Post on 11-Jun-2020
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.)
3 / 31
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.
5 / 31
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?
7 / 31
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
9 / 31
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
13 / 31
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.
14 / 31
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.
18 / 31
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]
19 / 31
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 .
20 / 31
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.
21 / 31
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.
22 / 31
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.
23 / 31
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.
24 / 31
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.
25 / 31
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.
26 / 31
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.
29 / 31
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.
30 / 31
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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