Generalities

The proof of this theorem uses the representation of W on a localization H_G.M.r/;k/⁰introduced in [37].

After our paper appeared in arXiv, A. Lauda informed us that there is some over-lap between our results and his ongoing projects with collaborators.

2. Generalities

Letkbe a commutative Noetherian ring.

2.1. The center and the trace of a category 2.1.1. Categories

All categories are assumed to be small. Ak-linearcategory is a category enriched over the tensor category ofk-modules, and ak-categoryis an additivek-linear cate-gory. For anyk-linear categoryC and anyk-algebrak⁰, letk⁰˝_kC be thek⁰-linear category whose objects are the same as those ofC but whose morphism spaces are given by

Hom_k⁰˝_kC.a; b/Dk⁰˝_kHomC.a; b/; 8a; b2C:

We denote the identity of an objectaby1aor by1if no confusion is possible. All the functorsF onC are assumed to be additive and/ork-linear. An additive andk-linear functor is called ak-functor. Let End.F /be the endomorphism ring ofF. We may denote the identity element in End.F /byF; 1F, or1, and the identity functor ofC by1C or1. The center ofCis defined as Z.C/DEnd.1C/. A composition of functors EandF is written asEF, while a composition of morphisms of functorsyandx is written asyıx.

An additive categoryC will be always equipped with itstrivialexact structure;

that is, the admissible exact sequences are the split short exact sequences. Therefore, a Serre subcategoryIC is a full additive subcategory which is stable under taking direct summands, and the quotient additive categoryBDC=Iis such that

HomB.a; b/DHomC.a; b/=X

c2I

HomC.c; b/ıHomC.a; c/; 8a; b2C: Ashort exact sequenceof additive categories is a sequence of functors which is equiv-alent to a sequence0!I!C!B!0as above.

Fix an integer`. By an.`Z/-graded k-category we will mean a k-categoryC equipped with a strictk-automorphismŒ`, which we call ashift of the grading. Unless specified otherwise, a functorF of.`Z/-gradedk-categories is always assumed to be graded; that is, it is ak-functorF with an isomorphismF ıŒ`'Œ`ıF. For each integer k2N\.`Z/we will abbreviateŒkDŒ`ıŒ`ı ıŒ`(jk=`j times) and ŒkDŒk¹.

LetC=`Zbe the category enriched over the tensor category of.`Z/-gradedk -modules whose objects are the same as those ofC but whose morphism spaces are given by

HomC=`Z.a; b/DM

k2`Z

HomC

a; bŒk :

Note that the center Z.C=`Z/is a graded ring whose degreekcomponent is equal to Hom.1; Œk/.

Given aZ-gradedk-moduleM, letM^dD ¹x2MIdeg.x/Ddºfor eachd 2Z. For any integer `, the `-twist of M is the .`Z/-graded k-module M<sup>Œ`</sup> such that .M^{Œ`</sup>/<sup>d</sup> DM<sup>d=`} if `jd and 0 otherwise. Then, for eachZ-gradedk-categoryC there is a canonical .`Z/-gradedk-categoryC<sup>Œ`</sup> called the `-twist of C such that C<sup>Œ`</sup>DC as ak-category and the shift of the gradingŒ`inC^Œ` is the same as the shift of the gradingŒ1inC. We have

Hom_CŒ`=`Z.a; b/DHomC=Z.a; b/^Œ`; 8a; b:

Finally, for any categoryCwe denote byC^cthe idempotent completion.

2.1.2. Trace and center

LetC be ak-linear category, and letHH.C/be the Hochschild homology ofC(see [20, Section 3.1]). It is aZ-gradedk-module. We set tr.C/DHH0.C/and CF.C/D Hom_k.tr.C/;k/. We call tr.C/thecocenteror thetraceofC and CF.C/the set of central forms onC. Recall that

tr.C/D M

a2Ob.C/

EndC.a/

=X

f;g

kŒf; g for anyf Wa!b; gWb!a:

For any morphismf inC, let tr.f /denote its image in tr.C/.

Now, letAbe anyk-algebra. Unless specified otherwise, all algebras are assumed to be unital. Let Z.A/be the center ofA, and letHH.A/be its Hochschild homology.

Define tr.A/and CF.A/as above; that is, tr.A/DA=ŒA; A, where ŒA; AA is thek-submodule spanned by the commutators of elements of A. For any element a2A, let tr.a/denote its imageaCŒA; Ain tr.A/. LetA-mod andA-proj be the categories of finitely generated modules and finitely generated projective modules.

For any commutativek-algebraRand anyk-moduleM we abbreviateRMDR˝_k M. The following is well known.

PROPOSITION2.1

LetA; B bek-algebras, and letB;C bek-linear categories.

(a) IfBC is full and any object ofC is isomorphic to a direct summand of a direct sum of objects ofB, then the embeddingBCyields an isomorphism tr.B/!tr.C/.

(b) If C DA-mod or A-proj, then Z.A/DZ.C/. If C DA-proj, then tr.A/D tr.C/.

(c) For any commutativek-algebraR, we havetr.RA/DRtr.A/.

(d) We havetr.A˝_kB/Dtr.A/˝_ktr.B/andZ.A˝_kB/DZ.A/˝_kZ.B/.

(e) Z.C/acts ontr.C/via the mapZ.C/!End_k.tr.C//,a7!.tr.a⁰/7!tr.aa⁰//.

(f) A short exact sequence ofk-categories0!I!C!B!0yields an exact sequence ofk-linear mapstr.I/!tr.C/!tr.B/!0.

For a future use, let us give some details on part.f /. Assume thatCDC^c. For any object X, let add.X /C be the smallest k-subcategory containing X which is closed under taking direct summands. Then, the functor HomC.X;/ yields an equivalence add.X /!EndC.X /^op-proj. In particular, if C has a finite number of indecomposable objects X1; X2; : : : ; Xn (up to isomorphisms) and XDLd

iD0Xi, then we have an equivalenceC'EndC.X /^op-proj.

Now, assume thatCDA-proj, whereAis a finitely generatedk-algebra. Given a Serrek-subcategoryIC, there is an idempotent e2Asuch thatIDeAe-proj and the functorI!Cis given byM 7!Ae˝_eAeM. SetBDC=I. Then, we have B^cDB-proj, whereBDA=AeAand the composed functorC!B!B^c is given byM 7!B˝AM. We must prove that taking the trace we get an exact sequence ofk -modules tr.I/!tr.C/!tr.B/!0. Equivalently, we must check that the following complex is exact:

eAe=ŒeAe; eAe ⁱ A=ŒA; A

j

B=ŒB; B 0:

Note that kerj D.AeACŒA; A/=ŒA; Aand imiD.eAeCŒA; A/=ŒA; A. Since aebDebaeCŒae; eb for all a; b2A, we deduce that kerj Dimi, proving the claim.

2.1.3. Operators on the trace Definition 2.2

Given a functorF WC!C⁰ between twok-categories and a morphism of functors x2End.F /, thetraceofF onxis the linear map

trF.x/Wtr.C/!tr.C⁰/; tr.f /7!tr

x.a/ıF .f /

; wheref 2End.a/andx.a/ıF .f /2End.F .a//.

Note thatx.a/ıF .f /DF .f /ıx.a/ by functoriality. Below are some basic properties of the trace map, whose proofs are standard and are left to the reader.

LEMMA2.3

(a) For each F1, F2 WC !C⁰, x 2End.F1 ˚F2/, we have trF1˚F2.x/D trF₁.x11/CtrF₂.x22/, wherex112End.F1/,x222End.F2/are the diago-nal coordinates ofx.

(b) For two morphismsWF1!F2, WF2!F1, we havetrF₁. ı/DtrF₂.ı /. In particular, ifWF1!F2 is an isomorphism of functors, then for any x2End.F₁/we havetrF2.ıxı¹/DtrF1.x/.

(c) For eachF WC !C⁰,GWC⁰!C⁰⁰,x2End.F /andy2End.G/, we have trGF.yx/DtrG.y/ıtrF.x/.

2.1.4. Adjunction

Given twok-categoriesC1,C2, apair of adjoint functors.E; F /fromC1 toC2 is the datum .E; F; E; "E/of functors E WC1 !C2, F WC2 !C1 and morphisms of functors EW1_C1!FE and "E WEF !1_C2, calledunit andcounit, such that ."EE/ı.EE/DE and.F "E/ı.EF /DF, where we abbreviateED1E and F D1_F.

A pair of biadjoint functors C1!C2 is the datum .E; F; E; "E; F; "F/ of functors EWC1!C2, F WC2 !C1 and morphisms of functorsE W1C₁ !FE,

"EWEF !1C₂such that.E; F; E; "E/and.F; E; F; "F/are adjoint pairs.

Example 2.4

Given two pairs of adjoint functors.E; F /,.E⁰; F⁰/fromC1 toC2, the direct sum .E˚E⁰; F ˚F⁰/is an adjoint pair such that

E˚E⁰D.E; 0; 0; E⁰/W1C₁!FE˚FE⁰˚F⁰E˚F⁰E⁰;

"E˚E⁰D"EC"E⁰WEF ˚EF⁰˚E⁰F ˚E⁰F⁰!1C₂:

If E WC1!C2 and E⁰WC2 !C3, then .E⁰E; FF⁰/ is an adjoint pair such that E⁰ED.F E⁰E/ıE and"E⁰ED"E⁰ı.E⁰"EF⁰/.

Suppose that.E; F /,.E⁰; F⁰/are two pairs of adjoint functors fromC1 toC2. For any morphismxWE!E⁰, theleft transpose^_xWF⁰!F is the composition of the chain of morphisms

F⁰

_EF⁰

FEF^{0 F xF}

0

FE⁰F⁰

F "_E0

F:

For any morphismyWF⁰!F, theright transposey^_WE!E⁰is the composition E

E _E0

EF⁰E⁰

EyE⁰

EFE⁰

"_EE⁰

E⁰: 2.1.5. Operators on the center

Let C1,C2 be two k-categories, and let .E; F; E; "E; F; "F/ be a pair of biad-joint functors C1 !C2. The isomorphisms 1C₂EDEDE1C₁ yield a canonical .Z.C1/;Z.C2//-bimodule structure on End.E/. Let Z.C2/!End.E/,z7!zE and Z.C1/!End.E/,z7!Ezdenote the correspondingk-algebra homomorphisms.

Definition 2.5 (see [5])

For eachx2End.E/we define a map

ZE.x/WZ.C2/!Z.C1/ by sending an elementz2Z.C₂/to the composed morphism

1C_{1 E}

F 1C₂E ^{F zx} F 1C₂E

"_F

1C₁:

We defineZF.x/WZ.C1/!Z.C2/for eachx 2End.F /in the same manner but with the roles ofEandF exchanged.

The proof of the following proposition is standard and is left to the reader.

PROPOSITION2.6

Let.E; F; E; "E; F; "F/,.E⁰; F⁰; E⁰; "E⁰; F⁰; "F⁰/be two pairs of biadjoint func-tors. Letx2End.E/,x⁰2End.E⁰/. Then, we have the following:

(a) ZE.x/WZ.C2/!Z.C1/isk-linear.

(b) ZE⁰E.xı/DZE⁰.x⁰/ıZE.x/andZE˚E⁰.x˚x/DZE.x/CZE⁰.x⁰/.

(c) The mapZEWEnd.E/!Hom_k.Z.C2/;Z.C1//is.Z.C1/;Z.C2//-bilinear.

(d) LetWE!E⁰ be an isomorphism with^_D^_; thenZ_E⁰.ıxı¹/D ZE.x/.

2.2. Symmetric algebras LetA; B; C bek-algebras.

2.2.1. Kernels

There is an equivalence of categories between the category of.A; B/-bimodules and the categories of functors fromB-Mod toA-Mod. It associates an.A; B/-bimodule Kwith the functorˆKWB-Mod!A-Mod given byN 7!K˝BN. We say thatKis thekernelofˆK. SinceˆK.B/DK, the kernel is uniquely determined by the func-torˆ_K. For two.A; B/-bimodulesK,K⁰we have HomA;B.K; K⁰/'Hom.ˆ_K; ˆ_K⁰/ given byf 7!f ˝Bid.

2.2.2. Induction and restriction

We call aB-algebraak-algebraAwith ak-algebra homomorphismiWB!A. We consider the restriction and induction functors

Res^A_BWA-Mod!B-Mod; Ind^A_BDA˝_B WB-Mod!A-Mod:

The pair.Ind^A_B;Res^A_B/is adjoint with the counit"WInd^A_BRes^A_B!1represented by the.A; A/-bimodule homomorphismWA˝BA!Agiven by the multiplication, and the unitW1!Res^A_BInd^A_B is represented by the morphismi, which is.B; B/-bilinear. LetA^Bbe the centralizer ofBinA. For anyf 2A^Bwe set

f WA˝BA!A; a˝a⁰Daf a⁰: (1) 2.2.3. Frobenius and symmetrizing forms

We refer to [36] for more details on this section.

LetAbe aB-algebra that is projective and finite as aB-module. A morphism of .B; B/-bimodulest WA!B is called aFrobenius formif the morphism of.A; B/-bimodulestOWA!HomB.A; B/,a7!.a⁰7!t .a⁰a// is an isomorphism. If such a form exists, then we say thatAis aFrobeniusB-algebra. If we havet .aa⁰/Dt .a⁰a/

for eacha2A,a⁰2A^B, thent is called asymmetrizing formandAis asymmetric B-algebra.

Givent WA!B a Frobenius form, the composition of the isomorphismA˝_B A! HomB.A; B/˝Agiven bya˝a⁰7! Ot .a/˝a⁰and the canonical isomorphism HomB.A; B/˝BA! EndB.A/yields an isomorphismA˝BA! EndB.A/. The preimage of the identity under this map is theCasimir element2.A˝BA/^A. We have.t˝1/./D.1˝t /./D1.

There is a bijection between the set of Frobenius forms and the set of adjunctions .Res^A_B;Ind^A_B/given as follows. Given a Frobenius form t WA!B, the counit"OW Res^A_BInd^A_B!1B is represented by the .B; B/-linear mapt WA!B, and the unit

O

W1A!Ind^A_BRes^A_B is represented by the unique.A; A/-linear mapOWA!A˝BA such that.1O A/D. This yields an adjunction for.Res^A_B;Ind^A_B/. Conversely, if "O andO are counit and unit, respectively, for.Res^A_B;Ind^A_B/, then the.B; B/-linear map tWA!Bwhich represents"Ois a Frobenius form.

Recall that tr.A/ is a Z.A/-module. We equip CF.A/Dtr.A/ with the dual Z.A/-action. Let us recall a few basic facts.

PROPOSITION2.7

LetA; B; C bek-algebras which are projective and finite ask-modules.

(a) Ift WA!B andt⁰WB!C are symmetrizing forms, thent⁰ıt WA!C is again a symmetrizing form.

(b) A symmetrizing formtWA!kinduces aZ.A/-bilinear form tWZ.A/tr.A/!k; .a; b/7!t .ab/:

It is perfect onZ.A/; that is, it induces an isomorphism of Z.A/-modules tOWZ.A/! CF.A/which sendsztot .z/.

(c) IftWA!kis a symmetrizing form, thentr.A/is a faithfulZ.A/-module.

Proof

Part (a) is proved in [36, Lemma 2.10]. To prove part (b), note that the bilinear form is well defined, since multiplication by an element in Z.A/sendsŒA; Ato itself. The pairing is perfect on Z.A/, because the induced map tOWZ.A/!tr.A/DCF.A/

is given by taking.A; A/-invariants for the.A; A/-linear isomorphismtOWA!A; hence the result is again an isomorphism. The compatibility with the Z.A/-module structure follows from the definition. To prove part (c), we must show that the map Z.A/!End_k.tr.A// is injective. Indeed, if there exists z 2Z.A/ such that za2 ŒA; Afor alla2A, thent .zA/D0, and hencezD0.

3. The center of quiver Hecke algebras