A Jacobian criterion of smoothness for algebraic diamonds

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A Jacobian criterion of smoothness for algebraic diamonds

Laurent Fargues (CNRS/IMJ) - Joint with P. Scholze

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Motivation

Ï Problem : Give a meaning toBun_G is "smooth of dimension 0", U→Bun_G is a "smooth chart" onBun_G.

Ï BunG=v-stack ofG₋bundles / curve (stack on^PerfFp) Ï U=locally spatial diamond

no innitesimal crietrion (perfect world), no Jacobian criterion Ï Solution :cohomological smoothness("Étale cohomology of diamonds")

morphisms that satisfy relative Poincaré duality

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Cohomological smoothness

Ï Spaces:X=smallv-stack

Ï small= ∃v-presentation by perfectoid spaces Ï v−stack=Stack on^PerfFp+v-topology Ï Example : Beauville-Laszlo :^Gr^B_G^dR−→BunG

GrB_dR

G =B_dR-ane Grassmanian = ind-diamond v-surjective (analog of Drinfeld-Simpson known) Ï Coecients:Λ=Z/n_Z,(n,p)₌1

D_ét(X,Λ) =

nF∈D(X,Λ)

| {z }

cartesian v−sheaves

| ∀ U

|{z}

strictly tot disc perf space

−−→f X, f^∗F∈D(U_ét,Λ)

| {z }

=D(|U|,Λ) o

= Dcart(_|U_•_|,Λ)

for some simplicial strictly tot. disc. perf. spaceU_•

Ï Morphisms:X−−→^f Yrepresentable in loc. spatial diamonds (compactiable, dim trg f< +∞)

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Cohomological smoothness

f:X_−→Y

D_ét(X,Λ) f_! //

D_ét(Y,Λ) f^!

oo

adjoint functors Denition

f is cohomologically smooth iff^!_'f^∗_⊗_Luniversally/YwithLinvertible i.e.

v-locally'Λ[d]

Equivalent to :

1. f^!Λ⊗f^∗(−)−−→^∼ f^!(−)(canonical morphism=iso) 2. f^!_Λinvertible

Important property : coho. smoothnessv-local / base

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Examples of coho smooth morphisms

Ï f :X→Y morphism of "classical" rigid spaces/K

f smooth⇒f^¦:X^¦_→Y^¦ coho smooth (Huber's relative Poincaré duality) Ï B^d_S^,¹^/p^∞−→SwithS_∈Perf_F_p

Ï

f:X_−→Y morphism of loc. spatial diamonds K=pro-p group acts freely onX/Y

g:X/K_−→Y Thenf coho smooth⇒g coho smooth B:X→X/K notcoho smooth A₌pronite set

h:A_Spa(C_,_O

C)→Spa(C,OC) étale sheaves /A_Spa(C_,O

C)=C^∞(A,Λ)−modules h^!Λ=D(A,Λ) (distributions)

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Examples of coho smooth morphisms

Ï X_S₌curve/Spa(_Q_p)"parametrized byS_∈Perf_F_p"

E/X_S vector bundle >0 H.N. slopesberwise onS BC(E)=relativeH⁰ofE :T/S7−→H⁰(X_T,E_|X_T) Then :BC(_E)_→Sis coho smooth

Kedlaya-Liu⇒loc./S _∃0→E⁰→E⁰⁰→E→0 with

Ï E⁰s.s. slope 0 (berwise/S)

Ï E⁰⁰s.s. slope ¹n for somen≥1 (berwise/S) 0−→BC(E⁰)

| {z } pro-étale loc'Qp m

−→ BC(E⁰⁰)

| {z } loc/S'◦

Bd,1/p∞

univ coverp-div formal groupS

−→BC(E)_−→0

⇒BC(E)/S is coho smooth

Ï Idem withBC(E)=relativeH¹, ifE has<0 H.N. slopes berwise/S Example :BC(_O(₋1))₌(_G_a_,_S_])^¦/_Q_p

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An application of the examples

Starting point of discussions about smoothness with Scholze.

G/_Q_p ane alg. group then

£•/G(Qp)^¤=classifying stack of pro-étaleG(Qp)-torsors iscohomologically smooth of dimension 0. IfG reductive

£•/G(_Q_p)^¤₌Bun^ss^,^c¹⁼⁰ G

open,→ Bun_G particular case of chat we want to prove (Bun_G coho smooth)

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An application of the examples

Ï G₌GLn.U_⊂BC(_O(1))ⁿ open (6= ;) subset of surjections u:_OⁿO(1)

s.t.ker(u)has<0 H.N. slopes. GLn(Qp)=Aut(Oⁿ)actsfreelyonU.

K_⊂GLn(_Q_p)pro-p open

f:U/K_−→^£_•/GLn(_Q_p)^¤ viaU→ •andK,→GLn(Qp)

Ï U/K_{→ •}is coho smooth Ï f=locally trivial bration inU×

KGLn(Qp)=`

K\GLⁿ⁽^Q^p⁾U=coho smooth Ï AnyG. FixG,→GLn,K⊂G(Qp)pro-p open and consider

U/K−→£

•/G(Qp)^¤

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Statement of the main theorem

S₌Spa(R,R⁺)anoid perfectoid

Z_→X_S smoothadic space, adication ofZ_→ X_R

|{z}

schematical curve

quasi-projective

Denition

MZ=a moduli space of sections ofZ/X_S MZ:T/S_7−→ⁿf, Z

X_T

f //<<X_S

, s.t.f^∗T_Z_/X

S has>0 H.N. slopesberwise/T^o

Theorem

MZ−→Sis a cohomologically smooth morphism of locally spatial diamonds

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Examples of spaces M

Z

Ï linear case:Z₌_V(_E),E=vector bundle onX_S MZ=BC(_E)_×_SU

whereBC(E)=relativeH⁰ ofE,U=open subset ofSwhereE has>0 H.N. slopes

Ï projective space case:Z=P(E) MZ= a

d_∈Z

U_d/_Q_p^×

U_d⊂BC(E^∨(d))\ {0}open subset of surjections u:_EO(d) s.t.keru has H.N. slopes<d

Ï Gromov-Witten case:Z₌X_S_×_Spa(_Q

p)Z⁰. ThenMZ dened over Spa(_F_p)

= moduli of morphismsf:X_S_→Z⁰s.t.f^∗T_Z₀_/_Q

p is>0

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Examples of spaces M

Z

Ï Reduction of aG-bundle to a parabolic subgroup: P⊂G parabolic subgroup

Bun^>_P⁰⊂Bun_P open substack

Bun^>_P⁰=©

P-bundlesE |E×

Pg/phas >0 H.N. slopes^ª Ad:P→GL(g/p)(adjoint representation).

Theorem implies :

Bun^>_P⁰−→Bun_G is coho smooth

nice coho smooth charts onBun_G,πb:Mb→Bun_G (see Scholze's IHES lectures)

Example :G=GLn,P=Borel,Bun^>_P⁰=moduli of 0=E₀(E₁(· · ·(En=E full ag s.t.

deg(Ei/Ei₋1)<deg(Ei₊1/Ei)

"anti-HN ltration"

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Proof of the main theorem : formal smoothness

Denition

f:X_−→Y isformally smoothif for any diagram S₀ _

//X

f

S //

s

>>

Y

whereS₀,→Sis a Zariski closed embedding of a perf spaces,∃s (up to replacingSby an étale neighborhood ofS₀)

→More near from the topological notion ofEuclidean Neighborhood Retract (Borsuk) than Grothendieck's algebraic notion.

Example :X,Y a perf

X //

B^I_Y vvyy s

Y for some setI. Then

X/Y f.s. ⇔ ∃retractions (up to replacingB^I_Y by an étale neighborhood ofX)

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Proof of the main theorem : formal smoothness

Theorem

The morphismMZ−→Sis formally smooth Remarks:

1. "Classical" caseX/k smooth projective curve, replacing ">0 H.N. slopes"

in the def. ofMZ by "has noH¹", this is very easy (with Groth. def. of formal smoothness)

2. formally smooth does not imply coho smooth à priori. For example normal crossing divisor{xy₌0}⊂A²_Cis a topological retract but no coho smooth

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Proof of the main theorem : consequence of formal smoothness

f :MZ−→S

f formally smooth⇒Λisf-U.L.A. (see Scholze's talk)

⇒f^!_Λ_⊗f^∗(₋)_−−→^∼ f^!(₋) Thus it remains to prove thatf^!_Λis invertible.

Use :f formally smooth⇒the dualizing complexD_M_Z_/S:=f^!Λis compatible with base change: for anyS⁰_→S, if

MZ×SS⁰₌_M_Z_×

XSX_S0 h //

MZ

S⁰ //S

h^∗_D_M

Z/S=D_M_Z×

XSXS0/S⁰

⇒one can supposeMZ→S has a sections and we want to prove s^∗D_M_Z_/S

is invertible.

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Proof of the main theorem : deformation to the normal cone

s₌section ofMZ/S corresponds toσ=section ofZ/X_S σ:X_S,→Z.

Zâ_×_B¹₌blow-up ofZ_×_B¹ along(_σ,0)(deformation to the normal coneof σ:X_S,→Z)

Considering

(Z^â_×B¹)_×_B₁_B¹^,¹^/p^∞ one denes (BS:₌_B¹^,¹^/p^∞

S )

Mf

ρ

BS es

;;

satisfying

Ï ρ⁻¹(_B_S\ {0^})₌_M_Z_×_S(_B_S\ {0^})andes_↔(s,Id) Ï se_|ρ₋₁₍₀₎:S zero section,→ BC( _σ^∗T_Z_/X

S

| {z } conormal bundle toσ

) ^open_,_→ _ρ⁻¹(0)

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Proof of the main theorem : deformation to the normal cone

Mf

ρ

BS es

;;

$∈O(S)=p.u. element

×$:_B¹^,¹^/p^∞

S →B¹_S^,¹^/p^∞ multiplication by$.

K:₌s_e^∗_ρ^!_Λ₌complex of$^N-equivariant étale sheaves onBS s.t. if i:S,→B¹_S^,¹^/p^∞ (zero section)

i^∗K is invertible (coho smoothness ofBC(s^∗T_Z_/X

S)+ formal smoothness ofρ)

=⇒K is invertible using that×$:_B¹^,¹^/p^∞

S →B¹_S^,¹^/p^∞ is contracting.

=⇒ the result looking atK_|B_S_\{₀_}.

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A Jacobian criterion

d>0,P∈Qp[X₁, . . . ,Xn]homogeneous of degreeδ BC(O(d))=(B_cris⁺ )^ϕ=^p^d

Pe:BC(_O(d))ⁿ_−→BC(_O(d_δ)) Forx_∈BC(_O(d))ⁿ(C)

Jac

Pe,x:_O(d)ⁿ_−→_O(d_δ) linear morphism between v.b./XC.

U₌open subset whereJac

Pe,x is surjective with>0 kernel.

ThenU_→Spa(_Fp)is cohomologically smooth.

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Conclusion

Good notion = formally smooth + cohomologically smooth

Pattern : prove rst something is formally smooth much easier then to prove this is coho smooth