Derived algebraic geometrywith a view to quantisation I:

derived stacks

J.P.Pridham

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Building blocks for AG§ Classical AG built from affine schemes,

i.e pcommutative ringsqop

§ Derived AG from pCDGď0AQqop:

§ cochain complex

A0 δÐÝ A´1 δ

ÐÝ A´2 δÐÝ . . .,

§ graded-commutative multiplicationAi b Aj Ñ Ai`j , δ a derivation, 1 P A0.

§ Quasi-isomorphisms f : AÑ B withH˚pAq – H˚pBq.

Also simplicial or E8-rings (all

equivalent over Q).2 / 31

DG schemes (Kontsevich)

Pairs pX 0,O‚X q with:

§ X 0 a scheme,§ Oď0

X a sheaf of CDGAs on X 0,§ On

X quasi-coherent,§ O0

X “ OX 0.

§ X quasi-isomorphic to completion along

π0X “ SpecpX 0q H0OX if Noetherian.

§ Too few objects, far too few morphisms.

(Ambient scheme X 0 not really intrinsic.)

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Functor of points§ Want to associate functors to derived

affines.

§ How about HompR ,´q : CDGA Ñ Set?

Not quasi-IM invariant.

§ What about taking homotopy classes

rA,´s? Not left-exact; won’t glue.

§ Have to take derived Hom to topological

spaces/simplicial sets

mappR ,´q : CDGA Ñ sSet.4 / 31

ExampleA1 “ Spec krxs

§ Hompkrxs,Aq “ A0, not

homotopy-invariant.

§ Homotopy classes of maps krxs Ñ A

given by H0A, not left-exact.

§ mappkrxs,Aq a space with

πimappkrxs,Aq “ H´iA.

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From local to global§ Any geometry has simple building

blocks:§ Convex opens (DG)§ Affine schemes Aff (AG)

§ Have to glue/quotient in larger category

A to get global objects (manifolds,

orbifolds, schemes, algebraic stacks).

§ Conventionally in DAG, take A to be

simplicial presheaves. We’ll see smaller

alternative.6 / 31

§ Schemes Ă ringed spaces

§ Algebraic stacks Ă functors on Aff

§ Quasi-coherent sheaves Ă OX -modules

(enough injectives)

Nice categories, with many nasty objects.

§ Who cares about arbitrary sheaves on

the big affine site?

§ Or about arbitrary OX -modules?

Are there smaller ambient categories?

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Cech nervesManifold X , covered by nice open subspaces

U1, . . .Uk . For U :“š

Ui , the Cech nerve

Xn :“ U ˆX U ˆX . . .ˆX Uloooooooooooomoooooooooooon

n`1

“ž

i0,...,in

Ui0 X . . .X Uin.

X0//X1oo

oo ////X2

oojj

ttX3gg

ww¨¨

¨¨ ¨ ¨ff

xx¨¨

¨

recovers X , H˚pX ,Rq from simplicial

diagram of nice opens.8 / 31

Algebraic analogueQuasi-compact, semi-separated scheme X ,

affine cover tUiui , same expression gives

simplicial diagram X of affine schemes,

recovering X , H˚pX ,OX q.

More generally, for affine presentationU Ñ X of Artin stack with affine diagonal,

Xn :“ U ˆX U ˆX . . .ˆX Ulooooooooooomooooooooooon

n`1

Example: X “ rU{G s; U //U ˆ Goooo //

//U ˆ G 2 ¨ ¨ ¨oomm

qq

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Simplicial objects§ |∆n| :“ tx P Rn`1

` :řn

i“0 xi “ 1u.

§ For topological space X ,

SingpX qn :“ Homp|∆n|,X q, so

SingpX q0 σ0 //SingpX q1B0

oo

B1oo ////SingpX q2 . . . ,oo

nn

pp

relations like Biσi “ id.

§ Any diagram of this form is called

simplicial.10 / 31

Higher algebraic stacks§ Motivation: moduli problems.

§ 1-stacks keep track of automorphisms.

§ n-stacks have higher automorphisms.

§ n-groupoids: spaces with πąnX “ 0.

§ Example: E P PerfX pAq has (i ě 2)

πipPerfX pAq,E q “ Ext1´iXbApE ,E q.

§ n-truncated derived stack F has

πąnF pAq “ 0 for A underived.11 / 31

Technical assumption: from now on, everything is assumed

strongly quasi-compact (quasi-compact, quasi-separated . . . )

§ Every algebraic n-stack can be resolved

by a simplicial affine scheme (sAff).

X0//X1oo

oo ////X2

oojj

ttX3gg

ww¨¨

¨¨ ¨ ¨ ,ff

xx¨¨

¨

§ Equivalently cosimplicial ring

A0 //

//A1oo //

**

44A2

oooo ''

77¨¨

¨A3

((

66¨¨

¨¨ ¨ ¨ ,

§ Which simplicial affines arise this way?

§ What about morphisms?12 / 31

Derived n-stacks§ Every algebraic derived n-stack resolved

by a simplicial derived affine (sdAff).

§ dAff{Q » pCDGď0AQqopp.

§ Equivalently cosimplicial CDGA

A0,0 ////A

1,0oo //++

33A2,0

oooo ((

66¨¨

¨¨ ¨ ¨

stackyÝÝÝÑ

derived

��

A0,´1

δ

OO

////A

1,´1oo

δ

OO

//,,

22A2,´1

oooo

δ

OO

((66¨

¨

¨¨ ¨ ¨

δ

OO

...

δ

OO

////

...oo

δ

OO

//++

33...oo

ooδ

OO

**66¨

¨

¨

. . .

δ

OO

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Simplices and horns§ m-simplex ∆m is simplicial set with

HomsSetp∆m,X q “ Xm.

§ Boundary B∆m “Ťm

i“0 Bi∆m´1 Ă ∆m.

§ kth horn Λm,k “Ťm

i“0,i‰kBi∆m´1.

§ Partial matching objects

MΛm,kX :“ HomsSetpΛm,k ,X q “

tx Pź

0ďiďmi‰k

Xm´1 : Bixj “ Bjxi`1, @i ě ju.

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Duskin–Glenn n-hypergroupoids§ Horn-fillers Λm,k

� _

��

// X

∆m

==

Xm Ñ MΛm,kX

are surjective for all m, k , and

isomorphisms for m ą n.

§ Relative n-hgpds X {Y : Λm,k� _

��

// X

f��

∆m

==

// YXm Ñ MΛm,kXˆpMΛm,k

Y qYm

are surjective for all m, k , and

isomorphisms for m ą n.16 / 31

§ 1-hgpds are nerves of groupoids.

§ Relative 0-hgpds are Cartesian:

Xm – X0 ˆY0 Ym.

§ n-hgpd determined by Xďn`1, but have

to check conditions at Xn`2.

§ n “ 1 case: objects X0, morphisms X1,

composition Xď2, associativity Xď3.

§ Relative n “ 0 case: f0 : X0 Ñ Y0 gives

fibres, f1 gluing data, f2 cocycle

condition.17 / 31

n-stacks the Grothendieck way

§ Can define n-hypergroupoids in any

category A with finite limits and a class

C of covering maps.

§ Sets and surjections n-groupoids.

§ Affine schemes and smooth/etale

surjections Artin/DM n-hgpds.

§ In Grothendieck’s “Pursuing stacks”,

apparently.

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HAG2 defines n-geometric stacks inductively,

but:

Theorem (P)

n-geometric Artin/DM stacks Ø

hypersheafifications X 7 of Artin/DM

n-hypergroupoids X .

[HAG2 n-geometric stacks (XÑ XSn´1affine)

Ă Lurie n-stacks (X » XSn`1) Ă pn ` 2q-geom stacks]

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Derived n-geometric stacks§ Subtleties: replace isos with quasi-isos,

§ require Reedy fibrant: matching mapsXm Ñ MB∆mX fibrations (i.e. quasi-free)

§ alternatively, use homotopy limits.

§ Derived ArtinDM n-hgpd in sdAff is Reedy

fibrant with smoothetale horn-fillers.

§ (HAG2): A‚ Ñ B‚ smooth/etale if H0AÑ H0B is so,

and H˚B – H˚AbH0A H0B .

Theorem [P] Derived n-geometric ArtinDM

stacks Ø X 7 for derived ArtinDM n-hgpds X .

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Example: Reedy fibrant

replacement of A1

§ Need X‚ P sdAff with (i) A1 „ÝÑ Xm, and

(ii) Xm Ñ MB∆mX quasi-free (m “ 1 is

X1 Ñ X0 ˆ X0).

§ Let NC‚p∆m, kq be normalised chains

(gen’d by non-degenerate simplices).

§ Set Xm “ Spec krNC‚p∆m, kqs.

§ (i) NC‚p∆m, kq » k , and

(ii) NC‚pB∆m, kq ãÑ NC‚p∆m, kq.

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Example: dg schemes

§ Semi-separated dg scheme

X “ pX 0,OX q derived Zariski

1-hgpd, by Reedy fibrant replacement of

Xi :“ Spec ΓpX 0i ,OX q,

for Cech nerve X 0 of X 0.

§ X‚ quasi-isomorphic to completion along

π0X “ SpecpX 0q H0OX if Noetherian.

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Example: derived schemes§ Derived scheme is derived Artin/DM

n-stack X with underived truncation

π0X » Y , a scheme.§ No ambient scheme (unlike dg schemes).§ When Y semi-separated, X given by

CDGAs A ď0 on Y with H0A “ OY

and H˚A Cartesian/quasi-coherent.

Zariski 1-hgpd is fibrant replacement of

Xi :“ Spec ΓpYi ,A‚q.

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Trivial n-hypergroupoidsTo calculate morphisms or sheaves (functor

of points), we need to refine atlases.

X is a trivial n-hgpd over Y if the matching

maps B∆m� _

��

// X

f��

∆m

<<

// YXm Ñ MB∆mX ˆpMB∆mY q Ym

are surjective for all m and isos for m ě n.

[Thus determined by Xăn Ñ Yăn.]

SmoothEtale

surjections trivial ArtinDM n-hgpds.

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Main theorem

Theorem (P)

The 8-category of pn,Pq-geometric stacks is

the localisation of the category of

pn,Pq-hypergroupoids with respect to trivial

pn,Pq-hypergroupoids.

§ P any property like (derived) Artin/DM.

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MorphismsMore explicitly, for Y a (derived) Artin

n-hgpd, mapping space is

mappX 7,Y 7qm “ lim

ÝÑα

HompXα ˆ∆m,Y q

for tXα Ñ X uα any weakly initial system

(«universal cover/maximal atlas ) of trivial

(derived) DM n-hgpds.

N.B. Y 7pAq “ mappSpec A,Y q.

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Sheaves§ Complexes F‚ of OX -mods on X 7 with

qu-coh homology are quasi-Cartesian

complexes of qu-coh sheaves on X .§ Given by complexes F‚pXmq of

OpXmq-mods and compatible quasi-isos

Bi : B˚i F‚pXmq Ñ F‚pXm`1q.

§ Same definition works for any sheaves

satisfying descent w.r.t. smooth/etale

morphisms.27 / 31

Stacks in other settings§ Zhu’s Lie n-groupoids are n-hgpds in

manifolds, w.r.t. surj submersions.

§ C-manifolds work just as well.

§ C8-rings for singular Lie n-groupoids.

§ Simplicial/dg C8-rings for derived Lie

n-groupoids (Borisov–Noel).

§ Poisson DGAs for Poisson str. on DM

stacks (Artin more complicated).

§ Pro-Artinian rings for formal moduli.28 / 31

ReferencesJ. P. Pridham.

Notes characterising higher and derived stacks concretely.arXiv:1105.4853v3 [math.AG], 2011.

Chenchang Zhu.

n-groupoids and stacky groupoids. Int. Math. Res. Not.IMRN, (21):4087–4141, 2009. arXiv:0801.2057.

J. P. Pridham.

Presenting higher stacks as simplicial schemes. Adv.Math., 238:184–245, 2013. arXiv:0905.4044v4 [math.AG].

A. Grothendieck.

Pursuing stacks. unpublished manuscript, 1983.

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D. Borisov and J. Noel.

Simplicial approach to derived differential manifolds.arXiv:1112.0033v1 [math.DG], 2011.

J. P. Pridham.

Unifying derived deformation theories. Adv. Math.,224(3):772–826, 2010. arXiv:0705.0344v6 [math.AG].

J. Duskin.

Higher-dimensional torsors and the cohomology of topoi:the abelian theory. In Applications of sheaves (Proc. Res.Sympos. Appl. Sheaf Theory to Logic, Algebra and Anal.,Univ. Durham, Durham, 1977), volume 753 of LectureNotes in Math., pages 255–279. Springer, Berlin, 1979.

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Paul G. Glenn.

Realization of cohomology classes in arbitrary exactcategories. J. Pure Appl. Algebra, 25(1):33–105, 1982.

J. Lurie.

Derived Algebraic Geometry. PhD thesis, M.I.T., 2004.www.math.harvard.edu/„lurie/papers/DAG.pdf .

Bertrand Toen and Gabriele Vezzosi.

Homotopical algebraic geometry. II. Geometric stacks andapplications. Mem. Amer. Math. Soc., 193(902):x+224,2008. arXiv math.AG/0404373 v7.

D. Carchedi and D. Roytenberg

On theories of superalgebras of differentiable functionsarXiv:1211.6134v1 [math.DG], 2012.

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