Relative singular locus and matrix factorizations

0.34 MB·

by  Yuki Hirano

RELATIVE SINGULAR LOCUS AND MATRIX FACTORIZATIONS
YUKIHIRANO
Abstract. We introduce the notion of the relative singular locus Sing(T/S) of a closed im-
mersion T ֒→S of Noetherian schemes, and for a separated Noetherian scheme X with ample
family of line bundles and a non-zero-divisor W ∈ Γ(X,L) of a line bundle L on X, we clas-
sifycertain thick subcategories of the derived matrix factorization category DMF(X,L,W) by
meansofspecialization-closed subsets ofrelativesingularlocusSing(X0/X)ofthezeroscheme
7 X0 :=W−1(0)⊂X. Furthermore, weshow that the spectrum of the tensor triangulated cate-
1 gory(DMF(X,L,W),⊗21)ishomeomorphictotherelativesingularlocusSing(X0/X)byusing
0 theclassificationresultandthetheoryofBalmer’stensortriangulargeometry.
2

n
a
J 1. Introduction

0
1 1.1. Background. For a given category of algebraic objects associated to a scheme, it is ex-
pected that we can extract geometric information of the scheme or the scheme itself from the

] category. Gabriel reconstructed a Noetherian scheme X from the abelian category QcohX of
G
quasi-coherent sheaves on X [Gab], and later Rosenberg generalized the reconstruction theorem
A
forarbitraryschemes [Ros]. Althoughwe can’treconstructa smoothvarietyfromthe derivedcat-
h. egory of coherent sheaves in general, Balmer reconstructed arbitrary Noetherian scheme X from
t the tensor triangulated category (PerfX, ) of perfect complexes on X with the natural tensor
a ⊗
m structure [Bal]. Balmer’sideaistoassociatetoanytensortriangulatedcategory( , )aringed
⊗ T ⊗
space Spec( , )=(Spc( , ), ), and he provedan isomorphismX =Spec(PerfX, ) by using
[ T ⊗ T ⊗ O ∼ ⊗
Thomason’s result of classification of thick subcategories of perfect complexes PerfX which are

1 closed under -action of PerfX.
v ⊗
In addition to the Thomason’s result, classifications of thick subcategories of triangulated cat-
3
egories are studied in many articles. For example, Takahashi classified thick subcategories of the
7
stable category CM(R) of maximal Cohen-Macaulay modules over an abstract hypersurface local
5
2 ring R [Tak]. Stevenson proved a classification of certain thick subcategories of the singularity
0 category Dsg(X) of a hypersurface singularity X [Ste].
.
1
1.2. Relative singular locus. To state our main results, we introduce a new notion of relative
0
7 singularlocus. Let i:T ֒ S be a closedimmersionofNoetherianschemes. We define the relative
→
1 singular locus, denoted by Sing(T/S), of i as the following subset of T;
:
v Sing(T/S):= p T F cohT suchthat F / Perf and i (F) PerfS .
p T,p ∗
i { ∈ | ∃ ∈ ∈ O ∈ }
X We also consider the locally relative singular locus Singloc(T/S) of i defined by
r
a Singloc(T/S):= p T M mod T,p suchthat M / Perf T,p and ip∗(M) Perf S,p .
{ ∈ | ∃ ∈ O ∈ O ∈ O }
By definition, we have the inclusions
Sing(T/S) Singloc(T/S) Sing(T),
⊆ ⊆
whereSing(T)istheusualsingularlocusofT. Theselocicanbedifferenttoeachother,but,ifS is
regular,these lociareequal;Sing(T/S)=Singloc(T/S)=Sing(T). Roughlyspeaking,the relative
singular locus Sing(T/S) is a set of points p in T such that the mildness of the singularity of p in
T get worse than the mildness of the singularity of p in S. In fact, for a quasi-projective variety
X over C and a regular function f Γ(X, ) which is non-zero-divisor, we have the following
X
∈ O
equality of subsets of the associated complex analytic space (Xan, an);
OX
Singloc(f−1(0)/X) Xan =Crit(fan) Zero(fan),
∩ ∩
2010 Mathematics Subject Classification. Primary14F05,18E30; Secondary14B05,32S05.
Key words and phrases. relativesingularlocus,matrixfactorizations, tensortriangulargeometry.
1
2 Y.HIRANO
where Crit(fan) denotes the critical locus of the associated function fan Γ(Xan, an), which is
∈ OX
defined by
Crit(fan):= p Xan (fan) fan(p) m2 ,
{ ∈ | p− ∈ p}
and Zero(fan) is the zero locus of fan.
1.3. Main results. A data (X,L,W) is called Landau-Ginzburg model, or just LG-model, if X is
a scheme, L is a line bundle on X, and W Γ(X,L) is a section of L. To a LG-model (X,L,W)
∈
we associate a triangulated category DMF(X,L,W), called derived matrix factorization category,
introducedbyPositselski[Pos,EP]. Tensorproductsofmatrixfactorizationsdefinesthebifunctor;
( ) ( ):DMF(X,L,W ) DMF(X,L,W ) DMF(X,L,W +W ).
1 2 1 2
− ⊗ − × →
In particular, DMF(X,L,W) has a tensor action from DMF(X,L,0).
The followingis ourmainresultofclassificationofthick subcategoriesofderivedmatrixfactor-
ization categories.
Theorem 1.1 (Theorem 5.6). Let X be a separated Noetherian scheme with ample family of line
bundles, L be a line bundle on X, and let W Γ(X,L) be a non-zero-divisor. Denote by X the
0
∈
zero scheme of W. Then there is a bijective correspondence
unions of closed −→σ thick subcategories of DMF(X,L,W)thatare
subsetsof Sing(X0/X) ←−τ closedunder tensor actionfromDMF(X,L,0)
(cid:26) (cid:27) (cid:26) (cid:27)
The bijective map σ sends Y to the thick subcategory consisting of matrix factorizations F
∈
DMF(X,L,W) with Supp(F) Y. The inverse bijection τ sends to the union Supp(F).
⊆ T F∈T
If X is a regularseparatedNoetherianscheme, then X has anample family oflSine bundles and
DMF(X,L,W) is equivalent to the singularity category Dsg(X ). Furthermore, if X = SpecR is
0
affine withR regularlocal,DMF(X,L,W)is equivalentto the stablecategoryCM(R/W)ofmax-
imalCohen-MacaulaymodulesoverthehypersurfaceR/W. Hence Theorem1.1canbe considered
as a simultaneous generalizationof Stevenson’s result in [Ste] and Takahashi’s result in [Tak].
As an application of the above main result, we see that the closedness of the relative singular
locus Sing(X /X) is related to the existence of a -generator of DMF(X,L,W), where we say
0
⊗
thatanobjectG DMF(X,L,W)isa -generatorifthesmallestthicksubcategorythatisclosed
∈ ⊗
under tensor action from DMF(X,L,0) and contains G is DMF(X,L,W).
Corollary 1.2 (Corollary5.8). Notation is same as in Theorem 1.1. Then thesubset Sing(X /X)
0
of X is closed if and only if DMF(X,L,W) has a -generator.
0
⊗
Furthermore, we construct the relative singular loci from the derived matrix factorizationcate-
gories. If 2 Γ(X, ) is a unit in the ring Γ(X, ), the derived matrix factorization category
X X
∈ O O 1
DMF(X,L,W)has a natural(pseudo) tensor triangulatedstructure 2 onit. Using Theorem1.1
⊗
andthetheoryofBalmer’stensortriangulargeometry,weprovethatthespectrumofthe(pseudo)
tensor triangulated category (DMF(X,L,W), 21) is the relative singular locus Sing(X0/X).
⊗
Theorem 1.3 (Corollary 6.10). Let X be a separated Noetherian scheme with an ample ample
family of line bundles, and let W Γ(X,L) be a non-zero-divisor of a line bundle L. Assume that
∈
2 Γ(X, ) is a unit. Then we have a homeomorphism
X
∈ O
1
Spc(DMF(X,L,W),⊗2)∼=Sing(X0/X).
This result is a generalization of Yu’s result [Yu2, Theorem 1.2], where he proves Theorem 1.3
in the case when X is anaffine regularscheme of finite Krull dimension by using the classification
result due to Walker.
1.4. Plan of the paper. In section 2 we provide basic definitions and properties about derived
matrix factorization categories. In section 3 we give the definitions of globally/locally relative
singular loci and prove some properties about relative singular loci for zero schemes of regular
sectionsoflinebundles. Insection4weprovetensornilpotence propertiesofmatrixfactorizations
whichis keypropertiesforourclassificationresult. Insection5weprovethe mainresultTheorem
1.1. In section 6 we recall the theory of Balmer’s tensor triangular geometry, and we study the
natural tensor triangulated structure on derived matrix factorization categories.
3
1.5. Acknowledgements. The author would like to thank his supervisor Hokuto Uehara for
many useful comments and his continuous support. The author also thank Michael Wemyss and
Shinnosuke Okawa for valuable discussions and comments on a draft version of the paper. The
authorisaResearchFellowofJapanSocietyforthePromotionofScience. Heispartiallysupported
by Grant-in-Aid for JSPS Fellows #26-6240.
2. Derived matrix factorizations
2.1. Derived matrix factorization categories. In the first subsection, we recall the definition
of the derived matrix factorization category of a Landau-Ginzburg model, which is introduced by
Positselski(cf. [Pos], [EP]), and provide its basic properties.
Definition 2.1. A Landau-Ginzburg model, or LG model, is data (X,L,W) consisting of a
scheme X, an invertible sheaf L on X, and a section W Γ(X,L) of L.
∈
Notation 2.2. If L is isomorphic to the structure sheaf , we denote the LG model by (X,W).
X
O
If X = SpecR is an affine scheme, we denote the LG model by (R,L,W), where L is considered
as an invertible R-module and W L.
∈
”
For a LG model, we consider its factorizations which are twisted” complexes.
Definition 2.3. Let (X,L,W) be a LG model. A factorization F of (X,L,W) is a sequence
ϕF ϕF
F = F 1 F 0 F L ,
1 0 1
−−→ −−→ ⊗
whereeachF isacoherentsheafonX a(cid:16)ndeachϕF isahomomo(cid:17)rphismsuchthatϕF ϕF =W id
i i 0◦ 1 · F1
and (ϕF L) ϕF = W id . Coherent sheaves F and F in the above sequence are called
1 ⊗ ◦ 0 · F0 0 1
components of the factorizationF. If the components F of F are locally free sheaves,we call F
i
a matrix factorization of (X,L,W).
Notation 2.4. We can consider any coherent sheaf F cohX as a factorization of (X,L,0) of
∈
the following form
0 F 0 .
−→ −→
By abuse of notation, we will often deno(cid:16)te the above fac(cid:17)torization by the same notation F.
Definition 2.5. For a LG model (X,L,W), we define an exact category
coh(X,L,W)
whose objects are factorizations of (X,L,W), and whose set of morphisms are defined as follows:
For two objects E,F coh(X,L,W), we define Hom(E,F) as the set of pairs (f ,f ) of f
1 0 i
∈ ∈
Hom (E ,F ) such that the following diagram is commutative;
cohX i i
ϕE ϕE
E 1 // E 0 //E L
1 0 1
⊗
f1 f0 f1⊗L
(cid:15)(cid:15) ϕF (cid:15)(cid:15) ϕF (cid:15)(cid:15)
F 1 //F 0 // F L.
1 0 1
⊗
Note that if X is Noetherian, coh(X,L,W) is an abelian category. We define a full additive
subcategory
MF(X,L,W)
ofcoh(X,L,W)whoseobjects are matrix factorizations. By construction,MF(X,L,W)is alsoan
exact category.
”
Since factorizations are twisted” complexes, we can consider homotopy category of factoriza-
tions.
Definition 2.6. Two morphisms f = (f ,f ) and g = (g ,g ) in Hom (E,F) are ho-
1 0 1 0 coh(X,L,W)
motopy equivalent, denoted by f g, if there exist two homomorphisms in cohX
∼
h :E F and h :E L F
0 0 1 1 1 0
→ ⊗ →
such that f g =ϕFh +h ϕE and f L g L=ϕFh +(h L)(ϕE L).
0− 0 1 0 1 0 1⊗ − 1⊗ 0 1 0⊗ 1 ⊗
4 Y.HIRANO
The homotopy category of factorizations
Kcoh(X,L,W)
is defined as the category whose objects are same as coh(X,L,W), and the set of morphisms are
defined as the set of homotopy equivalence classes;
Hom (E,F):=Hom (E,F)/ .
Kcoh(X,L,W) coh(X,L,W)
∼
Similarly, we define the homotopy category of matrix factorizations KMF(X,L,W), i.e.
Ob(KMF(X,L,W)):=MF(X,L,W)
Hom (E,F):=Hom (E,F)/ .
KMF(X,L,W) MF(X,L,W)
∼
Next we define the totalization of a bounded complex of factorizations, which is an analogy of
the total complex of a double complex.
Definition 2.7. LetF• =( Fi δi Fi+1 ) be abounded complex ofcoh(X,L,W). For
···→ −→ →···
l=0,1, set
T := Fi L⊗⌈j/2⌉,
l j ⊗
i+j=−l
M
and let
t :T T
l l → l+1
be a homomorphism given by
t :=δi L⊗⌈j/2⌉+( 1)iϕFi L⊗⌈j/2⌉,
l|Fji⊗L⊗⌈j/2⌉ j ⊗ − j ⊗
where n is n modulo 2, and m is the minimum integer which is greater than or equal to a real
number m. We define the to⌈tal⌉ization Tot(F•) coh(X,L,W)) of F• as
∈
Tot(F•):= T t1 T t0 T L .
1 0 1
−→ −→ ⊗
(cid:16) (cid:17)
Inwhatfollows,wewillrecallthatthehomotopycategoriesKcoh(X,L,W)andKMF(X,L,W)
have structures of triangulated categories.
Definition 2.8. WedefineanautomorphismT onKcoh(X,L,W)),whichiscalledshiftfunctor,
as follows. For an object F Kcoh(X,L,W), we define an object T(F) as
∈
−ϕF −ϕF⊗L
T(F):= F 0 F L 1 F L ,
0 1 0
−−−→ ⊗ −−−−−→ ⊗
(cid:16) (cid:17)
and for a morphism f = (f ,f ) Hom(E,F) we set T(f) := (f ,f L) Hom(T(E),T(F)).
1 0 0 1
∈ ⊗ ∈
For any integer n Z, denote by ( )[n] the functor Tn( ).
∈ − −
Definition 2.9. Let f : E F be a morphism in coh(X,L,W). We define its mapping cone
→
Cone(f) to be the totalization of the complex
f
( 0 E F 0 )
···→ → −→ → →···
with F in degree zero.
A distinguished triangle is a sequence in Kcoh(X,L,W) which is isomorphic to a sequence
of the form
f i p
E F Cone(f) E[1],
−→ −→ −→
where i and p are natural injection and projection respectively.
The following proposition is well known to experts.
Proposition 2.10. The homotopy categories Kcoh(X,L,W) and KMF(X,L,W) are triangulated
categories with respect to the above shift functor and the above distinguished triangles.
Following Positselski ([Pos], [EP]), we define derived factorization categories.
5
Definition 2.11. Denote by Acoh(X,L,W) the smallest thick subcategory of Kcoh(X,L,W)
containing all totalizations of short exact sequences in coh(X,L,W). We define the derived
factorization category of (X,L,W) as the Verdier quotient
Dcoh(X,L,W):=Kcoh(X,L,W)/Acoh(X,L,W).
Similarly, we consider the thick subcategory AMF(X,L,W) containing all totalizations of short
exactsequencesintheexactcategoryMF(X,L,W),anddefinethederivedmatrixfactorization
category by
DMF(X,L,W):=KMF(X,L,W)/AMF(X,L,W).
The following proposition is a special case of [BDFIK, Lemma 2.24].
Proposition 2.12 (cf. [BDFIK, Lemma 2.24]). Assume that X = SpecR is an affine scheme.
For P KMF(R,L,W) and A Acoh(R,L,W), we have
∈ ∈
Hom (P,A)=0.
Kcoh(R,L,W)
In particular, the Verdier localizing functor
∼
KMF(R,L,W) DMF(R,L,W)
−→
is an equivalence.
Forlateruse,weconsiderlargercategoriesoffactorizations. DenotebySh(X,L,W)theabelian
category whose objects are factorizations whose components are -modules. More precisely,
X
O
objects of Sh(X,L,W) are sequences of the following form
ϕF ϕF
F = F 1 F 0 F L ,
1 0 1
−−→ −−→ ⊗
(cid:16) (cid:17)
where F are -modules and ϕF are homomorphisms such that ϕF ϕF = W id and ϕF
i OX i 0 ◦ 1 · F1 1 ⊗
L ϕF = W id . Denote by Qcoh(X,L,W), InjSh(X,L,W), and InjQcoh(X,L,W) the full
◦ 0 · F0
subcategories of Sh(X,L,W) consisting of factorizations whose components are quasi-coherent
sheaves, injective -modules, and injective quasi-coherent sheaves respectively.
X
O
Then, similarly to Kcoh(X,L,W), we can consider their homotopy categories KSh(X,L,W),
KQcoh(X,L,W), KInjSh(X,L,W), KInjQcoh(X,L,W) respectively, and these homotopy cate-
gories have natural triangulated structures similar to Kcoh(X,L,W).
Definition2.13. DenotebyAcoSh(X,L,W)(resp. AcoQcoh(X,L,W))thesmallestthicksubcat-
egoryofKSh(X,L,W)(resp. KQcoh(X,L,W))containingalltotalizationsofshortexactsequences
in Sh(X,L,W) (resp. Qcoh(X,L,W)) and closed under arbitrary direct sums. Following [Pos],
[EP],wedefinethecoderived factorization categoriesDcoSh(X,L,W)andDcoQcoh(X,L,W)
as the following Verdier quotients
DcoSh(X,L,W):=KSh(X,L,W)/AcoSh(X,L,W)
DcoQcoh(X,L,W):=KQcoh(X,L,W)/AcoQcoh(X,L,W).
Lemma 2.14 ([BDFIK], [EP]). Assume that X is Noetherian.
(1) The natural functor KInjSh(X,L,W) DcoSh(X,L,W) is an equivalence.
→
(2) The natural functor KInjQcoh(X,L,W) DcoQcoh(X,L,W) is an equivalence.
→
(3) The natural functor DcoQcoh(X,L,W) DcoSh(X,L,W) is fully faithful.
→
(4) The natural functor Dcoh(X,L,W) DcoQcoh(X,L,W) is fully faithful.
→
(5) The natural functor DMF(X,L,W) Dcoh(X,L,W) is fully faithful.
→
Proof. (1) and (2) follow from [BDFIK, Corollary 2.25]. (3) follows from (1) and (2). (4) and (5)
are [EP, Propostion 1.5.(d)] and [EP, Corollary 2.3.(i)] respectively. (cid:3)
6 Y.HIRANO
2.2. Case when W = 0. In this section, we consider cases when W = 0. Firstly, we will define
cohomologies of factorizations of (X,L,0).
Definition 2.15. For an object F Qcoh(X,L,0), we define its cohomologies H (F) QcohX
i
∈ ∈
as
H (F):=Ker(ϕF)/Im(ϕF ) for i Z/2Z
i i i−1 ∈
Lemma 2.16. Let k be any field. Then, for any object F KMF(k,0), there are two finite
∈
dimensional k-vector spaces V and V such that F is isomorphic to V V [1] in KMF(k,0),
1 2 1 2
⊕
where V denotes the factorization of the form (0 V 0) by Notation 2.4.
i i
→ →
Proof. By [BDFIK, Lemma 2.26], there are two finite dimensional k-vactor spaces V and V′, and
a triangle of the following form in Dcoh(k,0)=DMF(k,0)
V V′ F V[1].
→ → →
ButDMF(k,0)=KMF(k,0)byProposition2.12,sowehaveak-linearhomomorphismf :V V′
→
such that F is isomorphic to C(f), where f is the morphism in KMF(k,0) represented by the
following morphism in MF(k,0)
0 //V //0
f
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 //V′ //0
By construction of mapping cones, C(f) is isomorphic to the following matrix factorization
V f V′ 0 V .
−→ −→
(cid:16) (cid:17)
LetI :=Im(f)be theimageoff,andletK :=Ker(f)be thekerneloff. Thenthereisak-vector
space J such that V′ =I J. Since V =K I, we have the following isomorphism in MF(k,0)
⊕ ⊕
0 0 ∼ 0 0 0
F = K 0 K I I I 0 J 0
∼ −→ −→ ⊕ −→ −→ ⊕ −→ −→
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
But the object I ∼ I 0 I is zero in KMF(k,0). Hence F =J K[1] in KMF(k,0). (cid:3)
−→ −→ ∼ ⊕
(cid:16) (cid:17)
Corollary 2.17. Let k be a field. Any non-zero morphism f :(0 k 0) E in KMF(k,0) is
→ → →
a split mono.
Proof. This follows from Lemma 2.16. (cid:3)
2.3. Tensor products and sheaf Homs functors. Inthissubsection,werecalltensorproducts
and local homs on derived matrix factorization categories. Let (X,L,W) be a LG model, and
V Γ(X,L) be another global section.
∈
For E MF(X,L,V) and F MF(X,L,W), we define the tensor product
∈ ∈
E F MF(X,L,V +W)
⊗ ∈
of E and F as
(E F) := E F E F ,
1 1 0 0 1
⊗ ⊗ ⊕ ⊗
(E F)0 := E(cid:0)0 F0 (cid:1) E(cid:0)1 F1 (cid:1)L ,
⊗ ⊗ ⊕ ⊗ ⊗
(cid:0) (cid:1) (cid:0) (cid:1)
ϕE 1 1 ϕF
ϕE⊗F := 1 ⊗ ⊗ 1 ,
1 1 ϕF ϕE 1
(cid:18)− ⊗ 0 0 ⊗ (cid:19)
and
ϕE 1 1 ϕF
ϕ0E⊗F := 10⊗⊗ϕF0 −ϕE1⊗⊗11!.
This defines an additive functor ( ) ( ) : MF(X,L,V) MF(X,L,W) MF(X,L,V +W),
− ⊗ − × →
and it naturally induces an exact functor
( ) ( ):DMF(X,L,V) DMF(X,L,W) DMF(X,L,V +W).
− ⊗ − × →
7
We define the sheaf Hom
om(E,F) MF(X,L,W V)
H ∈ −
from E to F as
om(E,F) :=( om(E ,F ) L−1) om(E ,F ),
1 1 0 0 1
H H ⊗ ⊕H
om(E,F) := om(E ,F ) om(E ,F ),
0 0 0 1 1
H H ⊕H
( ) ϕE ϕF ( )
ϕHom(E,F) := ∗ ◦ 0 1 ◦ ∗ ,
1 (ϕF L−1) ( ) ( ) ϕE
(cid:18) 0 ⊗ ◦ ∗ ∗ ◦ 1(cid:19)
and
( ) ϕE ϕF ( )
ϕHom(E,F) := − ∗ ◦ 1 1 ◦ ∗ .
0 ϕF ( ) ( ) (ϕE L−1)
(cid:18) 0 ◦ ∗ − ∗ ◦ 0 ⊗ (cid:19)
Thisdefinesanadditivefunctor om( , ):MF(X,L,V)op MF(X,L,W) MF(X,L,W V),
H − − × → −
and it induces an exact functor
om( , ):DMF(X,L,V)op DMF(X,L,W) DMF(X,L,W V).
H − − × → −
The following is standard, so we skip the proof (see [BFK] or [LS] for details) .
Proposition 2.18. Let E MF(X,L,V), F MF(X,L,W), and G MF(X,L,V +W).
∈ ∈ ∈
(1) We have a natural isomorphism
Hom (E F,G)=Hom (E, om(F,G)).
DMF(X,L,V+W) ⊗ ∼ DMF(X,L,V) H
(2) There is a natural isomorphism in MF(X,L,V +W)
om(G,E F)= om(G,E) F.
H ⊗ ∼H ⊗
Recall that MF(X,L,0) denotes the matrix factorization of the form 0 0 by
X X
O ∈ →O →
Notation 2.4. For any object F MF(X,L,W), we define the dual (cid:16) (cid:17)
∈
F∨ := om(F, ) MF(X,L, W)
X
H O ∈ −
of F. By Proposition 2.18, the functors ( ) F : DMF(X,L,V) DMF(X,L,V +W) and
− ⊗ →
( ) F∨ :DMF(X,L,V +W) DMF(X,L,V) are adjoint;
− ⊗ →
( ) F ( ) F∨.
− ⊗ ⊣ − ⊗
2.4. Supports of matrix factorizations. We study the supports of objects in derived matrix
factorization categories. Let (X,L,W) be a LG model.
For any point p X, we denote by X :=Spec( ) the stalk of X at p, and we consider the
p X,p
∈ O
functor of taking stalks at p,
( ) :DMF(X,L,W) KMF(X ,W ),
p p p
− →
given by
F := (F ) (ϕ1)p (F ) (ϕ0)p (F ) .
p 1 p 0 p 1 p
−−−→ −−−→
(cid:16) (cid:17)
Definition 2.19. For an object F DMF(X,L,W), we define its support as
∈
Supp(F):= p X F =0 KMF(X ,W ) .
p p p
{ ∈ | 6 ∈ }
Proposition 2.20. Let F DMF(X,L,W) be an object.
∈
(1) If X = U is an open covering of X, we have the equality of subsets of X
i∈I i
S Supp(F)= Supp(F ),
|Ui
i∈I
[
where F is the restriction of F to DMF(U ,L ,W ).
|Ui i |Ui |Ui
(2) Supp(F) is a closed subset of X.
8 Y.HIRANO
Proof. (1) This follows from isomorphisms F =(F ) for any p U .
p ∼ |Ui p ∈ i
(2) We show the following equality
Supp(F)c = U,
U∈U
[
where := U U is an open subscheme of X such that F = 0 in DMF(U,L ,W ) . The
U U U
U { | | | | }
inclusion Supp(F)c U is obvious. We verify that Supp(F)c U. By definition, for
⊃ U∈U ⊆ U∈U
anyp Supp(F)c, F =0in KMF(X ,W ). Leth=(h ,h )be a homotopygivingthe homotopy
p p p 1 0
∈ S S
equivalence id 0. Then there is a neighborhood U of p such that there exist morphisms
Fp ∼
h :F L F and h :F F in cohU with h =h and h =h . Furthermore,
1 1|U ⊗ U → 0|U 0 0|U → 1|U 1p 1 0p 0
sinceid (ϕ ) h h (ϕ ) =0incohU ,thereexistsanopenneighborhoodV U ofpsuch
(F0)p− 1 p 0− 1 0 p p ⊆
that id ϕ h ϕ h = 0. Then h = (h ,h ) gives a homotopy equivalence
F0|V − 1|V 1|V − 0|V 0|V V 1|V 0|V
id 0. Hence F = 0 in KMF(V,L ,W ), in particular, so is in DMF(V,L ,W ).
ThFe|Vref∼ore, we have V |V , which implies tha|Vt p |V U. |V |V(cid:3)
∈U ∈ U∈U
Definition 2.21. LetF DMF(X,L,W)be anobSject. Foranypointp X,letιp :Speck(p)
X be a naturalmorphism∈, where k(p):= /m is the residue field of t∈he localring ( ,m→).
X,p p X,p p
O O
Then we denote by W k(p) the pull-back ι∗W, and we set
⊗ p
F k(p):=ι∗F = ι∗F ι∗pϕ1 ι∗F ι∗pϕ0 ι∗F KMF(k(p),W k(p)).
⊗ p p 1 −−−→ p 0 −−−→ p 1 ∈ ⊗
(cid:16) (cid:17)
Then ( ) k(p) defines an exact functor
− ⊗
( ) k(p):DMF(X,L,W) KMF(k(p),W k(p)).
− ⊗ → ⊗
The following lemma is a version of Nakayama’s lemma for matrix factorizations.
Lemma 2.22. Let F DMF(X,L,W) be an object, and let p X be a point. Then
∈ ∈
p Supp(F) if andonlyif F k(p)=0 in KMF(k(p),W k(p)).
∈ ⊗ 6 ⊗
Proof. Note that the following diagram of functors is commutative
(−)⊗k(p)
DMF(X,L,W) // KMF(k(p),W k(p))
❯❯❯❯❯❯(❯−❯)❯p❯❯❯❯❯❯❯** ❤❤❤❤❤❤❤(−❤❤)⊗❤❤k❤(m❤❤p❤)❤❤❤44 ⊗
KMF(X ,W )
p p
where m X is the unique closed point. Hence, if F =0 in KMF(X ,W ), then F k(p)=0
p p p p p
∈ ⊗
in KMF(k(p),W k(p)).
Forthe otheri⊗mplication,itsufficesto showthatforalocalring(R,m),anelementw R,and
anobject E KMF(R,w), if E R/m=0 in KMF(R/m,w R/m), then E =0 in KM∈F(R,w).
R
∈ ⊗ ⊗
SinceRislocal,anylocallyfreemodulesarefree. Hence,the objectE canberepresentedbysome
matrix factorization of the following form
R⊕n1 ϕ1 R⊕n0 ϕ0 R⊕n1 .
−→ −→
If E R R/m = 0, there exist ho(cid:16)motopies h0 : (R/m)⊕n0 (cid:17) (R/m)⊕n1 and h1 : (R/m)⊕n1
(R/m⊗)⊕n0 suchthatid(R/m)⊕n0 =(ϕ1⊗R/m)h0+h1(ϕ0⊗R/→m)andid(R/m)⊕n1 =(ϕ0⊗R/m)h1→+
h (ϕ R/m). Since h and h can be represented by a matrix of units in R, there exist homo-
0 1 0 1
⊗
morphisms h0 : R⊕n0 R⊕n1 and h1 : R⊕n1 R⊕n0 such that hi R R/m = hi for i = 0,1.
→ → ⊗
Set
α :=ϕ h +h ϕ :R⊕n1 R⊕n1
1 0 1 0 1
−→
α :=ϕ h +h ϕ :R⊕n0 R⊕n0.
0 1 0 1 0
−→
Then the pair α := (α ,α ) defines an endomorphism of E in the exact category MF(R,w). By
1 0
construction, α = 0 in KMF(R,w). To show that E = 0 in KMF(R,w), it is enough to show
that α : E E is an automorphism in MF(R,w). For each i 0,1 , we only need to show
that α is an→automorphism. Since the tensor product ( ) R∈/m{is a}right exact functor and
i R
α R/m=id, we have − ⊗
i R
⊗
Cok(α ) R/m=Cok(α R/m)=0.
i ⊗R ∼ i⊗R
9
By Nakayama’s lemma, the above implies that Cok(α ) = 0. Hence α is an automorphism by
i i
[Mat2, Theorem 2.4]. (cid:3)
For later use, we provide the following lemma.
Lemma 2.23. Let R be a ring, and let F = R f1 R f0 R KMF(R,f f ) be an object. Then
0 1
−→ −→ ∈
we have the following equality of subsets of S(cid:16)pecR: (cid:17)
Supp(F)= Z(f ),
i
i=0,1
\
where Z(f ):= p SpecR f k(p)=0 .
i i
{ ∈ | ⊗ }
Proof. ( )Ifp Supp(F),thenF k(p)=0inKMF(k(p),(f f ) k(p))byLemma2.22. Suppose
0 1
⊆ ∈ ⊗ 6 ⊗
p / Z(f ) for some i. Then f k(p) is a unit in k(p), and hence F k(p) is homotopic to zero.
i i
∈ ⊗ ⊗
This contradicts to F k(p)=0. Hence p Z(f ).
⊗ 6 ∈ i=0,1 i
( ) If p Z(f ). Then f k(p) = 0 for i = 0,1, and so H (F k(p)) = k(p) = 0. Hence
⊇ ∈ i=0,1 i i⊗ T i ⊗ 6
by [LS, Proposition 2.30], F k(p)=0. Again by Lemma 2.22, p Supp(F). (cid:3)
T ⊗ 6 ∈
Thefollowinglemmaisusefultocomputethesupportoftensorproductsofmatrixfactorizations.
Lemma 2.24. Let V,W Γ(X,L) be any global sections of L, and let E DMF(X,L,V) and
∈ ∈
F DMF(X,L,W) be objects. Let p X be a point such that V k(p) = W k(p) = 0. Then
∈ ∈ ⊗ ⊗
p Supp(E F) if and only if p Supp(E) Supp(F).
∈ ⊗ ∈ ∩
Proof. We have (E F) k(p) = (E k(p)) (F k(p)) in KMF(k(p),0). By Lemma 2.22, it
⊗ ⊗ ∼ ⊗ ⊗ ⊗
is enough to show that for a field k, and for objects M,N KMF(k,0), M N = 0 if and only
∈ ⊗ 6
if M = 0 and N = 0. By Lemma 2.16, for i = 0,1, we may assume ϕM = ϕN = 0, and then
6 6 i i
H (M)=M and H (N)=N . Then, since ϕM⊗N =0 for i=0,1, we have
i i i i i
H (M N)= H (M) H (N) H (M) H (N)
1 1 0 0 1
⊗ ⊗ ⊕ ⊗
(cid:16) (cid:17) (cid:16) (cid:17)
H (M N)= H (M) H (N) H (M) H (N) .
0 0 0 1 1
⊗ ⊗ ⊕ ⊗
(cid:16) (cid:17) (cid:16) (cid:17)
Hence, by [LS, Proposition 2.30], we see that M N =0 if and only if M =0 and N =0. (cid:3)
⊗ 6 6 6
At the end of this section, we organize fundamental properties of supports of matrix factoriza-
tions.
Lemma 2.25. Let E,F,G DMF(X,L,W) be objects. We have the following.
∈
(1) Supp(E F)=Supp(E) Supp(F).
⊕ ∪
(2) Supp(F[1])=Supp(F).
(3) Supp(E) Supp(F) Supp(G) for any distinguished triangle E F G E[1].
⊆ ∪ → → →
(4) Supp(E F)=Supp(E) Supp(F).
⊗ ∩
Proof. (1), (2), and (3) are obvious. If p Supp(M) for some object M DMF(X,L,W), then
∈ ∈
W k(p) = 0, since KMF(k(p),W k(p)) = 0 if W k(p) = 0. Hence (4) follows from Lemma
2.24⊗. ⊗ ⊗ 6 (cid:3)
3. Relative singular locus and singularity category
In this section, we define relative singular loci and prove some properties about it. Let S be a
NoetherianschemeandletF Db(cohS)beaboundedcomplexofcoherentsheaves. Thecomplex
∈
F is called perfect if it is locally quasi-isomorphic to a bounded complex of locally free sheaves
of finite rank. PerfS Db(cohS) denotes the thick subcategory of perfect complexes.
⊂
We define globally/locallyrelative singular locus. Recall our notationS :=Spec( ) for any
p S,p
O
point p S.
∈
Definition 3.1. Let S be a Noetherian scheme, and let i:T ֒ S be a closed immersion.
→
(1) The subset Sing(T/S) T, called the singular locus of T globally relative (or just
⊂
relative) to S, is defined by
Sing(T/S):= p T F cohT suchthat F / PerfT and i (F) PerfS
p p ∗
{ ∈ | ∃ ∈ ∈ ∈ }
10 Y.HIRANO
(2) The subset Singloc(T/S) T, called the singular locus of T locally relative to S, is
⊂
defined by
Singloc(T/S):= p T F cohT suchthat F / PerfT and i (F) PerfS
{ ∈ | ∃ ∈ p ∈ p p∗ ∈ p}
where i :T ֒ S is the closed immersion induced by i:T ֒ S
p p p
→ →
Proposition 3.2. Let S be a Noetherian scheme, and let i:T ֒ S be a closed immersion. Then
→
we have
Sing(T/S) Singloc(T/S) Sing(T).
⊆ ⊆
Furthermore, if S is regular, globally and locally relative singular loci coincide with usual singular
locus;
Sing(T/S)=Singloc(T/S)=Sing(T).
Proof. The first assertion Sing(T/S) Singloc(T/S) Sing(T) is obvious.
⊆ ⊆
For the latter assertion, assume that S is regular. If Sing(T)= , Sing(T/S)=Singloc(T/S)=
∅
Sing(T) = by the former assertion. Assume that Sing(T) = , and let p Sing(T) be a
∅ 6 ∅ ∈
singular point. It is enough to show that p Sing(T/S). Since the projective dimension, denoted
∈
by pd k(p), of k(p) as -module coincides with the global dimension of , we have
OT,p OT,p OT,p
pd (k(p)) = . This implies that k(p) / PerfT . Let a ,...,a is a generator of
theOmT,paximal ide∞al m of the local ring ∈. Then tphere ex{ist1 a smarll}o⊂peOnTa,pffine neighborhood
p T,p
O
U =SpecR T of p and elements b ,...,b R such that (b ) =a . Let I := b ,...,b be the
1 r i p i 1 r
ideal of R ge⊂neratedby b . Then I =m and∈(R/I) =k(p). Take an extensionhF cohTi of the
i p ∼ p p ∼ ∈
coherent sheaf R/I cohU, i.e. F = R/I. Then F = k(p) / PerfT . Since S is regular, we
have Db(cohS)=Pe∈rfS, and so i (|FU)∼ PerfS. Hencepp∼ Sing∈(T/S). p (cid:3)
∗
∈ ∈
g g
The locally relative singular locus has a local property.
Lemma 3.3. Let i:T ֒ S be a closed immersion of Noetherian schemes. Then we have
→
Singloc(T/S)= Singloc(T /S ),
p p
p∈T
[
where the sets Singloc(T /S ) on the right hand side are considered as the subsets of T via the
p p
natural injective maps j :T ֒ T.
p p
→
Proof. If p Singloc(T/S), then m Singloc(T /S ), and j (m ) = p. This means that p
p p p p p
∈ ∈ ∈
Singloc(T /S ), andso Singloc(T/S) Singloc(T /S ). If q Singloc(T /S )for some p T,
p p ⊆ p∈T p p ∈ p p ∈
then, since (T ) =T , j (q) Singloc(T/S). Hence Singloc(T /S ) Singloc(T/S). (cid:3)
p q ∼ jp(q) p ∈ S p∈T p p ⊆
S
Next we recall singularity categories. Let X be a separated Noetherian scheme with resolution
property, i.e. for any F cohX, there exist a locally free coherent sheaf E and a surjective
homomorphism E ։ F.∈Following [Orl1], we define the triangulated category of singularities
Dsg(X) as the Verdier quotient
Dsg(X)=Db(cohX)/PerfX.
Inourassumption,PerfX coincideswiththicksubcategoryofcomplexeswhicharequasi-isomorphic
to a bounded complex of locally free sheaves of finite rank.
We recall that derived matrix factorization categories can be embedded into singularity cate-
gories. Let L be a line bundle on X, and W Γ(X,L) be a non-zero-divisor, i.e. the induced
∈
homomorphism W : L is injective, and denote by X the zero scheme of W. Denote by
X 0
O →
j :X ֒ X the closed immersion. Since the direct image j :Db(cohX ) Db(cohX) preserves
0 ∗ 0
→ →
perfect complexes by [TT, Proposition 2.7.(a)], it induces an exact functor
j :Dsg(X ) Dsg(X).
◦ 0
→
Asin[Orl2],thecokernelfunctorΣ:MF(X,L,W) cohX definedbyΣ(F):=Cok(ϕF)induces
→ 0 1
an exact functor
Σ:DMF(X,L,W) Dsg(X ).
0
→