Introduction to derived algebraic geometryweb.math.unifi.it/gruppi/algebraic-geometry/slides...Plan of the talk 1 A quick introduction to Derived Algebraic Geometry 2 An example {

Introduction to derived algebraic geometry

Gabriele Vezzosi

Firenze - 10 Ottobre, 2012

Gabriele Vezzosi () Introduction to derived algebraic geometry Firenze - 10 Ottobre, 2012 1 / 26

Plan of the talk

1 A quick introduction to Derived Algebraic Geometry

2 An example – the derived stack of vector bundles

3 Derived symplectic structures


Why derived geometry?

Hidden smoothness philosophy (Deligne-Drinfel’d-Kontsevich):singular moduli spaces are truncations of ’true’ moduli spaces whichare smooth ; good intersection theory.

Conjecture on elliptic cohomology (V, ∼ 2003; then proved andgeneralized by J. Lurie): Topological Modular Forms (TMF) areglobal sections of a natural sheaf on a version of Mell ≡M1,1

defined as a derived moduli space modeled over commutative (a.k.aE∞) ring spectra.

Understand more geometrically and functorially obstruction theoryand virtual fundamental classes (Li-Tian, Behrend-Fantechi) andmore generally deformation theory for schemes, stacks etc. (e.g. givea geometric interpretation of the full cotangent complex, a questionposed by A. Grothendieck in 1968 !).

Realize C∞-intersection theory without transversality ; C∞-derivedcobordism (realized by D. Spivak (2009)).



Hidden smoothness philosophy (Deligne-Drinfel’d-Kontsevich):singular moduli spaces are truncations of ’true’ moduli spaces whichare smooth

; good intersection theory.















Conjecture on elliptic cohomology (V, ∼ 2003; then proved andgeneralized by J. Lurie):

Topological Modular Forms (TMF) areglobal sections of a natural sheaf on a version of Mell ≡M1,1
















Understand more geometrically and functorially obstruction theoryand virtual fundamental classes (Li-Tian, Behrend-Fantechi)

andmore generally deformation theory for schemes, stacks etc. (e.g. givea geometric interpretation of the full cotangent complex, a questionposed by A. Grothendieck in 1968 !).







Understand more geometrically and functorially obstruction theoryand virtual fundamental classes (Li-Tian, Behrend-Fantechi) andmore generally deformation theory for schemes, stacks etc.

(e.g. givea geometric interpretation of the full cotangent complex, a questionposed by A. Grothendieck in 1968 !).















Realize C∞-intersection theory without transversality

; C∞-derivedcobordism (realized by D. Spivak (2009)).









What should derived geometry be? A path through hiddensmoothness

X - smooth projective variety /CVectn(X ): moduli stack classifying rank n vector bundles on XxE : SpecC→ Vectn(X ) ⇔ E → X ;

TxE Vectn(X ) ' τ≤1(RΓ(XZar,End(E ))[1])

• If dim X = 1 there is no truncation ; dim TE is locally constant ;

Vectn(X ) is smooth.• if dim X ≥ 2, truncation is effective ; dim TE is not locally constant; Vectn(X ) is not smooth (in general).

Upshot - smoothness would be assured for any X if Vectn(X ) was a’space’ whose tangent complex was the full RΓ(XZar,End(E ))[1] (i.e. notruncation).BUT (for arbitrary X ) RΓ(XZar,End(E ))[1] is a perfect complex inarbitrary positive degrees ; it cannot be the tangent space of any 1-stack(nor of any n-stack for n ≥ 1).



X - smooth projective variety /C

Vectn(X ): moduli stack classifying rank n vector bundles on XxE : SpecC→ Vectn(X ) ⇔ E → X ;







X - smooth projective variety /CVectn(X ): moduli stack classifying rank n vector bundles on X

xE : SpecC→ Vectn(X ) ⇔ E → X ;







X - smooth projective variety /CVectn(X ): moduli stack classifying rank n vector bundles on XxE : SpecC→ Vectn(X ) ⇔ E → X

;









• If dim X = 1 there is no truncation

; dim TE is locally constant ;







• If dim X = 1 there is no truncation ; dim TE is locally constant

;








Vectn(X ) is smooth.

• if dim X ≥ 2, truncation is effective ; dim TE is not locally constant; Vectn(X ) is not smooth (in general).







Vectn(X ) is smooth.• if dim X ≥ 2, truncation is effective

; dim TE is not locally constant; Vectn(X ) is not smooth (in general).







Vectn(X ) is smooth.• if dim X ≥ 2, truncation is effective ; dim TE is not locally constant

; Vectn(X ) is not smooth (in general).















Upshot - smoothness would be assured for any X if Vectn(X ) was a’space’ whose tangent complex was the full RΓ(XZar,End(E ))[1] (i.e. notruncation).

BUT (for arbitrary X ) RΓ(XZar,End(E ))[1] is a perfect complex inarbitrary positive degrees ; it cannot be the tangent space of any 1-stack(nor of any n-stack for n ≥ 1).







Upshot - smoothness would be assured for any X if Vectn(X ) was a’space’ whose tangent complex was the full RΓ(XZar,End(E ))[1] (i.e. notruncation).BUT (for arbitrary X ) RΓ(XZar,End(E ))[1] is a perfect complex inarbitrary positive degrees

; it cannot be the tangent space of any 1-stack(nor of any n-stack for n ≥ 1).







Upshot - smoothness would be assured for any X if Vectn(X ) was a’space’ whose tangent complex was the full RΓ(XZar,End(E ))[1] (i.e. notruncation).BUT (for arbitrary X ) RΓ(XZar,End(E ))[1] is a perfect complex inarbitrary positive degrees ; it cannot be the tangent space of any 1-stack

(nor of any n-stack for n ≥ 1).










So we need a new kind of spaces to accommodate tangent spaces T indegrees [0,∞).

To guess heuristically the local structure of this spaces

require smoothness (i.e. uncover hidden smoothness)

then, locally at any point, should look like Spec(Sym(T∨))

; local models for these spaces are cdga’s i.e. commutative differentialgraded C-algebras in degrees ≤ 0 (equivalently, simplicial commutativeC-algebras) and T is only defined up to quasi-isomorphisms (isos incohomology)So local/affine objects of derived algebraic geometry are cdga’s defined upto quasi-isomorphism.



















require smoothness

(i.e. uncover hidden smoothness)























; local models for these spaces are cdga’s i.e. commutative differentialgraded C-algebras in degrees ≤ 0

(equivalently, simplicial commutativeC-algebras) and T is only defined up to quasi-isomorphisms (isos incohomology)So local/affine objects of derived algebraic geometry are cdga’s defined upto quasi-isomorphism.







; local models for these spaces are cdga’s i.e. commutative differentialgraded C-algebras in degrees ≤ 0 (equivalently, simplicial commutativeC-algebras)

and T is only defined up to quasi-isomorphisms (isos incohomology)So local/affine objects of derived algebraic geometry are cdga’s defined upto quasi-isomorphism.







; local models for these spaces are cdga’s i.e. commutative differentialgraded C-algebras in degrees ≤ 0 (equivalently, simplicial commutativeC-algebras) and

T is only defined up to quasi-isomorphisms (isos incohomology)So local/affine objects of derived algebraic geometry are cdga’s defined upto quasi-isomorphism.







; local models for these spaces are cdga’s i.e. commutative differentialgraded C-algebras in degrees ≤ 0 (equivalently, simplicial commutativeC-algebras) and T is only defined up to quasi-isomorphisms (isos incohomology)

So local/affine objects of derived algebraic geometry are cdga’s defined upto quasi-isomorphism.







; local models for these spaces are cdga’s i.e. commutative differentialgraded C-algebras in degrees ≤ 0 (equivalently, simplicial commutativeC-algebras) and T is only defined up to quasi-isomorphisms (isos incohomology)So

local/affine objects of derived algebraic geometry are cdga’s defined upto quasi-isomorphism.









Derived affine schemes and homotopy theory

So derived affine schemes (i.e. the opposite category of cdga’s) have to beconsidered up to quasi-isomorphisms: i.e. we want to glue them alongquasi-isomorphisms not isomorphisms. Recall that a scheme is built out ofaffine schemes glued along isomorphisms.

So we need a theory enabling us to treat quasi-isomorphisms on the samefooting as isomorphisms, i.e. to make them essentially invertible.(Essentially? Formally inverting q-isos is too rough for gluing purposese.g. derived categories or objects in derived categories of a cover do notglue! )

Thanks to Quillen, we know how to do it properly:cdga’s together with q-isos constitute a homotopy theory (technicallyspeaking Quillen model category structure).



So derived affine schemes (i.e. the opposite category of cdga’s) have to beconsidered up to quasi-isomorphisms:

i.e. we want to glue them alongquasi-isomorphisms not isomorphisms. Recall that a scheme is built out ofaffine schemes glued along isomorphisms.





So derived affine schemes (i.e. the opposite category of cdga’s) have to beconsidered up to quasi-isomorphisms: i.e. we want to glue them alongquasi-isomorphisms not isomorphisms.

Recall that a scheme is built out ofaffine schemes glued along isomorphisms.











So we need a theory enabling us to treat quasi-isomorphisms on the samefooting as isomorphisms, i.e. to make them essentially invertible.

(Essentially? Formally inverting q-isos is too rough for gluing purposese.g. derived categories or objects in derived categories of a cover do notglue! )





So we need a theory enabling us to treat quasi-isomorphisms on the samefooting as isomorphisms, i.e. to make them essentially invertible.(Essentially? Formally inverting q-isos is too rough for gluing purposes

e.g. derived categories or objects in derived categories of a cover do notglue! )











Thanks to Quillen, we know how to do it properly:

cdga’s together with q-isos constitute a homotopy theory (technicallyspeaking Quillen model category structure).





Thanks to Quillen, we know how to do it properly:cdga’s together with q-isos constitute a homotopy theory

(technicallyspeaking Quillen model category structure).








What is a ’homotopy theory’ ? Roughly, a category M together with adistinguished class of maps w in M, such that we can define not onlyHom-set (between objects in M) up to maps in w (i.e. Homw−1M(−,−))but a whole mapping space (top. space or simpl. set) of maps up to mapsin w (i.e. Map(M,w)(−,−)) & homotopy ve rsions of lim/colim

Examples of homotopy theories

(M = Top,w = weak homotopy eq.ces)(M = SimplSets,w = weak homotopy eq.ces)

k : comm. ring, (Chk ,w = q-isos) (here πi ’s of mapping spaces arethe Ext-groups)

(cdgak ,w = q-isos) (char k = 0)(SimplCommAlgk ,w = weak htpy eq.ces) (any k).

w−1M := Ho(M) : homotopy category of the hom. theory (M,w).But the htpy theory (M,w) strictly enhance Ho(M) !



What is a ’homotopy theory’ ?

Roughly, a category M together with adistinguished class of maps w in M, such that we can define not onlyHom-set (between objects in M) up to maps in w (i.e. Homw−1M(−,−))but a whole mapping space (top. space or simpl. set) of maps up to mapsin w (i.e. Map(M,w)(−,−)) & homotopy ve rsions of lim/colim








What is a ’homotopy theory’ ? Roughly, a category M together with adistinguished class of maps w in M

, such that we can define not onlyHom-set (between objects in M) up to maps in w (i.e. Homw−1M(−,−))but a whole mapping space (top. space or simpl. set) of maps up to mapsin w (i.e. Map(M,w)(−,−)) & homotopy ve rsions of lim/colim








What is a ’homotopy theory’ ? Roughly, a category M together with adistinguished class of maps w in M, such that we can define not onlyHom-set (between objects in M) up to maps in w (i.e. Homw−1M(−,−))

but a whole mapping space (top. space or simpl. set) of maps up to mapsin w (i.e. Map(M,w)(−,−)) & homotopy ve rsions of lim/colim








What is a ’homotopy theory’ ? Roughly, a category M together with adistinguished class of maps w in M, such that we can define not onlyHom-set (between objects in M) up to maps in w (i.e. Homw−1M(−,−))but a whole mapping space (top. space or simpl. set) of maps up to mapsin w (i.e. Map(M,w)(−,−))

& homotopy ve rsions of lim/colim


















(M = Top,w = weak homotopy eq.ces)

(M = SimplSets,w = weak homotopy eq.ces)

















k : comm. ring, (Chk ,w = q-isos)

(here πi ’s of mapping spaces arethe Ext-groups)

















(cdgak ,w = q-isos) (char k = 0)

(SimplCommAlgk ,w = weak htpy eq.ces) (any k).

















w−1M := Ho(M)

: homotopy category of the hom. theory (M,w).But the htpy theory (M,w) strictly enhance Ho(M) !








w−1M := Ho(M) : homotopy category of the hom. theory (M,w).

But the htpy theory (M,w) strictly enhance Ho(M) !










What is derived algebraic geometry?

(underived) Algebraic Geometry

schemes, algebraic spaces ; 1-stacks ;∞-stacks

CommAlgk

1-stacks

))

∞-stacks

$$

schemes // Sets

right derivation

��

Grpds

π0

OO

SimplSets

Π1

OO

right derivation ≡ adjoining homotopy colimits (⇒ can take quotients) ;

promote the target categories to a homotopy theory (that of SimplSetsor, eq.ly, topological spaces).





CommAlgk

1-stacks

))

∞-stacks

$$

schemes // Sets

right derivation

��

Grpds

π0

OO

SimplSets

Π1

OO







CommAlgk

1-stacks

))

∞-stacks

$$

schemes // Sets

right derivation

��

Grpds

π0

OO

SimplSets

Π1

OO






schemes, algebraic spaces

; 1-stacks ;∞-stacks

CommAlgk

1-stacks

))

∞-stacks

$$

schemes // Sets

right derivation

��

Grpds

π0

OO

SimplSets

Π1

OO






schemes, algebraic spaces ; 1-stacks

;∞-stacks

CommAlgk

1-stacks

))

∞-stacks

$$

schemes // Sets

right derivation

��

Grpds

π0

OO

SimplSets

Π1

OO







CommAlgk

1-stacks

))

∞-stacks

$$

schemes // Sets

right derivation

��

Grpds

π0

OO

SimplSets

Π1

OO







CommAlgk

1-stacks

))

∞-stacks

$$

schemes // Sets

right derivation

��

Grpds

π0

OO

SimplSets

Π1

OO







CommAlgk

1-stacks

))

∞-stacks

$$

schemes // Sets

right derivation

��

Grpds

π0

OO

SimplSets

Π1

OO





if we derive also to the left ;

left deriv.

��

CommAlgk

1-stacks

**

∞-stacks

%%

schemes // Ens

right deriv.

��

Grpds

π0

OO

SimplCommAlgkderived ∞-stacks//

π0

OO

SimplSets

Π1

OO

; derived Algebraic Geometry: source and target are nontrivialhomotopy theories.

It is a kind of algebraic geometry where affine objects are simplicialcommutative algebras (or k-cdga if char(k) = 0)



if we derive also to the left

;

left deriv.

��

CommAlgk

1-stacks

**

∞-stacks

%%

schemes // Ens

right deriv.

��

Grpds

π0

OO


π0

OO

SimplSets

Π1

OO






left deriv.

��

CommAlgk

1-stacks

**

∞-stacks

%%

schemes // Ens

right deriv.

��

Grpds

π0

OO


π0

OO

SimplSets

Π1

OO






left deriv.

��

CommAlgk

1-stacks

**

∞-stacks

%%

schemes // Ens

right deriv.

��

Grpds

π0

OO


π0

OO

SimplSets

Π1

OO






left deriv.

��

CommAlgk

1-stacks

**

∞-stacks

%%

schemes // Ens

right deriv.

��

Grpds

π0

OO


π0

OO

SimplSets

Π1

OO

; derived Algebraic Geometry:

source and target are nontrivialhomotopy theories.





left deriv.

��

CommAlgk

1-stacks

**

∞-stacks

%%

schemes // Ens

right deriv.

��

Grpds

π0

OO


π0

OO

SimplSets

Π1

OO






left deriv.

��

CommAlgk

1-stacks

**

∞-stacks

%%

schemes // Ens

right deriv.

��

Grpds

π0

OO


π0

OO

SimplSets

Π1

OO


It is a kind of algebraic geometry where affine objects are simplicialcommutative algebras

(or k-cdga if char(k) = 0)




left deriv.

��

CommAlgk

1-stacks

**

∞-stacks

%%

schemes // Ens

right deriv.

��

Grpds

π0

OO


π0

OO

SimplSets

Π1

OO




Derived Algebraic Geometry (DAG) in two steps

Recall - A scheme, algebraic space, stack etc. is a functor as above whichmoreover

satisfies a sheaf condition (descent) with respect to some chosentopology defined on commutative algebras

admits a (Zariski, etale, flat, smooth) atlas of affine schemes

Example - A functor X : CommAlgk −→ Sets is a scheme iff

is an etale sheaf: for any comm. k-algebra A, for any etale coveringfamily {A→ Ai}i of A, the canonical map

X (A) −→ limjX (Aj)

is a bijection;

it admits a Zariski atlas∐

i Ui → X (Ui = Spec Ri , Ri ∈ CommAlg).



Recall -

A scheme, algebraic space, stack etc. is a functor as above whichmoreover






is a bijection;











is a bijection;











is a bijection;











is a bijection;











is a bijection;











is a bijection;











is a bijection;











is a bijection;





To translate this into DAG, we thus need two steps

we first need a notion of derived topology and derived sheaf theory

then we need to make sense of (Zariski, etale, flat, smooth) derivedatlases.

Just as schemes, algebraic spaces and stacks are (simplicial) sheavesadmitting some kind of atlases,the first step will give us up-to-homotopy (simplicial) sheaves, amongwhich the second step will single out the derived spaces studied by derivedalgebraic geometry.
























Just as schemes, algebraic spaces and stacks are (simplicial) sheavesadmitting some kind of atlases,

the first step will give us up-to-homotopy (simplicial) sheaves, amongwhich the second step will single out the derived spaces studied by derivedalgebraic geometry.






Just as schemes, algebraic spaces and stacks are (simplicial) sheavesadmitting some kind of atlases,the first step will give us up-to-homotopy (simplicial) sheaves,

amongwhich the second step will single out the derived spaces studied by derivedalgebraic geometry.








Derived Algebraic Geometry (DAG) - 1st step: derivedsheaf theory

First step (Toen-V., 2004) – develop a sheaf theory over homotopytheories (= Quillen model categories) endowed with a up-to-homotopytopology ; homotopy/higher topoi (model topoi in HAG I; thenreconsidered and generalized further by J. Lurie)

Derived topology on a homotopy theory/model category (M,w) ⇒Grothendieck topology on Ho(M) = w−1M.

Examples of homotopy theories we consider

Simplicial commutative k-algebras (k any commutative ring)

differential graded commutative k-algebras (char k = 0)

commutative ring spectra (E∞−ring spectra)

(more generally: commutative ring objects in a symmetric monoidalQuillen model category (M,w ,⊗) ; in such a general setting the derivedgeometry we get is called Homotopical Algebraic Geometry - HAG - )



First step (Toen-V., 2004) –

develop a sheaf theory over homotopytheories (= Quillen model categories) endowed with a up-to-homotopytopology ; homotopy/higher topoi (model topoi in HAG I; thenreconsidered and generalized further by J. Lurie)









First step (Toen-V., 2004) – develop a sheaf theory over homotopytheories (= Quillen model categories) endowed with a up-to-homotopytopology

; homotopy/higher topoi (model topoi in HAG I; thenreconsidered and generalized further by J. Lurie)









First step (Toen-V., 2004) – develop a sheaf theory over homotopytheories (= Quillen model categories) endowed with a up-to-homotopytopology ; homotopy/higher topoi

(model topoi in HAG I; thenreconsidered and generalized further by J. Lurie)



















Derived topology on a homotopy theory/model category (M,w)

⇒Grothendieck topology on Ho(M) = w−1M.




































































(more generally:

commutative ring objects in a symmetric monoidalQuillen model category (M,w ,⊗) ; in such a general setting the derivedgeometry we get is called Homotopical Algebraic Geometry - HAG - )









(more generally: commutative ring objects in a symmetric monoidalQuillen model category (M,w ,⊗)

; in such a general setting the derivedgeometry we get is called Homotopical Algebraic Geometry - HAG - )











Derived Algebraic Geometry (DAG) - 1st step derived sheaftheory

An example - etale derived topology on SimplCommAlgk :{A→ Bi} is an etale covering family for derived etale topology if

{π0A→ π0Bi} is an etale covering family (in the usual sense)

for any i and any n ≥ 0 , πnA⊗π0A π0B → πnBi is an isomorphism

The intuition is:

everything is as usual on the classical part/truncation π0(−),

on the higher πn everything is just a pullback along π0A→ π0B

Rmk. This is not an ad hoc definition: it is an elementary characterizationof a more conceptual definition (via derived infinitesimal lifting property).



An example - etale derived topology on SimplCommAlgk :

{A→ Bi} is an etale covering family for derived etale topology if



The intuition is:









The intuition is:









The intuition is:









The intuition is:









The intuition is:









The intuition is:









The intuition is:









The intuition is:






Choice of a derived topology (e.g. etale) on dAffk := SimplCommAlgopk

;

Homotopy theory of derived stacks

induces a homotopy theory (Quillen model category) on the categorydSPrk of simplicial presheaves on dAffk

SimplCommAlgk = dAffopk → SimplSets

weak equivalences f : F → G inducing πi (F , x) ' πi (G , f (x)) for anyi ≥ 0 and any x , as sheaves on the usual site Ho(dAffk).

The category of derived stacks is dStk := Ho(dSPrk)




;









;









;









;









;








Therefore, a derived stack, i.e. an object in dStk , is a functorF : SimplCommAlgk → SimplSets such that

F preserves sends weak equivalences in SimplCommAlgk to weakequivalences in SimplSetskF has descent with respect to etale homotopy-hypercoverings , i.e.

F (A)→ holimF (B•)

is an iso in Ho(SimplSets), for any A and any etale h-hypercoveringB• de A

Rmk. Don’t worry about hypercoverings, just think of Cech nervesassociated to covers.



Therefore,

a derived stack, i.e. an object in dStk , is a functorF : SimplCommAlgk → SimplSets such that















F preserves sends weak equivalences in SimplCommAlgk to weakequivalences in SimplSetsk

F has descent with respect to etale homotopy-hypercoverings , i.e.







F preserves sends weak equivalences in SimplCommAlgk to weakequivalences in SimplSetskF has descent with respect to etale homotopy-hypercoverings

, i.e.
















is an iso in Ho(SimplSets),

for any A and any etale h-hypercoveringB• de A








Rmk. Don’t worry about hypercoverings,

just think of Cech nervesassociated to covers.










Derived Yoneda:

RSpec : AlgCommSimplk → dStk , A 7→ MapAlgCommSimplk(A,−)

is fully faithful (up to homotopy).

dStk has internal HOM’s: F ,G ∈ dStk ;

MAPdStk (F ,G ) = RHOMdStk (F ,G )

and also homotopy limits and colimits e.g. homotopy fibered productis locally given by the derived tensor product

RSpecB ×hRSpecA RSpecC ' RSpec(B ⊗L

A C ).



Derived Yoneda:







A C ).



Derived Yoneda:







A C ).



Derived Yoneda:







A C ).



Derived Yoneda:







A C ).



There is a truncation/inclusion adjunction:

dStkt0 //

Stkioo

i is fully faithful (hence usually omitted in notations)t0(RSpecA) = Specπ0Athe adjunction map i(t0X ) ↪→ X is a closed immersioni preserves homotopy colimits but not homotopy limits nor internalHOM’s ; derived tangent spaces and derived fibered products ofschemes are not the usual tangent spaces and fibered products !

Geometric intuition - X like a formal thickening of its truncationt0(X ), (as if t0(X ) was the ’reduced’ subscheme of X ). In particular,the small etale sites of X and t0(X ) are equivalent.




dStkt0 //

Stkioo






dStkt0 //

Stkioo






dStkt0 //

Stkioo

i is fully faithful

(hence usually omitted in notations)t0(RSpecA) = Specπ0Athe adjunction map i(t0X ) ↪→ X is a closed immersioni preserves homotopy colimits but not homotopy limits nor internalHOM’s ; derived tangent spaces and derived fibered products ofschemes are not the usual tangent spaces and fibered products !





dStkt0 //

Stkioo

i is fully faithful (hence usually omitted in notations)

t0(RSpecA) = Specπ0Athe adjunction map i(t0X ) ↪→ X is a closed immersioni preserves homotopy colimits but not homotopy limits nor internalHOM’s ; derived tangent spaces and derived fibered products ofschemes are not the usual tangent spaces and fibered products !





dStkt0 //

Stkioo

i is fully faithful (hence usually omitted in notations)t0(RSpecA) = Specπ0A

the adjunction map i(t0X ) ↪→ X is a closed immersioni preserves homotopy colimits but not homotopy limits nor internalHOM’s ; derived tangent spaces and derived fibered products ofschemes are not the usual tangent spaces and fibered products !





dStkt0 //

Stkioo

i is fully faithful (hence usually omitted in notations)t0(RSpecA) = Specπ0Athe adjunction map i(t0X ) ↪→ X is a closed immersion

i preserves homotopy colimits but not homotopy limits nor internalHOM’s ; derived tangent spaces and derived fibered products ofschemes are not the usual tangent spaces and fibered products !





dStkt0 //

Stkioo

i is fully faithful (hence usually omitted in notations)t0(RSpecA) = Specπ0Athe adjunction map i(t0X ) ↪→ X is a closed immersioni preserves homotopy colimits but not homotopy limits nor internalHOM’s

; derived tangent spaces and derived fibered products ofschemes are not the usual tangent spaces and fibered products !





dStkt0 //

Stkioo


Geometric intuition -

X like a formal thickening of its truncationt0(X ), (as if t0(X ) was the ’reduced’ subscheme of X ). In particular,the small etale sites of X and t0(X ) are equivalent.




dStkt0 //

Stkioo


Geometric intuition - X like a formal thickening of its truncationt0(X ),

(as if t0(X ) was the ’reduced’ subscheme of X ). In particular,the small etale sites of X and t0(X ) are equivalent.




dStkt0 //

Stkioo


Geometric intuition - X like a formal thickening of its truncationt0(X ), (as if t0(X ) was the ’reduced’ subscheme of X ).

In particular,the small etale sites of X and t0(X ) are equivalent.




dStkt0 //

Stkioo




Derived Algebraic Geometry (DAG) - 2nd step: derivedgeometric stacks

• 2 notions of derived smooth maps between simpl. comm algebras:

A→ B smooth if π0A→ π0B is smooth and πnA⊗π0A π0B ' πnBfor any n ≥ 0

A→ B is p-smooth if the relative cotangent complex LB/A is perfect

Geometric types of derived stacks

F a derived stack

A derived atlas for F is a map∐

i RSpecAi → F surjective on π0

(and satisfying some representability conditions)

if the map is smooth (resp. etale, Zariski) we have a derived Artinstack (resp. Deligne-Mumford stack, scheme)

The truncation preserves the type of the stack.

Using atlases (and representability) ; extend notion of smooth, etale,flat, etc. to maps between geometric derived stacks.







F a derived stack













F a derived stack













F a derived stack













F a derived stack













F a derived stack













F a derived stack













F a derived stack




if the map is smooth (resp. etale, Zariski)

we have a derived Artinstack (resp. Deligne-Mumford stack, scheme)









F a derived stack













F a derived stack













F a derived stack






Using atlases (and representability)

; extend notion of smooth, etale,flat, etc. to maps between geometric derived stacks.







F a derived stack








Derived Algebraic Geometry (DAG) - Main properties

X ∈ dStk ; derived tangent stack

TX := MAP(Spec k[ε],X ) ' Spec Sym(L∨X )

LX - cotangent complex of X (it classifies derived derivations) ;

Geometric/modular interpretation of cotangent complex

X (underived) scheme ⇒ Li(X ) is Grothendieck-Illusie cotangent complex ofX (so it corresponds to the cotgt space of some geom. space)

Ext i (LX ,x , k) ' HomdStk, ∗(Spec k[εi ], (X , x)), x ∈ X (k)

(k[εi ] - trivial extension of k by K(k, i))

; the full cotangent complex is uniquely geometrically characterizedin DAG (this answers Grothendieck’s question in Categories cofibreesadditives et complexe cotangent relatif, 1968).



X ∈ dStk

; derived tangent stack






















LX - cotangent complex of X (it classifies derived derivations)

;






















X (underived) scheme

⇒ Li(X ) is Grothendieck-Illusie cotangent complex ofX (so it corresponds to the cotgt space of some geom. space)










X (underived) scheme ⇒ Li(X ) is Grothendieck-Illusie cotangent complex ofX

(so it corresponds to the cotgt space of some geom. space)






















(k[εi ] - trivial extension of k by K(k , i))











; the full cotangent complex is uniquely geometrically characterizedin DAG

(this answers Grothendieck’s question in Categories cofibreesadditives et complexe cotangent relatif, 1968).













The (full) cotangent complex of a derived stack has a universalmoduli property ; it is computable ! (this also explains why thestacky cotangent complex of some underived stacks is not known); deformation theory is functorial and ’easy’ in DAG.

For any derived DM stack X , the closed immersion i : t0(X ) −→ Xinduces a canonical obstruction theory on t0(X ) (in the sense ofBehrend-Fantechi)

i∗(LX ) −→ Lt0(X ).

; if i∗(LX ) is of perfect amplitude in [−1, 0] ; virtual fundamentalclass [X ]vir on t0(X ), which is moreover natural in X (unlike inBehrend-Fantechi’s approach).



The (full) cotangent complex of a derived stack has a universalmoduli property

; it is computable ! (this also explains why thestacky cotangent complex of some underived stacks is not known); deformation theory is functorial and ’easy’ in DAG.


i∗(LX ) −→ Lt0(X ).




The (full) cotangent complex of a derived stack has a universalmoduli property ; it is computable !

(this also explains why thestacky cotangent complex of some underived stacks is not known); deformation theory is functorial and ’easy’ in DAG.


i∗(LX ) −→ Lt0(X ).




The (full) cotangent complex of a derived stack has a universalmoduli property ; it is computable ! (this also explains why thestacky cotangent complex of some underived stacks is not known)

; deformation theory is functorial and ’easy’ in DAG.


i∗(LX ) −→ Lt0(X ).






i∗(LX ) −→ Lt0(X ).






i∗(LX ) −→ Lt0(X ).






i∗(LX ) −→ Lt0(X ).






i∗(LX ) −→ Lt0(X ).

; if i∗(LX ) is of perfect amplitude in [−1, 0]

; virtual fundamentalclass [X ]vir on t0(X ), which is moreover natural in X (unlike inBehrend-Fantechi’s approach).





i∗(LX ) −→ Lt0(X ).

; if i∗(LX ) is of perfect amplitude in [−1, 0] ; virtual fundamentalclass [X ]vir on t0(X ),

which is moreover natural in X (unlike inBehrend-Fantechi’s approach).





i∗(LX ) −→ Lt0(X ).

; if i∗(LX ) is of perfect amplitude in [−1, 0] ; virtual fundamentalclass [X ]vir on t0(X ), which is moreover natural in X

(unlike inBehrend-Fantechi’s approach).





i∗(LX ) −→ Lt0(X ).






i∗(LX ) −→ Lt0(X ).




All moduli problems have some (maybe more than one) naturalderived version. ; All known moduli spaces have derivedenhancements: Hilbert scheme, moduli of curves, of stable maps, oflocal systems, of coherent sheaves, . . .

M; RM

where M' t0(RM) ↪→ RM (most often a strict inclusion).Thechoice of a derived enhancement RM yields additional structures onM: obstruction theory (for arbitrary geometric n-stacks), virtualstructure sheaf, virtual fundamental class (when it exists) . . .; inparticular Gromov-Witten and Donaldson-Thomas invariants can becompletely reconstructed form these derived enhancements(invariants of the enhancement not of the truncation).Conversely, a (nice) underived stack endowed with an obstructiontheory essentially reconstructs a particular derived enhancement (e.g.reduced obstruction theory for stable maps to a K 3-surface).



All moduli problems have some (maybe more than one) naturalderived version.

; All known moduli spaces have derivedenhancements: Hilbert scheme, moduli of curves, of stable maps, oflocal systems, of coherent sheaves, . . .

M; RM





M; RM





M; RM

where M' t0(RM) ↪→ RM (most often a strict inclusion).

Thechoice of a derived enhancement RM yields additional structures onM: obstruction theory (for arbitrary geometric n-stacks), virtualstructure sheaf, virtual fundamental class (when it exists) . . .; inparticular Gromov-Witten and Donaldson-Thomas invariants can becompletely reconstructed form these derived enhancements(invariants of the enhancement not of the truncation).Conversely, a (nice) underived stack endowed with an obstructiontheory essentially reconstructs a particular derived enhancement (e.g.reduced obstruction theory for stable maps to a K 3-surface).




M; RM

where M' t0(RM) ↪→ RM (most often a strict inclusion).Thechoice of a derived enhancement RM yields additional structures onM:

obstruction theory (for arbitrary geometric n-stacks), virtualstructure sheaf, virtual fundamental class (when it exists) . . .; inparticular Gromov-Witten and Donaldson-Thomas invariants can becompletely reconstructed form these derived enhancements(invariants of the enhancement not of the truncation).Conversely, a (nice) underived stack endowed with an obstructiontheory essentially reconstructs a particular derived enhancement (e.g.reduced obstruction theory for stable maps to a K 3-surface).




M; RM

where M' t0(RM) ↪→ RM (most often a strict inclusion).Thechoice of a derived enhancement RM yields additional structures onM: obstruction theory (for arbitrary geometric n-stacks),

virtualstructure sheaf, virtual fundamental class (when it exists) . . .; inparticular Gromov-Witten and Donaldson-Thomas invariants can becompletely reconstructed form these derived enhancements(invariants of the enhancement not of the truncation).Conversely, a (nice) underived stack endowed with an obstructiontheory essentially reconstructs a particular derived enhancement (e.g.reduced obstruction theory for stable maps to a K 3-surface).




M; RM

where M' t0(RM) ↪→ RM (most often a strict inclusion).Thechoice of a derived enhancement RM yields additional structures onM: obstruction theory (for arbitrary geometric n-stacks), virtualstructure sheaf,

virtual fundamental class (when it exists) . . .; inparticular Gromov-Witten and Donaldson-Thomas invariants can becompletely reconstructed form these derived enhancements(invariants of the enhancement not of the truncation).Conversely, a (nice) underived stack endowed with an obstructiontheory essentially reconstructs a particular derived enhancement (e.g.reduced obstruction theory for stable maps to a K 3-surface).




M; RM

where M' t0(RM) ↪→ RM (most often a strict inclusion).Thechoice of a derived enhancement RM yields additional structures onM: obstruction theory (for arbitrary geometric n-stacks), virtualstructure sheaf, virtual fundamental class (when it exists) . . .

; inparticular Gromov-Witten and Donaldson-Thomas invariants can becompletely reconstructed form these derived enhancements(invariants of the enhancement not of the truncation).Conversely, a (nice) underived stack endowed with an obstructiontheory essentially reconstructs a particular derived enhancement (e.g.reduced obstruction theory for stable maps to a K 3-surface).




M; RM

where M' t0(RM) ↪→ RM (most often a strict inclusion).Thechoice of a derived enhancement RM yields additional structures onM: obstruction theory (for arbitrary geometric n-stacks), virtualstructure sheaf, virtual fundamental class (when it exists) . . .; inparticular Gromov-Witten and Donaldson-Thomas invariants can becompletely reconstructed form these derived enhancements

(invariants of the enhancement not of the truncation).Conversely, a (nice) underived stack endowed with an obstructiontheory essentially reconstructs a particular derived enhancement (e.g.reduced obstruction theory for stable maps to a K 3-surface).




M; RM

where M' t0(RM) ↪→ RM (most often a strict inclusion).Thechoice of a derived enhancement RM yields additional structures onM: obstruction theory (for arbitrary geometric n-stacks), virtualstructure sheaf, virtual fundamental class (when it exists) . . .; inparticular Gromov-Witten and Donaldson-Thomas invariants can becompletely reconstructed form these derived enhancements(invariants of the enhancement not of the truncation).

Conversely, a (nice) underived stack endowed with an obstructiontheory essentially reconstructs a particular derived enhancement (e.g.reduced obstruction theory for stable maps to a K 3-surface).




M; RM

where M' t0(RM) ↪→ RM (most often a strict inclusion).Thechoice of a derived enhancement RM yields additional structures onM: obstruction theory (for arbitrary geometric n-stacks), virtualstructure sheaf, virtual fundamental class (when it exists) . . .; inparticular Gromov-Witten and Donaldson-Thomas invariants can becompletely reconstructed form these derived enhancements(invariants of the enhancement not of the truncation).Conversely, a (nice) underived stack endowed with an obstructiontheory essentially reconstructs a particular derived enhancement

(e.g.reduced obstruction theory for stable maps to a K 3-surface).




M; RM




X ∈ dStk , OX - structural (up-to-homotopy) sheaf of simplicialcommutative rings ; πi (OX ) are quasi-coherent on X and supportedon t0(X ) ; a sheaf of graded commutative rings π∗(OX ) on t0(X )called the virtual structure sheaf on X . If πi (OX ) are coherent, and 0for i >> 0 (equivalent to i∗(LX ) ∈ Perf [−1,0]), we get a class

[OX ]vir :=∑i

(−1)i [πi (OX )] ∈ G0(t0(X ))

a K -theoretic version of the virtual fundamental class.



X ∈ dStk ,

OX - structural (up-to-homotopy) sheaf of simplicialcommutative rings ; πi (OX ) are quasi-coherent on X and supportedon t0(X ) ; a sheaf of graded commutative rings π∗(OX ) on t0(X )called the virtual structure sheaf on X . If πi (OX ) are coherent, and 0for i >> 0 (equivalent to i∗(LX ) ∈ Perf [−1,0]), we get a class

[OX ]vir :=∑i

(−1)i [πi (OX )] ∈ G0(t0(X ))




X ∈ dStk , OX - structural (up-to-homotopy) sheaf of simplicialcommutative rings

; πi (OX ) are quasi-coherent on X and supportedon t0(X ) ; a sheaf of graded commutative rings π∗(OX ) on t0(X )called the virtual structure sheaf on X . If πi (OX ) are coherent, and 0for i >> 0 (equivalent to i∗(LX ) ∈ Perf [−1,0]), we get a class

[OX ]vir :=∑i

(−1)i [πi (OX )] ∈ G0(t0(X ))




X ∈ dStk , OX - structural (up-to-homotopy) sheaf of simplicialcommutative rings ; πi (OX ) are quasi-coherent on X and supportedon t0(X )

; a sheaf of graded commutative rings π∗(OX ) on t0(X )called the virtual structure sheaf on X . If πi (OX ) are coherent, and 0for i >> 0 (equivalent to i∗(LX ) ∈ Perf [−1,0]), we get a class

[OX ]vir :=∑i

(−1)i [πi (OX )] ∈ G0(t0(X ))




X ∈ dStk , OX - structural (up-to-homotopy) sheaf of simplicialcommutative rings ; πi (OX ) are quasi-coherent on X and supportedon t0(X ) ; a sheaf of graded commutative rings π∗(OX ) on t0(X )called the virtual structure sheaf on X .

If πi (OX ) are coherent, and 0for i >> 0 (equivalent to i∗(LX ) ∈ Perf [−1,0]), we get a class

[OX ]vir :=∑i

(−1)i [πi (OX )] ∈ G0(t0(X ))




X ∈ dStk , OX - structural (up-to-homotopy) sheaf of simplicialcommutative rings ; πi (OX ) are quasi-coherent on X and supportedon t0(X ) ; a sheaf of graded commutative rings π∗(OX ) on t0(X )called the virtual structure sheaf on X . If πi (OX ) are coherent, and 0for i >> 0 (equivalent to i∗(LX ) ∈ Perf [−1,0]),

we get a class

[OX ]vir :=∑i

(−1)i [πi (OX )] ∈ G0(t0(X ))





[OX ]vir :=∑i

(−1)i [πi (OX )] ∈ G0(t0(X ))





[OX ]vir :=∑i

(−1)i [πi (OX )] ∈ G0(t0(X ))





[OX ]vir :=∑i

(−1)i [πi (OX )] ∈ G0(t0(X ))




The base-change formula for quasi-coherent coefficients is satisfiedeven without flatness conditions for derived stacks

Quasicoherent base-change

For any homotopy cartesian square of derived stacks

X ′f ′ //

p′

��

X

p

��S ′

f// S

the canonical mapp∗ ◦ f∗ −→ f ′∗ ◦ p′∗

is a q-iso in ’most’ cases (e.g. for all quasi-compact derived schemes).

; in derived algebraic geometry objects are very much transverse (nomoving-lemmas needed).






X ′f ′ //

p′

��

X

p

��S ′

f// S









X ′f ′ //

p′

��

X

p

��S ′

f// S









X ′f ′ //

p′

��

X

p

��S ′

f// S









X ′f ′ //

p′

��

X

p

��S ′

f// S









X ′f ′ //

p′

��

X

p

��S ′

f// S


is a q-iso in ’most’ cases

(e.g. for all quasi-compact derived schemes).







X ′f ′ //

p′

��

X

p

��S ′

f// S









X ′f ′ //

p′

��

X

p

��S ′

f// S



; in derived algebraic geometry objects are very much transverse (nomoving-lemmas needed).Gabriele Vezzosi () Introduction to derived algebraic geometry Firenze - 10 Ottobre, 2012 23 / 26

Derived Algebraic Geometry (DAG) - An example

- Derived moduli stack of vector bundles on a sm. proj. variety X/C -

For A ∈ SimplCommAlgC, ; Modder(X ,A) category with objectspresheaves of OX ⊗ A-dg-modules on X and morphisms inducingquasi-isomorphisms on stalks (maps called equivalences).

RVectn : CommSimplAlgC −→ SimplSets

A 7−→ Nerve(Vectdern (X ,A))

where Vectdern (X ,A) is the full sub-category of Modder(X ,A) of rk n

derived vector bundles on X i.e. OX ⊗ A-dg-modules M on X which are

locally on XZar × Aet equivalent to (OX ⊗ A)n

flat over A (more precisely, M(U) is a cofibrant A-dg-module, for anyopen U ⊂ X )














For A ∈ SimplCommAlgC,

; Modder(X ,A) category with objectspresheaves of OX ⊗ A-dg-modules on X and morphisms inducingquasi-isomorphisms on stalks (maps called equivalences).

































where Vectdern (X ,A)

is the full sub-category of Modder(X ,A) of rk nderived vector bundles on X i.e. OX ⊗ A-dg-modules M on X which are










derived vector bundles on X

i.e. OX ⊗ A-dg-modules M on X which are













































Theorem (Toen-V.)

RVectn is a p-smooth Artin derived 1-stack

If E → X is a rk n vector bundle over X ,

TE (RVectn(X )) ' CZar(X ,End(E ))[1]

t0(RVectn(X )) ' Vectn(X )

; this is a global realization of Kontsevich hidden smoothness philosophy.



Theorem (Toen-V.)








Theorem (Toen-V.)








Theorem (Toen-V.)








Theorem (Toen-V.)








Theorem (Toen-V.)







Derived symplectic structures

I’ll use the blackboard if I’ll get to this...


Derived symplectic structures

I’ll use the blackboard if I’ll get to this...


Introduction to derived algebraic geometryweb.math.unifi.it/gruppi/algebraic-geometry/slides...Plan of the talk 1 A quick introduction to Derived Algebraic Geometry 2 An example {

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