A Gersten Complex on Real Schemes

A Gersten complex on real schemes

Fangzhou Jin joint work with H. Xie

February 5, 2021

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Table of contents

1 Duality in algebraic geometry 2 Witt groups and real schemes 3 Gersten complex and duality

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Coherent duality: introduced by Grothendieck (1963) as a generalization of Serre duality Serre duality Coherent duality regular (Cohen-Macaulay) schemes singular schemes sheaves complexes of sheaves dualizing sheaf dualizing complex Grothendieck also adapts the relative point of view: projective varieties 7→ proper morphisms Coherent duality is a prototype for several other duality theories

Grothendieck’s coherent Duality

All schemes are assumed noetherian of ﬁnite dimension

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Serre duality Coherent duality regular (Cohen-Macaulay) schemes singular schemes sheaves complexes of sheaves dualizing sheaf dualizing complex Grothendieck also adapts the relative point of view: projective varieties 7→ proper morphisms Coherent duality is a prototype for several other duality theories

Grothendieck’s coherent Duality

All schemes are assumed noetherian of ﬁnite dimension Coherent duality: introduced by Grothendieck (1963) as a generalization of Serre duality

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Grothendieck also adapts the relative point of view: projective varieties 7→ proper morphisms Coherent duality is a prototype for several other duality theories

Grothendieck’s coherent Duality

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Coherent duality is a prototype for several other duality theories

Grothendieck’s coherent Duality

All schemes are assumed noetherian of ﬁnite dimension Coherent duality: introduced by Grothendieck (1963) as a generalization of Serre duality Serre duality Coherent duality regular (Cohen-Macaulay) schemes singular schemes sheaves complexes of sheaves dualizing sheaf dualizing complex Grothendieck also adapts the relative point of view: projective varieties 7→ proper morphisms

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Grothendieck’s coherent Duality

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Verdier duality (1966), based on Borel-Moore (1960) 1 A -homotopic duality (∼00’s) Duality Spaces Category b coherent duality Schemes Dcoh(Qcoh(X )) b Etale duality Schemes D (Xet , Λ) b Verdier duality Topological spaces D (X , Z) 1 A -homotopy Schemes Motivic categories

The duality theorems can be expressed in terms of the Grothendieck six functors formalism Major diﬀerences between Coherent duality and other duality theories

Other dualities

Etale duality (SGA4-5)

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes 1 A -homotopic duality (∼00’s) Duality Spaces Category b coherent duality Schemes Dcoh(Qcoh(X )) b Etale duality Schemes D (Xet , Λ) b Verdier duality Topological spaces D (X , Z) 1 A -homotopy Schemes Motivic categories

The duality theorems can be expressed in terms of the Grothendieck six functors formalism Major diﬀerences between Coherent duality and other duality theories

Other dualities

Etale duality (SGA4-5) Verdier duality (1966), based on Borel-Moore (1960)

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Duality Spaces Category b coherent duality Schemes Dcoh(Qcoh(X )) b Etale duality Schemes D (Xet , Λ) b Verdier duality Topological spaces D (X , Z) 1 A -homotopy Schemes Motivic categories

The duality theorems can be expressed in terms of the Grothendieck six functors formalism Major diﬀerences between Coherent duality and other duality theories

Other dualities

Etale duality (SGA4-5) Verdier duality (1966), based on Borel-Moore (1960) 1 A -homotopic duality (∼00’s)

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes The duality theorems can be expressed in terms of the Grothendieck six functors formalism Major diﬀerences between Coherent duality and other duality theories

Other dualities

Etale duality (SGA4-5) Verdier duality (1966), based on Borel-Moore (1960) 1 A -homotopic duality (∼00’s) Duality Spaces Category b coherent duality Schemes Dcoh(Qcoh(X )) b Etale duality Schemes D (Xet , Λ) b Verdier duality Topological spaces D (X , Z) 1 A -homotopy Schemes Motivic categories

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Major diﬀerences between Coherent duality and other duality theories

Other dualities

The duality theorems can be expressed in terms of the Grothendieck six functors formalism

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Other dualities

The duality theorems can be expressed in terms of the Grothendieck six functors formalism Major diﬀerences between Coherent duality and other duality theories

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Similarities: Poincar´eduality: for a smooth morphism f , f ! diﬀers from f ∗ by an “orientation sheaf”

Atiyah duality: Rf∗ for f a smooth proper morphism preserves dualizable objects Notion of dualizing objects and local (bi)duality Two dualizing objects on a given space diﬀer by a ⊗-invertible object up to a shift Dualizing objects are preserved by f !

Similarities between dualities

∗ ! L Six functors: f , Rf∗, Rf!, f , ⊗ , RHom

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Poincar´eduality: for a smooth morphism f , f ! diﬀers from f ∗ by an “orientation sheaf”

Similarities between dualities

∗ ! L Six functors: f , Rf∗, Rf!, f , ⊗ , RHom Similarities:

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Atiyah duality: Rf∗ for f a smooth proper morphism preserves dualizable objects Notion of dualizing objects and local (bi)duality Two dualizing objects on a given space diﬀer by a ⊗-invertible object up to a shift Dualizing objects are preserved by f !

Similarities between dualities

∗ ! L Six functors: f , Rf∗, Rf!, f , ⊗ , RHom Similarities: Poincar´eduality: for a smooth morphism f , f ! diﬀers from f ∗ by an “orientation sheaf”

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Notion of dualizing objects and local (bi)duality Two dualizing objects on a given space diﬀer by a ⊗-invertible object up to a shift Dualizing objects are preserved by f !

Similarities between dualities

∗ ! L Six functors: f , Rf∗, Rf!, f , ⊗ , RHom Similarities: Poincar´eduality: for a smooth morphism f , f ! diﬀers from f ∗ by an “orientation sheaf”

Atiyah duality: Rf∗ for f a smooth proper morphism preserves dualizable objects

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Two dualizing objects on a given space diﬀer by a ⊗-invertible object up to a shift Dualizing objects are preserved by f !

Similarities between dualities

∗ ! L Six functors: f , Rf∗, Rf!, f , ⊗ , RHom Similarities: Poincar´eduality: for a smooth morphism f , f ! diﬀers from f ∗ by an “orientation sheaf”

Atiyah duality: Rf∗ for f a smooth proper morphism preserves dualizable objects Notion of dualizing objects and local (bi)duality

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Dualizing objects are preserved by f !

Similarities between dualities

∗ ! L Six functors: f , Rf∗, Rf!, f , ⊗ , RHom Similarities: Poincar´eduality: for a smooth morphism f , f ! diﬀers from f ∗ by an “orientation sheaf”

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Similarities between dualities

∗ ! L Six functors: f , Rf∗, Rf!, f , ⊗ , RHom Similarities: Poincar´eduality: for a smooth morphism f , f ! diﬀers from f ∗ by an “orientation sheaf”

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Coherent duality: very general base changes (e.g. ﬂat base change) Other dualities: basically only smooth base change Purity: compare i ∗ and i ! for i a regular closed immersion Coherent duality: Fundamental local isomorphism ∗ L −1 ! i (−) ⊗ (det(Ni )) ' i (−)

Other dualities: purity transformation (D´eglise-J.-Khan) ∗ L −1 ! i (−) ⊗ Th(Ni ) → i (−) not an isomorphism in general (see absolute purity) In coherent duality, the functor Rf! does not exist Projection formula: f proper, ! Rf∗RHom(F, f G) ' RHom(Rf∗F, G) In coherent duality, no suitable subcategory of constructible objects preserved by 6 functors

Diﬀerences between coherent and ´etale/motivic dualities Base changes:

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Other dualities: basically only smooth base change Purity: compare i ∗ and i ! for i a regular closed immersion Coherent duality: Fundamental local isomorphism ∗ L −1 ! i (−) ⊗ (det(Ni )) ' i (−)

Differences between coherent and étale/motivic dualities Base changes: Coherent duality: very general base changes (e.g. flat base change)

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Purity: compare i ∗ and i ! for i a regular closed immersion Coherent duality: Fundamental local isomorphism ∗ L −1 ! i (−) ⊗ (det(Ni )) ' i (−)

Differences between coherent and étale/motivic dualities Base changes: Coherent duality: very general base changes (e.g. flat base change) Other dualities: basically only smooth base change

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Coherent duality: Fundamental local isomorphism ∗ L −1 ! i (−) ⊗ (det(Ni )) ' i (−)

Differences between coherent and étale/motivic dualities Base changes: Coherent duality: very general base changes (e.g. flat base change) Other dualities: basically only smooth base change Purity: compare i ∗ and i ! for i a regular closed immersion

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Other dualities: purity transformation (D´eglise-J.-Khan) ∗ L −1 ! i (−) ⊗ Th(Ni ) → i (−) not an isomorphism in general (see absolute purity) In coherent duality, the functor Rf! does not exist Projection formula: f proper, ! Rf∗RHom(F, f G) ' RHom(Rf∗F, G) In coherent duality, no suitable subcategory of constructible objects preserved by 6 functors

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes In coherent duality, the functor Rf! does not exist Projection formula: f proper, ! Rf∗RHom(F, f G) ' RHom(Rf∗F, G) In coherent duality, no suitable subcategory of constructible objects preserved by 6 functors

Other dualities: purity transformation (D´eglise-J.-Khan) ∗ L −1 ! i (−) ⊗ Th(Ni ) → i (−) not an isomorphism in general (see absolute purity)

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Projection formula: f proper, ! Rf∗RHom(F, f G) ' RHom(Rf∗F, G) In coherent duality, no suitable subcategory of constructible objects preserved by 6 functors

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Differences between coherent and étale/motivic dualities Base changes: Coherent duality: very general base changes (e.g. flat base change) Other dualities: basically only smooth base change Purity: compare i ∗ and i ! for i a regular closed immersion Coherent duality: Fundamental local isomorphism ∗ L −1 ! i (−) ⊗ (det(Ni )) ' i (−)

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes X Gorenstein ⇔ OX is a dualizing complex In particular if X regular ⇒ OX is a dualizing complex f : Y → X quasi-projective, K dualizing complex over X ⇒ f !K dualizing complex over Y If K and K 0 are two dualizing complexes over X , then 0 K = K ⊗ L[n], where n ∈ Z, L invertible OX -module

Dualizing complexes

A (coherent) dualizing complex over a scheme X is a complex b K ∈ Dcoh(Qcoh(X )) quasi-isomorphic to a bounded complex of b injective OX -modules, such that for any F ∈ Dcoh(Qcoh(X )), F' RHom(RHom(F, K), K)

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes In particular if X regular ⇒ OX is a dualizing complex f : Y → X quasi-projective, K dualizing complex over X ⇒ f !K dualizing complex over Y If K and K 0 are two dualizing complexes over X , then 0 K = K ⊗ L[n], where n ∈ Z, L invertible OX -module

Dualizing complexes

X Gorenstein ⇔ OX is a dualizing complex

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes If K and K 0 are two dualizing complexes over X , then 0 K = K ⊗ L[n], where n ∈ Z, L invertible OX -module

Dualizing complexes

X Gorenstein ⇔ OX is a dualizing complex In particular if X regular ⇒ OX is a dualizing complex f : Y → X quasi-projective, K dualizing complex over X ⇒ f !K dualizing complex over Y

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Dualizing complexes

X Gorenstein ⇔ OX is a dualizing complex In particular if X regular ⇒ OX is a dualizing complex f : Y → X quasi-projective, K dualizing complex over X ⇒ f !K dualizing complex over Y If K and K 0 are two dualizing complexes over X , then 0 K = K ⊗ L[n], where n ∈ Z, L invertible OX -module

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes K dualizing complex over X , x ∈ X b Kx ∈ D (OX ,x ) stalk of K at x

πx : Spec(k(x)) → Spec(OX ,x ) ! b πx Kx ∈ D (k(x)) is a dualizing complex over k(x) ⇒ there exist a unique µK (x) ∈ Z and a 1-dimensional ! k(x)-vector space Kk(x) such that πx Kx ' Kk(x)[−µK (x)] The map x 7→ µK (x) deﬁnes a codimension function on X In particular, a scheme with a dualizing complex is universally catenary and has a codimension function

Dualizing complexes (II)

A dualizing complex carries local information at each point:

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes b Kx ∈ D (OX ,x ) stalk of K at x

Dualizing complexes (II)

A dualizing complex carries local information at each point: K dualizing complex over X , x ∈ X

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes πx : Spec(k(x)) → Spec(OX ,x ) ! b πx Kx ∈ D (k(x)) is a dualizing complex over k(x) ⇒ there exist a unique µK (x) ∈ Z and a 1-dimensional ! k(x)-vector space Kk(x) such that πx Kx ' Kk(x)[−µK (x)] The map x 7→ µK (x) deﬁnes a codimension function on X In particular, a scheme with a dualizing complex is universally catenary and has a codimension function

Dualizing complexes (II)

A dualizing complex carries local information at each point: K dualizing complex over X , x ∈ X b Kx ∈ D (OX ,x ) stalk of K at x

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes ! b πx Kx ∈ D (k(x)) is a dualizing complex over k(x) ⇒ there exist a unique µK (x) ∈ Z and a 1-dimensional ! k(x)-vector space Kk(x) such that πx Kx ' Kk(x)[−µK (x)] The map x 7→ µK (x) deﬁnes a codimension function on X In particular, a scheme with a dualizing complex is universally catenary and has a codimension function

Dualizing complexes (II)

A dualizing complex carries local information at each point: K dualizing complex over X , x ∈ X b Kx ∈ D (OX ,x ) stalk of K at x

πx : Spec(k(x)) → Spec(OX ,x )

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes ⇒ there exist a unique µK (x) ∈ Z and a 1-dimensional ! k(x)-vector space Kk(x) such that πx Kx ' Kk(x)[−µK (x)] The map x 7→ µK (x) deﬁnes a codimension function on X In particular, a scheme with a dualizing complex is universally catenary and has a codimension function

Dualizing complexes (II)

A dualizing complex carries local information at each point: K dualizing complex over X , x ∈ X b Kx ∈ D (OX ,x ) stalk of K at x

πx : Spec(k(x)) → Spec(OX ,x ) ! b πx Kx ∈ D (k(x)) is a dualizing complex over k(x)

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes The map x 7→ µK (x) deﬁnes a codimension function on X In particular, a scheme with a dualizing complex is universally catenary and has a codimension function

Dualizing complexes (II)

A dualizing complex carries local information at each point: K dualizing complex over X , x ∈ X b Kx ∈ D (OX ,x ) stalk of K at x

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes In particular, a scheme with a dualizing complex is universally catenary and has a codimension function

Dualizing complexes (II)

A dualizing complex carries local information at each point: K dualizing complex over X , x ∈ X b Kx ∈ D (OX ,x ) stalk of K at x

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Dualizing complexes (II)

A dualizing complex carries local information at each point: K dualizing complex over X , x ∈ X b Kx ∈ D (OX ,x ) stalk of K at x

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes b L invertible OX -module ⇒ (D (Vect(X )), RHom(−, L)) is a triangulated category with duality K dualizing complex on X ⇒ same for (Db(Coh(X )), RHom(−, K)) T triangulated category with duality W (T ) = (⊕{symmetric spaces})/{metabolic spaces}

W (X , L) = W (Db(Vect(X )), RHom(−, L)) (triangulated) Witt group Wf(X , K) = W (Db(Coh(X )), RHom(−, K)) coherent Witt group

Triangulated Witt groups

X scheme, for A = Vect(X ) or Coh(X ), Db(A) = K b(A)/quasi − isomorphisms

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes K dualizing complex on X ⇒ same for (Db(Coh(X )), RHom(−, K)) T triangulated category with duality W (T ) = (⊕{symmetric spaces})/{metabolic spaces}

W (X , L) = W (Db(Vect(X )), RHom(−, L)) (triangulated) Witt group Wf(X , K) = W (Db(Coh(X )), RHom(−, K)) coherent Witt group

Triangulated Witt groups

X scheme, for A = Vect(X ) or Coh(X ), Db(A) = K b(A)/quasi − isomorphisms

b L invertible OX -module ⇒ (D (Vect(X )), RHom(−, L)) is a triangulated category with duality

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes T triangulated category with duality W (T ) = (⊕{symmetric spaces})/{metabolic spaces}

W (X , L) = W (Db(Vect(X )), RHom(−, L)) (triangulated) Witt group Wf(X , K) = W (Db(Coh(X )), RHom(−, K)) coherent Witt group

Triangulated Witt groups

X scheme, for A = Vect(X ) or Coh(X ), Db(A) = K b(A)/quasi − isomorphisms

b L invertible OX -module ⇒ (D (Vect(X )), RHom(−, L)) is a triangulated category with duality K dualizing complex on X ⇒ same for (Db(Coh(X )), RHom(−, K))

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes W (X , L) = W (Db(Vect(X )), RHom(−, L)) (triangulated) Witt group Wf(X , K) = W (Db(Coh(X )), RHom(−, K)) coherent Witt group

Triangulated Witt groups

X scheme, for A = Vect(X ) or Coh(X ), Db(A) = K b(A)/quasi − isomorphisms

b L invertible OX -module ⇒ (D (Vect(X )), RHom(−, L)) is a triangulated category with duality K dualizing complex on X ⇒ same for (Db(Coh(X )), RHom(−, K)) T triangulated category with duality W (T ) = (⊕{symmetric spaces})/{metabolic spaces}

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Wf(X , K) = W (Db(Coh(X )), RHom(−, K)) coherent Witt group

Triangulated Witt groups

X scheme, for A = Vect(X ) or Coh(X ), Db(A) = K b(A)/quasi − isomorphisms

W (X , L) = W (Db(Vect(X )), RHom(−, L)) (triangulated) Witt group

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Triangulated Witt groups

X scheme, for A = Vect(X ) or Coh(X ), Db(A) = K b(A)/quasi − isomorphisms

W (X , L) = W (Db(Vect(X )), RHom(−, L)) (triangulated) Witt group Wf(X , K) = W (Db(Coh(X )), RHom(−, K)) coherent Witt group

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes C(X , Z/2) = continuous (=locally constant) functions (reduced) rank homomorphism rk : W (X ) → C(X , Z/2) fundamental ideal I (X ) = ker(rk) I j (X ) −→2 I j+1(X ) induces

W (X ) = I 0(X ) −→2 I 1(X ) −→2 I 2(X ) −→·2 · · −→ I ∞(X ) = colim I j (X )

We have W [1/2] ' I [1/2] '···' I ∞[1/2] Witt sheaf: W = sheaf associated to the presheaf U 7→ W (U) Similarly we deﬁne I j , I ∞ j j+1 For j > dim(X ) + 1, I ' I .

Fundamental ideal and Witt sheaf

W (X ) = W (X , OX )

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes (reduced) rank homomorphism rk : W (X ) → C(X , Z/2) fundamental ideal I (X ) = ker(rk) I j (X ) −→2 I j+1(X ) induces

W (X ) = I 0(X ) −→2 I 1(X ) −→2 I 2(X ) −→·2 · · −→ I ∞(X ) = colim I j (X )

We have W [1/2] ' I [1/2] '···' I ∞[1/2] Witt sheaf: W = sheaf associated to the presheaf U 7→ W (U) Similarly we deﬁne I j , I ∞ j j+1 For j > dim(X ) + 1, I ' I .

Fundamental ideal and Witt sheaf

W (X ) = W (X , OX ) C(X , Z/2) = continuous (=locally constant) functions

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes fundamental ideal I (X ) = ker(rk) I j (X ) −→2 I j+1(X ) induces

W (X ) = I 0(X ) −→2 I 1(X ) −→2 I 2(X ) −→·2 · · −→ I ∞(X ) = colim I j (X )

We have W [1/2] ' I [1/2] '···' I ∞[1/2] Witt sheaf: W = sheaf associated to the presheaf U 7→ W (U) Similarly we deﬁne I j , I ∞ j j+1 For j > dim(X ) + 1, I ' I .

Fundamental ideal and Witt sheaf

W (X ) = W (X , OX ) C(X , Z/2) = continuous (=locally constant) functions (reduced) rank homomorphism rk : W (X ) → C(X , Z/2)

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes I j (X ) −→2 I j+1(X ) induces

W (X ) = I 0(X ) −→2 I 1(X ) −→2 I 2(X ) −→·2 · · −→ I ∞(X ) = colim I j (X )

We have W [1/2] ' I [1/2] '···' I ∞[1/2] Witt sheaf: W = sheaf associated to the presheaf U 7→ W (U) Similarly we deﬁne I j , I ∞ j j+1 For j > dim(X ) + 1, I ' I .

Fundamental ideal and Witt sheaf

W (X ) = W (X , OX ) C(X , Z/2) = continuous (=locally constant) functions (reduced) rank homomorphism rk : W (X ) → C(X , Z/2) fundamental ideal I (X ) = ker(rk)

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes We have W [1/2] ' I [1/2] '···' I ∞[1/2] Witt sheaf: W = sheaf associated to the presheaf U 7→ W (U) Similarly we deﬁne I j , I ∞ j j+1 For j > dim(X ) + 1, I ' I .

Fundamental ideal and Witt sheaf

W (X ) = W (X , OX ) C(X , Z/2) = continuous (=locally constant) functions (reduced) rank homomorphism rk : W (X ) → C(X , Z/2) fundamental ideal I (X ) = ker(rk) I j (X ) −→2 I j+1(X ) induces

W (X ) = I 0(X ) −→2 I 1(X ) −→2 I 2(X ) −→·2 · · −→ I ∞(X ) = colim I j (X )

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Witt sheaf: W = sheaf associated to the presheaf U 7→ W (U) Similarly we deﬁne I j , I ∞ j j+1 For j > dim(X ) + 1, I ' I .

Fundamental ideal and Witt sheaf

W (X ) = W (X , OX ) C(X , Z/2) = continuous (=locally constant) functions (reduced) rank homomorphism rk : W (X ) → C(X , Z/2) fundamental ideal I (X ) = ker(rk) I j (X ) −→2 I j+1(X ) induces

W (X ) = I 0(X ) −→2 I 1(X ) −→2 I 2(X ) −→·2 · · −→ I ∞(X ) = colim I j (X )

We have W [1/2] ' I [1/2] '···' I ∞[1/2]

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Similarly we deﬁne I j , I ∞ j j+1 For j > dim(X ) + 1, I ' I .

Fundamental ideal and Witt sheaf

W (X ) = W (X , OX ) C(X , Z/2) = continuous (=locally constant) functions (reduced) rank homomorphism rk : W (X ) → C(X , Z/2) fundamental ideal I (X ) = ker(rk) I j (X ) −→2 I j+1(X ) induces

W (X ) = I 0(X ) −→2 I 1(X ) −→2 I 2(X ) −→·2 · · −→ I ∞(X ) = colim I j (X )

We have W [1/2] ' I [1/2] '···' I ∞[1/2] Witt sheaf: W = sheaf associated to the presheaf U 7→ W (U)

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes j j+1 For j > dim(X ) + 1, I ' I .

Fundamental ideal and Witt sheaf

W (X ) = W (X , OX ) C(X , Z/2) = continuous (=locally constant) functions (reduced) rank homomorphism rk : W (X ) → C(X , Z/2) fundamental ideal I (X ) = ker(rk) I j (X ) −→2 I j+1(X ) induces

W (X ) = I 0(X ) −→2 I 1(X ) −→2 I 2(X ) −→·2 · · −→ I ∞(X ) = colim I j (X )

We have W [1/2] ' I [1/2] '···' I ∞[1/2] Witt sheaf: W = sheaf associated to the presheaf U 7→ W (U) Similarly we deﬁne I j , I ∞

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Fundamental ideal and Witt sheaf

W (X ) = W (X , OX ) C(X , Z/2) = continuous (=locally constant) functions (reduced) rank homomorphism rk : W (X ) → C(X , Z/2) fundamental ideal I (X ) = ker(rk) I j (X ) −→2 I j+1(X ) induces

W (X ) = I 0(X ) −→2 I 1(X ) −→2 I 2(X ) −→·2 · · −→ I ∞(X ) = colim I j (X )

We have W [1/2] ' I [1/2] '···' I ∞[1/2] Witt sheaf: W = sheaf associated to the presheaf U 7→ W (U) Similarly we deﬁne I j , I ∞ j j+1 For j > dim(X ) + 1, I ' I .

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Topology generated by opens, for a ∈ A D(a) = {ξ ∈ Sper(A)|a(ξ) > 0}

The real scheme Xr associated to a scheme X is a topological space obtained by gluing real spectra of rings

Xr = {(x, P)|x ∈ X , P ordering on k(x)}

X quasi-compact quasi-separated ⇒ Xr is a spectral space (Hochster: S is a spectral space ⇔ S homeomorphic to a Zariski spectrum ⇔ S is a limit of ﬁnite T0 spaces) Xr depends functorially on X : f : X → Y ⇒ fr : Xr → Yr

Real schemes

The real spectrum of a ring A is a topological space Sper(A) Sper(A) = {ξ = (p, P)|p ∈ Spec(A), P ordering on k(p)}

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes The real scheme Xr associated to a scheme X is a topological space obtained by gluing real spectra of rings

Xr = {(x, P)|x ∈ X , P ordering on k(x)}

Real schemes

The real spectrum of a ring A is a topological space Sper(A) Sper(A) = {ξ = (p, P)|p ∈ Spec(A), P ordering on k(p)}

Topology generated by opens, for a ∈ A D(a) = {ξ ∈ Sper(A)|a(ξ) > 0}

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes X quasi-compact quasi-separated ⇒ Xr is a spectral space (Hochster: S is a spectral space ⇔ S homeomorphic to a Zariski spectrum ⇔ S is a limit of ﬁnite T0 spaces) Xr depends functorially on X : f : X → Y ⇒ fr : Xr → Yr

Real schemes

The real spectrum of a ring A is a topological space Sper(A) Sper(A) = {ξ = (p, P)|p ∈ Spec(A), P ordering on k(p)}

Topology generated by opens, for a ∈ A D(a) = {ξ ∈ Sper(A)|a(ξ) > 0}

The real scheme Xr associated to a scheme X is a topological space obtained by gluing real spectra of rings

Xr = {(x, P)|x ∈ X , P ordering on k(x)}

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes (Hochster: S is a spectral space ⇔ S homeomorphic to a Zariski spectrum ⇔ S is a limit of ﬁnite T0 spaces) Xr depends functorially on X : f : X → Y ⇒ fr : Xr → Yr

Real schemes

The real spectrum of a ring A is a topological space Sper(A) Sper(A) = {ξ = (p, P)|p ∈ Spec(A), P ordering on k(p)}

Topology generated by opens, for a ∈ A D(a) = {ξ ∈ Sper(A)|a(ξ) > 0}

The real scheme Xr associated to a scheme X is a topological space obtained by gluing real spectra of rings

Xr = {(x, P)|x ∈ X , P ordering on k(x)}

X quasi-compact quasi-separated ⇒ Xr is a spectral space

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Xr depends functorially on X : f : X → Y ⇒ fr : Xr → Yr

Real schemes

The real spectrum of a ring A is a topological space Sper(A) Sper(A) = {ξ = (p, P)|p ∈ Spec(A), P ordering on k(p)}

Topology generated by opens, for a ∈ A D(a) = {ξ ∈ Sper(A)|a(ξ) > 0}

The real scheme Xr associated to a scheme X is a topological space obtained by gluing real spectra of rings

Xr = {(x, P)|x ∈ X , P ordering on k(x)}

X quasi-compact quasi-separated ⇒ Xr is a spectral space (Hochster: S is a spectral space ⇔ S homeomorphic to a Zariski spectrum ⇔ S is a limit of ﬁnite T0 spaces)

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Real schemes

The real spectrum of a ring A is a topological space Sper(A) Sper(A) = {ξ = (p, P)|p ∈ Spec(A), P ordering on k(p)}

Topology generated by opens, for a ∈ A D(a) = {ξ ∈ Sper(A)|a(ξ) > 0}

The real scheme Xr associated to a scheme X is a topological space obtained by gluing real spectra of rings

Xr = {(x, P)|x ∈ X , P ordering on k(x)}

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes The constructible subsets of Sper(A) are boolean combinations of the D(a)’s (ﬁnite unions, ﬁnite intersections, complements)

The constructible subsets U ⊂ Xr are such that U ∩ Sper(A) is constructible in Sper(A) for any aﬃne open Sper(A) of Xr Example: If X is separated of ﬁnite type over the real numbers R, then the constructible subsets of Xr are precisely the semi-algebraic subsets of X (R)

Real schemes and real algebraic geometry

The real schemes are important in real algebraic geometry:

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes The constructible subsets U ⊂ Xr are such that U ∩ Sper(A) is constructible in Sper(A) for any aﬃne open Sper(A) of Xr Example: If X is separated of ﬁnite type over the real numbers R, then the constructible subsets of Xr are precisely the semi-algebraic subsets of X (R)

Real schemes and real algebraic geometry

The real schemes are important in real algebraic geometry: The constructible subsets of Sper(A) are boolean combinations of the D(a)’s (ﬁnite unions, ﬁnite intersections, complements)

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Example: If X is separated of ﬁnite type over the real numbers R, then the constructible subsets of Xr are precisely the semi-algebraic subsets of X (R)

Real schemes and real algebraic geometry

The real schemes are important in real algebraic geometry: The constructible subsets of Sper(A) are boolean combinations of the D(a)’s (ﬁnite unions, ﬁnite intersections, complements)

The constructible subsets U ⊂ Xr are such that U ∩ Sper(A) is constructible in Sper(A) for any aﬃne open Sper(A) of Xr

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Real schemes and real algebraic geometry

The real schemes are important in real algebraic geometry: The constructible subsets of Sper(A) are boolean combinations of the D(a)’s (ﬁnite unions, ﬁnite intersections, complements)

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes It originates from the study of relations between étale cohomology with Z/2-coefficients and orderings of residue fields (Colliot-Thélène-Parimala, Coste-Roy, Scheiderer,...) The real étaletopology is finer than the Nisnevich topology, but not comparable with the étaletopology

Theorem (Coste-Roy, Scheiderer)

There is a canonical equivalence of sites Xr ' Xret .

The real ´etalesite

The real ´etalesite Xret is the category of ´etale X -schemes endowed with the Grothendieck topology where covering families are those which induce a surjection on real schemes

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes The real étaletopology is finer than the Nisnevich topology, but not comparable with the étaletopology

Theorem (Coste-Roy, Scheiderer)

There is a canonical equivalence of sites Xr ' Xret .

The real ´etalesite

The real étalesite Xret is the category of étale X -schemes endowed with the Grothendieck topology where covering families are those which induce a surjection on real schemes It originates from the study of relations between étale cohomology with Z/2-coefficients and orderings of residue fields (Colliot-Thélène-Parimala, Coste-Roy, Scheiderer,...)

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Theorem (Coste-Roy, Scheiderer)

There is a canonical equivalence of sites Xr ' Xret .

The real ´etalesite

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes The real ´etalesite

Theorem (Coste-Roy, Scheiderer)

There is a canonical equivalence of sites Xr ' Xret .

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes In particular, W [1/2] ' Z[1/2]Xr Sketch of proof: reduce to local rings, then use Hoobler’s trick to reduce to ﬁelds, then apply the Arason-Knebusch theorem Jacobson’s theorem can be extended to the twisted setting

Jacobson’s theorem

Global signature map

Sign : W (X ) → C(Xr , Z) ∗ [φ] 7→ ((x, P) 7→ SignP ([ix φ]))

Theorem (Jacobson) If 2 ∈ O(X )×, this induces an isomorphism of ret-sheaves

∞ I ' ZXr

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Sketch of proof: reduce to local rings, then use Hoobler’s trick to reduce to ﬁelds, then apply the Arason-Knebusch theorem Jacobson’s theorem can be extended to the twisted setting

Jacobson’s theorem

Global signature map

Sign : W (X ) → C(Xr , Z) ∗ [φ] 7→ ((x, P) 7→ SignP ([ix φ]))

Theorem (Jacobson) If 2 ∈ O(X )×, this induces an isomorphism of ret-sheaves

∞ I ' ZXr

In particular, W [1/2] ' Z[1/2]Xr

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Jacobson’s theorem can be extended to the twisted setting

Jacobson’s theorem

Global signature map

Sign : W (X ) → C(Xr , Z) ∗ [φ] 7→ ((x, P) 7→ SignP ([ix φ]))

Theorem (Jacobson) If 2 ∈ O(X )×, this induces an isomorphism of ret-sheaves

∞ I ' ZXr

In particular, W [1/2] ' Z[1/2]Xr Sketch of proof: reduce to local rings, then use Hoobler’s trick to reduce to ﬁelds, then apply the Arason-Knebusch theorem

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Jacobson’s theorem

Global signature map

Sign : W (X ) → C(Xr , Z) ∗ [φ] 7→ ((x, P) 7→ SignP ([ix φ]))

Theorem (Jacobson) If 2 ∈ O(X )×, this induces an isomorphism of ret-sheaves

∞ I ' ZXr

In particular, W [1/2] ' Z[1/2]Xr Sketch of proof: reduce to local rings, then use Hoobler’s trick to reduce to ﬁelds, then apply the Arason-Knebusch theorem Jacobson’s theorem can be extended to the twisted setting

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes A trivialization of L induces F(L) 'F, in particular this always holds locally

Similarly we deﬁne ZXr (L) locally constant sheaf on Xr

Theorem (Hornbostel-Wendt-Xie-Zibrowius) There are canonical isomorphisms

∞ ∞ I (−, L) ' I (L) ' ZXr (L)

{twisted Witt sheaves} ↔ {twisted constant sheaves on Xr }

Twisting by invertible sheaves

(X , OX ) ringed space, L invertible OX -module, F sheaf on X × with an action of OX , define F(L) as the sheafification of

× U 7→ F(U) ⊗ × Z[L(U) ] Z[OX (U)]

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Similarly we deﬁne ZXr (L) locally constant sheaf on Xr

Theorem (Hornbostel-Wendt-Xie-Zibrowius) There are canonical isomorphisms

∞ ∞ I (−, L) ' I (L) ' ZXr (L)

{twisted Witt sheaves} ↔ {twisted constant sheaves on Xr }

Twisting by invertible sheaves

(X , OX ) ringed space, L invertible OX -module, F sheaf on X × with an action of OX , define F(L) as the sheafification of

× U 7→ F(U) ⊗ × Z[L(U) ] Z[OX (U)]

A trivialization of L induces F(L) 'F, in particular this always holds locally

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Theorem (Hornbostel-Wendt-Xie-Zibrowius) There are canonical isomorphisms

∞ ∞ I (−, L) ' I (L) ' ZXr (L)

{twisted Witt sheaves} ↔ {twisted constant sheaves on Xr }

Twisting by invertible sheaves

(X , OX ) ringed space, L invertible OX -module, F sheaf on X × with an action of OX , define F(L) as the sheafification of

× U 7→ F(U) ⊗ × Z[L(U) ] Z[OX (U)]

A trivialization of L induces F(L) 'F, in particular this always holds locally

Similarly we deﬁne ZXr (L) locally constant sheaf on Xr

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes {twisted Witt sheaves} ↔ {twisted constant sheaves on Xr }

Twisting by invertible sheaves

(X , OX ) ringed space, L invertible OX -module, F sheaf on X × with an action of OX , define F(L) as the sheafification of

× U 7→ F(U) ⊗ × Z[L(U) ] Z[OX (U)]

A trivialization of L induces F(L) 'F, in particular this always holds locally

Similarly we deﬁne ZXr (L) locally constant sheaf on Xr

Theorem (Hornbostel-Wendt-Xie-Zibrowius) There are canonical isomorphisms

∞ ∞ I (−, L) ' I (L) ' ZXr (L)

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Twisting by invertible sheaves

(X , OX ) ringed space, L invertible OX -module, F sheaf on X × with an action of OX , define F(L) as the sheafification of

× U 7→ F(U) ⊗ × Z[L(U) ] Z[OX (U)]

A trivialization of L induces F(L) 'F, in particular this always holds locally

Similarly we deﬁne ZXr (L) locally constant sheaf on Xr

Theorem (Hornbostel-Wendt-Xie-Zibrowius) There are canonical isomorphisms

∞ ∞ I (−, L) ' I (L) ' ZXr (L)

{twisted Witt sheaves} ↔ {twisted constant sheaves on Xr }

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes 1 1 The A -derived category DA (X ) is obtained from 1 PSh(Sm/X , C(Ab)) by Nisnevich-A localization and 1 P -stabilization ρ : Z → Gm is induced by −1 ∈ Gm In particular, X 7→ D(Xr ) is a motivic category and satisﬁes the six functors formalism Constructible objects Dc (Xr ) ⊂ D(Xr ) thick subcategory ! generated by Rf!f ZXr for f : Y → X smooth Theorem (J.)

A ∈ Dc (Xr ) ⇔ ∃ ﬁnite stratiﬁcation of Xr into constructible

subsets Xi such that A|Xi is constant with perfect stalks

Real schemes and motivic homotopy

Real schemes are closely related to motivic homotopy: Theorem (Bachmann) 1 −1 There is a canonical equivalence D(Xr ) ' DA (X )[ρ ]

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes ρ : Z → Gm is induced by −1 ∈ Gm In particular, X 7→ D(Xr ) is a motivic category and satisﬁes the six functors formalism Constructible objects Dc (Xr ) ⊂ D(Xr ) thick subcategory ! generated by Rf!f ZXr for f : Y → X smooth Theorem (J.)

A ∈ Dc (Xr ) ⇔ ∃ ﬁnite stratiﬁcation of Xr into constructible

subsets Xi such that A|Xi is constant with perfect stalks

Real schemes and motivic homotopy

Real schemes are closely related to motivic homotopy: Theorem (Bachmann) 1 −1 There is a canonical equivalence D(Xr ) ' DA (X )[ρ ]

1 1 The A -derived category DA (X ) is obtained from 1 PSh(Sm/X , C(Ab)) by Nisnevich-A localization and 1 P -stabilization

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes In particular, X 7→ D(Xr ) is a motivic category and satisﬁes the six functors formalism Constructible objects Dc (Xr ) ⊂ D(Xr ) thick subcategory ! generated by Rf!f ZXr for f : Y → X smooth Theorem (J.)

A ∈ Dc (Xr ) ⇔ ∃ ﬁnite stratiﬁcation of Xr into constructible

subsets Xi such that A|Xi is constant with perfect stalks

Real schemes and motivic homotopy

Real schemes are closely related to motivic homotopy: Theorem (Bachmann) 1 −1 There is a canonical equivalence D(Xr ) ' DA (X )[ρ ]

1 1 The A -derived category DA (X ) is obtained from 1 PSh(Sm/X , C(Ab)) by Nisnevich-A localization and 1 P -stabilization ρ : Z → Gm is induced by −1 ∈ Gm

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Constructible objects Dc (Xr ) ⊂ D(Xr ) thick subcategory ! generated by Rf!f ZXr for f : Y → X smooth Theorem (J.)

A ∈ Dc (Xr ) ⇔ ∃ ﬁnite stratiﬁcation of Xr into constructible

subsets Xi such that A|Xi is constant with perfect stalks

Real schemes and motivic homotopy

Real schemes are closely related to motivic homotopy: Theorem (Bachmann) 1 −1 There is a canonical equivalence D(Xr ) ' DA (X )[ρ ]

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Theorem (J.)

A ∈ Dc (Xr ) ⇔ ∃ ﬁnite stratiﬁcation of Xr into constructible

subsets Xi such that A|Xi is constant with perfect stalks

Real schemes and motivic homotopy

Real schemes are closely related to motivic homotopy: Theorem (Bachmann) 1 −1 There is a canonical equivalence D(Xr ) ' DA (X )[ρ ]

1 1 The A -derived category DA (X ) is obtained from 1 PSh(Sm/X , C(Ab)) by Nisnevich-A localization and 1 P -stabilization ρ : Z → Gm is induced by −1 ∈ Gm In particular, X 7→ D(Xr ) is a motivic category and satisﬁes the six functors formalism Constructible objects Dc (Xr ) ⊂ D(Xr ) thick subcategory ! generated by Rf!f ZXr for f : Y → X smooth

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Real schemes and motivic homotopy

Real schemes are closely related to motivic homotopy: Theorem (Bachmann) 1 −1 There is a canonical equivalence D(Xr ) ' DA (X )[ρ ]

A ∈ Dc (Xr ) ⇔ ∃ ﬁnite stratiﬁcation of Xr into constructible

subsets Xi such that A|Xi is constant with perfect stalks

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes M ∂ M ∂ W (k(x), Kk(x)) −→ W (k(x), Kk(x)) −→· · ·

µK (x)=m µK (x)=m+1

Several ways to construct boundary maps ∂: b Via the ﬁltration by codimension of support on Dcoh(Qcoh(X )) (Balmer-Walter) Via the Rost-Schmid boundary (Schmid, Morel) Theorem (Gille): if 2 ∈ O(X )×, these two coincide Pass to I ∞ and sheaﬁfy ⇒ complex C(X , I ∞, K )

M ∞ ∂ M ∞ ∂ I (k(x), Kk(x)) −→ I (k(x), Kk(x)) −→· · ·

µK (x)=m µK (x)=m+1

The Gersten-Witt complex

K dualizing complex on X , the Gersten-Witt complex (Gille, Plowman, J.-Xie)

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Several ways to construct boundary maps ∂: b Via the ﬁltration by codimension of support on Dcoh(Qcoh(X )) (Balmer-Walter) Via the Rost-Schmid boundary (Schmid, Morel) Theorem (Gille): if 2 ∈ O(X )×, these two coincide Pass to I ∞ and sheaﬁfy ⇒ complex C(X , I ∞, K )

M ∞ ∂ M ∞ ∂ I (k(x), Kk(x)) −→ I (k(x), Kk(x)) −→· · ·

µK (x)=m µK (x)=m+1

The Gersten-Witt complex

K dualizing complex on X , the Gersten-Witt complex (Gille, Plowman, J.-Xie)

M ∂ M ∂ W (k(x), Kk(x)) −→ W (k(x), Kk(x)) −→· · ·

µK (x)=m µK (x)=m+1

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes b Via the ﬁltration by codimension of support on Dcoh(Qcoh(X )) (Balmer-Walter) Via the Rost-Schmid boundary (Schmid, Morel) Theorem (Gille): if 2 ∈ O(X )×, these two coincide Pass to I ∞ and sheaﬁfy ⇒ complex C(X , I ∞, K )

M ∞ ∂ M ∞ ∂ I (k(x), Kk(x)) −→ I (k(x), Kk(x)) −→· · ·

µK (x)=m µK (x)=m+1

The Gersten-Witt complex

K dualizing complex on X , the Gersten-Witt complex (Gille, Plowman, J.-Xie)

M ∂ M ∂ W (k(x), Kk(x)) −→ W (k(x), Kk(x)) −→· · ·

µK (x)=m µK (x)=m+1

Several ways to construct boundary maps ∂:

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Via the Rost-Schmid boundary (Schmid, Morel) Theorem (Gille): if 2 ∈ O(X )×, these two coincide Pass to I ∞ and sheaﬁfy ⇒ complex C(X , I ∞, K )

M ∞ ∂ M ∞ ∂ I (k(x), Kk(x)) −→ I (k(x), Kk(x)) −→· · ·

µK (x)=m µK (x)=m+1

The Gersten-Witt complex

K dualizing complex on X , the Gersten-Witt complex (Gille, Plowman, J.-Xie)

M ∂ M ∂ W (k(x), Kk(x)) −→ W (k(x), Kk(x)) −→· · ·

µK (x)=m µK (x)=m+1

Several ways to construct boundary maps ∂: b Via the ﬁltration by codimension of support on Dcoh(Qcoh(X )) (Balmer-Walter)

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Theorem (Gille): if 2 ∈ O(X )×, these two coincide Pass to I ∞ and sheaﬁfy ⇒ complex C(X , I ∞, K )

M ∞ ∂ M ∞ ∂ I (k(x), Kk(x)) −→ I (k(x), Kk(x)) −→· · ·

µK (x)=m µK (x)=m+1

The Gersten-Witt complex

K dualizing complex on X , the Gersten-Witt complex (Gille, Plowman, J.-Xie)

M ∂ M ∂ W (k(x), Kk(x)) −→ W (k(x), Kk(x)) −→· · ·

µK (x)=m µK (x)=m+1

Several ways to construct boundary maps ∂: b Via the ﬁltration by codimension of support on Dcoh(Qcoh(X )) (Balmer-Walter) Via the Rost-Schmid boundary (Schmid, Morel)

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Pass to I ∞ and sheaﬁfy ⇒ complex C(X , I ∞, K )

M ∞ ∂ M ∞ ∂ I (k(x), Kk(x)) −→ I (k(x), Kk(x)) −→· · ·

µK (x)=m µK (x)=m+1

The Gersten-Witt complex

K dualizing complex on X , the Gersten-Witt complex (Gille, Plowman, J.-Xie)

M ∂ M ∂ W (k(x), Kk(x)) −→ W (k(x), Kk(x)) −→· · ·

µK (x)=m µK (x)=m+1

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes The Gersten-Witt complex

K dualizing complex on X , the Gersten-Witt complex (Gille, Plowman, J.-Xie)

M ∂ M ∂ W (k(x), Kk(x)) −→ W (k(x), Kk(x)) −→· · ·

µK (x)=m µK (x)=m+1

M ∞ ∂ M ∞ ∂ I (k(x), Kk(x)) −→ I (k(x), Kk(x)) −→· · ·

µK (x)=m µK (x)=m+1

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Question: Analog of Jacobson’s theorem for dualizing objects? The real ´etaleduality is quite similar to Verdier duality, via the following purity results in D(Xr ): Poincar´eduality (J.-Xie): f smooth of relative dimension d ⇒

∗ ! fr (−) ⊗ Z(det(Tf ))[d] ' fr (−)

Absolute purity (Scheiderer, J.-Xie): Z, X regular, i : Z → X closed immersion of codimension c, N normal bundle of i, F locally constant constructible (lcc) sheaf on Xr

! ∗ −1 ir F' ir F ⊗ Z(det(N) )[−c]

The Gersten-Witt complex and real schemes

sheaves on X sheaves on Xr twisted Witt sheaf twisted constant sheaves (sheaﬁﬁed) Gersten-Witt complex ???

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes The real ´etaleduality is quite similar to Verdier duality, via the following purity results in D(Xr ): Poincar´eduality (J.-Xie): f smooth of relative dimension d ⇒

∗ ! fr (−) ⊗ Z(det(Tf ))[d] ' fr (−)

Absolute purity (Scheiderer, J.-Xie): Z, X regular, i : Z → X closed immersion of codimension c, N normal bundle of i, F locally constant constructible (lcc) sheaf on Xr

! ∗ −1 ir F' ir F ⊗ Z(det(N) )[−c]

The Gersten-Witt complex and real schemes

sheaves on X sheaves on Xr twisted Witt sheaf twisted constant sheaves (sheaﬁﬁed) Gersten-Witt complex ???

Question: Analog of Jacobson’s theorem for dualizing objects?

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Poincar´eduality (J.-Xie): f smooth of relative dimension d ⇒

∗ ! fr (−) ⊗ Z(det(Tf ))[d] ' fr (−)

Absolute purity (Scheiderer, J.-Xie): Z, X regular, i : Z → X closed immersion of codimension c, N normal bundle of i, F locally constant constructible (lcc) sheaf on Xr

! ∗ −1 ir F' ir F ⊗ Z(det(N) )[−c]

The Gersten-Witt complex and real schemes

sheaves on X sheaves on Xr twisted Witt sheaf twisted constant sheaves (sheaﬁﬁed) Gersten-Witt complex ???

Question: Analog of Jacobson’s theorem for dualizing objects? The real ´etaleduality is quite similar to Verdier duality, via the following purity results in D(Xr ):

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Absolute purity (Scheiderer, J.-Xie): Z, X regular, i : Z → X closed immersion of codimension c, N normal bundle of i, F locally constant constructible (lcc) sheaf on Xr

! ∗ −1 ir F' ir F ⊗ Z(det(N) )[−c]

The Gersten-Witt complex and real schemes

sheaves on X sheaves on Xr twisted Witt sheaf twisted constant sheaves (sheaﬁﬁed) Gersten-Witt complex ???

Question: Analog of Jacobson’s theorem for dualizing objects? The real ´etaleduality is quite similar to Verdier duality, via the following purity results in D(Xr ): Poincar´eduality (J.-Xie): f smooth of relative dimension d ⇒

∗ ! fr (−) ⊗ Z(det(Tf ))[d] ' fr (−)

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes The Gersten-Witt complex and real schemes

sheaves on X sheaves on Xr twisted Witt sheaf twisted constant sheaves (sheaﬁﬁed) Gersten-Witt complex ???

∗ ! fr (−) ⊗ Z(det(Tf ))[d] ' fr (−)

Absolute purity (Scheiderer, J.-Xie): Z, X regular, i : Z → X closed immersion of codimension c, N normal bundle of i, F locally constant constructible (lcc) sheaf on Xr

! ∗ −1 ir F' ir F ⊗ Z(det(N) )[−c]

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Coniveau spectral sequence (Grothendieck, Bloch-Ogus, Scheiderer) degenerates at E2 p,q M p+q p+q E1 = Hxr (Xr , F) =⇒ H (Xr , F) x∈X (p)

By absolute purity, the E1-page is the Cousin complex M 0 M 0 H (xr , F(ωx/X )) → H (xr , F(ωx/X )) → · · · x∈X (0) x∈X (1)

where ωx/X = determinant of the normal bundle of x → Spec(OX ,x ) By sheaﬁfying we get a canonical resolution of F by acyclic sheaves (Scheiderer, J.-Xie) M M (ixr )∗F(ωx/X ) → (ixr )∗F(ωx/X ) → · · · x∈X (0) x∈X (1)

Coniveau spectral sequence and the Cousin complex

X regular excellent, F lcc sheaf on Xr

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes By absolute purity, the E1-page is the Cousin complex M 0 M 0 H (xr , F(ωx/X )) → H (xr , F(ωx/X )) → · · · x∈X (0) x∈X (1)

Coniveau spectral sequence and the Cousin complex

X regular excellent, F lcc sheaf on Xr Coniveau spectral sequence (Grothendieck, Bloch-Ogus, Scheiderer) degenerates at E2 p,q M p+q p+q E1 = Hxr (Xr , F) =⇒ H (Xr , F) x∈X (p)

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes By sheaﬁfying we get a canonical resolution of F by acyclic sheaves (Scheiderer, J.-Xie) M M (ixr )∗F(ωx/X ) → (ixr )∗F(ωx/X ) → · · · x∈X (0) x∈X (1)

Coniveau spectral sequence and the Cousin complex

X regular excellent, F lcc sheaf on Xr Coniveau spectral sequence (Grothendieck, Bloch-Ogus, Scheiderer) degenerates at E2 p,q M p+q p+q E1 = Hxr (Xr , F) =⇒ H (Xr , F) x∈X (p)

By absolute purity, the E1-page is the Cousin complex M 0 M 0 H (xr , F(ωx/X )) → H (xr , F(ωx/X )) → · · · x∈X (0) x∈X (1)

where ωx/X = determinant of the normal bundle of x → Spec(OX ,x )

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Coniveau spectral sequence and the Cousin complex

X regular excellent, F lcc sheaf on Xr Coniveau spectral sequence (Grothendieck, Bloch-Ogus, Scheiderer) degenerates at E2 p,q M p+q p+q E1 = Hxr (Xr , F) =⇒ H (Xr , F) x∈X (p)

By absolute purity, the E1-page is the Cousin complex M 0 M 0 H (xr , F(ωx/X )) → H (xr , F(ωx/X )) → · · · x∈X (0) x∈X (1)

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Twisted residue: R =DVR, F =fraction ﬁeld, m =maximal ideal, k = R/m, H = free R-module of rank 1

2 ∗ C(Fr , Z(HF )) → C(kr , Z(Hk ⊗ (m/m ) ))

is a twisted variant of the sign homomorphism Twisted transfer: L/F ﬁnite ﬁeld extension, H = one-dimensional F -vector space

C(Lr , Z(HomF (L, H))) → C(Fr , Z(H))

is deﬁned using the trace form L → HomF (L, F )

Twisted Gersten complex

This suggests that a candidate for (???) should be a complex of similar form

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Twisted transfer: L/F ﬁnite ﬁeld extension, H = one-dimensional F -vector space

C(Lr , Z(HomF (L, H))) → C(Fr , Z(H))

is deﬁned using the trace form L → HomF (L, F )

Twisted Gersten complex

This suggests that a candidate for (???) should be a complex of similar form Twisted residue: R =DVR, F =fraction ﬁeld, m =maximal ideal, k = R/m, H = free R-module of rank 1

2 ∗ C(Fr , Z(HF )) → C(kr , Z(Hk ⊗ (m/m ) ))

is a twisted variant of the sign homomorphism

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Twisted Gersten complex

This suggests that a candidate for (???) should be a complex of similar form Twisted residue: R =DVR, F =fraction ﬁeld, m =maximal ideal, k = R/m, H = free R-module of rank 1

2 ∗ C(Fr , Z(HF )) → C(kr , Z(Hk ⊗ (m/m ) ))

is a twisted variant of the sign homomorphism Twisted transfer: L/F ﬁnite ﬁeld extension, H = one-dimensional F -vector space

C(Lr , Z(HomF (L, H))) → C(Fr , Z(H))

is deﬁned using the trace form L → HomF (L, F )

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes We show that it is a complex, by comparing to the Witt case

Sheaﬁfying the complex C(Xr , K) we obtain a complex C(Xr , K) ∈ D(Xr ) M M (ixr )∗Z(Kk(x)) → (ixr )∗Z(Kk(x)) → · · · µK (x)=m µK (x)=m+1

Here ixr : xr → Xr is the canonical map, and the terms

(ixr )∗Z(Kk(x)) look like skyscraper sheaves

Twisted Gersten complex (II)

For K dualizing complex on X , we obtain a complex C(Xr , K) M M C(k(x)r , Z(Kk(x))) → C(k(x)r , Z(Kk(x))) → · · · µK (x)=m µK (x)=m+1

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Sheaﬁfying the complex C(Xr , K) we obtain a complex C(Xr , K) ∈ D(Xr ) M M (ixr )∗Z(Kk(x)) → (ixr )∗Z(Kk(x)) → · · · µK (x)=m µK (x)=m+1

Here ixr : xr → Xr is the canonical map, and the terms

(ixr )∗Z(Kk(x)) look like skyscraper sheaves

Twisted Gersten complex (II)

For K dualizing complex on X , we obtain a complex C(Xr , K) M M C(k(x)r , Z(Kk(x))) → C(k(x)r , Z(Kk(x))) → · · · µK (x)=m µK (x)=m+1

We show that it is a complex, by comparing to the Witt case

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Here ixr : xr → Xr is the canonical map, and the terms

(ixr )∗Z(Kk(x)) look like skyscraper sheaves

Twisted Gersten complex (II)

For K dualizing complex on X , we obtain a complex C(Xr , K) M M C(k(x)r , Z(Kk(x))) → C(k(x)r , Z(Kk(x))) → · · · µK (x)=m µK (x)=m+1

We show that it is a complex, by comparing to the Witt case

Sheaﬁfying the complex C(Xr , K) we obtain a complex C(Xr , K) ∈ D(Xr ) M M (ixr )∗Z(Kk(x)) → (ixr )∗Z(Kk(x)) → · · · µK (x)=m µK (x)=m+1

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Twisted Gersten complex (II)

For K dualizing complex on X , we obtain a complex C(Xr , K) M M C(k(x)r , Z(Kk(x))) → C(k(x)r , Z(Kk(x))) → · · · µK (x)=m µK (x)=m+1

We show that it is a complex, by comparing to the Witt case

Sheaﬁfying the complex C(Xr , K) we obtain a complex C(Xr , K) ∈ D(Xr ) M M (ixr )∗Z(Kk(x)) → (ixr )∗Z(Kk(x)) → · · · µK (x)=m µK (x)=m+1

Here ixr : xr → Xr is the canonical map, and the terms

(ixr )∗Z(Kk(x)) look like skyscraper sheaves

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes ! (B) The complex C(Xr , K) is preserved by f : X , Y excellent, f : X → Y quasi-projective

! ! C(Xr , f K ) ' fr C(Yr , K )

(C) X excellent, the complex C(Xr , K ) ∈ D(Xr ) is a dualizing object, i.e. C(Xr , K ) ∈ Dc (Xr ) is constructible, and the endofunctor DK = RHom(−, C(Xr , K )) of Dc (Xr ) satisﬁes DK ◦ DK = id

Consequently, K 7→ C(Xr , K ) gives rise to a map {dualizing complexes over X } → {dualizing objects in D(Xr )}

Main theorem

Theorem (J.-Xie)

(A) X excellent regular, L is an invertible OX -module, then C(Xr , L) ' Z(L).

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes (C) X excellent, the complex C(Xr , K ) ∈ D(Xr ) is a dualizing object, i.e. C(Xr , K ) ∈ Dc (Xr ) is constructible, and the endofunctor DK = RHom(−, C(Xr , K )) of Dc (Xr ) satisﬁes DK ◦ DK = id

Consequently, K 7→ C(Xr , K ) gives rise to a map {dualizing complexes over X } → {dualizing objects in D(Xr )}

Main theorem

Theorem (J.-Xie)

(A) X excellent regular, L is an invertible OX -module, then C(Xr , L) ' Z(L). ! (B) The complex C(Xr , K) is preserved by f : X , Y excellent, f : X → Y quasi-projective

! ! C(Xr , f K ) ' fr C(Yr , K )

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Consequently, K 7→ C(Xr , K ) gives rise to a map {dualizing complexes over X } → {dualizing objects in D(Xr )}

Main theorem

Theorem (J.-Xie)

(A) X excellent regular, L is an invertible OX -module, then C(Xr , L) ' Z(L). ! (B) The complex C(Xr , K) is preserved by f : X , Y excellent, f : X → Y quasi-projective

! ! C(Xr , f K ) ' fr C(Yr , K )

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Main theorem

Theorem (J.-Xie)

(A) X excellent regular, L is an invertible OX -module, then C(Xr , L) ' Z(L). ! (B) The complex C(Xr , K) is preserved by f : X , Y excellent, f : X → Y quasi-projective

! ! C(Xr , f K ) ' fr C(Yr , K )

Consequently, K 7→ C(Xr , K ) gives rise to a map {dualizing complexes over X } → {dualizing objects in D(Xr )}

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes For (B), we may assume f is a closed immersion or smooth f closed immersion: reduce to the Witt case, and prove a devissage-type result f smooth: use Poincar´eduality and (A), and compare ∗ C(Xr , f K ) with the total complex of a double complex For (C),

To show C(Xr , K ) ∈ Dc (Xr ) is constructible, use (A), (B) and noetherian induction To show biduality, reduce to show that for any f : Y → X projective with Y regular

! ! Z ' RHom(RHom(Z, f C(Xr , K )), f C(Xr , K )) using resolution of singularities and a lemma of Cisinski-D´eglise. Then apply (A) and (B)

Sketch of the proof

For (A), compare C(Xr , L) with the Cousin complex and use a result of Jacobson

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes f closed immersion: reduce to the Witt case, and prove a devissage-type result f smooth: use Poincar´eduality and (A), and compare ∗ C(Xr , f K ) with the total complex of a double complex For (C),

To show C(Xr , K ) ∈ Dc (Xr ) is constructible, use (A), (B) and noetherian induction To show biduality, reduce to show that for any f : Y → X projective with Y regular

! ! Z ' RHom(RHom(Z, f C(Xr , K )), f C(Xr , K )) using resolution of singularities and a lemma of Cisinski-D´eglise. Then apply (A) and (B)

Sketch of the proof

For (A), compare C(Xr , L) with the Cousin complex and use a result of Jacobson For (B), we may assume f is a closed immersion or smooth

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes f smooth: use Poincar´eduality and (A), and compare ∗ C(Xr , f K ) with the total complex of a double complex For (C),

To show C(Xr , K ) ∈ Dc (Xr ) is constructible, use (A), (B) and noetherian induction To show biduality, reduce to show that for any f : Y → X projective with Y regular

! ! Z ' RHom(RHom(Z, f C(Xr , K )), f C(Xr , K )) using resolution of singularities and a lemma of Cisinski-D´eglise. Then apply (A) and (B)

Sketch of the proof

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes For (C),

To show C(Xr , K ) ∈ Dc (Xr ) is constructible, use (A), (B) and noetherian induction To show biduality, reduce to show that for any f : Y → X projective with Y regular

! ! Z ' RHom(RHom(Z, f C(Xr , K )), f C(Xr , K )) using resolution of singularities and a lemma of Cisinski-D´eglise. Then apply (A) and (B)

Sketch of the proof

For (A), compare C(Xr , L) with the Cousin complex and use a result of Jacobson For (B), we may assume f is a closed immersion or smooth f closed immersion: reduce to the Witt case, and prove a devissage-type result f smooth: use Poincar´eduality and (A), and compare ∗ C(Xr , f K ) with the total complex of a double complex

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes To show C(Xr , K ) ∈ Dc (Xr ) is constructible, use (A), (B) and noetherian induction To show biduality, reduce to show that for any f : Y → X projective with Y regular

! ! Z ' RHom(RHom(Z, f C(Xr , K )), f C(Xr , K )) using resolution of singularities and a lemma of Cisinski-D´eglise. Then apply (A) and (B)

Sketch of the proof

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes To show biduality, reduce to show that for any f : Y → X projective with Y regular

! ! Z ' RHom(RHom(Z, f C(Xr , K )), f C(Xr , K )) using resolution of singularities and a lemma of Cisinski-D´eglise. Then apply (A) and (B)

Sketch of the proof

To show C(Xr , K ) ∈ Dc (Xr ) is constructible, use (A), (B) and noetherian induction

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes using resolution of singularities and a lemma of Cisinski-D´eglise. Then apply (A) and (B)

Sketch of the proof

To show C(Xr , K ) ∈ Dc (Xr ) is constructible, use (A), (B) and noetherian induction To show biduality, reduce to show that for any f : Y → X projective with Y regular

! ! Z ' RHom(RHom(Z, f C(Xr , K )), f C(Xr , K ))

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Then apply (A) and (B)

Sketch of the proof

To show C(Xr , K ) ∈ Dc (Xr ) is constructible, use (A), (B) and noetherian induction To show biduality, reduce to show that for any f : Y → X projective with Y regular

! ! Z ' RHom(RHom(Z, f C(Xr , K )), f C(Xr , K )) using resolution of singularities and a lemma of Cisinski-D´eglise.

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Sketch of the proof

To show C(Xr , K ) ∈ Dc (Xr ) is constructible, use (A), (B) and noetherian induction To show biduality, reduce to show that for any f : Y → X projective with Y regular

! ! Z ' RHom(RHom(Z, f C(Xr , K )), f C(Xr , K )) using resolution of singularities and a lemma of Cisinski-D´eglise. Then apply (A) and (B)

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Borel-Moore homology: f : X → Spec(R) quasi-projective, L invertible OX -module

−i ! BM H (C(Xr , f OSpec(R) ⊗ L)) ' H (X (R), L(R))

where L(R) is the associated real line bundle on X (R), and the right-hand side is the (topological) Borel-Moore homology If we replace R by a real closed ﬁeld, we obtain Delfs’ semi-algebraic Borel-Moore homology

Relation with topology

Hypercohomology:

−i ∞ −i −i H (C(X , I , K )) ' H (C(Xr , K )) ' H (C(Xr , K ))

The ﬁrst part is a counterpart of Jacobson’s theorem for complexes, and the second part is because C(Xr , K ) is a complex of acyclic sheaves

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes If we replace R by a real closed ﬁeld, we obtain Delfs’ semi-algebraic Borel-Moore homology

Relation with topology

Hypercohomology:

−i ∞ −i −i H (C(X , I , K )) ' H (C(Xr , K )) ' H (C(Xr , K ))

The ﬁrst part is a counterpart of Jacobson’s theorem for complexes, and the second part is because C(Xr , K ) is a complex of acyclic sheaves Borel-Moore homology: f : X → Spec(R) quasi-projective, L invertible OX -module

−i ! BM H (C(Xr , f OSpec(R) ⊗ L)) ' H (X (R), L(R))

where L(R) is the associated real line bundle on X (R), and the right-hand side is the (topological) Borel-Moore homology

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Relation with topology

Hypercohomology:

−i ∞ −i −i H (C(X , I , K )) ' H (C(Xr , K )) ' H (C(Xr , K ))

−i ! BM H (C(Xr , f OSpec(R) ⊗ L)) ' H (X (R), L(R))

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes Thank you!

Fangzhou Jin joint work with H. Xie A Gersten complex on real schemes