Introduction to derived algebraic geometry

Introduction to derived algebraic geometry Gabriele Vezzosi

Firenze - 10 Ottobre, 2012

Gabriele Vezzosi ()

Introduction to derived algebraic geometry


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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

Gabriele Vezzosi ()



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Why derived geometry?

Gabriele Vezzosi ()



3 / 26

Why derived geometry? Hidden smoothness philosophy (Deligne-Drinfel’d-Kontsevich): singular moduli spaces are truncations of ’true’ moduli spaces which are smooth

Gabriele Vezzosi ()



3 / 26

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

Gabriele Vezzosi ()



3 / 26

Why derived geometry? Hidden smoothness philosophy (Deligne-Drinfel’d-Kontsevich): singular moduli spaces are truncations of ’true’ moduli spaces which are smooth ; good intersection theory. Conjecture on elliptic cohomology (V, ∼ 2003; then proved and generalized by J. Lurie):

Gabriele Vezzosi ()



3 / 26

Why derived geometry? Hidden smoothness philosophy (Deligne-Drinfel’d-Kontsevich): singular moduli spaces are truncations of ’true’ moduli spaces which are smooth ; good intersection theory. Conjecture on elliptic cohomology (V, ∼ 2003; then proved and generalized by J. Lurie): Topological Modular Forms (TMF) are global sections of a natural sheaf on a version of Mell ≡ M1,1 defined as a derived moduli space modeled over commutative (a.k.a E∞ ) ring spectra.

Gabriele Vezzosi ()



3 / 26

Why derived geometry? Hidden smoothness philosophy (Deligne-Drinfel’d-Kontsevich): singular moduli spaces are truncations of ’true’ moduli spaces which are smooth ; good intersection theory. Conjecture on elliptic cohomology (V, ∼ 2003; then proved and generalized by J. Lurie): Topological Modular Forms (TMF) are global sections of a natural sheaf on a version of Mell ≡ M1,1 defined as a derived moduli space modeled over commutative (a.k.a E∞ ) ring spectra. Understand more geometrically and functorially obstruction theory and virtual fundamental classes (Li-Tian, Behrend-Fantechi)

Gabriele Vezzosi ()



3 / 26

Why derived geometry? Hidden smoothness philosophy (Deligne-Drinfel’d-Kontsevich): singular moduli spaces are truncations of ’true’ moduli spaces which are smooth ; good intersection theory. Conjecture on elliptic cohomology (V, ∼ 2003; then proved and generalized by J. Lurie): Topological Modular Forms (TMF) are global sections of a natural sheaf on a version of Mell ≡ M1,1 defined as a derived moduli space modeled over commutative (a.k.a E∞ ) ring spectra. Understand more geometrically and functorially obstruction theory and virtual fundamental classes (Li-Tian, Behrend-Fantechi) and more generally deformation theory for schemes, stacks etc.

Gabriele Vezzosi ()



3 / 26

Why derived geometry? Hidden smoothness philosophy (Deligne-Drinfel’d-Kontsevich): singular moduli spaces are truncations of ’true’ moduli spaces which are smooth ; good intersection theory. Conjecture on elliptic cohomology (V, ∼ 2003; then proved and generalized by J. Lurie): Topological Modular Forms (TMF) are global sections of a natural sheaf on a version of Mell ≡ M1,1 defined as a derived moduli space modeled over commutative (a.k.a E∞ ) ring spectra. Understand more geometrically and functorially obstruction theory and virtual fundamental classes (Li-Tian, Behrend-Fantechi) and more generally deformation theory for schemes, stacks etc. (e.g. give a geometric interpretation of the full cotangent complex, a question posed by A. Grothendieck in 1968 !).

Gabriele Vezzosi ()



3 / 26

Why derived geometry? Hidden smoothness philosophy (Deligne-Drinfel’d-Kontsevich): singular moduli spaces are truncations of ’true’ moduli spaces which are smooth ; good intersection theory. Conjecture on elliptic cohomology (V, ∼ 2003; then proved and generalized by J. Lurie): Topological Modular Forms (TMF) are global sections of a natural sheaf on a version of Mell ≡ M1,1 defined as a derived moduli space modeled over commutative (a.k.a E∞ ) ring spectra. Understand more geometrically and functorially obstruction theory and virtual fundamental classes (Li-Tian, Behrend-Fantechi) and more generally deformation theory for schemes, stacks etc. (e.g. give a geometric interpretation of the full cotangent complex, a question posed by A. Grothendieck in 1968 !). Realize C ∞ -intersection theory without transversality

Gabriele Vezzosi ()



3 / 26

Why derived geometry? Hidden smoothness philosophy (Deligne-Drinfel’d-Kontsevich): singular moduli spaces are truncations of ’true’ moduli spaces which are smooth ; good intersection theory. Conjecture on elliptic cohomology (V, ∼ 2003; then proved and generalized by J. Lurie): Topological Modular Forms (TMF) are global sections of a natural sheaf on a version of Mell ≡ M1,1 defined as a derived moduli space modeled over commutative (a.k.a E∞ ) ring spectra. Understand more geometrically and functorially obstruction theory and virtual fundamental classes (Li-Tian, Behrend-Fantechi) and more generally deformation theory for schemes, stacks etc. (e.g. give a geometric interpretation of the full cotangent complex, a question posed by A. Grothendieck in 1968 !). Realize C ∞ -intersection theory without transversality ; C ∞ -derived cobordism (realized by D. Spivak (2009)). Gabriele Vezzosi ()



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What should derived geometry be? A path through hidden smoothness

Gabriele Vezzosi ()



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What should derived geometry be? A path through hidden smoothness X - smooth projective variety /C

Gabriele Vezzosi ()



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What should derived geometry be? A path through hidden smoothness X - smooth projective variety /C Vectn (X ): moduli stack classifying rank n vector bundles on X

Gabriele Vezzosi ()



4 / 26

What should derived geometry be? A path through hidden smoothness X - smooth projective variety /C Vectn (X ): moduli stack classifying rank n vector bundles on X xE : Spec C → Vectn (X ) ⇔ E → X

Gabriele Vezzosi ()



4 / 26

What should derived geometry be? A path through hidden smoothness X - smooth projective variety /C Vectn (X ): moduli stack classifying rank n vector bundles on X xE : Spec C → Vectn (X ) ⇔ E → X ; TxE Vectn (X ) ' τ≤1 (RΓ(XZar , End(E ))[1]) • If dim X = 1 there is no truncation

Gabriele Vezzosi ()



4 / 26

What should derived geometry be? A path through hidden smoothness X - smooth projective variety /C Vectn (X ): moduli stack classifying rank n vector bundles on X xE : Spec C → 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

Gabriele Vezzosi ()



4 / 26

What should derived geometry be? A path through hidden smoothness X - smooth projective variety /C Vectn (X ): moduli stack classifying rank n vector bundles on X xE : Spec C → 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.

Gabriele Vezzosi ()



4 / 26

What should derived geometry be? A path through hidden smoothness X - smooth projective variety /C Vectn (X ): moduli stack classifying rank n vector bundles on X xE : Spec C → 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

Gabriele Vezzosi ()



4 / 26

What should derived geometry be? A path through hidden smoothness X - smooth projective variety /C Vectn (X ): moduli stack classifying rank n vector bundles on X xE : Spec C → 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

Gabriele Vezzosi ()



4 / 26

What should derived geometry be? A path through hidden smoothness X - smooth projective variety /C Vectn (X ): moduli stack classifying rank n vector bundles on X xE : Spec C → 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).

Gabriele Vezzosi ()



4 / 26

What should derived geometry be? A path through hidden smoothness X - smooth projective variety /C Vectn (X ): moduli stack classifying rank n vector bundles on X xE : Spec C → 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. no truncation).

Gabriele Vezzosi ()



4 / 26

What should derived geometry be? A path through hidden smoothness X - smooth projective variety /C Vectn (X ): moduli stack classifying rank n vector bundles on X xE : Spec C → 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. no truncation). BUT (for arbitrary X ) RΓ(XZar , End(E ))[1] is a perfect complex in arbitrary positive degrees Gabriele Vezzosi ()



4 / 26

What should derived geometry be? A path through hidden smoothness X - smooth projective variety /C Vectn (X ): moduli stack classifying rank n vector bundles on X xE : Spec C → 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. no truncation). BUT (for arbitrary X ) RΓ(XZar , End(E ))[1] is a perfect complex in arbitrary positive degrees ; it cannot be the tangent space of any 1-stack Gabriele Vezzosi ()



4 / 26

What should derived geometry be? A path through hidden smoothness X - smooth projective variety /C Vectn (X ): moduli stack classifying rank n vector bundles on X xE : Spec C → 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. no truncation). BUT (for arbitrary X ) RΓ(XZar , End(E ))[1] is a perfect complex in arbitrary positive degrees ; it cannot be the tangent space of any 1-stack (nor of any n-stack for n ≥ 1). Gabriele Vezzosi ()



4 / 26

What should derived geometry be? A path through hidden smoothness

Gabriele Vezzosi ()



5 / 26

What should derived geometry be? A path through hidden smoothness So we need a new kind of spaces to accommodate tangent spaces T in degrees [0, ∞).

Gabriele Vezzosi ()



5 / 26

What should derived geometry be? A path through hidden smoothness So we need a new kind of spaces to accommodate tangent spaces T in degrees [0, ∞). To guess heuristically the local structure of this spaces

Gabriele Vezzosi ()



5 / 26

What should derived geometry be? A path through hidden smoothness So we need a new kind of spaces to accommodate tangent spaces T in degrees [0, ∞). To guess heuristically the local structure of this spaces require smoothness

Gabriele Vezzosi ()



5 / 26

What should derived geometry be? A path through hidden smoothness So we need a new kind of spaces to accommodate tangent spaces T in degrees [0, ∞). To guess heuristically the local structure of this spaces require smoothness (i.e. uncover hidden smoothness)

Gabriele Vezzosi ()



5 / 26

What should derived geometry be? A path through hidden smoothness So we need a new kind of spaces to accommodate tangent spaces T in degrees [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∨ ))

Gabriele Vezzosi ()



5 / 26

What should derived geometry be? A path through hidden smoothness So we need a new kind of spaces to accommodate tangent spaces T in degrees [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 differential graded C-algebras in degrees ≤ 0

Gabriele Vezzosi ()



5 / 26

What should derived geometry be? A path through hidden smoothness So we need a new kind of spaces to accommodate tangent spaces T in degrees [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 differential graded C-algebras in degrees ≤ 0 (equivalently, simplicial commutative C-algebras)

Gabriele Vezzosi ()



5 / 26

What should derived geometry be? A path through hidden smoothness So we need a new kind of spaces to accommodate tangent spaces T in degrees [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 differential graded C-algebras in degrees ≤ 0 (equivalently, simplicial commutative C-algebras) and

Gabriele Vezzosi ()



5 / 26

What should derived geometry be? A path through hidden smoothness So we need a new kind of spaces to accommodate tangent spaces T in degrees [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 differential graded C-algebras in degrees ≤ 0 (equivalently, simplicial commutative C-algebras) and T is only defined up to quasi-isomorphisms (isos in cohomology)

Gabriele Vezzosi ()



5 / 26

What should derived geometry be? A path through hidden smoothness So we need a new kind of spaces to accommodate tangent spaces T in degrees [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 differential graded C-algebras in degrees ≤ 0 (equivalently, simplicial commutative C-algebras) and T is only defined up to quasi-isomorphisms (isos in cohomology) So

Gabriele Vezzosi ()



5 / 26

What should derived geometry be? A path through hidden smoothness So we need a new kind of spaces to accommodate tangent spaces T in degrees [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 differential graded C-algebras in degrees ≤ 0 (equivalently, simplicial commutative C-algebras) and T is only defined up to quasi-isomorphisms (isos in cohomology) So local/affine objects of derived algebraic geometry are cdga’s defined up to quasi-isomorphism. Gabriele Vezzosi ()



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Derived affine schemes and homotopy theory

Gabriele Vezzosi ()



6 / 26

Derived affine schemes and homotopy theory So derived affine schemes (i.e. the opposite category of cdga’s) have to be considered up to quasi-isomorphisms:

Gabriele Vezzosi ()



6 / 26

Derived affine schemes and homotopy theory So derived affine schemes (i.e. the opposite category of cdga’s) have to be considered up to quasi-isomorphisms: i.e. we want to glue them along quasi-isomorphisms not isomorphisms.

Gabriele Vezzosi ()



6 / 26

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

Gabriele Vezzosi ()



6 / 26

Derived affine schemes and homotopy theory So derived affine schemes (i.e. the opposite category of cdga’s) have to be considered up to quasi-isomorphisms: i.e. we want to glue them along quasi-isomorphisms not isomorphisms. Recall that a scheme is built out of affine schemes glued along isomorphisms. So we need a theory enabling us to treat quasi-isomorphisms on the same footing as isomorphisms, i.e. to make them essentially invertible.

Gabriele Vezzosi ()



6 / 26

Derived affine schemes and homotopy theory So derived affine schemes (i.e. the opposite category of cdga’s) have to be considered up to quasi-isomorphisms: i.e. we want to glue them along quasi-isomorphisms not isomorphisms. Recall that a scheme is built out of affine schemes glued along isomorphisms. So we need a theory enabling us to treat quasi-isomorphisms on the same footing as isomorphisms, i.e. to make them essentially invertible. (Essentially? Formally inverting q-isos is too rough for gluing purposes

Gabriele Vezzosi ()



6 / 26

Derived affine schemes and homotopy theory So derived affine schemes (i.e. the opposite category of cdga’s) have to be considered up to quasi-isomorphisms: i.e. we want to glue them along quasi-isomorphisms not isomorphisms. Recall that a scheme is built out of affine schemes glued along isomorphisms. So we need a theory enabling us to treat quasi-isomorphisms on the same footing 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 not glue! )

Gabriele Vezzosi ()



6 / 26

Derived affine schemes and homotopy theory So derived affine schemes (i.e. the opposite category of cdga’s) have to be considered up to quasi-isomorphisms: i.e. we want to glue them along quasi-isomorphisms not isomorphisms. Recall that a scheme is built out of affine schemes glued along isomorphisms. So we need a theory enabling us to treat quasi-isomorphisms on the same footing 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 not glue! ) Thanks to Quillen, we know how to do it properly:

Gabriele Vezzosi ()



6 / 26

Derived affine schemes and homotopy theory So derived affine schemes (i.e. the opposite category of cdga’s) have to be considered up to quasi-isomorphisms: i.e. we want to glue them along quasi-isomorphisms not isomorphisms. Recall that a scheme is built out of affine schemes glued along isomorphisms. So we need a theory enabling us to treat quasi-isomorphisms on the same footing 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 not glue! ) Thanks to Quillen, we know how to do it properly: cdga’s together with q-isos constitute a homotopy theory

Gabriele Vezzosi ()



6 / 26

Derived affine schemes and homotopy theory So derived affine schemes (i.e. the opposite category of cdga’s) have to be considered up to quasi-isomorphisms: i.e. we want to glue them along quasi-isomorphisms not isomorphisms. Recall that a scheme is built out of affine schemes glued along isomorphisms. So we need a theory enabling us to treat quasi-isomorphisms on the same footing 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 not glue! ) Thanks to Quillen, we know how to do it properly: cdga’s together with q-isos constitute a homotopy theory (technically speaking Quillen model category structure).

Gabriele Vezzosi ()



6 / 26

Derived affine schemes and homotopy theory

Gabriele Vezzosi ()



7 / 26

Derived affine schemes and homotopy theory What is a ’homotopy theory’ ?

Gabriele Vezzosi ()



7 / 26

Derived affine schemes and homotopy theory What is a ’homotopy theory’ ? Roughly, a category M together with a distinguished class of maps w in M

Gabriele Vezzosi ()



7 / 26

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

Gabriele Vezzosi ()



7 / 26

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

Gabriele Vezzosi ()



7 / 26

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

Gabriele Vezzosi ()



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

Gabriele Vezzosi ()



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

Gabriele Vezzosi ()



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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)

Gabriele Vezzosi ()



7 / 26


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 are the Ext-groups)

Gabriele Vezzosi ()



7 / 26


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 are the Ext-groups) (cdgak , w = q-isos) (char k = 0)

Gabriele Vezzosi ()



7 / 26


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

Gabriele Vezzosi ()



7 / 26


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 are the Ext-groups) (cdgak , w = q-isos) (char k = 0) (SimplCommAlgk , w = weak htpy eq.ces) (any k). w −1 M := Ho(M) Gabriele Vezzosi ()



7 / 26


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 are the Ext-groups) (cdgak , w = q-isos) (char k = 0) (SimplCommAlgk , w = weak htpy eq.ces) (any k). w −1 M := Ho(M) : homotopy category of the hom. theory (M, w ). Gabriele Vezzosi ()



7 / 26


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 are the Ext-groups) (cdgak , w = q-isos) (char k = 0) (SimplCommAlgk , w = weak htpy eq.ces) (any k). w −1 M := Ho(M) : homotopy category of the hom. theory (M, w ). But the htpy theory (M, w ) strictly enhance Ho(M) ! Gabriele Vezzosi ()



7 / 26

What is derived algebraic geometry?

Gabriele Vezzosi ()



8 / 26


Gabriele Vezzosi ()



8 / 26

What is derived algebraic geometry? (underived) Algebraic Geometry

Gabriele Vezzosi ()



8 / 26

What is derived algebraic geometry? (underived) Algebraic Geometry schemes, algebraic spaces

Gabriele Vezzosi ()



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What is derived algebraic geometry? (underived) Algebraic Geometry schemes, algebraic spaces ; 1-stacks

Gabriele Vezzosi ()



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What is derived algebraic geometry? (underived) Algebraic Geometry schemes, algebraic spaces ; 1-stacks ;∞-stacks

Gabriele Vezzosi ()



8 / 26

What is derived algebraic geometry? (underived) Algebraic Geometry schemes, algebraic spaces ; 1-stacks ;∞-stacks CommAlgk

/ Sets O

schemes 1-stacks

π0

) ∞-stacks

Grpds O $

Π1

SimplSets

Gabriele Vezzosi ()

right derivation



8 / 26

What is derived algebraic geometry? (underived) Algebraic Geometry schemes, algebraic spaces ; 1-stacks ;∞-stacks CommAlgk

/ Sets O

schemes 1-stacks

π0

) ∞-stacks

Grpds O $

right derivation

Π1

SimplSets

right derivation ≡ adjoining homotopy colimits (⇒ can take quotients) ; promote the target categories to a homotopy theory (that of SimplSets or, eq.ly, topological spaces). Gabriele Vezzosi ()



8 / 26


Gabriele Vezzosi ()



9 / 26

What is derived algebraic geometry? if we derive also to the left

Gabriele Vezzosi ()



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What is derived algebraic geometry? if we derive also to the left ;

Gabriele Vezzosi ()



9 / 26

What is derived algebraic geometry? if we derive also to the left ; / Ens O

schemes

CommAlgk O

1-stacks

left deriv.

π0

π0

* ∞-stacks

SimplCommAlgk

Grpds O %

right deriv.

Π1

/ SimplSets

derived ∞-stacks

Gabriele Vezzosi ()



9 / 26


schemes

CommAlgk O

1-stacks left deriv.

π0

π0

* ∞-stacks

SimplCommAlgk

Grpds O %

right deriv.

Π1

/ SimplSets

derived ∞-stacks

; derived Algebraic Geometry:

Gabriele Vezzosi ()



9 / 26


schemes

CommAlgk O


π0

π0

* ∞-stacks

SimplCommAlgk

Grpds O %

right deriv.

Π1

/ SimplSets

derived ∞-stacks

; derived Algebraic Geometry: source and target are nontrivial homotopy theories.

Gabriele Vezzosi ()



9 / 26


schemes

CommAlgk O


π0

π0

* ∞-stacks

SimplCommAlgk

Grpds O %

right deriv.

Π1

/ SimplSets

derived ∞-stacks

; derived Algebraic Geometry: source and target are nontrivial homotopy theories. It is a kind of algebraic geometry where affine objects are simplicial commutative algebras Gabriele Vezzosi ()



9 / 26


schemes

CommAlgk O


π0

π0

* ∞-stacks

SimplCommAlgk

Grpds O %

right deriv.

Π1

/ SimplSets

derived ∞-stacks

; derived Algebraic Geometry: source and target are nontrivial homotopy theories. It is a kind of algebraic geometry where affine objects are simplicial commutative algebras (or k-cdga if char (k) = 0) Gabriele Vezzosi ()



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Derived Algebraic Geometry (DAG) in two steps

Gabriele Vezzosi ()



10 / 26

Derived Algebraic Geometry (DAG) in two steps Recall -

Gabriele Vezzosi ()



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Derived Algebraic Geometry (DAG) in two steps Recall - A scheme, algebraic space, stack etc. is a functor as above which moreover

Gabriele Vezzosi ()



10 / 26

Derived Algebraic Geometry (DAG) in two steps Recall - A scheme, algebraic space, stack etc. is a functor as above which moreover satisfies a sheaf condition (descent) with respect to some chosen topology defined on commutative algebras

Gabriele Vezzosi ()



10 / 26

Derived Algebraic Geometry (DAG) in two steps Recall - A scheme, algebraic space, stack etc. is a functor as above which moreover satisfies a sheaf condition (descent) with respect to some chosen topology defined on commutative algebras admits a (Zariski, étale, flat, smooth) atlas of affine schemes

Gabriele Vezzosi ()



10 / 26

Derived Algebraic Geometry (DAG) in two steps Recall - A scheme, algebraic space, stack etc. is a functor as above which moreover satisfies a sheaf condition (descent) with respect to some chosen topology defined on commutative algebras admits a (Zariski, étale, flat, smooth) atlas of affine schemes Example - A functor X : CommAlgk −→ Sets is a scheme iff

Gabriele Vezzosi ()



10 / 26

Derived Algebraic Geometry (DAG) in two steps Recall - A scheme, algebraic space, stack etc. is a functor as above which moreover satisfies a sheaf condition (descent) with respect to some chosen topology defined on commutative algebras admits a (Zariski, étale, flat, smooth) atlas of affine schemes Example - A functor X : CommAlgk −→ Sets is a scheme iff is an étale sheaf: for any comm. k-algebra A, for any étale covering family {A → Ai }i of A, the canonical map X (A) −→ limj X (Aj ) is a bijection;

Gabriele Vezzosi ()



10 / 26

Derived Algebraic Geometry (DAG) in two steps Recall - A scheme, algebraic space, stack etc. is a functor as above which moreover satisfies a sheaf condition (descent) with respect to some chosen topology defined on commutative algebras admits a (Zariski, étale, flat, smooth) atlas of affine schemes Example - A functor X : CommAlgk −→ Sets is a scheme iff is an étale sheaf: for any comm. k-algebra A, for any étale covering family {A → Ai }i of A, the canonical map X (A) −→ limj X (Aj ) is a bijection; it admits a Zariski atlas

Gabriele Vezzosi ()

`

i

Ui → X (Ui = Spec Ri , Ri ∈ CommAlg).



10 / 26

Derived Algebraic Geometry (DAG) in two steps Recall - A scheme, algebraic space, stack etc. is a functor as above which moreover satisfies a sheaf condition (descent) with respect to some chosen topology defined on commutative algebras admits a (Zariski, étale, flat, smooth) atlas of affine schemes Example - A functor X : CommAlgk −→ Sets is a scheme iff is an étale sheaf: for any comm. k-algebra A, for any étale covering family {A → Ai }i of A, the canonical map X (A) −→ limj X (Aj ) is a bijection; it admits a Zariski atlas

Gabriele Vezzosi ()

`

i

Ui → X (Ui = Spec Ri , Ri ∈ CommAlg).



10 / 26


Gabriele Vezzosi ()



11 / 26


To translate this into DAG, we thus need two steps

Gabriele Vezzosi ()



11 / 26


To translate this into DAG, we thus need two steps we first need a notion of derived topology and derived sheaf theory

Gabriele Vezzosi ()



11 / 26


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, étale, flat, smooth) derived atlases.

Gabriele Vezzosi ()



11 / 26


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, étale, flat, smooth) derived atlases. Just as schemes, algebraic spaces and stacks are (simplicial) sheaves admitting some kind of atlases,

Gabriele Vezzosi ()



11 / 26


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, étale, flat, smooth) derived atlases. Just as schemes, algebraic spaces and stacks are (simplicial) sheaves admitting some kind of atlases, the first step will give us up-to-homotopy (simplicial) sheaves,

Gabriele Vezzosi ()



11 / 26


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, étale, flat, smooth) derived atlases. Just as schemes, algebraic spaces and stacks are (simplicial) sheaves admitting some kind of atlases, the first step will give us up-to-homotopy (simplicial) sheaves, among which the second step will single out the derived spaces studied by derived algebraic geometry.

Gabriele Vezzosi ()



11 / 26

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

Gabriele Vezzosi ()



12 / 26

Derived Algebraic Geometry (DAG) - 1st step: derived sheaf theory First step (Toën-V., 2004) –

Gabriele Vezzosi ()



12 / 26

Derived Algebraic Geometry (DAG) - 1st step: derived sheaf theory First step (Toën-V., 2004) – develop a sheaf theory over homotopy theories (= Quillen model categories) endowed with a up-to-homotopy topology

Gabriele Vezzosi ()



12 / 26

Derived Algebraic Geometry (DAG) - 1st step: derived sheaf theory First step (Toën-V., 2004) – develop a sheaf theory over homotopy theories (= Quillen model categories) endowed with a up-to-homotopy topology ; homotopy/higher topoi

Gabriele Vezzosi ()



12 / 26

Derived Algebraic Geometry (DAG) - 1st step: derived sheaf theory First step (Toën-V., 2004) – develop a sheaf theory over homotopy theories (= Quillen model categories) endowed with a up-to-homotopy topology ; homotopy/higher topoi (model topoi in HAG I; then reconsidered and generalized further by J. Lurie)

Gabriele Vezzosi ()



12 / 26

Derived Algebraic Geometry (DAG) - 1st step: derived sheaf theory First step (Toën-V., 2004) – develop a sheaf theory over homotopy theories (= Quillen model categories) endowed with a up-to-homotopy topology ; homotopy/higher topoi (model topoi in HAG I; then reconsidered and generalized further by J. Lurie) Derived topology on a homotopy theory/model category (M, w )

Gabriele Vezzosi ()



12 / 26

Derived Algebraic Geometry (DAG) - 1st step: derived sheaf theory First step (Toën-V., 2004) – develop a sheaf theory over homotopy theories (= Quillen model categories) endowed with a up-to-homotopy topology ; homotopy/higher topoi (model topoi in HAG I; then reconsidered and generalized further by J. Lurie) Derived topology on a homotopy theory/model category (M, w ) ⇒ Grothendieck topology on Ho(M) = w −1 M.

Gabriele Vezzosi ()



12 / 26

Derived Algebraic Geometry (DAG) - 1st step: derived sheaf theory First step (Toën-V., 2004) – develop a sheaf theory over homotopy theories (= Quillen model categories) endowed with a up-to-homotopy topology ; homotopy/higher topoi (model topoi in HAG I; then reconsidered and generalized further by J. Lurie) Derived topology on a homotopy theory/model category (M, w ) ⇒ Grothendieck topology on Ho(M) = w −1 M. Examples of homotopy theories we consider

Gabriele Vezzosi ()



12 / 26

Derived Algebraic Geometry (DAG) - 1st step: derived sheaf theory First step (Toën-V., 2004) – develop a sheaf theory over homotopy theories (= Quillen model categories) endowed with a up-to-homotopy topology ; homotopy/higher topoi (model topoi in HAG I; then reconsidered and generalized further by J. Lurie) Derived topology on a homotopy theory/model category (M, w ) ⇒ Grothendieck topology on Ho(M) = w −1 M. Examples of homotopy theories we consider Simplicial commutative k-algebras (k any commutative ring)

Gabriele Vezzosi ()



12 / 26

Derived Algebraic Geometry (DAG) - 1st step: derived sheaf theory First step (Toën-V., 2004) – develop a sheaf theory over homotopy theories (= Quillen model categories) endowed with a up-to-homotopy topology ; homotopy/higher topoi (model topoi in HAG I; then reconsidered and generalized further by J. Lurie) Derived topology on a homotopy theory/model category (M, w ) ⇒ Grothendieck topology on Ho(M) = w −1 M. Examples of homotopy theories we consider Simplicial commutative k-algebras (k any commutative ring) differential graded commutative k-algebras (char k = 0)

Gabriele Vezzosi ()



12 / 26

Derived Algebraic Geometry (DAG) - 1st step: derived sheaf theory First step (Toën-V., 2004) – develop a sheaf theory over homotopy theories (= Quillen model categories) endowed with a up-to-homotopy topology ; homotopy/higher topoi (model topoi in HAG I; then reconsidered and generalized further by J. Lurie) Derived topology on a homotopy theory/model category (M, w ) ⇒ Grothendieck topology on Ho(M) = w −1 M. 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)

Gabriele Vezzosi ()



12 / 26

Derived Algebraic Geometry (DAG) - 1st step: derived sheaf theory First step (Toën-V., 2004) – develop a sheaf theory over homotopy theories (= Quillen model categories) endowed with a up-to-homotopy topology ; homotopy/higher topoi (model topoi in HAG I; then reconsidered and generalized further by J. Lurie) Derived topology on a homotopy theory/model category (M, w ) ⇒ Grothendieck topology on Ho(M) = w −1 M. 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)

Gabriele Vezzosi ()



12 / 26

Derived Algebraic Geometry (DAG) - 1st step: derived sheaf theory First step (Toën-V., 2004) – develop a sheaf theory over homotopy theories (= Quillen model categories) endowed with a up-to-homotopy topology ; homotopy/higher topoi (model topoi in HAG I; then reconsidered and generalized further by J. Lurie) Derived topology on a homotopy theory/model category (M, w ) ⇒ Grothendieck topology on Ho(M) = w −1 M. 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:

Gabriele Vezzosi ()



12 / 26

Derived Algebraic Geometry (DAG) - 1st step: derived sheaf theory First step (Toën-V., 2004) – develop a sheaf theory over homotopy theories (= Quillen model categories) endowed with a up-to-homotopy topology ; homotopy/higher topoi (model topoi in HAG I; then reconsidered and generalized further by J. Lurie) Derived topology on a homotopy theory/model category (M, w ) ⇒ Grothendieck topology on Ho(M) = w −1 M. 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 monoidal Quillen model category (M, w , ⊗) Gabriele Vezzosi ()



12 / 26

Derived Algebraic Geometry (DAG) - 1st step: derived sheaf theory First step (Toën-V., 2004) – develop a sheaf theory over homotopy theories (= Quillen model categories) endowed with a up-to-homotopy topology ; homotopy/higher topoi (model topoi in HAG I; then reconsidered and generalized further by J. Lurie) Derived topology on a homotopy theory/model category (M, w ) ⇒ Grothendieck topology on Ho(M) = w −1 M. 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 monoidal Quillen model category (M, w , ⊗) ; in such a general setting the derived geometry we get is called Homotopical Algebraic Geometry - HAG - ) Gabriele Vezzosi ()



12 / 26

Derived Algebraic Geometry (DAG) - 1st step derived sheaf theory

Gabriele Vezzosi ()



13 / 26


An example - étale derived topology on SimplCommAlgk :

Gabriele Vezzosi ()



13 / 26


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

Gabriele Vezzosi ()



13 / 26


An example - étale derived topology on SimplCommAlgk : {A → Bi } is an étale covering family for derived étale topology if {π0 A → π0 Bi } is an étale covering family (in the usual sense)

Gabriele Vezzosi ()



13 / 26


An example - étale derived topology on SimplCommAlgk : {A → Bi } is an étale covering family for derived étale topology if {π0 A → π0 Bi } is an étale covering family (in the usual sense) for any i and any n ≥ 0 , πn A ⊗π0 A π0 B → πn Bi is an isomorphism

Gabriele Vezzosi ()



13 / 26


An example - étale derived topology on SimplCommAlgk : {A → Bi } is an étale covering family for derived étale topology if {π0 A → π0 Bi } is an étale covering family (in the usual sense) for any i and any n ≥ 0 , πn A ⊗π0 A π0 B → πn Bi is an isomorphism The intuition is:

Gabriele Vezzosi ()



13 / 26


An example - étale derived topology on SimplCommAlgk : {A → Bi } is an étale covering family for derived étale topology if {π0 A → π0 Bi } is an étale covering family (in the usual sense) for any i and any n ≥ 0 , πn A ⊗π0 A π0 B → πn Bi is an isomorphism The intuition is: everything is as usual on the classical part/truncation π0 (−),

Gabriele Vezzosi ()



13 / 26


An example - étale derived topology on SimplCommAlgk : {A → Bi } is an étale covering family for derived étale topology if {π0 A → π0 Bi } is an étale covering family (in the usual sense) for any i and any n ≥ 0 , πn A ⊗π0 A π0 B → πn Bi 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 π0 A → π0 B

Gabriele Vezzosi ()



13 / 26


An example - étale derived topology on SimplCommAlgk : {A → Bi } is an étale covering family for derived étale topology if {π0 A → π0 Bi } is an étale covering family (in the usual sense) for any i and any n ≥ 0 , πn A ⊗π0 A π0 B → πn Bi 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 π0 A → π0 B Rmk. This is not an ad hoc definition: it is an elementary characterization of a more conceptual definition (via derived infinitesimal lifting property).

Gabriele Vezzosi ()



13 / 26


Gabriele Vezzosi ()



14 / 26

Derived Algebraic Geometry (DAG) - 1st step: derived sheaf theory Choice of a derived topology (e.g. étale) on dAff k := SimplCommAlgop k

Gabriele Vezzosi ()



14 / 26

Derived Algebraic Geometry (DAG) - 1st step: derived sheaf theory Choice of a derived topology (e.g. étale) on dAff k := SimplCommAlgop k ;

Homotopy theory of derived stacks

Gabriele Vezzosi ()



14 / 26


Homotopy theory of derived stacks induces a homotopy theory (Quillen model category) on the category dSPrk of simplicial presheaves on dAff k SimplCommAlgk = dAff op k → SimplSets

Gabriele Vezzosi ()



14 / 26



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

Gabriele Vezzosi ()



14 / 26



weak equivalences f : F → G inducing πi (F , x) ' πi (G , f (x)) for any i ≥ 0 and any x, as sheaves on the usual site Ho(dAff k ). The category of derived stacks is dStk := Ho(dSPrk )

Gabriele Vezzosi ()



14 / 26


Gabriele Vezzosi ()



15 / 26

Derived Algebraic Geometry (DAG) - 1st step: derived sheaf theory Therefore,

Gabriele Vezzosi ()



15 / 26

Derived Algebraic Geometry (DAG) - 1st step: derived sheaf theory Therefore, a derived stack, i.e. an object in dStk , is a functor F : SimplCommAlgk → SimplSets such that

Gabriele Vezzosi ()



15 / 26

Derived Algebraic Geometry (DAG) - 1st step: derived sheaf theory Therefore, a derived stack, i.e. an object in dStk , is a functor F : SimplCommAlgk → SimplSets such that F preserves sends weak equivalences in SimplCommAlgk to weak equivalences in SimplSetsk

Gabriele Vezzosi ()



15 / 26

Derived Algebraic Geometry (DAG) - 1st step: derived sheaf theory Therefore, a derived stack, i.e. an object in dStk , is a functor F : SimplCommAlgk → SimplSets such that F preserves sends weak equivalences in SimplCommAlgk to weak equivalences in SimplSetsk F has descent with respect to étale homotopy-hypercoverings

Gabriele Vezzosi ()



15 / 26

Derived Algebraic Geometry (DAG) - 1st step: derived sheaf theory Therefore, a derived stack, i.e. an object in dStk , is a functor F : SimplCommAlgk → SimplSets such that F preserves sends weak equivalences in SimplCommAlgk to weak equivalences in SimplSetsk F has descent with respect to étale homotopy-hypercoverings , i.e. F (A) → holimF (B• )

Gabriele Vezzosi ()



15 / 26

Derived Algebraic Geometry (DAG) - 1st step: derived sheaf theory Therefore, a derived stack, i.e. an object in dStk , is a functor F : SimplCommAlgk → SimplSets such that F preserves sends weak equivalences in SimplCommAlgk to weak equivalences in SimplSetsk F has descent with respect to étale homotopy-hypercoverings , i.e. F (A) → holimF (B• ) is an iso in Ho(SimplSets),

Gabriele Vezzosi ()



15 / 26

Derived Algebraic Geometry (DAG) - 1st step: derived sheaf theory Therefore, a derived stack, i.e. an object in dStk , is a functor F : SimplCommAlgk → SimplSets such that F preserves sends weak equivalences in SimplCommAlgk to weak equivalences in SimplSetsk F has descent with respect to étale homotopy-hypercoverings , i.e. F (A) → holimF (B• ) is an iso in Ho(SimplSets), for any A and any étale h-hypercovering B• de A Rmk. Don’t worry about hypercoverings,

Gabriele Vezzosi ()



15 / 26

Derived Algebraic Geometry (DAG) - 1st step: derived sheaf theory Therefore, a derived stack, i.e. an object in dStk , is a functor F : SimplCommAlgk → SimplSets such that F preserves sends weak equivalences in SimplCommAlgk to weak equivalences in SimplSetsk F has descent with respect to étale homotopy-hypercoverings , i.e. F (A) → holimF (B• ) is an iso in Ho(SimplSets), for any A and any étale h-hypercovering B• de A Rmk. Don’t worry about hypercoverings, just think of Cech nerves associated to covers.

Gabriele Vezzosi ()



15 / 26


Gabriele Vezzosi ()



16 / 26

Derived Algebraic Geometry (DAG) - 1st step: derived sheaf theory Derived Yoneda:

Gabriele Vezzosi ()



16 / 26

Derived Algebraic Geometry (DAG) - 1st step: derived sheaf theory Derived Yoneda: RSpec : AlgCommSimplk → dStk , A 7→ MapAlgCommSimplk (A, −) is fully faithful (up to homotopy).

Gabriele Vezzosi ()



16 / 26

Derived Algebraic Geometry (DAG) - 1st step: derived sheaf theory 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 product is locally given by the derived tensor product RSpecB ×hRSpecA RSpecC ' RSpec(B ⊗LA C ).

Gabriele Vezzosi ()



16 / 26

Derived Algebraic Geometry (DAG) - 1st step: derived sheaf theory 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 product is locally given by the derived tensor product RSpecB ×hRSpecA RSpecC ' RSpec(B ⊗LA C ).

Gabriele Vezzosi ()



16 / 26


Gabriele Vezzosi ()



17 / 26

Derived Algebraic Geometry (DAG) - 1st step: derived sheaf theory There is a truncation/inclusion adjunction:

Gabriele Vezzosi ()



17 / 26

Derived Algebraic Geometry (DAG) - 1st step: derived sheaf theory There is a truncation/inclusion adjunction: t0

dStk o

Gabriele Vezzosi ()

/

Stk

i



17 / 26


dStk o

/

Stk

i

i is fully faithful

Gabriele Vezzosi ()



17 / 26


dStk o

/

Stk

i

i is fully faithful (hence usually omitted in notations)

Gabriele Vezzosi ()



17 / 26


dStk o

/

Stk

i

i is fully faithful (hence usually omitted in notations) t0 (RSpecA) = Spec π0 A

Gabriele Vezzosi ()



17 / 26


dStk o

/

Stk

i

i is fully faithful (hence usually omitted in notations) t0 (RSpecA) = Spec π0 A the adjunction map i(t0 X ) ,→ X is a closed immersion

Gabriele Vezzosi ()



17 / 26


dStk o

/

Stk

i

i is fully faithful (hence usually omitted in notations) t0 (RSpecA) = Spec π0 A the adjunction map i(t0 X ) ,→ X is a closed immersion i preserves homotopy colimits but not homotopy limits nor internal HOM’s

Gabriele Vezzosi ()



17 / 26


dStk o

/

Stk

i

i is fully faithful (hence usually omitted in notations) t0 (RSpecA) = Spec π0 A the adjunction map i(t0 X ) ,→ X is a closed immersion i preserves homotopy colimits but not homotopy limits nor internal HOM’s ; derived tangent spaces and derived fibered products of schemes are not the usual tangent spaces and fibered products !

Geometric intuition -

Gabriele Vezzosi ()



17 / 26


dStk o

/

Stk

i


Geometric intuition - X like a formal thickening of its truncation t0 (X ), Gabriele Vezzosi ()



17 / 26


dStk o

/

Stk

i


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



17 / 26


dStk o

/

Stk

i


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



17 / 26

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

Gabriele Vezzosi ()



18 / 26

Derived Algebraic Geometry (DAG) - 2nd step: derived geometric stacks • 2 notions of derived smooth maps between simpl. comm algebras:

Gabriele Vezzosi ()



18 / 26

Derived Algebraic Geometry (DAG) - 2nd step: derived geometric stacks • 2 notions of derived smooth maps between simpl. comm algebras: A → B smooth if π0 A → π0 B is smooth and πn A ⊗π0 A π0 B ' πn B for any n ≥ 0

Gabriele Vezzosi ()



18 / 26

Derived Algebraic Geometry (DAG) - 2nd step: derived geometric stacks • 2 notions of derived smooth maps between simpl. comm algebras: A → B smooth if π0 A → π0 B is smooth and πn A ⊗π0 A π0 B ' πn B for any n ≥ 0 A → B is p-smooth if the relative cotangent complex LB/A is perfect

Gabriele Vezzosi ()



18 / 26


Geometric types of derived stacks

Gabriele Vezzosi ()



18 / 26


Geometric types of derived stacks F a derived stack

Gabriele Vezzosi ()



18 / 26


Geometric types of derived stacks F a derived stack ` A derived atlas for F is a map i RSpec Ai → F surjective on π0 (and satisfying some representability conditions)

Gabriele Vezzosi ()



18 / 26


Geometric types of derived stacks F a derived stack ` A derived atlas for F is a map i RSpec Ai → F surjective on π0 (and satisfying some representability conditions) if the map is smooth (resp. étale, Zariski)

Gabriele Vezzosi ()



18 / 26


Geometric types of derived stacks F a derived stack ` A derived atlas for F is a map i RSpec Ai → F surjective on π0 (and satisfying some representability conditions) if the map is smooth (resp. étale, Zariski) we have a derived Artin stack (resp. Deligne-Mumford stack, scheme)

Gabriele Vezzosi ()



18 / 26


Geometric types of derived stacks F a derived stack ` A derived atlas for F is a map i RSpec Ai → F surjective on π0 (and satisfying some representability conditions) if the map is smooth (resp. étale, Zariski) we have a derived Artin stack (resp. Deligne-Mumford stack, scheme) The truncation preserves the type of the stack.

Gabriele Vezzosi ()



18 / 26


Geometric types of derived stacks F a derived stack ` A derived atlas for F is a map i RSpec Ai → F surjective on π0 (and satisfying some representability conditions) if the map is smooth (resp. étale, Zariski) we have a derived Artin stack (resp. Deligne-Mumford stack, scheme) The truncation preserves the type of the stack. Using atlases (and representability) Gabriele Vezzosi ()



18 / 26


Geometric types of derived stacks F a derived stack ` A derived atlas for F is a map i RSpec Ai → F surjective on π0 (and satisfying some representability conditions) if the map is smooth (resp. étale, Zariski) we have a derived Artin stack (resp. Deligne-Mumford stack, scheme) The truncation preserves the type of the stack. Using atlases (and representability) ; extend notion of smooth, étale, flat, etc. to maps between geometric derived stacks. Gabriele Vezzosi ()



18 / 26

Derived Algebraic Geometry (DAG) - Main properties

Gabriele Vezzosi ()



19 / 26

Derived Algebraic Geometry (DAG) - Main properties X ∈ dStk

Gabriele Vezzosi ()



19 / 26

Derived Algebraic Geometry (DAG) - Main properties X ∈ dStk ; derived tangent stack TX := MAP(Spec k[], X ) ' Spec Sym(L∨ X)

Gabriele Vezzosi ()



19 / 26

Derived Algebraic Geometry (DAG) - Main properties X ∈ dStk ; derived tangent stack TX := MAP(Spec k[], X ) ' Spec Sym(L∨ X) LX - cotangent complex of X (it classifies derived derivations)

Gabriele Vezzosi ()



19 / 26

Derived Algebraic Geometry (DAG) - Main properties X ∈ dStk ; derived tangent stack TX := MAP(Spec k[], X ) ' Spec Sym(L∨ X) LX - cotangent complex of X (it classifies derived derivations) ;

Gabriele Vezzosi ()



19 / 26


Geometric/modular interpretation of cotangent complex X (underived) scheme

Gabriele Vezzosi ()



19 / 26


Geometric/modular interpretation of cotangent complex X (underived) scheme ⇒ Li(X ) is Grothendieck-Illusie cotangent complex of X

Gabriele Vezzosi ()



19 / 26


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

Gabriele Vezzosi ()



19 / 26


Geometric/modular interpretation of cotangent complex X (underived) scheme ⇒ Li(X ) is Grothendieck-Illusie cotangent complex of X (so it corresponds to the cotgt space of some geom. space) Ext i (LX ,x , k) ' HomdStk, ∗ (Spec k[i ], (X , x)), x ∈ X (k) (k[i ] - trivial extension of k by K(k, i))

Gabriele Vezzosi ()



19 / 26



; the full cotangent complex is uniquely geometrically characterized in DAG Gabriele Vezzosi ()



19 / 26



; the full cotangent complex is uniquely geometrically characterized in DAG (this answers Grothendieck’s question in Catégories cofibrées additives et complexe cotangent relatif, 1968). Gabriele Vezzosi ()



19 / 26


Gabriele Vezzosi ()



20 / 26


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

Gabriele Vezzosi ()



20 / 26


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

Gabriele Vezzosi ()



20 / 26


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

Gabriele Vezzosi ()



20 / 26


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

Gabriele Vezzosi ()



20 / 26


The (full) cotangent complex of a derived stack has a universal moduli property ; it is computable ! (this also explains why the stacky 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 ) −→ X induces a canonical obstruction theory on t0 (X ) (in the sense of Behrend-Fantechi)

Gabriele Vezzosi ()



20 / 26


The (full) cotangent complex of a derived stack has a universal moduli property ; it is computable ! (this also explains why the stacky 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 ) −→ X induces a canonical obstruction theory on t0 (X ) (in the sense of Behrend-Fantechi) i ∗ (LX ) −→ Lt0 (X ) .

Gabriele Vezzosi ()



20 / 26


The (full) cotangent complex of a derived stack has a universal moduli property ; it is computable ! (this also explains why the stacky 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 ) −→ X induces a canonical obstruction theory on t0 (X ) (in the sense of Behrend-Fantechi) i ∗ (LX ) −→ Lt0 (X ) . ; if i ∗ (LX ) is of perfect amplitude in [−1, 0]

Gabriele Vezzosi ()



20 / 26


The (full) cotangent complex of a derived stack has a universal moduli property ; it is computable ! (this also explains why the stacky 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 ) −→ X induces a canonical obstruction theory on t0 (X ) (in the sense of Behrend-Fantechi) i ∗ (LX ) −→ Lt0 (X ) . ; if i ∗ (LX ) is of perfect amplitude in [−1, 0] ; virtual fundamental class [X ]vir on t0 (X ),

Gabriele Vezzosi ()



20 / 26


The (full) cotangent complex of a derived stack has a universal moduli property ; it is computable ! (this also explains why the stacky 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 ) −→ X induces a canonical obstruction theory on t0 (X ) (in the sense of Behrend-Fantechi) i ∗ (LX ) −→ Lt0 (X ) . ; if i ∗ (LX ) is of perfect amplitude in [−1, 0] ; virtual fundamental class [X ]vir on t0 (X ), which is moreover natural in X

Gabriele Vezzosi ()



20 / 26


The (full) cotangent complex of a derived stack has a universal moduli property ; it is computable ! (this also explains why the stacky 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 ) −→ X induces a canonical obstruction theory on t0 (X ) (in the sense of Behrend-Fantechi) i ∗ (LX ) −→ Lt0 (X ) . ; if i ∗ (LX ) is of perfect amplitude in [−1, 0] ; virtual fundamental class [X ]vir on t0 (X ), which is moreover natural in X (unlike in Behrend-Fantechi’s approach).

Gabriele Vezzosi ()



20 / 26


The (full) cotangent complex of a derived stack has a universal moduli property ; it is computable ! (this also explains why the stacky 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 ) −→ X induces a canonical obstruction theory on t0 (X ) (in the sense of Behrend-Fantechi) i ∗ (LX ) −→ Lt0 (X ) . ; if i ∗ (LX ) is of perfect amplitude in [−1, 0] ; virtual fundamental class [X ]vir on t0 (X ), which is moreover natural in X (unlike in Behrend-Fantechi’s approach).

Gabriele Vezzosi ()



20 / 26


Gabriele Vezzosi ()



21 / 26

Derived Algebraic Geometry (DAG) - Main properties All moduli problems have some (maybe more than one) natural derived version.

Gabriele Vezzosi ()



21 / 26

Derived Algebraic Geometry (DAG) - Main properties All moduli problems have some (maybe more than one) natural derived version. ; All known moduli spaces have derived enhancements: Hilbert scheme, moduli of curves, of stable maps, of local systems, of coherent sheaves, . . .

Gabriele Vezzosi ()



21 / 26

Derived Algebraic Geometry (DAG) - Main properties All moduli problems have some (maybe more than one) natural derived version. ; All known moduli spaces have derived enhancements: Hilbert scheme, moduli of curves, of stable maps, of local systems, of coherent sheaves, . . . M ; RM where M ' t0 (RM) ,→ RM (most often a strict inclusion).

Gabriele Vezzosi ()



21 / 26

Derived Algebraic Geometry (DAG) - Main properties All moduli problems have some (maybe more than one) natural derived version. ; All known moduli spaces have derived enhancements: Hilbert scheme, moduli of curves, of stable maps, of local systems, of coherent sheaves, . . . M ; RM where M ' t0 (RM) ,→ RM (most often a strict inclusion).The choice of a derived enhancement RM yields additional structures on M:

Gabriele Vezzosi ()



21 / 26

Derived Algebraic Geometry (DAG) - Main properties All moduli problems have some (maybe more than one) natural derived version. ; All known moduli spaces have derived enhancements: Hilbert scheme, moduli of curves, of stable maps, of local systems, of coherent sheaves, . . . M ; RM where M ' t0 (RM) ,→ RM (most often a strict inclusion).The choice of a derived enhancement RM yields additional structures on M: obstruction theory (for arbitrary geometric n-stacks),

Gabriele Vezzosi ()



21 / 26

Derived Algebraic Geometry (DAG) - Main properties All moduli problems have some (maybe more than one) natural derived version. ; All known moduli spaces have derived enhancements: Hilbert scheme, moduli of curves, of stable maps, of local systems, of coherent sheaves, . . . M ; RM where M ' t0 (RM) ,→ RM (most often a strict inclusion).The choice of a derived enhancement RM yields additional structures on M: obstruction theory (for arbitrary geometric n-stacks), virtual structure sheaf,

Gabriele Vezzosi ()



21 / 26

Derived Algebraic Geometry (DAG) - Main properties All moduli problems have some (maybe more than one) natural derived version. ; All known moduli spaces have derived enhancements: Hilbert scheme, moduli of curves, of stable maps, of local systems, of coherent sheaves, . . . M ; RM where M ' t0 (RM) ,→ RM (most often a strict inclusion).The choice of a derived enhancement RM yields additional structures on M: obstruction theory (for arbitrary geometric n-stacks), virtual structure sheaf, virtual fundamental class (when it exists) . . .

Gabriele Vezzosi ()



21 / 26

Derived Algebraic Geometry (DAG) - Main properties All moduli problems have some (maybe more than one) natural derived version. ; All known moduli spaces have derived enhancements: Hilbert scheme, moduli of curves, of stable maps, of local systems, of coherent sheaves, . . . M ; RM where M ' t0 (RM) ,→ RM (most often a strict inclusion).The choice of a derived enhancement RM yields additional structures on M: obstruction theory (for arbitrary geometric n-stacks), virtual structure sheaf, virtual fundamental class (when it exists) . . . ; in particular Gromov-Witten and Donaldson-Thomas invariants can be completely reconstructed form these derived enhancements

Gabriele Vezzosi ()



21 / 26

Derived Algebraic Geometry (DAG) - Main properties All moduli problems have some (maybe more than one) natural derived version. ; All known moduli spaces have derived enhancements: Hilbert scheme, moduli of curves, of stable maps, of local systems, of coherent sheaves, . . . M ; RM where M ' t0 (RM) ,→ RM (most often a strict inclusion).The choice of a derived enhancement RM yields additional structures on M: obstruction theory (for arbitrary geometric n-stacks), virtual structure sheaf, virtual fundamental class (when it exists) . . . ; in particular Gromov-Witten and Donaldson-Thomas invariants can be completely reconstructed form these derived enhancements (invariants of the enhancement not of the truncation).

Gabriele Vezzosi ()



21 / 26

Derived Algebraic Geometry (DAG) - Main properties All moduli problems have some (maybe more than one) natural derived version. ; All known moduli spaces have derived enhancements: Hilbert scheme, moduli of curves, of stable maps, of local systems, of coherent sheaves, . . . M ; RM where M ' t0 (RM) ,→ RM (most often a strict inclusion).The choice of a derived enhancement RM yields additional structures on M: obstruction theory (for arbitrary geometric n-stacks), virtual structure sheaf, virtual fundamental class (when it exists) . . . ; in particular Gromov-Witten and Donaldson-Thomas invariants can be completely reconstructed form these derived enhancements (invariants of the enhancement not of the truncation). Conversely, a (nice) underived stack endowed with an obstruction theory essentially reconstructs a particular derived enhancement Gabriele Vezzosi ()



21 / 26

Derived Algebraic Geometry (DAG) - Main properties All moduli problems have some (maybe more than one) natural derived version. ; All known moduli spaces have derived enhancements: Hilbert scheme, moduli of curves, of stable maps, of local systems, of coherent sheaves, . . . M ; RM where M ' t0 (RM) ,→ RM (most often a strict inclusion).The choice of a derived enhancement RM yields additional structures on M: obstruction theory (for arbitrary geometric n-stacks), virtual structure sheaf, virtual fundamental class (when it exists) . . . ; in particular Gromov-Witten and Donaldson-Thomas invariants can be completely reconstructed form these derived enhancements (invariants of the enhancement not of the truncation). Conversely, a (nice) underived stack endowed with an obstruction theory essentially reconstructs a particular derived enhancement (e.g. reduced obstruction theory for stable maps to a K 3-surface). Gabriele Vezzosi ()



21 / 26


Gabriele Vezzosi ()



22 / 26


X ∈ dStk ,

Gabriele Vezzosi ()



22 / 26


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

Gabriele Vezzosi ()



22 / 26


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

Gabriele Vezzosi ()



22 / 26


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

Gabriele Vezzosi ()



22 / 26


X ∈ dStk , OX - structural (up-to-homotopy) sheaf of simplicial commutative rings ; πi (OX ) are quasi-coherent on X and supported on 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 0 for i >> 0 (equivalent to i ∗ (LX ) ∈ Perf [−1,0] ),

Gabriele Vezzosi ()



22 / 26


X ∈ dStk , OX - structural (up-to-homotopy) sheaf of simplicial commutative rings ; πi (OX ) are quasi-coherent on X and supported on 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 0 for i >> 0 (equivalent to i ∗ (LX ) ∈ Perf [−1,0] ), we get a class X [OX ]vir := (−1)i [πi (OX )] ∈ G0 (t0 (X )) i

Gabriele Vezzosi ()



22 / 26



a K -theoretic version of the virtual fundamental class.

Gabriele Vezzosi ()



22 / 26



a K -theoretic version of the virtual fundamental class.

Gabriele Vezzosi ()



22 / 26


Gabriele Vezzosi ()



23 / 26

Derived Algebraic Geometry (DAG) - Main properties The base-change formula for quasi-coherent coefficients is satisfied even without flatness conditions for derived stacks

Gabriele Vezzosi ()



23 / 26


Quasicoherent base-change

Gabriele Vezzosi ()



23 / 26


Quasicoherent base-change For any homotopy cartesian square of derived stacks X0 p0

/X p

S0

Gabriele Vezzosi ()

f0

f

/S



23 / 26



f0

p

S0

/X

f

/S

the canonical map p ∗ ◦ f∗ −→ f∗0 ◦ p 0∗

Gabriele Vezzosi ()



23 / 26



f0

p

S0

/X

f

/S

the canonical map p ∗ ◦ f∗ −→ f∗0 ◦ p 0∗ is a q-iso in ’most’ cases

Gabriele Vezzosi ()



23 / 26



f0

p

S0

/X

f

/S

the canonical map p ∗ ◦ f∗ −→ f∗0 ◦ p 0∗ is a q-iso in ’most’ cases (e.g. for all quasi-compact derived schemes).

Gabriele Vezzosi ()



23 / 26



f0

p

S0

/X

f

/S

the canonical map p ∗ ◦ f∗ −→ f∗0 ◦ p 0∗ is a q-iso in ’most’ cases (e.g. for all quasi-compact derived schemes). ; in derived algebraic geometry objects are very much transverse (no moving-lemmas needed). Gabriele Vezzosi ()



23 / 26

Derived Algebraic Geometry (DAG) - An example

Gabriele Vezzosi ()



24 / 26

Derived Algebraic Geometry (DAG) - An example - Derived moduli stack of vector bundles on a sm. proj. variety X /C -

Gabriele Vezzosi ()



24 / 26

Derived Algebraic Geometry (DAG) - An example - Derived moduli stack of vector bundles on a sm. proj. variety X /C For A ∈ SimplCommAlgC ,

Gabriele Vezzosi ()



24 / 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 objects presheaves of OX ⊗ A-dg-modules on X and morphisms inducing quasi-isomorphisms on stalks (maps called equivalences).

Gabriele Vezzosi ()



24 / 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 objects presheaves of OX ⊗ A-dg-modules on X and morphisms inducing quasi-isomorphisms on stalks (maps called equivalences). RVectn : CommSimplAlgC −→ SimplSets A 7−→ Nerve(Vectder n (X , A))

Gabriele Vezzosi ()



24 / 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 objects presheaves of OX ⊗ A-dg-modules on X and morphisms inducing quasi-isomorphisms on stalks (maps called equivalences). RVectn : CommSimplAlgC −→ SimplSets A 7−→ Nerve(Vectder n (X , A)) where Vectder n (X , A)

Gabriele Vezzosi ()



24 / 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 objects presheaves of OX ⊗ A-dg-modules on X and morphisms inducing quasi-isomorphisms on stalks (maps called equivalences). RVectn : CommSimplAlgC −→ SimplSets A 7−→ Nerve(Vectder n (X , A)) der where Vectder (X , A) of rk n n (X , A) is the full sub-category of Mod derived vector bundles on X

Gabriele Vezzosi ()



24 / 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 objects presheaves of OX ⊗ A-dg-modules on X and morphisms inducing quasi-isomorphisms on stalks (maps called equivalences). RVectn : CommSimplAlgC −→ SimplSets A 7−→ Nerve(Vectder n (X , A)) der where Vectder (X , A) of rk n n (X , A) is the full sub-category of Mod derived vector bundles on X i.e. OX ⊗ A-dg-modules M on X which are

Gabriele Vezzosi ()



24 / 26


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

Gabriele Vezzosi ()



24 / 26


locally on XZar × Aét equivalent to (OX ⊗ A)n flat over A (more precisely, M(U) is a cofibrant A-dg-module, for any open U ⊂ X ) Gabriele Vezzosi ()



24 / 26


locally on XZar × Aét equivalent to (OX ⊗ A)n flat over A (more precisely, M(U) is a cofibrant A-dg-module, for any open U ⊂ X ) Gabriele Vezzosi ()



24 / 26


Gabriele Vezzosi ()



25 / 26


Theorem (Toën-V.)

Gabriele Vezzosi ()



25 / 26


Theorem (Toën-V.) RVectn is a p-smooth Artin derived 1-stack

Gabriele Vezzosi ()



25 / 26


Theorem (Toën-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]

Gabriele Vezzosi ()



25 / 26



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

Gabriele Vezzosi ()



25 / 26



t0 (RVectn (X )) ' Vectn (X ) ; this is a global realization of Kontsevich hidden smoothness philosophy.

Gabriele Vezzosi ()



25 / 26


Gabriele Vezzosi ()



26 / 26


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

Gabriele Vezzosi ()



26 / 26