Arithmetic mirror symmetry for the 2-torus 1 Introduction

95 Pages·2012·1.24 MB·English

Arithmetic mirror symmetry for the 2-torus
YANKILEKILIANDTIMOTHYPERUTZ
This paper explores a refinement of homological mirror symmetry which relates
exactsymplectictopologytoarithmeticalgebraicgeometry. Weestablishaderived
equivalence of the Fukaya category of the 2-torus, relative to a basepoint, with
the category of perfect complexes of coherent sheaves on the Tate curve over
the formal disc SpecZ[[q]]. It specializes to a derived equivalence, over Z, of
the Fukaya category of the punctured torus with perfect complexes on the curve
y2 + xy = x3 over SpecZ, the central fibre of the Tate curve; and, over the
‘punctureddisc’ SpecZ((q)),toanintegralrefinementoftheknownstatementof
homological mirror symmetry for the 2-torus. We also prove that the wrapped
Fukayacategoryofthepuncturedtorusisderived-equivalentover Z tocoherent
sheavesonthecentralfiberoftheTatecurve.
1 Introduction
This paper explores a basic case of what we believe is a general connection between
exact Lagrangian submanifolds in the complement to an ample divisor D in a com-
plexCalabi–Yaumanifold X—weview X \D asanexactsymplecticmanifold—and
coherentsheavesonaschemedefinedover SpecZ,the‘mirror’to X\D. Wetake X
to be an elliptic curve; its complex structure is irrelevant, so it is really a 2-torus T.
Wetake D tobeapoint z. ThemirroristheWeierstrasscubic Y2Z +XYZ = X3,the
restrictionto q = 0 oftheTatecurve T → SpecZ[[q]].
Kontsevich’s1994homologicalmirrorsymmetry(HMS)conjecture[31]claimsthatthe
Fukaya A -category F(X) ofapolarizedCalabi–Yaumanifoldshouldhaveaformal
∞
enlargement—precisely formulated a little later as the closure twπF(X) under taking
mappingconesandpassingtoidempotentsummands—whichis A -quasi-equivalent
∞
toadgenhancementforthederivedcategoryofcoherentsheavesonthe‘mirror’ Xˇ,a
Calabi–Yau variety over the field of complex Novikov series.1 The HMS conjecture
has inspired a great deal of work in symplectic geometry, algebraic geometry and
mathematicalphysics;theHMSparadigmhasbeenadaptedsoastoapplynotonlyto
varieties whose canonical bundle K is trivial, but also to those where either K−1 or
1Beware: thecircumstancesunderwhichoneexpectstofindsuchanXˇ aremoresubtlethan
thoseclaimedbyourone-sentencepre´cisofKontsevich’sconjecture.
2 YankıLekiliandTimothyPerutz
K is ample, with such varieties playing either symplectic or algebro-geometric roles.
Meanwhile, progress on the original case of Calabi–Yau manifolds has been slow.
There are currently complete mirror-symmetric descriptions of the Fukaya category
onlyforthe2-torus R2/Z2 andofthesquare4-torus R4/Z4 [6]. ThecaseofCalabi–
Yau hypersurfaces in projective space has been solved up to a certain ambiguity in
identifying the mirror variety [46, 53]. There are significant partial results for linear
symplectictoriofarbitrarydimension[33].
Our contention is that even in the solved cases, there is more to be said about HMS.
TheFukayacategoryforthe2-torushasanaturalmodelwhichisdefinedoverZ[[q]],a
subringofthecomplexNovikovfield. Thismodelhasamirror-symmetricdescription
astheperfectcomplexesontheTatecurve T over Z[[q]]. Thesymplecticgeometryof
thetorusistherebyconnectedtothearithmeticalgebraicgeometryof T. Establishing
thisconnectionisthetaskofthisarticle.
Experts have certainly been aware that, in principle, homological mirror symmetry
should have an arithmetic-geometric dimension (cf. Kontsevich’s lecture [34], for
instance), but we believe that this article is the first to treat this idea in detail. Whilst
refining existing proofs of HMS for the 2-torus might be a viable option, our method
isalsonew: weidentifyageneratingsubalgebra A oftheFukayacategory,andshow
that Weierstrass cubic curves precisely parametrize the possible A -structures on it
∞
(TheoremC).Themirrorto (T,z) isthentheuniqueWeierstrasscurvecorresponding
to the A -structure belonging to the Fukaya category. Our identification of this
∞
mirror parallels an argument of Gross [23] but also has a novel aspect, relating the
multiplication rules for theta-functions on the Tate curve to counts of lattice points in
triangles (not areas of triangles). Our identification of the wrapped Fukaya category
ofthepuncturedtoruswithcoherentcomplexeson T| appearstobeabasiccaseof
q=0
anunexploredaspectofmirrorsymmetryforCalabi–Yaumanifolds.
1.1 Statement
Let T be a closed, orientable surface of genus 1; ω a symplectic form on T; z ∈ T
a basepoint; T = T \ {z}; and θ a primitive for ω on T . Fix also a grading for
0 0
the symplectic manifold T, that is, an unoriented line-field (cid:96). These data suffice to
specify the relative Fukaya category F(T,z) up to quasi-isomorphism. It is an A -
∞
categorylinearoverZ[[q]]whoseobjectsareembeddedcirclesγ ⊂ T whichareexact
0
(cid:82)
( θ = 0)andareequippedwithorientations, doublecovers γ˜ → γ andgradings(a
γ
gradingisahomotopyfrom (cid:96)| to Tγ in T(T )| ).
γ 0 γ
LetT → SpecZ[[q]]denotetheTatecurve,thecubiccurveinP2(Z[[q]])withequation
(1) Y2Z+XYZ = X3+a (q)XZ2+a (q)Z3,
4 6
Arithmeticmirrorsymmetryforthe2-torus 3
where
(cid:88) n3qn 1 (cid:88) (5n3+7n5)qn
(2) a (q) = −5 , a (q) = −
4 1−qn 6 12 1−qn
n>0 n>0
(notethat n2(5+7n2) isalwaysdivisibleby12).
Letvect(T)denotetheZ[[q]]-lineardifferentialgraded(dg)categorywhoseobjectsare
locallyfreesheavesoffiniterankover T,andwhosemorphismspacesareCˇechcom-
plexeswithrespecttoafixedaffineopencover: homvect(T)(E,F) = Cˇ•(Hom(E,F)).
TheoremA Achoiceofbasis (α,β) for H (T),with α·β = 1,determines,canon-
1
ically up to an overall shift and up to natural quasi-equivalence, a Z[[q]]-linear A -
∞
functor
ψ: F(T,z) → tw(vect(T))
fromtherelativeFukayacategorytothedgcategoryoftwistedcomplexesin vect(T).
Moreover,
(i) thefunctorψ mapsanobjectL# representingβ tothestructuresheafO. Itmaps
0
anobject L# representing α tothecomplex [O → O(σ)],where σ = [0 : 1 : 0]
∞
is the section at infinity of T, and the map is the inclusion. (This complex is
quasi-isomorphic to the skyscraper sheaf Oσ = σ∗OSpecZ[[q]] at the section at
infinity.) It is an embedding on the full subcategory A on {L#,L# }; and is
0 ∞
characterized,uptonaturalequivalence,byitsrestrictionto A. SeeFigure1.
(ii) ψ extendstoanequivalence
DπF(T,z) → Perf(T) (cid:39) H0(twvect(T))
fromtheidempotent-closedderivedFukayacategorytothetriangulatedcategory
ofperfectcomplexeson T.
(iii) Thespecializationof ψ to q = 0 isa Z-linearfunctor
ψ : F(T ) → twvect(T| )
0 0 q=0
fromtheexactFukayacategoryof(T ,θ)tothecategoryofperfectcomplexeson
0
thecentralfiberoftheTatecurve,inducinganequivalenceonderivedcategories
Dψ : DF(T ) → Perf(T| )
0 0 q=0
(bothofthesederivedcategoriesarealreadyidempotent-closed).
(iv) Dψ extendstoanequivalenceoftriangulatedcategories
0
DW(T ) → DbCoh(T| )
0 q=0
from the derived wrapped Fukaya category to the bounded derived category
of coherent sheaves on T| (these derived categories are again idempotent-
q=0
closed).
4 YankıLekiliandTimothyPerutz
Remark. Thefunctorψhasanadditionalproperty,whichisthatitis‘trace-preserving’,
inasensetobediscussedlater.
Clause(ii)hasthefollowingcorollary:
Corollary 1.1 There is an A quasi-equivalence Mod-F(T,z) → QC(T) from the
∞
category of cohomologically unital F(T,z)-modules to a DG enhancement of the
derivedcategoryofunboundedquasi-coherentcomplexesontheTatecurve.
Indeed, QC(T)isquasi-equivalenttoMod-vect(T)asaninstanceofthegeneraltheory
of[59]or[9].
starsindicatenon-triviality rotatelinefieldtogradeaLagrangian
ofthedoublecover
horizognratadlelsinTefield L∞# ←→Oσ
L# ←→O
0
L# (slopeα) L#
∞ (1,−5)
L# ←→O(np)
(1,−n)
z
L#(slopeβ)
0
Figure 1: The torus T and the mirror correspondence ψ, for one possible choice of the line
field (cid:96).
Comparisontothestandardformulation. TheA -structureinthe‘relative’Fukaya
∞
category F(T,z) isbasedoncountingholomorphicpolygonsweightedbypowers qs,
where s counts how many times the polygon passes through the basepoint z. The
‘absolute’ Fukaya category F(T), in the version most popular for mirror symmetry,
has as objects Lagrangian branes L# in T equipped with U(1) local systems E → L.
In the latter version, holomorphic polygons are weighted by (holonomy)qarea. The
coefficient-ringforF(T)isusuallytakentobeΛC,thefieldofcomplexNovikovseries
(cid:80)k>0akqrk: here ak ∈ C, rk ∈ R,and rk → ∞.
Arithmeticmirrorsymmetryforthe2-torus 5
Toexplaintherelationbetweentherelativeandabsoluteversions,notefirstthatthere
isanequationofcurrents ω = δ +dΘ,where Θ isa1-current. Wetake θ tobethe
D
(smooth)restrictionof Θ to M.
Lemma1.2 Thereisafullyfaithful‘inclusion’functor
e: F(T,z)⊗Z[[q]]ΛC → F(T),
linear over ΛC and acting as the identity on objects. For each exact Lagrangian L,
selectafunctionK ∈ C∞(L)suchthatdK = θ| . Thendefineeonmorphism-spaces
L L L
hom(L#,L#) = CF(φ(L#),L#) by
0 1 0 1
e(x) = qA(x)x, x ∈ φ(L )∩L ,
0 1
where A(x) = A (x) is the symplectic action, defined via the K , and φ is the
φ(L0),L1 L
exact symplectomorphism used to obtain transversality. The higher A -terms for e
∞
areidenticallyzero.
Proof The symplectic action is defined as follows. For a path γ: ([0,1];0,1) →
(M;L ,L ) (forinstance,aconstantpathatanintersectionpoint)weput
0 1
(cid:90) 1
A (γ) = − γ∗θ−K (γ(0))+K (γ(1)).
L ,L L L
0 1 0 1
0
Foranydisc u: (D,∂D) → (X,L),wehave
(cid:90) (cid:90) (cid:90) (cid:90)
u∗ω−D·u = u∗(ω−δ ) = u|∗ θ = d(u|∗ K ) = 0.
D ∂D ∂D L
D D ∂D ∂D
Similarly,ifu: D → XisapolygonattachedtoasequenceofLagrangians(L ,L ,...,L )
0 1 d
(where d ≥ 1)atcorners x ∈ L ∩L ,...,x ∈ L ∩L ,then
1 0 1 d+1 d 0
(cid:90) (cid:90) d+1
(cid:88)
u∗ω−D·u = u∗(ω−δ ) = A (x )+ A (x).
D Ld+1,L0 d+1 Li−1,Li i
D D
i=1
Fromthisitfollowsthat e◦µd (x ,...,x ) = µd ◦(ex ,...,ex ),whichproves
F(T,z) 1 d F(T) 1 d
that e is a functor. Note that the perturbations that are used to define hom-spaces in
F(T,z) serveequallywellin F(T). Itisclearthat e isfullyfaithful.
The ‘standard’ statement of mirror symmetry is as follows. Let TΛC = T ×Z[[q]] ΛC;
(cid:82)
itisanellipticcurveoverthefield ΛC. When ω isnormalizedsothat ω = 1,there
T
isafunctor
b
Φ: F(T) → D(cid:101) Coh(TΛC),
6 YankıLekiliandTimothyPerutz
b
where D(cid:101) Coh istheuniquedgenhancementoftheboundedderivedcategory DbCoh
[36], inducing a derived equivalence; and that this functor is again canonically de-
termined by a choice of basis for H (T): see [41, 39, 40, 6] for one proof; [23] for
1
an expository account of another, occasionally missing technical details (e.g. certain
signs);and[51]foryetanother. Ourresultisanarithmeticrefinementofthisstandard
one:
Theorem1.3 Thediagram
F(T,z)⊗ΛC ψ⊗1(cid:47)(cid:47)twvect(T)⊗ΛC
e (cid:15)(cid:15)i
(cid:15)(cid:15)
F(T) Φ (cid:47)(cid:47)D(cid:101)bCoh(TΛC).
ishomotopy-commutativeundercompositionof A -functors.
∞
SinceT×Z[[q]]ΛC isanon-singularvarietyoverthefieldΛC,wemaytaketwvect(TΛC)
asourdgenhancementofDCoh(T ). Theniisthebase-changefunctortwvect(T) →
ΛC
twvect(T ). Forthistheoremtomakesense,ψ andΦmustbesetupsothati◦(ψ⊗1)
ΛC
and Φ◦e agreeprecisely(notjustuptoquasi-isomorphism)onobjects.
1.2 TheTatecurve
Usefulreferencesforthismaterialinclude[56,26,14,23]. TheTatecurveistheplane
projectivecurve T over Z[[q]] whoseaffineequationistheWeierstrasscubic
(3) y2+xy = x3+a x+a ,
4 6
where a and a are as at (2). So T is a projective curve in P2(Z[[q]]). Like any
4 6
Weierstrasscurve w(x,y) = 0, T comeswithacanonicaldifferentialwithpolesatthe
singularities,
Ω = dx/w = −dy/w = dx/(2y+x) = −dy/(y−3x2−a ).
y x 4
Notation:
(4) Tˆ = T specializedto Z((q))(= Z[[q]][q−1]).
The analytic significance of the Tate curve is the following. Consider the Riemann
surface E = C/(cid:104)1,τ(cid:105), where Imτ > 0. The exponential map z (cid:55)→ q := exp(2πiz)
τ
identifies E with C∗/qZ. As q variesoverthepuncturedunitdisc D∗,theRiemann
τ
Arithmeticmirrorsymmetryforthe2-torus 7
surfaces C∗/qZ formaholomorphicfamily E → D∗. TheWeierstrassfunction ℘ for
q
themodularparameter q definesanembedding
E → CP2×D∗; (z,q) (cid:55)→ ([(2πi)−2℘ (z) : (2πi)−3℘(cid:48)(z) : 1],q).
q q
This embedding is cut out by an equation y2 = 4x3 − g (q)x − g (q), which is a
2 3
Weierstrass cubic in (x,y) varying holomorphically with q. The functions g and g
2 3
areholomorphicat q = 0,andmoreoveraredefinedover Z[1][[q]] (makingthissois
6
thepurposeofthepowersof 2πi inthedefinitionoftheembedding). Wecanchange
coordinates, writing x(cid:48) = x− 1 and 2y(cid:48) +x(cid:48) = y, so as to put the equation in the
12
form y(cid:48)2 +x(cid:48)y(cid:48) = x(cid:48)3 +a (q)x(cid:48) +a (q). The benefit of the coordinate-change is that
4 6
thecoefficientsnowliein Z[[q]]. Theseries a and a arethosegivenabove—sothe
4 6
algebraiccurve y(cid:48)2+x(cid:48)y(cid:48) = x(cid:48)3+a (q)x(cid:48)+a (q) istheTatecurve T.
4 6
We conclude, then, that the specialized Tate curve Tˆ is an elliptic curve, analytically
isomorphicover C tothefamily Z((q))∗/qZ when 0 < |q| < 1.
Itsintegralityisoneinterestingfeatureof T,butanotheristhattheabsenceofnegative
powers of q. One can therefore specialize T to q = 0. The result is the curve
T = T| in P2(Z) givenby
0 q=0
(5) y2+xy = x3.
WecancharacterizethisWeierstrasscurveasfollows:
Lemma 1.4 The curve T → SpecZ has a section s = [0 : 0 : 1] which is a node
0
of T0 ×Z Fp, the mod p reduction of T0, for every prime p. Any Weierstrass curve
C → SpecZ possessingasection s withthispropertycanbetransformedbyintegral
changesofvariableto T .
0
Proof Consider a general Weierstrass curve C = [a ,a ,a ,a ,a ], given as the
1 2 3 4 6
projectiveclosureof
(6) y2+a xy+a y = x3+a x2+a x+a , a ∈ Z.
1 3 2 4 6 i
Integral points of C ⊂ P2, other than [0 : 1 : 0], can represented as rational points
Z
on the affine curve. The point [0 : 1 : 0] is regular over any field, and is the unique
point of C with Z = 0. Suppose [X : Y : Z] is an integral point that is nodal mod p
forallprimes p. Then Z mustbenon-zeromod p foreveryprime p,andhence Z isa
unitof Z. Considerthe Z-point (x ,y ) = (X/Z,Y/Z) oftheaffinecurve. Thepartial
0 0
derivativesvanish,sincetheyvanishmod p forall p:
(7) 2y +a x +a = 0, a y = 3x2+2a x +a .
0 1 0 3 1 0 0 2 0 4
8 YankıLekiliandTimothyPerutz
ThenodalconditionisthattheHessianisnon-singular,thatis,
(8) a2+2(6x +2a ) (cid:54)= 0 mod p.
1 0 2
(We note in passing that conditions (7, 8) hold for the point [0 : 0 : 1] of T| at all
0
primes p.) Since(8)holdsforall p,wehave
(9) a2+12x +4a = ±1.
1 0 2
We shall use the criterion (9) to make three changes of variable, successively making
a , a and a equaltotheircounterpartsfor T .
1 2 3 0
First, (9) tells us that a is odd. Hence by a change of variable x = x(cid:48), y = y(cid:48) +c,
1
we may assume that a = 1, whereupon 6x + 2a is either 0 or −1. The latter
1 0 2
possibility is absurd, so 3x + a = 0. Being divisible by 3, a can be removed
0 2 2
altogether by a change of variable x = x(cid:48) +dy, y = y(cid:48) without interfering with a .
1
Thus we can assume additionally that a = 0. We now find from (9) that x = 0.
2 0
Hence 2y +a = 0, so a is even. It follows that a can be set to zero by a change
0 3 3 3
of variable x = x(cid:48), y = y(cid:48) +e, leaving a and a untouched. Equations (7) now tell
1 2
usthat y = 0 = a ,whiletheequation(6)for C tellsusthat a = a2 = 0.
0 4 6 4
More abstractly, if we define a curve π: C → SpecZ by taking P1 and identifying
Z
the sections [0 : 1] and [1 : 1], so as to make every geometric fiber nodal, then the
parametrization P1 → P2 given by [s : t] (cid:55)→ [st(s−t) : s(s−t)2 : t3] identifies C
Z Z
with T .
0
Outline of method and algebraic results. This article is long partly because it
contains rather more than a single proof of Theorem A, and partly because working
over Z presents significant technicalities beyond those that would be present if one
worked over fields (or in some cases, of fields in which 6 is invertible). Part I—a
large chunk—is purely algebraic; it refines and elaborates the method of [35]. The
basic point is that for any Weierstrass curve C, one has a 2-object subcategory B
C
of PerfC—the dg category of perfect complexes of coherent sheaves—with objects
O (the structure sheaf) and O (the skyscraper sheaf at the point at infinity), and
p
this subcategory split-generates PerfC. The cohomology category A = H∗B is
C
independent of C, but the dg structure of B knows C. One can transfer the dg
C
structure to a minimal A -structure on A. This procedure defines a functor from the
∞
category of Weierstrass curves to the category of minimal A -structures on A. We
∞
proveinTheoremCthatthisfunctorisanequivalence. Aslightlycoarsenedstatement
ofTheoremCisasfollows:
Arithmeticmirrorsymmetryforthe2-torus 9
Theorem 1.5 Let R be an integral domain which is either noetherian and normal
of characteristic zero, or an arbitrary field. Let (B,µ∗) be an R-linear A -category
B ∞
togetherwithaCalabi–Yaustructureofdimension1. Assumethat B isminimal, has
just two objects a and b, both spherical of dimension 1 and forming an A -chain
2
(i.e. hom(a,a) ∼= Λ∗(R[−1]) ∼= hom(b,b) asgraded R-algebras; and hom(a,b) ∼= R,
hom(b,a) ∼= R[−1] asgraded R-modules;and µ1 = 0). Then B istrace-preservingly
B
quasi-equivalent to B for a unique Weierstrass curve C → SpecR, where B has
C C
theCalabi–YaustructurearisingfromitsWeierstrassdifferential Ω ∈ Ω1 .
C/SpecR
The proof of Theorem C invokes the Hochschild cohomology HH∗(A,A). We com-
puted this cohomology additively in [35], but here we give a complete calculation,
as a Gerstenhaber algebra, by interpreting HH∗(A,A) as the Hochschild cohomology
HH∗(C ) ofacuspidalWeierstrasscurve C (TheoremB).
cusp cusp
InPartII,weidentifytheuniquecurve C forwhich A isquasi-isomorphic
mirror Cmirror
tothe2-objectsubcategory A oftheFukayacategory F(T,z) onobjectsofslopes
symp
0 and −∞,equippedwithnon-trivialdoublecoverings. In[35],weusedAbouzaid’s
plumbingmodel[3]toprovethatA | isnotformal,whichimpliesthatC is
symp q=0 mirror
notcuspidal. Hereweidentify C precisely. Infact,weidentifythespecialization
mirror
C | in three independent ways: (i) by eliminating the possibility that C
mirror q=0 mirror
issmoothorcuspidalafterreductiontoanarbitraryprime p,bymeansofthe‘closed
open string map’ from symplectic cohomology to Hochschild cohomology of the
Fukaya category; (ii) by calculating “Seidel’s mirror map” [64], or more precisely,
by determining the affine coordinate ring of C | via a calculation in the exact
mirror q=0
Fukaya category; and (iii) via theta-functions. The third proof extends to a proof of
mirrorsymmetryforF(T,z),notjustitsrestrictiontoq = 0. Weuseanintrinsicmodel
fortheTatecurve,andtheintegraltheta-functionsforthiscurvewhichplayedamajor
roleinGross’sproof[23]. Thenubisthemultiplicationruleforthesetheta-functions
and its relation to counts of lattice-points in triangles. The proof of mirror symmetry
forthewrappedcategoryisaratherformalextensionofthatfortheexactcategory.
Weshouldperhapsmakeonemoreremarkaboutexposition. Theauthors’background
is in symplectic topology. We imagine that typical readers will have an interest in
mirror symmetry, perhaps with a bias towards the symplectic, algebro-geometric or
physicalaspects,but,likeus,willnotbeexpertinarithmeticgeometry. Wewouldbe
delightedtohavereaderswhodocomefromanarithmeticgeometrybackground, but
askfortheirpatienceinanexpositionwhichwefearbelaborswhatisobvioustothem
andrushesthroughwhatisnot.
Higher dimensions? We believe that there should be an arithmetic refinement to
homological mirror symmetry for Calabi–Yau manifolds in higher dimensions, but
10 YankıLekiliandTimothyPerutz
will leave the formulation of such conjectures for elsewhere; the 2-torus is, we think,
far from being an isolated case. The case of 2-tori with several basepoints can be
treated rather straightforwardly starting from the one-pointed case, but we shall also
leavethatforanotherarticle.
Acknowledgements. YL was partly supported by the Herchel Smith Fund and by
Marie Curie grant EU-FP7-268389; TP by NSF grant DMS 1049313. Paul Seidel
provided notes outlining the algebro-geometric approach to computing Hochschild
cohomology for A. Conversations with several mathematicians proved useful as this
work evolved, and we would particularly like to thank Mohammed Abouzaid, David
Ben-Zvi, Mirela C¸iperiani, Brian Conrad, Kevin Costello, and Paul Seidel. We thank
theSimonsCenterforGeometryandPhysicsforitsgeneroushospitality.
Part I
Algebraic aspects
2 Background material
2.1 Derivedcategoriesand A -categories
∞
Our conventions and definitions are those of [49, chapter 1]; see [23] for an informal
introduction. Fornow,weworkoveragroundfield K(commutativeandunital),butwe
shalldiscusspresentlymoregeneralgroundrings. Allour A -categoriesandfunctors
∞
arecohomologicallyunital.
Triangulatedenvelopes. AnyA -categoryChasatriangulatedenvelope,aminimal
∞
formalenlargementthatisatriangulated A -category, i.e.,everymorphismin C has
∞
amappingconein C. Thetwistedcomplexes twC ofan A -category C formamodel
∞
for the triangulated envelope. The cohomological category H0(twC) is known as the
derivedcategoryanddenoted DC.
Split closure. One can formally enlarge twC further to another triangulated A -
∞
category twπC which is additionally split-closed (also known as idempotent-closed
or Karoubi complete). An idempotent in the A -category twC is defined to be an
∞