall that every matrix factorization of q_{n} over Q̂ is isomorphic in [MF(Q̂, q_{n})] to one with (linear) polynomial entries ([Yos90] Proposition 14.3); hence, the functor is essentially surjective. Also, one has a commutative diagram

\xymatrix{[MF(Q, q_{n})] \ar[r]^{≅} \ar[d] & \underline{MCM}(Q/(q_{n})) \ar[d] \\ [MF(Q̂, q_{n})] \ar[r]^{≅} & \underline{MCM}(Q̂/(q_{n}))}

The morphism sets in \underline{MCM}(Q/(q_{n})) are Artinian modules, and hence complete. Thus, the functor on the right is fully faithful, and so the functor on the left is as well.

It now follows from Theorem 4.11 in [Dyc11] that the functor

[MF(k[x_{1}, …, x_{n}], q_{n})] → [MF(Q̂, q_{n})]

is fully faithful.

Suppose q′ : W → k is another non-degenerate quadratic form; choose a basis of W with respect to which q′ is diagonal, and let y_{1}, …, y_{m} denote the corresponding basis of W^{∗}. As above, we may think of q′ as an element of S^{2}(W^{∗}). Set Q′ to be the localization of S(W^{∗}) at the ideal (y_{1}, …, y_{m}). It is well-known that the ℤ/2ℤ-graded tensor product of Cliff_{k}(q) and Cliff_{k}(q′) over k is canonically isomorphic to Cliff_{k}(q ⊕ q′). Further, by Remark 1.3 in [Yos98], the ℤ/2ℤ-graded tensor product of Clifford modules is compatible, via this canonical isomorphism and the equivalence in Theorem 2.17, with the tensor product ST_{MF} in Proposition 2.11. That is, one has a commutative diagram of k-linear categories

\xymatrix{mod_{ℤ/2ℤ}(Cliff_{k}(q)) × mod_{ℤ/2ℤ}(Cliff_{k}(q′)) \ar[r] \ar[d]^{Θ×Θ} & mod_{ℤ/2ℤ}(Cliff_{k}(q ⊕ q′)) \ar[d]^{Θ} \\ [MF(Q, q)] × [MF(Q′, q′)] \ar[r]^{_{ST_{MF}}} & [MF(Q ⊗_{k} Q′, q ⊕ q′)]}

Let C be a rank 1 free ℤ/2ℤ-graded Cliff_{k}(q)-module. If dim(V) = 1 and q = x^{2}, it is easy to check that the isomorphism class of Θ(C) is k^{stab}, where k^{stab} is as defined in Section 2.1.3. Further, E_{q} ⊗_{MF} E_{q′} ≅ E_{q⊕q′}, where E_{q}, E_{q′}, and E_{q⊕q′} are as in Section 2.1.3 ([Dyc11] Section 6.1). Thus, we have:

Proposition 2.19. If a_{i} = 1 for 1 ⩽ i ⩽ n, the isomorphism class of Θ(C) is k^{stab}.

2.4 Periodicity

Following [Dyc11], given a commutative algebra Q over a field k and an element f of Q, we define MF^{∞}(Q, f) to be the dg category of possibly infinitelygenerated matrix factorizations; that is, objects of MF^{∞}(Q, f) are defined in the same way as MF(Q, f), except the projective ℤ/2ℤ-graded Q-module P need not be finitely generated. A version of Knörrer periodicity (Theorem 1.1) for isolated hypersurface singularities may be deduced from the following proposition:

Proposition 2.20. Suppose Q and Q′ are commutative algebras over a field k. Let f ∈ Q and f′ ∈ Q′, and suppose Q/(f) and Q′/(f′) are IHS. If there exists an object X in MF(Q′, f′) such that

(a) X is a compact generator of [MF^{∞}(Q′, f′)], and

(b) the inclusion k ↪ End_{MF(Q̂′,f′)}(X) is a quasi-isomorphism then the dg functor

K_{X} : MF(Q̂, f) → MF({ \atop Q ⊗_{k} Q′}, f ⊕ f′)

given by

P ↦ P ⊗_{MF} X

on objects and

α ↦ α ⊗ id_{X}

on morphisms is a quasi-equivalence.

Proof. By Theorems 4.11, 5.1, and 5.7 in [Dyc11], the inclusion

End_{MF(Q̂′,f′)}(X) ↪ MF(Q̂′, f′)

is a Morita equivalence. We have a chain of Morita equivalences

MF(Q̂, f) ⊗_{k} k ↪ MF(Q̂, f) ⊗_{k} End_{MF(Q̂′,f′)}(X) ↪ MF(Q̂, f) ⊗_{k} MF(Q̂′, f′).

Composing with ŜT_{MF}, Proposition 2.14 yields a Morita equivalence

MF(Q̂, f) → MF({ \atop Q ⊗_{k} Q′}, f ⊕ f′).

This composition is clearly the functor K_{X}; thus, K_{X} is a Morita equivalence. Since both MF(Q̂, f) and MF({ \atop Q ⊗_{k} Q′}, f ⊕ f′) are triangulated in the dg sense by Lemma 5.6 in [Dyc11], we may apply Theorem 3.2.1 in [Toë11] and Theorem 1.2.10 in [Hov07] to conclude that K_{X} is a quasi-equivalence.

To deduce a version of Knörrer periodicity for isolated hypersurface singularities, assume k to be an algebraically closed field such that char(k) ≠ 2, set Q′ = k[u, v] and f′ = u^{2} + v^{2}, and take X to be the matrix factorization

k[u, v] \underleftarrow{−}=\overrightarrow{−}_{u−iv}^{u+iv} k[u, v].

This is the approach taken in Section 5.3 of [Dyc11]. We point out that k is not assumed to be algebraically closed in Proposition 2.20, and no assumptions on the characteristic of k are made, either. In particular, we may use Proposition 2.20 to prove an 8-periodic version of Knörrer periodicity over ℝ (this result implies Theorem 1.2 from the introduction):

Theorem 2.21. Suppose Q is an ℝ-algebra. Let f ∈ Q, and suppose Q/(f) is IHS. Set Q′ := ℝ[u_{1}, …, u_{8}]. Then there exists a matrix factorization X of −u_{1}^{2} − ⋯ − u_{8}^{2} over Q′ such that the dg functor

MF(Q̂, f) → MF({ \atop Q ⊗_{ℝ} Q′}, f − u_{1}^{2} − ⋯ − u_{8}^{2})

given by

P ↦ P ⊗_{MF} X

on objects and

α ↦ α ⊗ id_{X}

on morphisms is a quasi-equivalence.

Remark 2.22. One may replace −u_{1}^{2} − ⋯ − u_{8}^{2} with u_{1}^{2} + ⋯ + u_{8}^{2} and obtain a similar result; the proof is the same.

Proof. Set q := −u_{1}^{2} − ⋯ − u_{8}^{2} ∈ Q′. We equip the matrix algebra Mat_{16}(ℝ) of 16×16 of matrices over ℝ with a ℤ/2ℤ-grading in the following way: A = (a_{ij}) is homogeneous of even degree if a_{ij} = 0 whenever i + j is odd, and A is homogeneous of odd degree if a_{ij} = 0 whenever i + j is even. By Proposition V.4.2 in [Lam05],

Cliff_{ℝ}(q) ≅ Mat_{16}(ℝ)

as ℤ/2ℤ-graded algebras. In particular, by Theorem 2.17,

[MF(Q̂′, q)] ≅ mod_{ℤ/2ℤ}(Mat_{16}(ℝ)),

where the right hand side is the category of finitely generated ℤ/2ℤ-graded left Mat_{16}(ℝ)-modules. Let M ∈ mod_{ℤ/2ℤ}(Mat_{16}(ℝ)) be the module consisting of elements of Mat_{16}(ℝ) with nonzero entries only in the first column. Recall that, by Remark 2.18, the canonical map

[MF(Q′, q)] → [MF(Q̂′, q)]

is an equivalence; let X be an object of [MF(Q′, q)] corresponding to M. Let 𝔪 := (u_{1}, …, u_{8}) ⊆ Q′, and let E_{q} ∈ MF(Q′_{𝔪}, q) be as in Section 2.1.3. Notice that, by Proposition 2.19, (X ⊕ X[1])^{⊕8} ≅ E_{q} in [MF(Q′_{𝔪}, q)]. In particular, it follows from Theorems 4.1 and 4.11 of [Dyc11] that X is a compact generator of [MF^{∞}(Q′, q)]. Since End_{Mat_{16}(ℝ)}(M) ≅ ℝ as ℤ/2ℤ-graded ℝ-algebras, where ℝ is concentrated in even degree, we have H^{0}(End_{MF}(X)) ≅ ℝ. We now show H^{1}(End_{MF}(X)) = 0. By Section 5.5 of [Dyc11], H^{0}(End_{MF}(E_{q}))⊕H^{1}(End_{MF}(E_{q})) is isomorphic, as a ℤ/2ℤ-graded ℝ-vector space, to Cliff_{ℝ}(q), and so H^{1}(End_{MF}(E_{q})) has rank 128. Also, we have isomorphisms

H^{1}(End_{MF}(E_{q})) ≅ H^{1}(End_{MF}((X ⊕ X[1])^{⊕8}))
≅ H^{0}(End_{MF}(X))^{128} ⊕ H^{1}(End_{MF}(X))^{128}.

Thus, H^{1}(End_{MF}(X)) = 0, and so the inclusion

ℝ ↪ End_{MF}(X)

is a quasi-isomorphism. Now apply Proposition 2.20.

Remark 2.23. Theorem 2.21 implies the existence of a Knörrer-type periodicity for matrix factorizations over ℝ of period at most 8. We point out that the period is exactly 8, since the Brauer-Wall group of ℝ is the cyclic group ℤ/8ℤ generated by the class of Cliff_{ℝ}(x^{2}).

3 Matrix factorizations and the topological K-theory of the Milnor fiber

We have demonstrated that matrix factorization categories exhibit 2- and 8periodic versions of Knörrer periodicity over ℂ and ℝ, respectively. This pattern resembles Bott periodicity in topological K-theory; the goal of this section is to explain this resemblance. We give a rough sketch of our approach. The classical link between the periodicity of Clifford algebras up to ℤ/2ℤ-graded Morita equivalence and Bott periodicity in topological K-theory is the Atiyah-Bott-Shapiro construction, which first appeared in Part III of [ABS64] (and, in fact, a proof of Bott periodicity using Clifford algebras is provided by Wood in [Woo66]). Loosely speaking, the Atiyah-Bott-Shapiro construction is a way of mapping a finitely generated ℤ/2ℤ-graded module over a real or complex Clifford algebra to a class in the K-theory of a sphere. Composing the Buchweitz-Eisenbud-Herzog equivalence (Theorem 2.17) with the Atiyah-Bott-Shapiro construction, we have a way of assigning a class in the topological K-theory of a sphere to a matrix factorization of a non-degenerate quadratic form over ℝ or ℂ:

mf’s of real/complex quadratics K-theory of spheres

ABS ∘ BEH

The idea is to lift this composition; that is, we wish to associate a space X_{f} to a real or complex polynomial f and construct a map from matrix factorizations of f to the topological K-theory of X_{f} so that the diagram

mf’s of real/complex quadratics K-theory of spheres mf’s of real/complex polynomials K-theory of spaces of the form X_{f}

ABS ∘ BEH

commutes. It turns out that the right choice of X_{f} is the Milnor fiber (positive or negative Milnor fiber) associated to the complex (real) polynomial. We begin this section with discussions of known results concerning the Milnor fiber and relative topological K-theory. Then, using the work of Atiyah-Bott-Shapiro in [ABS64] as a guide, we will complete the above diagram, and we will use the bottom arrow to explain a precise sense in which Knörrer periodicity and Bott periodicity are compatible phenomena.

3.1 The real and complex Milnor fibers

Let f ∈ ℂ[x_{1}, …, x_{n}], and suppose f(0) = 0. We begin this section by describing the construction of the Milnor fiber associated to f, following the exposition in Section 1 of [BvS12]. We then discuss various properties of the Milnor fiber that we will make use of later on.

3.1.1 Construction of the Milnor fibration and some properties of the Milnor fiber

For ϵ > 0, define B_{ϵ} to be the closed ball centered at the origin of radius ϵ in ℂ^{n}, and for δ > 0, set D_{δ}^{∗} to be the open punctured disk centered at the origin in ℂ of radius δ. Choose ϵ > 0 so that, for 0 < ϵ′ ⩽ ϵ, ∂B_{ϵ′} intersects f^{−1}(0) transversely. Upon choosing such a number ϵ, choose δ ∈ (0, ϵ) such that f^{−1}(t) intersects ∂B_{ϵ} transversely for all t ∈ D_{δ}^{∗}. Then the map

ψ : B_{ϵ} ∩ f^{−1}(D_{δ}^{∗}) → D_{δ}^{∗}

given by ψ(x) = f(x) is a locally trivial fibration. The map ψ depends, of course, on our choices of ϵ and δ. However, if ϵ′, δ′ is another pair of positive numbers satisfying the above conditions, the fibration associated to these choices is fiber homotopy equivalent to the one above (see Definition 1.5 in Chapter 3, §1 of [Dim92] for the definition of a fiber homotopy equivalence). We are thus justified in calling ψ the Milnor fibration associated to f.

Remark 3.1. The Milnor fibration was originally introduced in [Mil68]. The above construction is not the same as the construction of the Milnor fibration in [Mil68] and is due to Lê ([Lê76]). The two constructions yield fiber homotopy equivalent fibrations ([Dim92] Chapter 3, §1).

Choose t ∈ D_{δ}^{∗}. The fiber of ψ over t is called the Milnor fiber of f over t; we will denote it by F_{f}. F_{f} is independent of our choices of ϵ, δ, and t up to homotopy equivalence, so we suppress these choices in the notation, and we will often refer to F_{f} as just the Milnor fiber of f. However, these choices will be significant at several points later on. If ℂ[x_{1}, …, x_{n}]_{(x_{1},...,x_{n})}/(f) is IHS (see Definition 2.9), set

μ := dim_{ℂ} \frac{ℂ[[x_{1}, …, x_{n}]]}{(\frac{∂f}{∂x_{1}} , …, \frac{∂f}{∂x_{n}})} < ∞.

Theorem 3.2 ([Mil68] Theorem 6.5). If ℂ[x_{1}, …, x_{n}]_{(x_{1},...,x_{n})}/(f) is IHS, F_{f} is homotopy equivalent to a wedge sum of μ copies of S^{n−1}.

Remark 3.3. Since ψ restricts to a fibration over a circle, F_{f} comes equipped with a monodromy homeomorphism

h : F_{f} ⟶^{≅} F_{f}.

3.1.2 The Sebastiani-Thom homotopy equivalence

We recall the definition of the join of two topological spaces:

Definition 3.4. Let X and Y be compact Hausdorff spaces. The join of X and Y, denoted X ∗ Y, is the quotient of X × Y × I by the relations

(x_{1}, y, 0) ∼ (x_{2}, y, 0)
(x, y_{1}, 1) ∼ (x, y_{2}, 1)

equipped with the quotient topology.

Remark 3.5. We express the cone CX over a compact Hausdorff space X explicitly as the quotient of

X × [0, 1]

by the relation

(x_{1}, 0) ∼ (x_{2}, 0)

for all x_{1}, x_{2} ∈ X. When X and Y are compact Hausdorff, X ∗ Y is homeomorphic to (CX × Y) ∪ (X × CY) ⊆ CX × CY; here, we identify X and Y with the subsets X × {1} and Y × {1} of CX and CY, respectively. By [Bro06] 5.7.4, an explicit homeomorphism

CX × CY ⟶^{≅} C(X ∗ Y)

is given by

(x, t, y, t′) ↦ ((x, y, \frac{t}{2t′}), t′), if t′ ⩾ t, t′ ≠ 0

(x, t, y, t′) ↦ ((x, y, 1 − \frac{t′}{2t}), t), if t ⩾ t′, t ≠ 0

(x, 0, y, 0) ↦ ((x, y, 0), 0),

and this map restricts to a homeomorphism

w : (CX × Y) ∪ (X × CY) ⟶^{≅} X ∗ Y.

Now, suppose f ∈ ℂ[x_{1}, …, x_{n}], f′ ∈ ℂ[y_{1}, …, y_{m}], and f(0) = 0 = f′(0). Assume R := ℂ[x_{1}, …, x_{n}]_{(x_{1},...,x_{n})}/(f) and R′ := ℂ[y_{1}, …, y_{m}]_{(y_{1},...,y_{m})}/(f′) are IHS (see Definition 2.9). Let f ⊕ f′ denote the sum of f and f′ thought of as an element of ℂ[x_{1}, …, x_{n}, y_{1}, …, y_{m}]. The following theorem of Sebastiani-Thom relates the Milnor fibers of f, f′, and f ⊕ f′:

Theorem 3.6 ([ST71]). There is a homotopy equivalence

ST : F_{f} ∗ F_{f′} → F_{f⊕f′}

that is compatible with monodromy; that is, the square

F_{f} ∗ F_{f′} ⟶^{ST} F_{f⊕f′}
h∗h↓  ^{h}↓
F_{f} ∗ F_{f′} ⟶^{ST} F_{f⊕f′}

commutes up to homotopy.

Remark 3.7. By results of Oka in [Oka73], the assumption in Theorem 3.6 that R and R′ are IHS is not necessary if f and f′ are quasi-homogeneous. We refer the reader to Section 2.7 of [AGZV12] and §3 of Chapter 3 in [Dim92] for discussions related to Theorem 3.6. We now exhibit an explicit map realizing the homotopy equivalence in Theorem 3.6, following Section 2.7 of [AGZV12]. Choose real numbers ϵ′′, δ′′, such that the map

B_{ϵ′′} ∩ (f ⊕ f′)^{−1}(D_{δ′′}^{∗}) → D_{δ′′}^{∗}

given by x ↦ (f ⊕ f′)(x) is a locally trivial fibration, as above. Similarly, choose ϵ, δ and ϵ′, δ′, as well as t′′ ∈ D_{δ′′}^{∗}, so that the analogous maps

B_{ϵ} ∩ f^{−1}(D_{δ}^{∗}) → D_{δ}^{∗}
B_{ϵ′} ∩ (f′)^{−1}(D_{δ′}^{∗}) → D_{δ′}^{∗}

are locally trivial fibrations, and also so that

(a) ϵ, ϵ′ are sufficiently small so that B_{ϵ} × B_{ϵ′} ⊆ B_{ϵ′′}.

(b) |t′′| < min{δ, δ′}.

Set F_{f}, F_{f′}, and F_{f⊕f′} to be the Milnor fibers of f, f′, and f ⊕ f′ over t′′. Applying Lemma 2.10 in [AGZV12], choose a continuous map

H : CF_{f} → B_{ϵ}

such that

• H(x, 1) = x ∈ F_{f} ⊆ B_{ϵ},

• H(−, s) : F_{f} → B_{ϵ} maps into the Milnor fiber B_{ϵ} ∩f^{−1}(st′′) for s ∈ (0, 1), and

• H(x, 0) = 0 for all x ∈ F_{f}

Example 3.8. If f is quasi-homogeneous of degree d with weights w_{1}, …, w_{d}, such a map H may be given by

(x, s) ↦ (s^{\frac{w_{1}}{d}} x_{1}, …, s^{\frac{wn}{d}} x_{n}).

Notice our assumption that R is IHS is not needed here.

Choose H′ similarly for the Milnor fiber F_{f′}. By the discussion on pages 54-55 of [AGZV12] and Remark 3.5, there is a homotopy equivalence

g : CF_{f} × F_{f′} ∪ F_{f} × CF_{f′} → F_{f⊕f′}

given by

(x, s, y, s′) ↦ (H(x, \frac{1 + s − s′}{2} ), H′(y, \frac{1 − s + s′}{2} )).

Composing, one has a homotopy equivalence

g ∘ w^{−1} : F_{f} ∗ F_{f′} → F_{f⊕f′},

where w is the homeomorphism in Remark 3.5. The homotopy equivalence g ∘ w^{−1} enjoys the same properties as the map ST in Theorem 2.14.

Remark 3.9. g extends to a homotopy equivalence of pairs

G : (CF_{f} × CF_{f′}, CF_{f} × F_{f′} ∪ F_{f} × CF_{f′}) → (B_{ϵ′′}, F_{f⊕f′})

that maps a point (x, s, y, s′) to

(H(x, \frac{s}{2}), H′(y, \frac{2s′ − s}{2}), if s ⩽ s′, s′ ≠ 0

(H(x, \frac{2s − s′}{2} ), H′(y, \frac{s′}{2} ), if s′ ⩽ s, s ≠ 0

0, if s = 0 = s′.

Remark 3.10. When f and f′ are quasi-homogeneous (and R, R′ are not necessarily IHS), we may use Example 3.8 to build a homotopy equivalence g : CF_{f} × F_{f′} ∪ F_{f} × CF_{f′} → F_{f⊕f′} in the same way as above ([Dim92] Chapter 3, Remark 3.19’).

3.1.3 An analogue of the Milnor fibration over ℝ

Now, suppose f ∈ ℝ[x_{1}, …, x_{n}] and f(0) = 0. One may construct a locally trivial fibration

ψ : B_{ϵ} ∩ f^{−1}((−δ, 0) ∪ (0, δ)) → (−δ, 0) ∪ (0, δ)

for some ϵ > 0 and δ such that 0 < δ << ϵ in the same way as above, where B_{ϵ} is now the closed ball of radius ϵ centered at the origin in ℝ^{n}. But now, fibers over (−δ, 0) and (0, δ) need not be homotopy equivalent. For instance, if f = x_{1}^{2} + ⋯ + x_{n}^{2}, the positive fibers of ψ are homeomorphic to S^{n−1}, while the negative fibers are empty. Choose t ∈ (0, δ) and t′ ∈ (−δ, 0). The fiber of ψ over t is called the positive Milnor fiber of f over t, denoted by F_{f}^{+}, and the fiber of ψ over t′ is called the negative Milnor fiber of f over t′, denoted F_{f}^{−}. As in the complex case, F_{f}^{+} and F_{f}^{−} are independent of our choices of ϵ, δ, t, and t′ up to homotopy equivalence, so we suppress these choices in our notation, and we will often refer to F_{f}^{+} and F_{f}^{−} as just the positive and negative Milnor fibers of f. The topology of the real Milnor fibers is more complicated than that of the complex Milnor fiber. However, there is a version of Theorem 3.6 for real Milnor fibers of quasi-homogeneous polynomials. Suppose f ∈ ℝ[x_{1}, …, x_{n}], f′ ∈ ℝ[y_{1}, …, y_{m}] are quasi-homogeneous and nonconstant. If F_{f}^{+} and F_{f′}^{+} are nonempty, there is a homotopy equivalence

F_{f}^{+} ∗ F_{f′}^{+} → F_{f⊕f′}^{+}

([DP92] Remark 11). Moreover, the homotopy equivalence may be constructed as in Remark 3.10; that is, one has a homotopy equivalence of pairs

G : (CF_{f}^{+} × CF_{f′}^{+}, CF_{f}^{+} × F_{f′}^{+} ∪ F_{f}^{+} × CF_{f′}^{+}) → (B_{ϵ′′}, F_{f⊕f′}^{+}).

Since F_{f}^{−} = F_{−f}^{+}, one has a similar result for negative Milnor fibers.

3.2 Relative topological K-theory

We introduce some facts concerning relative topological K-theory. All of the results in this section are essentially due to Atiyah-Bott-Shapiro in [ABS64], but we modify their exposition at several points to suit our purposes. Let X be a compact topological space, and let Y be a closed subspace of X such that there exists a homotopy equivalence of pairs between (X, Y) and a finite CW pair; we construct a category C_{1}(X, Y) from (X, Y) in the following way:

• An object of C_{1}(X, Y) is a pair of real vector bundles V_{1}, V_{0} over X equipped with an isomorphism

V_{1}|_{Y} ⟶^{σ} V_{0}|_{Y} .

Denote objects of C_{1}(X, Y) by (V_{1}, V_{0}; σ).

• Morphisms in C_{1}(X, Y) are pairs of morphisms of vector bundles over X

α_{1} : V_{1} → V′_{1}, α_{0} : V_{0} → V′_{0}

such that the following diagram of maps of vector bundles over Y commutes:

V_{1}|_{Y} ⟶^{σ} V_{0}|_{Y}
α_{1}|_{Y} ↓  ^{α_{0}|_{Y}} ↓
V′_{1}|_{Y} ⟶^{σ′} V′_{0}|_{Y}

We write morphisms in C_{1}(X, Y) as ordered pairs (α_{1}, α_{0}). Remark 3.11. The reason for the subscript in the notation C_{1}(X, Y) is that, for any n ⩾ 1, one may similarly build a category C_{n}(X, Y) with objects given by ordered (n + 1)-tuples of vector spaces on X whose restrictions to Y fit into an exact sequence (cf. [ABS64] §7).

Remark 3.12. We will work with real vector bundles throughout this section; however, there is an analogous version of every result in this section for complex vector bundles. The following facts about C_{1}(X, Y) are easily verified:

• If (V_{1}, V_{0}; σ) and (V′_{1}, V′_{0}; σ′) are objects in C_{1}(X, Y), (V_{1}⊕V′_{1}, V_{0}⊕V′_{0}, σ⊕ σ′) is their coproduct.

• C_{1}(X, Y) is an additive category.

• A map g : (X_{1}, Y_{1}) → (X_{2}, Y_{2}) of pairs of spaces as above induces a

functor

g^{∗} : C_{1}(X_{2}, Y_{2}) → C_{1}(X_{1}, Y_{1})

via pullback.

• A morphism (α_{1}, α_{0}) in C_{1}(X, Y) is an isomorphism (resp. monomorphism, epimorphism) if and only if α_{1} and α_{0} are isomorphisms (resp. monomorphisms, epimorphisms) of vector bundles over X.

We shall call an object of C_{1}(X, Y) elementary if it is isomorphic to an object of the form (V, V; id_{V |_{Y}}). It is easy to check that (V_{1}, V_{0}; σ) is elementary if and only if σ can be extended to an isomorphism σ̃ : V_{1} → V_{0}. If V and V′ are objects in C_{1}(X, Y), we will say V ∼ V′ if and only if there exist elementary objects E, E′ such that

V ⊕ E ≅ V′ ⊕ E′.

The relation ∼ is an equivalence relation. Let L_{1}(X, Y) denote the commutative monoid of equivalence classes under ∼ with operation ⊕. We shall denote by [V_{1}, V_{0}; σ] the class in L_{1}(X, Y) represented by (V_{1}, V_{0}, σ). Remark 3.13. Let (X_{1}, Y_{1}), (X_{2}, Y_{2}) be pairs of spaces as above, and let g : (X_{1}, Y_{1}) → (X_{2}, Y_{2}) be a map of pairs. Then the functor

g^{∗} : C_{1}(X_{2}, Y_{2}) → C_{1}(X_{1}, Y_{1})

applied to an elementary object is again elementary. Hence, g^{∗} induces a map of monoids

L_{1}(X_{2}, Y_{2}) → L_{1}(X_{1}, Y_{1}).

The main reason we are interested in the monoid L_{1}(X, Y) is the following result:

Proposition 3.14 (Atiyah-Bott-Shapiro, [ABS64]). There exists a unique natural homomorphism

χ : L_{1}(X, Y) → KO^{0}(X, Y)

which, when Y = ∅, is given by

χ(E) = [V_{0}] − [V_{1}].

Moreover, χ is an isomorphism.

In particular, L_{1}(X, Y) is an abelian group. Let (X, Y), (X′, Y′) be pairs as above. We conclude this section by exhibiting a product map

L_{1}(X, Y) ⊗ L_{1}(X′, Y′) → L_{1}(X × X′, X × Y′ ∪ Y × X′)

that agrees, via χ, with the usual product on relative K-theory. Let V = (V_{1}, V_{0}; σ) ∈ Ob(C_{1}(X, Y)) and V′ = (V′_{1}, V′_{0}; σ′) ∈ Ob(C_{1}(X′, Y′)). By Proposition 10.1 in [ABS64], we may lift σ, σ′ to maps σ̃, σ′ of bundles over X and X′, respectively. Thinking of

0 → V_{1} \underrightarrow{σ̃} V_{0} → 0
0 → V′_{1} \underrightarrow{σ′} V′_{0} → 0

as complexes of bundles with V_{1}, V′_{1} in degree 1 and V_{0}, V′_{0} in degree 0, we may take their tensor product

0 → V_{1} ⊗ V′_{1} ⟶^{τ_{2}} (V_{1} ⊗ V′_{0}) ⊕ (V_{0} ⊗ V′_{1}) ⟶^{τ_{1}} V_{0} ⊗ V′_{0} → 0,

where

τ_{1} = (σ̃ ⊗ id_{V′_{0}} id_{V_{0}} ⊗ σ′)

τ_{2} = (\begin{matrix}−id_{V_{1}} ⊗ σ′ \\ σ̃ ⊗ id_{V′_{1}} \end{matrix})

The result is a complex of vector bundles over X × X′ that is exact upon restriction to X × Y′ ∪ Y × X′. Choose a splitting π of τ_{2}|_{X×Y′∪Y ×X′}. Then,

[(V_{1} ⊗ V′_{0}) ⊕ (V_{0} ⊗ V′_{1}), (V_{0} ⊗ V′_{0}) ⊕ (V_{1} ⊗ V′_{1}); (\begin{matrix}τ_{1}|_{X×Y′∪Y ×X′} \\ π \end{matrix})]

is an element of L_{1}(X × X′, X × Y′ ∪ Y × X′). One may define monoids L_{n}(X, Y) involving longer sequences of bundles; see [ABS64] Definition 7.1 for details. Denote elements of L_{n}(X, Y) by

[V_{n}, …, V_{0}; σ_{n}, …, σ_{1}].

There is a map

j_{n} : L_{1}(X, Y) → L_{n}(X, Y)

given by

[V_{1}, V_{0}; σ] ↦ [0, …, 0, V_{1}, V_{0}; 0, …, 0, σ],

and, by Proposition 7.4 in [ABS64], j_{n} is an isomorphism for all n. We will need the following technical lemma:

Lemma 3.15. Let (X, Y) be a pair as above, and let [V_{2}, V_{1}, V_{0}; σ_{2}, σ_{1}] ∈ L_{2}(X, Y). If π is a splitting of σ_{2},

j_{2}([V_{1}, V_{0} ⊕ V_{2}; (\begin{matrix}σ_{1} \\ π \end{matrix})]) = [V_{2}, V_{1}, V_{0}; σ_{2}, σ_{1}].

Proof. First, suppose dim(V_{1}) > dim(V_{2}) + dim(X). Apply Lemma 7.2 in [ABS64] to construct a monomorphism

h : V_{2} → V_{1}

that extends σ_{2}. By the proof of Lemma 7.3 in [ABS64],

j_{2}([coker(h), V_{0}; \overline{σ_{1}}]) = [V_{2}, V_{1}, V_{0}; σ_{2}, σ_{1}],

and so

j_{2}([coker(h) ⊕ V_{2}, V_{0} ⊕ V_{2}; A]) = [V_{2}, V_{1}, V_{0}; σ_{2}, σ_{1}],

where

A = (\begin{matrix}\overline{σ_{1}} & 0 \\ 0 & id_{V_{2}}|_{Y} \end{matrix} ) .

Hence, it suffices to show

[coker(h) ⊕ V_{2}, V_{0} ⊕ V_{2}; A] = [V_{1}, V_{0} ⊕ V_{2}; (\begin{matrix}σ_{1} \\ π \end{matrix})]

Choose a splitting s of h, and let

p : V_{1} → coker(h)

denote the canonical map. Then we have an isomorphism

(\begin{matrix}p \\ s \end{matrix}) : V_{1} → coker(h) ⊕ V_{2}.

Since s|_{Y} is a splitting of σ_{2}, we also have an isomorphism

(\begin{matrix}σ_{1} \\ s|_{Y} \end{matrix} ) : V_{1}|_{Y} → V_{0}|_{Y} ⊕ V_{2}|_{Y} .

We have a commutative square

[figure: V_{1}|_{Y}; ^{(\begin{matrix}σ_{1} \\ s|_{Y} \end{matrix} )}; _{(\begin{matrix}p|_{Y} \\ s|_{Y} \end{matrix} )}; V_{0}|_{Y} ⊕ V_{2}|_{Y}; _{id_{V_{0}|_{Y} ⊕V_{2}|_{Y}}}; coker(h)|_{Y} ⊕ V_{2}|_{Y}; A; V_{0}|_{Y} ⊕ V_{2}|_{Y}]

Thus,

[coker(h) ⊕ V_{2}, V_{0} ⊕ V_{2}; A] = [V_{1}, V_{0} ⊕ V_{2}; (\begin{matrix}σ_{1} \\ s|_{Y} \end{matrix} )].

Notice that we have an object

[V_{1} × I, (V_{0} ⊕ V_{2}) × I; t (\begin{matrix}σ_{1} \\ s|_{Y} \end{matrix} ) + (1 − t) (\begin{matrix}σ_{1} \\ π \end{matrix})]

in C_{1}(X × I, Y × I) whose restrictions to X × {0} and X × {1} are [V_{1}, V_{0} ⊕ V_{2}; (\begin{matrix}σ_{1} \\ π \end{matrix})] and [V_{1}, V_{0} ⊕ V_{2}; (\begin{matrix}σ_{1} \\ s|_{Y} \end{matrix} )], respectively. It now follows from Proposition 9.2 in [ABS64] that

[V_{1}, V_{0} ⊕ V_{2}; (\begin{matrix}σ_{1} \\ s|_{Y} \end{matrix} )] = [V_{1}, V_{0} ⊕ V_{2}; (\begin{matrix}σ_{1} \\ π \end{matrix})].

This finishes the case where dim(V_{1}) > dim(V_{2}) + dim(X). For the general case, choose a bundle E such that

dim(E) + dim(V_{1}) > dim(V_{2}) + dim(X).

Define

U := [V_{2}, V_{1} ⊕ E, V_{0} ⊕ E; (\begin{matrix}σ_{2} \\ 0 \end{matrix}) , (\begin{matrix}σ_{1} & 0 \\ 0 & id_{E}|_{Y} \end{matrix} )],

U′ := [V_{1} ⊕ E, V_{0} ⊕ E ⊕ V_{2}; (\begin{matrix}σ_{1} & 0 \\ 0 & id_{E}|_{Y} \\ π & 0 \end{matrix} )]

Notice that

[V_{2}, V_{1}, V_{0}; σ_{2}, σ_{1}] = U,

and

[V_{1}, V_{0} ⊕ V_{2}; (\begin{matrix}σ_{1} \\ π \end{matrix})] = U′,

so that it suffices to show that j(U′) = U. Since (π 0) is a splitting of (\begin{matrix}σ_{2} \\ 0 \end{matrix}), this follows from the case we have already considered. Now, the pairing

L_{1}(X, Y) ⊗ L_{1}(X′, Y′) → L_{1}(X × X′, X × Y′ ∪ Y × X′)

described in Proposition 10.4 of [ABS64] is given by sending a simple tensor [V_{1}, V_{0}; σ] ⊗ [V′_{1}, V′_{0}; σ′]

Documenta Mathematica 21 (2016) 1459–1501

to

j_{2}^{−1}([V_{1} ⊗ V′_{1}, (V_{1} ⊗ V′_{0}) ⊕ (V_{0} ⊗ V′_{1}), V_{0} ⊗ V′_{0}; τ_{2}|_{X×Y′∪Y ×X′}, τ_{1}|_{X×Y′∪Y ×X′}]);

this follows from the proof of Proposition 10.4. Thus, by Lemma 3.15, the map

Ob(C_{1}(X, Y)) × Ob(C_{1}(X′, Y′)) → L_{1}(X × X′, X × Y′ ∪ Y × X′)

given by

(V, V′) ↦ [(V_{1} ⊗ V′_{0}) ⊕ (V_{0} ⊗ V′_{1}), (V_{0} ⊗ V′_{0}) ⊕ (V_{1} ⊗ V′_{1}); (\begin{matrix}τ_{1}|_{X×Y′∪Y ×X′} \\ π \end{matrix})]

determines

(a) a well-defined pairing on Ob(C_{1}(X, Y))×Ob(C_{1}(X′, Y′)) up to our choices of liftings σ̃, σ̃′ and splitting π, and

(b) a pairing

L_{1}(X, Y) ⊗ L_{1}(X′, Y′) → L_{1}(X × X′, X × Y′ ∪ Y × X′)

that coincides with the pairing in Proposition 10.4 of [ABS64].

Let [V],[V′] denote the classes represented by V and V′ in L_{1}(X, Y) and L_{1}(X′, Y′). Define

[V] ⊗_{L_{1}} [V′] := [(V_{1} ⊗ V′_{0}) ⊕ (V_{0} ⊗ V′_{1}), (V_{0} ⊗ V′_{0}) ⊕ (V_{1} ⊗ V′_{1}); (\begin{matrix}τ_{1}|_{X×Y′∪Y ×X′} \\ π \end{matrix})].

Remark 3.16. By Proposition 10.4 in [ABS64] and the above remarks,

χ([V]) ⊗ χ([V′]) = χ([V] ⊗_{L_{1}} [V′]).

3.3 A generalized Atiyah-Bott-Shapiro construction applied to matrix factorizations

In this section, we construct the maps ϕ_{f}^{ℂ} and ϕ_{f}^{ℝ} described in the introduction. We begin with a discussion of the Atiyah-Bott-Shapiro construction ([ABS64] Part III). Following Atiyah-Bott-Shapiro, we work with real Clifford algebras and KO-theory, and we point out that one may perform a similar construction involving complex Clifford algebras and KU-theory.

3.3.1 The Atiyah-Bott-Shapiro construction

Define

q_{n} := −x_{1}^{2} − ⋯ − x_{n}^{2} ∈ ℝ[x_{1}, …, x_{n}]

for all n ⩾ 1, and set C_{n} := Cliff_{ℝ}(q_{n}). We also set C_{0} := ℝ; we will think of C_{0} as a ℤ/2ℤ-graded algebra concentrated in degree 0. Let M(C_{n}) denote the free abelian group generated by isomorphism classes of finitely-generated, indecomposable ℤ/2ℤ-graded left C_{n}-modules. There are evident injective maps

i_{n} : C_{n} → C_{n+1}

for all n ⩾ 0; these injections induce homomorphisms

i_{n}^{∗} : M(C_{n+1}) → M(C_{n})

via restriction of scalars. Set

A_{n} := M(C_{n})/i_{n}^{∗}(M(C_{n+1})).

Define D^{n} to be the closed disk of radius 1 in ℝ^{n}. An important special case of the classical Atiyah-Bott-Shapiro construction is the group isomorphism

α_{n} : A_{n} ⟶^{≅} L_{1}(D^{n}, ∂D^{n})

that appears in [ABS64] Theorem 11.5. α_{n} is defined as follows: let M = M_{1} ⊕ M_{0} be a finitely generated ℤ/2ℤ-graded left C_{n}-module. We use the ℝ-vector spaces M_{1} and M_{0} to construct real vector bundles over D^{n}:

V_{1} := D^{n} × M_{1}
V_{0} := D^{n} × M_{0}

and we define a map

σ : V_{1} → V_{0}

given by (x, m) ↦ (x, x · m), where · denotes the action of C_{n} on M. Here, we are thinking of D^{n} ⊆ ℝ^{n} as a subset of C_{n}. Notice that σ restricts to an isomorphism of bundles over ∂D^{n}. Thus, we have constructed an element [V_{1}, V_{0}; σ] ∈ L_{1}(D^{n}, ∂D^{n}). Define

α_{n}([M]) = [V_{1}, V_{0}; σ].

We refer the reader to [ABS64] for verification that the mapping

[M] ↦ [V_{1}, V_{0}; σ]

is well-defined on the quotient A_{n} and determines an isomorphism.

3.3.2 A more general construction

Let f ∈ (x_{1}, …, x_{n}) ⊆ Q := ℝ[x_{1}, …, x_{n}]. Choose real numbers ϵ, δ, and t such that ϵ > 0, 0 < δ << ϵ, and t ∈ (−δ, 0) in such a way that we may construct a negative Milnor fiber F_{f}^{−} as in Section 3.1.3. Denote by B_{ϵ} the closed ball of radius ϵ in ℝ^{n} centered at the origin. We now construct a map

Ob(MF(Q, f)) → L_{1}(B_{ϵ}, F_{f}^{−})

that (a) recovers the Atiyah-Bott-Shapiro construction via the Buchweitz-

Eisenbud-Herzog equivalence (Theorem 2.17) when f = q_{n}, and

(b) descends to a group homomorphism

K_{0}[MF(Q, f)] → L_{1}(B_{ϵ}, F_{f}^{−}).

We emphasize that a similar construction involving complex polynomials and their Milnor fibers may be performed mutatis mutandis. One may also perform the following construction using the positive Milnor fiber F_{f}^{+} of f.

Let P = (P_{1} \underleftarrow{−}=\overrightarrow{−}_{d_{0}}^{d_{1}} P_{0}) be a matrix factorization of f over Q. Denote by C(B_{ϵ})

the ring of ℝ-valued continuous functions on B_{ϵ}. Applying extension of scalars along the inclusion

Q ↪ C(B_{ϵ}),

we obtain a map

P_{1} ⊗_{Q} C(B_{ϵ}) ⟶^{d_{1}⊗id} P_{0} ⊗_{Q} C(B_{ϵ})

of finitely generated projective C(B_{ϵ})-modules. The category of real vector bundles over B_{ϵ} is equivalent to the category of finitely generated projective C(B_{ϵ})-modules; on objects, the equivalence sends a bundle to its space of sections. Let

V_{1} ⟶^{d_{1}} V_{0}

be a map of real vector bundles over B_{ϵ} corresponding to the above map d_{1} ⊗id under this equivalence. Since d_{1} ∘ d_{0} = f · id_{P_{0}} and d_{0} ∘ d_{1} = f · id_{P_{1}}, and since the restriction of the polynomial f, thought of as a map ℝ^{n} → ℝ, to F_{f}^{−} = B_{ϵ} ∩ f^{−1}(t) is constant with value t ≠ 0, d_{1}|_{F_{f}^{−}} is an isomorphism of

vector bundles on F_{f}^{−}. Its inverse is the restriction to F_{f}^{−} of the map V_{0} → V_{1} determined by

P_{0} ⊗_{Q} C(B_{ϵ}) ⟶^{\frac{1}{t} (d_{0}⊗id)} P_{1} ⊗_{Q} C(B_{ϵ}).

Define Φ_{f}^{ℝ}(P_{1} \underleftarrow{−}=\overrightarrow{−}_{d_{0}}^{d_{1}} P_{0}) = (V_{1}, V_{0}; d_{1}|_{F_{f}^{−}}) ∈ Ob(C_{1}(B_{ϵ}, F_{f}^{−})).

Remark 3.17. The map analogous to Φ_{f}^{ℝ} in the setting of polynomials over ℂ and KU-theory appears in [BvS12]; we discuss this in detail in Section 3.3.3. A morphism in Z^{0}MF(Q, f) determines a morphism in C_{1}(B_{ϵ}, F_{f}^{−}) in an obvious way (see Section 2.1.1 for the definition of the category Z^{0}MF(Q, f)). Hence, we have shown:

Proposition 3.18. There is an additive functor

Φ_{f}^{ℝ} : Z^{0}MF(Q, f) → C_{1}(B_{ϵ}, F_{f}^{−})

given, on objects, by

(P_{1} \underleftarrow{−}=\overrightarrow{−}_{d_{0}}^{d_{1}} P_{0}) ↦ [V_{1}, V_{0}; d_{1}|_{F_{f}^{−}}].

In particular, we have a map

Ob(MF(Q, f)) → L_{1}(B_{ϵ}, F_{f}^{−}).

Suppose f = q_{n}. Then ϵ can be chosen to be 1 in the construction of the negative Milnor fiber F_{f}^{−}, and the fiber can be chosen to be exactly S^{n−1} ⊆ ℝ^{n}. Let Iso([MF(Q, f)]) and Iso(mod_{ℤ/2ℤ}(Cliff_{ℝ}(q_{n}))) denote the sets of isomorphism classes of objects in [MF(Q, f)] and mod_{ℤ/2ℤ}(Cliff_{ℝ}(q_{n})). It is easy to check that one has a commutative triangle

\xymatrix{Iso(mod_{ℤ/2ℤ}(Cliff_{ℝ}(q_{n}))) \ar[r]^{ABS} \ar[d]^{[Θ]} & L_{1}(B_{1}, F_{f}^{−}) \\ Iso([MF(Q, f)]) \ar[ur]^{Φ_{f}^{ℝ}}}

where [Θ] denotes the bijection on isomorphism classes induced by the explicit construction Θ of the Buchweitz-Eisenbud-Herzog equivalence (Theorem 2.17) provided in the proof of Theorem 14.7 of [Yos90], and ABS denotes the Atiyah-Bott-Shapiro construction. Hence, our construction recovers the Atiyah-Bott-Shapiro construction via the Buchweitz-Eisenbud-Herzog equivalence when f = q_{n}. Our next goal is to show that Φ_{f}^{ℝ} induces a map on K-theory:

Proposition 3.19. Φ_{f}^{ℝ} induces a group homomorphism

ϕ_{f}^{ℝ} : K_{0}[MF(Q, f)] → L_{1}(B_{ϵ}, F_{f}^{−}).

We will adopt the following notational conventions for the purposes of the proof of Proposition 3.19:

(1) A pair (ϵ, t) is a good pair if ϵ > 0, t < 0, and the map

ψ : B_{ϵ} ∩ f^{−1}((−δ, 0) ∪ (0, δ)) → (−δ, 0) ∪ (0, δ)

from Section 3.1.3 is a locally trivial fibration for some δ > 0 such that

0 < |t| < δ << ϵ.

(2) If (ϵ, t) is a good pair, we denote the negative Milnor fiber B_{ϵ} ∩ f^{−1}(t) by F_{t}^{−}.

We will need the following technical lemma:

Lemma 3.20. Let (ϵ_{1}, t_{1}), (ϵ_{2}, t_{2}) be good pairs. Then there is an isomorphism

g : L_{1}(B_{ϵ_{1}}, F_{t_{1}}^{−}) ⟶^{≅} L_{1}(B_{ϵ_{2}}, F_{t_{2}}^{−})

yielding a commutative triangle

\xymatrix{Ob(MF(Q, f)) \ar[r]^{Φ_{f}^{ℝ}} \ar[d]^{Φ_{f}^{ℝ}} & L_{1}(B_{ϵ_{2}}, F_{t_{2}}^{−}) \\ L_{1}(B_{ϵ_{1}}, F_{t_{1}}^{−}) \ar[ur]^{g}}

Proof. The case where t_{1} = t_{2} is immediate, so we may assume t_{1} ≠ t_{2}. First, suppose ϵ_{1} = ϵ_{2}. Without loss, assume t_{2} < t_{1}. Set F_{[t_{2},t_{1}]}^{−} := f^{−1}([t_{2}, t_{1}]). Since the inclusions

F_{t_{1}}^{−} ↪ F_{[t_{2},t_{1}]}^{−}, F_{t_{2}}^{−} ↪ F_{[t_{2},t_{1}]}^{−}

are homotopy equivalences, the pullback maps

L_{1}(B_{ϵ_{1}}, F_{[t_{2},t_{1}]}) → L_{1}(B_{ϵ_{1}}, F_{t_{1}}), L_{1}(B_{ϵ_{1}}, F_{[t_{2},t_{1}]}) → L_{1}(B_{ϵ_{1}}, F_{t_{2}})

are isomorphisms. We have commuting triangles

\xymatrix{Ob(MF(Q, f)) \ar[r]^{Φ_{f}^{ℝ}} \ar[d]^{Φ_{f}^{ℝ}} & L_{1}(B_{ϵ_{1}}, F_{t_{1}}^{−}) \\ L_{1}(B_{ϵ_{1}}, F_{[t_{2},t_{1}]}^{−}) \ar@{-->}[ur]^{≅}}

for i = 1, 2. It follows that the result holds when ϵ_{1} = ϵ_{2}. For the general case, assume, without loss, that |t_{2}| < |t_{1}|. Then (ϵ_{1}, t_{2}) is also a good pair. By the cases we’ve already considered, the result holds for the pairs (ϵ_{1}, t_{1}) and (ϵ_{1}, t_{2}), and also for the pairs (ϵ_{1}, t_{2}) and (ϵ_{2}, t_{2}). Hence, the result holds for the pairs (ϵ_{1}, t_{1}), (ϵ_{2}, t_{2}).

We now prove Proposition 3.19:

Proof. It is not hard to see that Φ_{f}^{ℝ}(P ⊕ P′) = Φ_{f}^{ℝ}(P) ⊕ Φ_{f}^{ℝ}(P′); we need only show that ϕ_{f}^{ℝ} is well-defined. First, suppose P ≅ 0 in [MF(Q, f)]. Then id_{P} is a boundary in MF(Q, f), and so id_{P} factors through a trivial matrix factorization, by Proposition 2.4. Write P = (P_{1} \underleftarrow{−}=\overrightarrow{−}_{d_{0}}^{d_{1}} P_{0}). Since P is a summand of a trivial matrix fac-

torization, coker(d_{1}) is a projective Q/(f) module. Choose g ∈ Q such that g(0) ≠ 0 and coker(d_{1})_{g} is free over Q_{g}/(f), and choose ϵ′ ∈ (0, ϵ) such that B_{ϵ′} ∩ g^{−1}(0) = ∅. The inclusion Q ↪ Q_{g} induces a functor MF(Q, f) → MF(Q_{g}, f). Choose t′ such that (ϵ′, t′) is a good pair. Applying Lemma 3.20, we have a commutative diagram

\xymatrix{Ob(MF(Q, f)) \ar[r] \ar[d]^{Φ_{f}^{ℝ}} \ar[dr]^{Φ_{f}^{ℝ}} & Ob(MF(Q_{g}, f)) \ar[d]^{Φ_{f}^{ℝ}} \\ L_{1}(B_{ϵ}, F_{t}^{−}) \ar[r]^{≅} & L_{1}(B_{ϵ′}, F_{t′}^{−})}

It is easy to see that the ϕ_{f}^{ℝ} is well-defined when f = 0, so assume f ≠ 0. Then f is a non-zero-divisor in Q, so we may apply Proposition 2.5 to conclude that the image of P in Ob(MF(Q_{g}, f)) maps to 0 via Φ_{f}^{ℝ}. Hence, the map Φ_{f}^{ℝ} : Ob(MF(Q, f)) → L_{1}(B_{ϵ}, F_{t}^{−}) sends P to 0, as required. We now show that, if α : P → P′ is a morphism in Z^{0}MF(Q, f), Φ_{f}^{ℝ}(P) ⊕ Φ_{f}^{ℝ}(cone(α)) and Φ_{f}^{ℝ}(P′) represent the same class in L_{1}(B_{ϵ}, F_{t}^{−}). We start by showing Φ_{f}^{ℝ}(P [1]) = −Φ_{f}^{ℝ}(P) in L_{1}(B_{ϵ}, F_{t}^{−}). Write Φ_{f}^{ℝ}(P) = (V_{1}, V_{0}; d_{1}|_{F_{t}^{−}}), so that Φ_{f}^{ℝ}(P [1]) = (V_{0}, V_{1}; −d_{0}|_{F_{t}^{−}}). Since cone(id_{P}) is contractible, the class represented by

Φ_{f}^{ℝ}(cone(id_{P})) = (V_{0} ⊕ V_{1}, V_{1} ⊕ V_{0}; (\begin{matrix}d_{0}|_{F_{t}^{−}} & id \\ 0 & −d_{1}|_{F_{t}^{−}} \end{matrix}))

in L_{1}(B_{ϵ}, F_{t}^{−}) is 0. The object

((V_{0} ⊕ V_{1}) × I, (V_{1} ⊕ V_{0}) × I; (\begin{matrix}d_{0}|_{F_{t}^{−}} & s · id \\ 0 & −d_{1}|_{F_{t}^{−}} \end{matrix}))

of C_{1}(B_{ϵ} × I, F_{t}^{−} × I) restricts to Φ_{f}^{ℝ}(cone(id_{P})) at s = 1 and Φ_{f}^{ℝ}((P ⊕ P [1])[1]) at s = 0. Since (P ⊕P [1])[1] ≅ P ⊕P [1], we may use Proposition 9.2 in [ABS64] to conclude that Φ_{f}^{ℝ}(P [1]) = −Φ_{f}^{ℝ}(P) in L_{1}(B_{ϵ}, F_{t}^{−}). Now, we have

Φ_{f}^{ℝ}(cone(α)) = (V_{0} ⊕ V_{1}, V_{1} ⊕ V_{0}; (\begin{matrix}d_{0}|_{F_{t}^{−}} & α_{1} \\ 0 & −d′_{1}|_{F_{t}^{−}} \end{matrix})).

Using Proposition 9.2 in [ABS64] in the same manner as above, we may conclude that Φ_{f}^{ℝ}(cone(α)) and Φ_{f}^{ℝ}(P′) ⊕ Φ_{f}^{ℝ}(P [1]) represent the same class in L_{1}(B_{ϵ}, F_{t}^{−}). Finally, suppose α : P ⟶^{≅} P′ is an isomorphism in [MF(Q, f)]. Then cone(α) is contractible, and so the results we just established imply that Φ_{f}^{ℝ}(P) = Φ_{f}^{ℝ}(P′).

Since every distinguished triangle in [MF(Q, f)] is isomorphic to one of the form

P ⟶^{α} P′ → cone(α) → P [1],

and we have shown that Φ_{f}^{ℝ} preserves such triangles, we are done.

3.3.3 The kernel and image of ϕ_{f}^{ℂ}

Let Q := ℂ[x_{1}, …, x_{n}], and set 𝔪 := (x_{1}, …, x_{n}) ⊆ Q. Fix f ∈ 𝔪, and define R := Q/(f). Assume the hypersurface R has an isolated singularity at the origin in the sense of Definition 2.9. Choose ϵ, δ > 0 so that the map

B_{ϵ} ∩ f^{−1}(D_{δ}^{∗}) → D_{δ}^{∗}

given by x ↦ f(x) is a locally trivial fibration, as in Section 3.1; let F_{f} denote the Milnor fiber of f over some value t ∈ D_{δ}^{∗}. We wish to examine the kernel and image of the map

ϕ_{f}^{ℂ} : K_{0}[MF(Q, f)] → L_{1}(B_{ϵ}, F_{f}).

Recall that, by Theorem 3.2, F_{f} is homotopy equivalent to a wedge sum of μ copies of S^{n−1}, where μ is the Milnor number of f. Thus,

L_{1}(B_{ϵ}, F_{f}) ≅ KU^{0}(B_{ϵ}, F_{f}) ≅ KU^{−1}(F_{f})

≅ ⨁_{μ} KU^{−1}(S^{n−1}) ≅ \{ \begin{matrix}ℤ^{μ} & if n is even \\ 0 & if n is odd \end{matrix}

In particular, when n is odd, ϕ_{f}^{ℂ} = 0. As we noted in Section 3.1, F_{f} is equipped with a monodromy homeomorphism

h : F_{f} ⟶^{≅} F_{f}.

Let S ⊆ D_{δ}^{∗} denote the circle of radius |t| centered at the origin, and set E := B_{ϵ} ∩ f^{−1}(S). One has a long exact sequence, the Wang exact sequence ([Dim92] page 74)

⋯ → H^{i}(E) ⟶^{j^{∗}} H^{i}(F_{f}) ⟶^{h^{∗}−1} H^{i}(F_{f}) → H^{i+1}(E) → ⋯

where j : F_{f} ↪ E is the inclusion. One also has an automorphism T :

L_{1}(B_{ϵ}, F_{f}) ⟶^{≅} L_{1}(B_{ϵ}, F_{f}) induced by h.

We have the following result regarding the image of ϕ_{f}^{ℂ}:

Proposition 3.21. ϕ_{f}^{ℂ}(K_{0}[MF(Q, f)]) ⊆ ker(T − 1).

Proof. The result is obvious when n is odd, since L_{1}(B_{ϵ}, F_{f}) = 0 in this case. Suppose n is even. Notice that ϕ_{f}^{ℂ}(K_{0}[MF(Q, f)]) ⊆ l^{∗}(L_{1}(B_{ϵ}, E)), where l : (B_{ϵ}, F_{f}) ↪ (B_{ϵ}, E) is the inclusion of pairs. Thus, the result follows from the commutative diagram

\xymatrix{L_{1}(B_{ϵ}, E) ⊗ ℚ \ar[r]^{l^{∗}} \ar[d]^{≅} & L_{1}(B_{ϵ}, F_{f}) ⊗ ℚ \ar[d]^{≅} \ar[r]^{T −1} & L_{1}(B_{ϵ}, F_{f}) ⊗ ℚ \ar[d]^{≅} \\ KU^{−1}(E) ⊗ ℚ \ar[r]^{j^{∗}} \ar[d] & KU^{−1}(F_{f}) ⊗ ℚ \ar[d]^{≅} \ar[r]^{h^{∗}−1} & KU^{−1}(F_{f}) ⊗ ℚ \ar[d]^{≅} \\ H^{n−1}(E; ℚ) \ar[r]^{j^{∗}} & H^{n−1}(F_{f}; ℚ) \ar[r]^{h^{∗}−1} & H^{n−1}(F_{f}; ℚ)}

and the Wang exact sequence. The bottom-most vertical arrows are Chern class maps; the bottom-middle and bottom-right vertical maps are isomorphisms because F_{f} has nonzero odd cohomology only in degree n − 1.

The map Φ_{f}^{ℂ} : Ob(MF(Q, f)) → L_{1}(B_{ϵ}, F_{f}) is used in [BvS12] to study the Hochster theta pairing. We recall the definition of this pairing:

Definition 3.22. The Hochster theta pairing

θ : K_{0}[MF(Q_{𝔪}, f)] × K_{0}[MF(Q_{𝔪}, f)] → ℤ

sends a pair ([P_{1} \underleftarrow{−}=\overrightarrow{−}_{d_{0}}^{d_{1}} P_{0}], [P′_{1} \underleftarrow{−}=\overrightarrow{−}_{d′_{0}}^{d′_{1}} P′_{0}]) to

l(Tor_{2}^{R_{𝔪}}(coker(d_{1}), coker(d′_{1}))) − l(Tor_{1}^{R_{𝔪}}(coker(d_{1}), coker(d′_{1}))),

where l denotes length as an R_{𝔪}-module.

Remark 3.23. Our assumption that R is IHS guarantees that the lengths in Definition 3.22 are finite. The pairing θ was introduced in [Hoc81]; for more detailed discussions related to this pairing, we refer the reader to [BvS12], [Dao13], and [MPSW11].

Remark 3.24. Under our assumptions, by Theorem 4.11 of [Dyc11], the map

K_{0}[MF(Q, f)] → K_{0}[MF(Q_{𝔪}, f)]

induced by inclusion is an isomorphism, so we may think of θ as a pairing on K_{0}[MF(Q, f)].

Let P = (P_{1} \underleftarrow{−}=\overrightarrow{−}_{d_{0}}^{d_{1}} P_{0}) be a matrix factorization of f over Q. We observe that

the image of ϕ_{f}^{ℂ}([P]) under the isomorphism L_{1}(B_{ϵ}, F_{f}) ≅ KU^{−1}(F_{f}) coincides with α(coker(d_{1})_{𝔪})|_{F_{f}}, where α is as in Section 4 of [BvS12]. Thus, Proposition 4.1 and Theorem 4.2 of [BvS12] immediately imply:

Proposition 3.25. If X ∈ ker(ϕ_{f}^{ℂ}), θ(X, −) : K_{0}[MF(Q, f)] → ℤ is the zero map.

Set K_{0}[MF(Q, f)]_{tors} to be the torsion subgroup of K_{0}[MF(Q, f)]. We conclude this section with the following explicit description of ker(ϕ_{f}^{ℂ}) when n = 2:

Proposition 3.26. If f ∈ (x_{1}, x_{2}) ⊆ Q = ℂ[x_{1}, x_{2}], and the hypersurface Q/(f) has an isolated singularity at the origin in the sense of Definition 2.9, ker(ϕ_{f}^{ℂ}) = K_{0}[MF(Q, f)]_{tors}.

Proof. K_{0}[MF(Q, f)]_{tors} ⊆ ker(ϕ_{f}^{ℂ}) is obvious. Suppose [P] ∈ ker(ϕ_{f}^{ℂ}). By Proposition 3.25, the map θ([P], −) : K_{0}[MF(Q, f)] → ℤ is the zero map. Set R = Q/(f). Since K_{0}[MF(Q_{(x_{1},x_{2})}, f)] ≅ G_{0}(R_{(x_{1},x_{2})})/[R_{(x_{1},x_{2})}], an application of Proposition 3.3 in [Dao13] finishes the proof.

3.4 Knörrer periodicity and Bott periodicity

We now use our constructions ϕ_{f}^{ℝ} and ϕ_{f}^{ℂ} to exhibit a compatibility between Knörrer periodicity (Theorem 1.1) and Bott periodicity. Set

Q := ℝ[x_{1}, …, x_{n}], Q′ := ℝ[y_{1}, …, y_{m}]

and let

f ∈ (x_{1}, …, x_{n}) ⊆ Q, f′ ∈ (y_{1}, …, y_{m}) ⊆ Q′

be quasi-homogeneous polynomials.

Remark 3.27. We are assuming f and f′ are quasi-homogeneous so that the version of the Sebastiani-Thom homotopy equivalence for real polynomials is available to us (see Section 3.1.3). Analogous versions of every result in this section hold over ℂ when both f ∈ ℂ[x_{1}, …, x_{n}] and f′ ∈ ℂ[y_{1}, …, y_{m}] are either quasi-homogeneous or IHS.

We now construct the negative Milnor fibers of f and f′. Choose real numbers ϵ′′, δ′′, such that the map

B_{ϵ′′} ∩ (f ⊕ f′)^{−1}((−δ′′, 0)) → (−δ′′, 0)

given by x ↦ (f ⊕ f′)(x) is a locally trivial fibration. Similarly, choose ϵ, δ and ϵ′, δ′, as well as t′′ ∈ (−δ′′, 0), so that the analogous maps

B_{ϵ} ∩ f^{−1}((−δ, 0)) → (−δ, 0)
B_{ϵ′} ∩ (f′)^{−1}((−δ′, 0)) → (−δ′, 0)

are locally trivial fibrations, and also so that

(a) ϵ, ϵ′ are sufficiently small so that B_{ϵ} × B_{ϵ′} ⊆ B_{ϵ′′}.

(b) |t′′| < min{δ, δ′}.

Set F_{f}^{−}, F_{f′}^{−}, and F_{f⊕f′}^{−} to be the negative Milnor fibers of f, f′, and f ⊕ f′ over t′′. Assume they are nonempty.

Remark 3.28. We could proceed using positive Milnor fibers as well, but we use negative fibers to stay consistent with Section 3.3.2.

Recall from Remark 2.12 that we have a map

K_{0}[MF(Q, f)] ⊗ K_{0}[MF(Q′, f′)] → K_{0}[MF(Q ⊗_{ℝ} Q′, f ⊕ f′)]

given by [P]⊗[P′] ↦ [P ⊗_{MF} P′]. The following proposition is the key technical result in this section.

Proposition 3.29. There exists a map

ST_{L_{1}} : L_{1}(B_{ϵ}, F_{f}^{−}) ⊗ L_{1}(B_{ϵ′}, F_{f′}^{−}) → L_{1}(B_{ϵ′′}, F_{f⊕f′}^{−})

such that, given matrix factorizations P and P′ of f and f′, respectively,

ST_{L_{1}}(ϕ_{f}^{ℝ}([P]) ⊗ ϕ_{f′}^{ℝ}([P′])) = ϕ_{f⊕f′}^{ℝ}([P ⊗_{MF} P′]).

Proof. Write

P = (P_{1} \underleftarrow{−}=\overrightarrow{−}_{d_{0}}^{d_{1}} P_{0}), P′ = (P′_{1} \underleftarrow{−}=\overrightarrow{−}_{d′_{0}}^{d′_{1}} P′_{0})

and

Φ_{f}^{ℝ}(P) = [V_{1}, V_{0}; d_{1}|_{F_{f}^{−}}], Φ_{f′}^{ℝ}(P′) = [V′_{1}, V′_{0}; d′_{1}|_{F_{f′}^{−}}].

We note that

ϕ_{f⊕f′}^{ℝ}([P ⊗_{MF} P′]) = [(V_{1} ⊗ V′_{0}) ⊕ (V_{0} ⊗ V′_{1}), (V_{0} ⊗ V′_{0}) ⊕ (V_{1} ⊗ V′_{1}); A],

where A is the restriction of the matrix

(\begin{matrix}d_{1} ⊗ id & id ⊗ d′_{1} \\ −id ⊗ d′_{0} & d_{0} ⊗ id \end{matrix})

to F_{f⊕f′}^{−}. As in Section 3.1.2, choose a continuous injection H : CF_{f}^{−} → B_{ϵ} such that

• H(x, 1) = x ∈ F_{f}^{−} ⊆ B_{ϵ},

• H(−, s) : F_{f}^{−} → B_{ϵ} maps into the Milnor fiber B_{ϵ}∩f^{−1}(st′′) for s ∈ (0, 1), and

• H(x, 0) = 0 for all x ∈ F_{f}^{−}

Choose H′ : CF_{f′}^{−} → B_{ϵ′} similarly. The maps of pairs

l : (CF_{f}^{−}, F_{f}^{−}) → (B_{ϵ}, F_{f}^{−}), l′ : (CF_{f′}^{−}, F_{f′}^{−}) → (B_{ϵ′}, F_{f′}^{−})

induced by H and H′ yield isomorphisms on L_{1} upon pullback; this is immediate from the long exact sequence in KO-theory and the naturality of the map χ from Section 3.2 with respect to maps of pairs. Recall from Section 3.2 that we have a map

L_{1}(CF_{f}^{−}, F_{f}^{−}) ⊗ L_{1}(CF_{f′}^{−}, F_{f′}^{−}) → L_{1}(CF_{f}^{−} × CF_{f′}^{−}, CF_{f}^{−} × F_{f′}^{−} ∪ F_{f}^{−} × CF_{f′}^{−})

denoted by

[V] ⊗ [V′] ↦ [V] ⊗_{L_{1}} [V′].

Define

ST_{L_{1}} : L_{1}(B_{ϵ}, F_{f}^{−}) ⊗ L_{1}(B_{ϵ′}, F_{f′}^{−}) → L_{1}(B_{ϵ′′}, F_{f⊕f′}^{−})

to be given by

[V] ⊗ [V′] ↦ (G^{∗})^{−1}(l^{∗}([V]) ⊗_{L_{1}} (l′)^{∗}([V′])),

where

G : (CF_{f}^{−} × CF_{f′}^{−}, CF_{f}^{−} × F_{f′}^{−} ∪ F_{f}^{−} × CF_{f′}^{−}) → (B_{ϵ′′}, F_{f⊕f′}^{−})

is as in Section 3.1.3. Recall that G is an explicit formulation of the Sebastiani-Thom homotopy equivalence. We now compute l^{∗}(ϕ_{f}^{ℝ}(P)) ⊗_{L_{1}} (l′)^{∗}(ϕ_{f}^{ℝ}(P′)) explicitly. A splitting of the restriction of

(\begin{matrix}−id ⊗ (H′)^{∗}(d′_{1}) \\ H^{∗}(d_{1}) ⊗ id \end{matrix})

to CF_{f}^{−} × F_{f′}^{−} ∪ F_{f}^{−} × CF_{f′}^{−} is given, on the fiber over (x, s, y, s′), by

\frac{1}{f(H(x, s)) + f′(H′(y, s′))} (−id ⊗ (H′)^{∗}(d′_{0}) H^{∗}(d_{0}) ⊗ id)

(notice that f(H(x, s)) + f′(H′(y, s′)) = (s + s′)t′′ ≠ 0 when (x, s, y, s′) ∈ CF_{f}^{−} × F_{f′}^{−} ∪ F_{f}^{−} × CF_{f′}^{−}, since either s or s′ is equal to 1). Thus, by the discussion at the end of Section 3.2, the product

l^{∗}([V_{1}, V_{0}; d|_{F_{f}^{−}}]) ⊗_{L_{1}} (l′)^{∗}([V′_{1}, V′_{0}; d′|_{F_{f′}^{−}}])

is equal to

[(W_{1} ⊗ W′_{0}) ⊕ (W_{0} ⊗ W′_{1}), (W_{0} ⊗ W′_{0}) ⊕ (W_{1} ⊗ W′_{1}); B],

where W_{i} = H^{∗}(V_{i}) and W′_{i} = (H′)^{∗}(V′_{i}) for i = 0, 1, and B is given, on the fiber over (x, s, y, s′) ∈ CF_{f}^{−} × F_{f′}^{−} ∪ F_{f}^{−} × CF_{f′}^{−}, by the matrix

(\begin{matrix}H^{∗}(d_{1}) ⊗ id & id ⊗ (H′)^{∗}(d′_{1}) \\ \frac{1}{f(H(x,s))+f′(H′(y,s′))}(−id ⊗ (H′)^{∗}(d′_{0})) & \frac{1}{f(H(x,s))+f′(H′(y,s′))}(H^{∗}(d_{0}) ⊗ id) \end{matrix}) .

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Knörrer Periodicity and Bott Periodicity 1497

We wish to show that, upon applying (G^{∗})^{−1} to this class, one obtains

[(V_{1} ⊗ V′_{0}) ⊕ (V_{0} ⊗ V′_{1}), (V_{0} ⊗ V′_{0}) ⊕ (V_{1} ⊗ V′_{1}); C],

where C is the restriction of the matrix

(\begin{matrix}d_{1} ⊗ id & id ⊗ d′_{1} \\ \frac{1}{t′′} (−id ⊗ d′_{0}) & \frac{1}{t′′} (d_{0} ⊗ id) \end{matrix})

to F_{f⊕f′}^{−}. This will finish the proof, since the class

[(V_{1} ⊗ V′_{0}) ⊕ (V_{0} ⊗ V′_{1}), (V_{0} ⊗ V′_{0}) ⊕ (V_{1} ⊗ V′_{1}); C]

is clearly equal to

[(V_{1} ⊗ V′_{0}) ⊕ (V_{0} ⊗ V′_{1}), (V_{0} ⊗ V′_{0}) ⊕ (V_{1} ⊗ V′_{1}); A].

Observe that we have an object

[((W_{1} ⊗ W′_{0}) ⊕ (W_{0} ⊗ W′_{1})) × I, ((W_{0} ⊗ W′_{0}) ⊕ (W_{1} ⊗ W′_{1})) × I; D] in C_{1}(CF_{f}^{−} × CF_{f′}^{−} × I, (CF_{f}^{−} × F_{f′}^{−} ∪ F_{f}^{−} × CF_{f′}^{−}) × I), where D is given, on the fiber over

(x, s, y, s′, T) ∈ (CF_{f}^{−} × F_{f′}^{−} ∪ F_{f}^{−} × CF_{f′}^{−}) × I,

by the matrix

(\begin{matrix}H^{∗}(d_{1}) ⊗ id & id ⊗ (H′)^{∗}(d′_{1}) \\ \frac{1}{f(a(T))+f′(b(T))}(−id ⊗ (H′)^{∗}(d′_{0})) & \frac{1}{f(a(T))+f′(b(T))}(H^{∗}(d_{0}) ⊗ id) \end{matrix}) .

Here, f, f′, and the entries of d_{1}, d′_{1}, d_{0}, d′_{0} are evaluated at the point

(a(T), b(T)) := (H(x, \frac{T (1 − s′ − s) + 2s}{2}), H′(y, \frac{T (1 − s′ − s) + 2s′}{2} )).

Notice that f(a(T)) + f′(b(T)) ≠ 0 for all

(x, s, y, s′, T) ∈ (CF_{f}^{−} × F_{f′}^{−} ∪ F_{f}^{−} × CF_{f′}^{−}) × I,

so this matrix is indeed an isomorphism on every fiber over (CF_{f}^{−} ×F_{f′}^{−} ∪F_{f}^{−} × CF_{f′}^{−}) × I. Restricting to T = 0, one obtains the object

((W_{1} ⊗ W′_{0}) ⊕ (W_{0} ⊗ W′_{1}), (W_{0} ⊗ W′_{0}) ⊕ (W_{1} ⊗ W′_{1}); B).

Restricting to T = 1 and applying (G^{∗})^{−1}, one obtains

((V_{1} ⊗ V′_{0}) ⊕ (V_{0} ⊗ V′_{1}), (V_{0} ⊗ V′_{0}) ⊕ (V_{1} ⊗ V′_{1}); C).

Now apply Proposition 9.2 in [ABS64].

Documenta Mathematica 21 (2016) 1459–1501

Remark 3.30. It follows easily from the naturality of the map χ from Section 3.2 and Remark 3.16 that ST_{L_{1}} induces a map

ST_{KO} : KO^{0}(B_{ϵ}, F_{f}^{−}) ⊗ KO^{0}(B_{ϵ′}, F_{f′}^{−}) → KO^{0}(B_{ϵ′′}, F_{f⊕f′}^{−}).

Remark 3.31. We emphasize that the group homomorphism ST_{L_{1}} in Proposition 3.29 is given by the composition of the product in topological K-theory with the inverse of the pullback along a specific formulation of the Sebastiani-Thom homotopy equivalence. Hence, Proposition 3.29 yields a precise sense in which the tensor product of matrix factorizations is related to the Sebastiani-Thom homotopy equivalence. Let us now consider the case where Q′ = ℝ[u_{1}, …, u_{8}] and f′ = −u_{1}^{2} −⋯−u_{8}^{2}. By Theorem 2.21 and Remark 2.18, [MF(Q′, f′)] ≅ [MF(ℝ, 0)]. It follows that K_{0}[MF(ℝ[u_{1}, …, u_{8}], −u_{1}^{2} − ⋯ − u_{8}^{2})] is isomorphic to ℤ, generated by the class represented by the matrix factorization X constructed in the proof of Theorem 2.21. Also, F_{−u_{1}^{2}−···−u_{8}^{2}}^{−} is homeomorphic to S^{7}, and so L_{1}(B_{ϵ′}, F_{−u_{1}^{2}−···−u_{8}^{2}}^{−}) is isomorphic to ℤ. This group is generated by ϕ_{−u_{1}^{2}−···−u_{8}^{2}}^{ℝ}(X); thus, ϕ_{−u_{1}^{2}−···−u_{8}^{2}}^{ℝ}([X]) is

a Bott element in the group L_{1}(B_{ϵ′}, F_{−u_{1}^{2}−···−u_{8}^{2}}) ≅ K̃O^{0}(S^{8}); we shall denote by β_{ℝ} the map

KO^{0}(B_{ϵ}, F_{f}^{−}) → KO^{0}(B_{ϵ}, F_{f}^{−}) ⊗ KO^{0}(B_{ϵ′}, F_{−u_{1}^{2}−···−u_{8}^{2}})

given by (χ ⊗ χ) ∘ (− ⊗ ϕ_{−u_{1}^{2}−···−u_{8}^{2}}^{ℝ}([X])) ∘ χ^{−1}. β_{ℝ} is the Bott periodicity isomorphism. Since real Knörrer periodicity may be induced by tensoring with the matrix factorization X, we will denote by K_{ℝ} the map

K_{0}[MF(Q, f)] → K_{0}[MF(Q[u_{1}, …, u_{8}], f − u_{1}^{2} − ⋯ − u_{8}^{2})]

given by − ⊗_{MF} [X]. The following result gives a precise sense in which Bott periodicity and Knörrer periodicity are compatible; it follows immediately from Proposition 3.29. We emphasize that a virtually identical proof yields a similar result involving positive Milnor fibers.

Theorem 3.32. Let f ∈ Q = ℝ[x_{1}, …, x_{n}] be a quasi-homogeneous polynomial such that F_{f}^{−} ≠ ∅, and set q = −u_{1}^{2} − ⋯ − u_{8}^{2}. Then the diagram

\xymatrix{K_{0}[MF(Q, f)] \ar[r]^{χ∘ϕ_{f}^{ℝ}} \ar[dd]^{K_{ℝ}} & KO^{0}(B_{ϵ}, F_{f}^{−}) \ar[d]^{β_{ℝ}} \\  & KO^{0}(B_{ϵ}, F_{f}^{−}) ⊗ KO^{0}(B_{ϵ′}, F_{q}^{−}) \ar[d]^{_{ST_{KO}}} \\ K_{0}[MF(Q[u_{1}, …, u_{8}], f + q)] \ar[r]^{χ∘ϕ_{f+q}^{ℝ}} & KO^{0}(B_{ϵ′′}, F_{f+q}^{−})}

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commutes.

We state the analogous version of this result over the complex numbers. Let Y denote the matrix factorization ℂ[u, v] \underleftarrow{−}=\overrightarrow{−}_{u−iv}^{u+iv} ℂ[u, v] of u^{2} + v^{2}, and let

K : K_{0}[MF(Q, f)] → K_{0}[MF(Q[u, v], f + u^{2} + v^{2})]

be given by −⊗_{MF}[Y]. K_{0}[MF(ℂ[u, v], u^{2}+v^{2})] ≅ ℤ, and the group is generated by [Y]. Also, by Theorem 3.2, F_{u^{2}+v^{2}} is homotopy equivalent to S^{1}. Thus, L_{1}(B_{ϵ′}, F_{u^{2}+v^{2}}^{−}) is isomorphic to ℤ, generated by ϕ_{u^{2}+v^{2}}^{ℂ}([Y]). ϕ_{u^{2}+v^{2}}^{ℂ}([Y]) is a

Bott element in the group L_{1}(B_{ϵ′}, F_{u^{2}+v^{2}}) ≅ K̃U^{0}(S^{2}); we shall denote by β the map

KU^{0}(B_{ϵ}, F_{f}) → KU^{0}(B_{ϵ}, F_{f}) ⊗ KU^{0}(B_{ϵ′}, F_{u^{2}+v^{2}})

given by (χ ⊗ χ) ∘ (− ⊗ ϕ_{u^{2}+v^{2}}^{ℂ}([Y])) ∘ χ^{−1}. β is the complex Bott periodicity isomorphism. Let ST_{KU} denote the pairing on relative KU-theory induced by the complex version of the pairing ST_{L_{1}}. The following is a complex analogue of Theorem 3.32:

Theorem 3.33. Let f ∈ (x_{1}, …, x_{n}) ⊆ Q = ℂ[x_{1}, …, x_{n}] and suppose either

• The hypersurface ℂ[x_{1}, …, x_{n}]_{(x_{1},...,x_{n})}/(f) is IHS (see Definition 2.9), or

• f is quasi-homogeneous.

Then the diagram

\xymatrix{K_{0}[MF(Q, f)] \ar[r]^{χ∘ϕ_{f}^{ℂ}} \ar[dd]^{K} & KU^{0}(B_{ϵ}, F_{f}) \ar[d]^{β} \\  & KU^{0}(B_{ϵ}, F_{f}) ⊗ KU^{0}(B_{ϵ′}, F_{u^{2}+v^{2}}) \ar[d]^{_{ST_{KU}}} \\ K_{0}[MF(Q[u, v], f + u^{2} + v^{2})] \ar[r]^{χ∘ϕ_{f+u^{2}+v^{2}}^{ℂ}} & KU^{0}(B_{ϵ′′}, F_{f+u^{2}+v^{2}})}

commutes.

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Michael K. Brown

Department of Mathematics University of Nebraska-Lincoln Lincoln, NE 68588-0130 USA 1502

Documenta Mathematica 21 (2016)