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Functoriality of categorical traces in geometric setting

Chapter XIII: Geometric traces in sheaf theory

13.5 Functoriality of categorical traces in geometric setting

Example 13.4.4. Consider the case𝑍 =π‘ŠΓ—π‘‹for some𝑔: π‘Š β†’π‘Œ. Then we have a split augmented simplicial object

HH(π‘‹Γ—π‘Œ 𝑋 ,(π‘Š ×𝑋) Γ—π‘ŒΓ—π‘Œ (𝑋 ×𝑋))β€’β†’π‘ŠΓ—π‘Œ 𝑋

with the last map given by the action of π‘‹Γ—π‘Œ 𝑋 onπ‘ŠΓ—π‘Œ 𝑋. In this case, we reduce to the tautological equivalence

D (π‘ŠΓ—π‘‹ 𝑋) βŠ—D (π‘‹Γ—π‘Œπ‘‹) D (π‘‹Γ—π‘Œ 𝑋) βˆ’β†’ D (∼ π‘Š Γ—π‘Œ 𝑋).

and let

𝑀 =𝑋 Γ—π‘Œ π‘Š Γ—π‘Œβ€² 𝑋′.

We would like to know when D (𝑀) is dualizable as a D (𝑋

1)-D (𝑋′

1)-bimodule, with the dual of given byD (𝑁)where𝑁 = π‘‹β€²Γ—π‘Œβ€² π‘ŠΓ—π‘Œ 𝑋. We will assume that

β€’ π‘Š β†’π‘Œβ€²andπ‘Š β†’π‘ŠΓ—π‘Œβ€²π‘Š belong toCβ„Žπ‘œπ‘Ÿ 𝑖 𝑧;

β€’ π‘Š β†’π‘Œ andπ‘Š β†’π‘Š Γ—π‘Œ π‘Š belongs toC𝑣 π‘’π‘Ÿ 𝑑.

Then we have the morphism 𝑒

geo : π‘Œβ€² d π‘Š Γ—π‘Œ π‘Š in Corr(C)𝑣 π‘’π‘Ÿ 𝑑;β„Žπ‘œπ‘Ÿ 𝑖 𝑧 given by π‘ŠΓ—π‘Œπ‘Š β†π‘Š β†’π‘Œβ€²and𝑒

geo:π‘Š Γ—π‘Œβ€²π‘Š dπ‘Œ given byπ‘Œ β†π‘Š β†’π‘Š Γ—π‘Œβ€²π‘Š. They induce

D (𝑋′

1)

D (id×𝑒geoΓ—id)

D (𝑁) βŠ—D (𝑋

1) D (𝑀) //D (π‘‹β€²Γ—π‘Œβ€² π‘ŠΓ—π‘Œ π‘Š Γ—π‘Œβ€² 𝑋′).

(13.17)

and

D (𝑀) βŠ—D (𝑋′

1) D (𝑁) //

𝑒 **

D (𝑋 Γ—π‘Œ π‘Š Γ—π‘Œβ€² π‘ŠΓ—π‘Œ 𝑋)

D (id×𝑒

geoΓ—id)

D (𝑋

1).

(13.18)

Here𝑒is defined to be the composition.

Lemma 13.5.2. If the vertical morphism in (13.17) factors through a D (𝑋′

1)- bimodule morphism

𝑒 :D (𝑋′

1) β†’ D (𝑁) βŠ—D (𝑋

1) D (𝑀) (e.g. ifD (𝑁) βŠ—D (𝑋

1)D (𝑀) β†’ D (π‘‹β€²Γ—π‘Œβ€²π‘ŠΓ—π‘Œπ‘ŠΓ—π‘Œβ€² 𝑋′)is an equivalence), then 𝑒and𝑒from(13.18)give the duality datum ofD (𝑀)as a D (𝑋

1)-D (𝑋′

1)-module.

Proof. Write𝑅= π‘‹β€²Γ—π‘Œβ€²π‘ŠΓ—π‘Œπ‘ŠΓ—π‘Œβ€²π‘‹β€²and𝑆= π‘‹Γ—π‘Œπ‘ŠΓ—π‘Œβ€²π‘ŠΓ—π‘Œ 𝑋for simplicity.

Note that (using (13.16)) we have the following commutative diagram

D (𝑋′

1) βŠ—D (𝑋′

1)D (𝑁)

π‘’βŠ—id

D (𝑋′

1) βŠ—D (𝑋′

1)D (𝑁)

//D (𝑁)=D (π‘‹β€²Γ—π‘Œβ€²π‘Œβ€²Γ—π‘Œβ€²π‘ŠΓ—π‘Œ 𝑋)

D (id×𝑒

geoΓ—idΓ—id)

D (𝑅) βŠ—D (𝑋′

1)D (𝑁)

++D (𝑁) βŠ—D (𝑋

1)D (𝑀) βŠ—D (𝑋′

1)D (𝑁)

33

idβŠ—π‘’

++

D (π‘‹β€²Γ—π‘Œβ€²π‘ŠΓ—π‘Œπ‘ŠΓ—π‘Œβ€²π‘ŠΓ—π‘Œπ‘‹)

D (idΓ—id×𝑒geoΓ—id)

D (𝑁) βŠ—D (𝑋1)D (𝑆)

33

D (𝑁) βŠ—D (𝑋)D (𝑋) D (𝑁) βŠ—D (𝑋

1)D (𝑋

1) //D (𝑁)=D (π‘‹β€²Γ—π‘Œβ€²π‘Œβ€²Γ—π‘Œβ€²π‘ŠΓ—π‘Œ 𝑋)

The composition of functors in the right column is isomorphic to the identity functor by the base change isomorphism (11.2). It follows that (12.11) in the current setting holds. The same reasoning implies that (12.10) in the current setting also holds. β–‘ In practice, the assumption in Lemma 13.5.2 may not hold. But under some certain technical assumptions, we can still understand the duality datum.

Lemma 13.5.3. Suppose the sheaf theory D satisfies assumptions as in Corol- lary 13.4.2, and let 𝑓 : 𝑋 β†’π‘Œ and 𝑓′ : 𝑋′ β†’π‘Œβ€²be as in Corollary 13.4.2 (so in particularD (𝑋

1)andD (𝑋′

1)are rigid). SupposeD (𝑀) βŠ— D (𝑇) β†’ D (𝑀×𝑇)is an equivalence for every𝑇 ∈ C. Then

𝑒: D (𝑋′

1) β†’ D (π‘‹β€²Γ—π‘Œβ€² π‘ŠΓ—π‘Œ π‘ŠΓ—π‘Œβ€² 𝑋′) βˆ’β†’ D (𝑁) βŠ—D (𝑋

1) D (𝑀),

where the last functor is the right adjoint of the vertical morphism in(13.17)and𝑒 from(13.18)form a duality datum.

Proof. As in the proof of Lemma 13.5.2, it is enough to establish the following commutative diagram

D (𝑋′

1) βŠ—D (𝑋′

1) D (𝑁)

π‘’βŠ—id

D (𝑋′

1) βŠ—D (𝑋′

1)D (𝑁)

//D (𝑁)

D (𝑅) βŠ—D (𝑋′

1)D (𝑁)

(βˆ—βˆ—)

ss ++

D (𝑁) βŠ—D (𝑋

1)D (𝑀) βŠ—D (𝑋′

1)D (𝑁)

idβŠ—π‘’

++

D (π‘‹β€²Γ—π‘Œβ€²π‘ŠΓ—π‘Œπ‘ŠΓ—π‘Œβ€²π‘ŠΓ—π‘Œπ‘‹)

(βˆ—βˆ—)

ss

D (𝑁) βŠ—D (𝑋

1)D (𝑆)

D (𝑁) βŠ—D (𝑋)D (𝑋) D (𝑁) βŠ—D (𝑋

1) D (𝑋

1) //D (𝑁),

where the arrows labelled by (βˆ—βˆ—)are right adjoint of the corresponding arrows in the diagram from the proof of Lemma 13.5.2.

Only the commutativity of the middle parallelogram and the lower right trapzoid requires justification. For the middle parallelogram, first consider the commutative diagram

D (𝑁 ×𝑀×𝑁) //

D (𝑅×𝑁)

D (𝑁×𝑆) //D (π‘‹β€²Γ—π‘Œβ€²π‘Š Γ—π‘Œ π‘Š Γ—π‘Œβ€² π‘ŠΓ—π‘Œ 𝑋).

with horizontal morphisms are induced by the correspondenceπ‘Œ ← 𝑋 β†’ 𝑋×𝑋and vertical morphisms induced byπ‘Œβ€²β† 𝑋′→ 𝑋′×𝑋′. As 𝑓 , 𝑓′ ∈ Cπ‘π‘Ÿ π‘œ 𝑝andΔ𝑋,Δ𝑋′ ∈ Cπ‘ π‘š, the above diagram is right adjointable by the same proof as in Lemma 13.3.2.

Under our assumption thatD (𝑀) βŠ— D (𝑇) β†’ D (𝑀×𝑇)is an equivalence for every 𝑇 ∈ C, we may replaceD (𝑁×𝑀×𝑁)byD (𝑁) βŠ— D (𝑀) βŠ— D (𝑁),D (𝑅×𝑁) by D (𝑅) βŠ— D (𝑁) andD (𝑁 ×𝑆) by D (𝑁) βŠ— D (𝑆). Then as D (𝑋′

1) andD (𝑋

1) are rigid, using Lemma 12.3.1, we obtain the commutativity of the parallelogram.

Similarly, the lower right trapzoid is commutative by Lemma 13.5.1 and that D (𝑁) βŠ—D (𝑋

1) D (𝑆) D (𝑁) βŠ—geo

D (𝑋

1) D (𝑆) :=Trgeo(D (𝑋

1),D (𝑁×𝑆)). β–‘ Now suppose we are given aD (𝑋

1)-D (𝑋′

1)-bimodule homomorphism 𝛼:D (𝑀) βŠ—D (𝑋′

1) D (𝑄′) β†’ D (𝑄) βŠ—D (𝑋

1) D (𝑀).

Then as explained above, under certain dualizability assumption ofD (𝑀), there is a functor

Tr(D (𝑀), 𝛼) : Tr(D (𝑋′

1),D (𝑄′)) β†’Tr(D (𝑋

1),D (𝑄)). On the other hand, suppose we are given a correspondence

𝛼geo :π‘Š Γ—π‘Œβ€² 𝑍′d π‘Γ—π‘Œ π‘Š in Corr(C/π‘ŒΓ—π‘Œβ€²)𝑣 π‘’π‘Ÿ 𝑑;β„Žπ‘œπ‘Ÿ 𝑖 𝑧. One can form the correspondence

𝐢(π‘Š , 𝛼

geo) :π‘Œβ€²Γ—π‘Œβ€²Γ—π‘Œβ€² 𝑍′dπ‘Œ Γ—π‘ŒΓ—π‘Œ 𝑍 given by the composition

π‘Œβ€²Γ—π‘Œβ€²Γ—π‘Œβ€² 𝑍′

𝑒geoΓ—id

d (π‘ŠΓ—π‘Œ π‘Š)π‘Œβ€²Γ—π‘Œβ€²π‘β€²π‘Œ Γ—π‘ŒΓ—π‘Œ (π‘Š Γ—π‘Œβ€² π‘β€²Γ—π‘Œβ€²π‘Š)

id×𝛼geoΓ—id

d π‘Œ Γ—π‘ŒΓ—π‘Œ (π‘Γ—π‘Œ π‘ŠΓ—π‘Œβ€²π‘Š) idΓ—idd×𝑒geoπ‘Œ Γ—π‘ŒΓ—π‘Œ 𝑍 . (13.19)

The sheaf theoryD then induces a functor D (𝐢(π‘Š , 𝛼

geo)) :D (π‘Œβ€²Γ—π‘Œβ€²Γ—π‘Œβ€² 𝑍′) β†’ D (π‘Œ Γ—π‘ŒΓ—π‘Œ 𝑍).

We would like to relate Tr(D (𝑀), 𝛼) with the above functor under certain assump- tions.

Assumptions 13.5.4. (I) We assume that the following diagram is commutative D (𝑀) βŠ—D (𝑋′

1) D (𝑄′) //

𝛼

D (𝑋 Γ—π‘Œ π‘Š Γ—π‘Œβ€² π‘β€²Γ—π‘Œβ€² 𝑋′)

D (id×𝛼geoΓ—id)

D (𝑄) βŠ—D (𝑋

1) D (𝑀) //D (𝑋 Γ—π‘Œ 𝑍 Γ—π‘Œ π‘ŠΓ—π‘Œβ€² 𝑋′).

(13.20)

(II) We assume that the following diagram is commutative D (𝑀) βŠ—D (𝑋′

1) D (𝑄′)

𝛼

D (𝑋 Γ—π‘Œ π‘Š Γ—π‘Œβ€² π‘β€²Γ—π‘Œβ€² 𝑋′)

oo

D (id×𝛼geoΓ—id)

D (𝑄) βŠ—D (𝑋

1) D (𝑀)oo D (𝑋 Γ—π‘Œ 𝑍 Γ—π‘Œ π‘ŠΓ—π‘Œβ€² 𝑋′).

(13.21)

where the horizontal arrows are right adjoint of the natural ones.

Remark 13.5.5. Note that Assumptions 13.5.4 holds in the case 𝑍′ = π‘Œβ€² with 𝑔′

1 = πœ™π‘Œβ€² : π‘Œβ€² β†’ π‘Œ, 𝑍 = π‘Œ with 𝑔

2 = πœ™π‘Œ : π‘Œ β†’ π‘Œ and there is πœ™π‘Š : π‘Š β†’ π‘Š compatible withπœ™π‘Œβ€² andπœ™π‘Œ.

Proposition 13.5.6. Under the assumption in Lemma 13.5.2 and Assumptions 13.5.4 I, then the following diagram is commutative

Tr(D (𝑋′

1),D (𝑄′)) Tr(D (𝑀),𝛼) //

Tr(D (𝑋

1),D (𝑄))

D (π‘Œβ€²Γ—π‘Œβ€²Γ—π‘Œβ€² 𝑍′) D (𝐢(π‘Š ,𝛼geo)) //D (π‘Œ Γ—π‘ŒΓ—π‘Œ 𝑍).

Under the assumption in Lemma 13.5.3 and Assumptions 13.5.4 II, the following diagram is commutative

Tr(D (𝑋′

1),D (𝑄′)) Tr(D (𝑀),𝛼) //

Tr(D (𝑋

1),D (𝑄)) D (π‘Œβ€²Γ—π‘Œβ€²Γ—π‘Œβ€² 𝑍′) D (

𝐢(π‘Š ,𝛼

geo))

//D (π‘Œ Γ—π‘ŒΓ—π‘Œ 𝑍)

OO

Proof. The first case follows from the following commutative diagram

D (𝑋′

1) βŠ—D (𝑋′ 1)βŠ—D (𝑋′

1)rev D (𝑄′)

π‘’βŠ—1

//D (π‘Œβ€²Γ—π‘Œβ€²Γ—π‘Œβ€²π‘β€²)

D (𝑒geoΓ—id)

(D (𝑁) βŠ—D (𝑋

1)D (𝑀)) βŠ—D (𝑋′

1)βŠ—D (𝑋′

1)revD (𝑄′)

//D ( (π‘ŠΓ—π‘Œπ‘Š) Γ—π‘Œβ€²Γ—π‘Œβ€²π‘β€²)

D (𝑋

1) βŠ—D (𝑋1)βŠ—D (𝑋1)rev (D (𝑀) βŠ—D (𝑋′

1)D (𝑄′) βŠ—D (𝑋′

1)D (𝑁)) //

1βŠ—π›ΌβŠ—1

D (π‘ŒΓ—π‘ŒΓ—π‘Œ (π‘ŠΓ—π‘Œβ€²π‘β€²Γ—π‘Œβ€²π‘Š))

D (id×𝛼geoΓ—id)

D (𝑋

1) βŠ—D (𝑋1)βŠ—D (𝑋

1)rev (D (𝑄) βŠ—D (𝑋1)D (𝑀) βŠ—D (𝑋′

1)D (𝑁)) //

1βŠ—1βŠ—π‘’

D (π‘ŒΓ—π‘ŒΓ—π‘Œ (π‘Γ—π‘Œπ‘ŠΓ—π‘Œβ€²π‘Š))

D)idΓ—id×𝑒geo)

D (𝑋

1) βŠ—D (𝑋

1)βŠ—D (𝑋1)rev D (𝑄) //D (π‘Œ Γ—π‘ŒΓ—π‘Œ 𝑍)

(13.22) The second case follows from a similar diagram

D (𝑋′

1) βŠ—D (𝑋′ 1)βŠ—D (𝑋′

1)rev D (𝑄′)

π‘’βŠ—1

//D (π‘Œβ€²Γ—π‘Œβ€²Γ—π‘Œβ€²π‘β€²)

D (𝑒geoΓ—id)

(D (𝑁) βŠ—D (𝑋1)D (𝑀)) βŠ—D (𝑋′

1)βŠ—D (𝑋′

1)revD (𝑄′)

D ( (π‘ŠΓ—π‘Œπ‘Š) Γ—π‘Œβ€²Γ—π‘Œβ€²π‘β€²)

oo

D (𝑋

1) βŠ—D (𝑋

1)βŠ—D (𝑋1)rev (D (𝑀) βŠ—D (𝑋′

1)D (𝑄′) βŠ—D (𝑋′

1)D (𝑁))

1βŠ—π›ΌβŠ—1

D (π‘ŒΓ—π‘ŒΓ—π‘Œ (π‘ŠΓ—π‘Œβ€²π‘β€²Γ—π‘Œβ€²π‘Š))

oo

D (id×𝛼geoΓ—id)

D (𝑋

1) βŠ—D (𝑋1)βŠ—D (𝑋

1)rev (D (𝑄) βŠ—D (𝑋1)D (𝑀) βŠ—D (𝑋′

1)D (𝑁))

1βŠ—1βŠ—π‘’

D (π‘ŒΓ—π‘ŒΓ—π‘Œ (π‘Γ—π‘Œπ‘ŠΓ—π‘Œβ€²π‘Š))

oo

D (idΓ—id×𝑒 geo)

D (𝑋

1) βŠ—D (𝑋1)βŠ—D (𝑋1)rev D (𝑄)oo D (π‘ŒΓ—π‘ŒΓ—π‘Œ 𝑍),

(13.23) where the horizontal left arrows are obtained by the corresponding horizontal right arrows in (13.22) by passing to the right adjoint. We need to justify the commuta- tivity of this diagram. First we have the commutativity of the following diagram

D (𝑋′

1) βŠ—D (𝑋′ 1)βŠ—D (𝑋′

1)revD (𝑄′)

π‘’βŠ—1

rr //D (π‘Œβ€²Γ—π‘Œβ€²Γ—π‘Œβ€²π‘β€²)

D (𝑒 geoΓ—id) (D (𝑁) βŠ—D (𝑋

1)D (𝑀)) βŠ—D (𝑋′ 1)βŠ—D (𝑋′

1)revD (𝑄′) D (𝑅) βŠ—D (𝑋′ 1)βŠ—D (𝑋′

1)revD (𝑄′)

oo D ( (π‘ŠΓ—π‘Œπ‘Š) Γ—π‘Œβ€²Γ—π‘Œβ€²π‘β€²)oo

Indeed, the left triangle is commutative as we are in the case as in Lemma 13.5.3, and the right square is commutative as the natural functor D (𝑅) βŠ—D (𝑋′

1)βŠ—D (𝑋′

1)rev

D (𝑄′) β†’ D ( (π‘Š Γ—π‘Œ π‘Š) Γ—π‘Œβ€²Γ—π‘Œβ€² 𝑍′) is fully faithful by Corollary 13.4.2. This justifies the commutativity of the top square in (13.23).

For the commutativity of the third square in (13.23), by our assuption it is enough

to show that

D (𝑋

1) βŠ—D (𝑋

1)βŠ—D (𝑋1)rev(D (π‘‹Γ—π‘Œπ‘ŠΓ—π‘Œβ€²π‘β€²Γ—π‘Œβ€²π‘‹β€²) βŠ—D (𝑋′ 1)D (𝑁))

D (π‘ŒΓ—π‘ŒΓ—π‘Œ (π‘ŠΓ—π‘Œβ€²π‘β€²Γ—π‘Œβ€²π‘Š))

oo

D (𝑋

1) βŠ—D (𝑋

1)βŠ—D (𝑋1)rev(D (π‘‹Γ—π‘Œπ‘Γ—π‘Œπ‘ŠΓ—π‘Œβ€²π‘‹β€²) βŠ—D (𝑋′

1)D (𝑁))oo D (π‘ŒΓ—π‘ŒΓ—π‘Œ(π‘Γ—π‘Œπ‘ŠΓ—π‘Œβ€²π‘Š))

is commutative. Under the assumption that D (𝑀) βŠ— D (𝑇) β†’ D (𝑀 ×𝑇) is an equivalence for any𝑇 ∈ C, we can use Lemma 13.5.1 twice to conclude.

Similar argument also shows that the last square in (13.23) is commutative. β–‘ Example 13.5.7. Takeπ‘Œβ€²=ptandπ‘Š = 𝑋. Then the naive class identifies with the elementLπœ™(𝑓)†(𝑒𝑋) ofD (Lπœ™(π‘Œ))considered as a functor fromModΞ›.

Example 13.5.8. If π‘Œ β†’ pt is in Cβ„Žπ‘œπ‘Ÿ 𝑖 𝑧 we can take 𝑍 = π‘Œ and 𝑔 = idπ‘Œ. Then (Ξ”π‘Œ/π‘Œ)† =id. Then the geometric class inD (Lπœ™(π‘Œ))is equivalent toPTr

geo(𝑒L

πœ™(π‘Œ)).

BIBLIOGRAPHY

Beauville, Arnaud and Yves Laszlo (1995). β€œUn lemme de descente”. In:Comptes rendus de l’AcadΓ©mie des sciences. SΓ©rie 1, MathΓ©matique320.3, pp. 335–340.

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