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Absolute Poincaré duality in étale cohomology

Published online by Cambridge University Press:  10 November 2022

Adeel A. Khan*
Affiliation:
Institute of Mathematics, Academia Sinica, No. 1, Sec. 4, Roosevelt Road, 10617Taipei, Taiwan; E-mail: adeelkhan@gate.sinica.edu.tw.

Abstract

We extend Poincaré duality in étale cohomology from smooth schemes to regular ones. This is achieved via a formalism of trace maps for local complete intersection morphisms.

Type
Algebraic and Complex Geometry
Creative Commons
Creative Common License - CCCreative Common License - BYCreative Common License - NCCreative Common License - ND
This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives licence (https://creativecommons.org/licenses/by-nc-nd/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is unaltered and is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use or in order to create a derivative work.
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Introduction

Let S be a base scheme and let $\Lambda $ denote the constant sheaf $\mathbf {Z}/n\mathbf {Z}$ for an integer n which is invertible on S. For a locally of finite typeFootnote 1 S-scheme X, define the Borel–Moore homologyFootnote 2 of X (relative to S) as cohomology with coefficients in $K_X := f^!(\Lambda )$; that is,

$$ \begin{align*} \operatorname{H}_*(X/S, \Lambda) = \operatorname{H}^{-*}(X, K_X), \end{align*} $$

where $f : X \to S$ is the structural morphism. Our starting point is the following classical result.

Theorem 1 Poincaré duality

Let X be a smooth S-scheme of relative dimension d. Then there is a canonical isomorphism

$$ \begin{align*} f^!(\Lambda) \simeq \Lambda(d)[2d] \end{align*} $$

in the derived category $\operatorname {\mathbf {D}}(X_{\mathrm {\acute {e}t}}, \Lambda )$ of étale sheaves of $\Lambda $-modules on X, where $f : X \to S$ is the structural morphism. In particular, there is a canonical isomorphism

(0.1)$$ \begin{align} \operatorname{H}_*(X/S, \Lambda) \simeq \operatorname{H}^{2d-*}(X, \Lambda(d)). \end{align} $$

See [Reference Artin, Grothendieck and VerdierSGA4, Exposé XVIII, Thm. 3.2.5] and [Reference Liu and ZhengLZ, Thm. 0.1.4] in case X is not separated of finite type. Theorem 1 is proven by constructing a formalism of tracesFootnote 3

$$ \begin{align*} \mathrm{tr}_f : f_!\Lambda(d)[2d] \to \Lambda \end{align*} $$

for flat morphisms f whose geometric fibres are of dimension $\leqslant d$ (see [Reference Artin, Grothendieck and VerdierSGA4, Exposé XVIII, Thm. 2.9]). By adjunction, the trace gives rise to a fundamental class

$$ \begin{align*} [X] \in \operatorname{H}_{2d}(X/S, \Lambda)(-d), \end{align*} $$

and the isomorphism (0.1) can be realised as cap product with $[X]$.

In this article, our goal is to prove an ‘absolute’ version of Theorem 1, where X is a regular scheme over a regular base scheme S and the morphism $f : X \to S$ is only assumed to be locally of finite type. To that end, we construct a formalism of traces for local complete intersection morphisms.

Theorem A. Let X and Y be schemes on which n is invertible. For every local complete intersection morphism $f : X \to Y$ of relative virtual dimension d, there is a canonical morphism

$$ \begin{align*} \mathrm{tr}_f : f_!\Lambda(d)[2d] \to \Lambda, \end{align*} $$

in $\operatorname {\mathbf {D}}(Y_{\mathrm {\acute {e}t}}, \Lambda )$ satisfying the following properties:

  1. (i) Functoriality. Given another local complete intersection morphism $g : Y \to Z$ of relative virtual dimension e, the composite $g\circ f$ is a local complete intersection morphism of relative virtual dimension $d+e$ and there is a commutative diagram

    in $\operatorname {\mathbf {D}}(Z_{\mathrm {\acute {e}t}}, \Lambda )$. If $f=\mathrm {id}_X$, then $\mathrm {tr}_f = \mathrm {id} : \Lambda \to \Lambda $.
  2. (ii) Transverse base change. Given any morphism $q: Y' \to Y$, form the Cartesian square

    If this square is Tor-independent (e.g., if q is flat), then g is a local complete intersection morphism of relative virtual dimension d and there is a commutative square
    in $\operatorname {\mathbf {D}}(Y^{\prime }_{\mathrm {\acute {e}t}}, \Lambda )$.
  3. (iii) Purity. Denote by

    $$ \begin{align*} \mathrm{gys}_f : \Lambda(d)[2d] \to f^!\Lambda \end{align*} $$
    the morphism in $\operatorname {\mathbf {D}}(Y_{\mathrm {\acute {e}t}}, \Lambda )$ obtained from $\mathrm {tr}_f$ by transposition. If f is smooth, or if X and Y are regular, then $\mathrm {gys}_f$ is an isomorphism.
  4. (iv) If f is smooth, then $\mathrm {tr}_f$ agrees with the trace morphism of [Reference Artin, Grothendieck and VerdierSGA4, Exposé XVIII, Thm. 2.9] (or, rather, [Reference Liu and ZhengLZ, Thm. 0.1.4] in the noncompactifiable case).

  5. (v) If f is a regular closed immersion, then $\mathrm {gys}_f$ coincides with the Gysin morphism $\mathrm {Cl}_f : \Lambda \to f^!\Lambda (-d)[-2d]$ constructed in [Reference Illusie, Laszlo and OrgogozoILO, §2.3] and [Reference FujiwaraAzu, §1] (i.e., $\mathrm {gys}_f = \mathrm {Cl}_f(d)[2d]$). In particular, it refines the local cycle class of [Reference GrothendieckCycle, §2.2].

Here local complete intersection (lci) morphisms are defined as in [Reference Berthelot, Grothendieck and IllusieSGA6, Exposé VIII, §1, Déf. 1.1]. For us the relevant description will be as follows: a morphism of schemes is lci if and only if it is locally of finite presentation and has perfect relative cotangent complex of Tor-amplitude $[-1,0]$ (under cohomological grading conventions). See, for example, [Reference Khan and RydhKRy, Prop. 2.3.14] for this equivalence.Footnote 4

Remark 1. Fix a base scheme S on which n is invertible. From Theorem A we can now read off:

  1. (i) If X is an lci S-scheme of relative virtual dimension d, then it admits a fundamental class

    $$ \begin{align*} [X] \in \operatorname{H}_{2d}(X/S, \Lambda)(-d), \end{align*} $$
    given by the morphism $\mathrm {gys}_p(-d)[-2d] : \Lambda \to p^!\Lambda (-d)[-2d]$, where $p : X \to S$ is the structural morphism.Footnote 5
  2. (ii) If X and S are regular and $d = \dim (X) - \dim (S)$, then $\mathrm {gys}_p : p^!\Lambda \simeq \Lambda (d)[2d]$ gives rise to canonical isomorphisms (‘absolute Poincaré duality’)

    $$ \begin{align*} \cap [X] : \operatorname{H}^*(X, \Lambda) \to \operatorname{H}_{2d-*}(X/S, \Lambda)(-d). \end{align*} $$
  3. (iii) For any lci morphism $f : X \to Y$ between S-schemes, $\mathrm {gys}_f : \Lambda (d)[2d] \to f^!\Lambda $ gives rise to Gysin pullbacks

    $$ \begin{align*} f^! : \operatorname{H}_*(Y/S, \Lambda) \to \operatorname{H}_{*+2d}(X/S, \Lambda)(-d). \end{align*} $$
  4. (iv) For any proper lci morphism $f : X \to Y$ of relative virtual dimension d, $\mathrm {tr}_f$ gives rise to Gysin pushforwards in cohomology

    $$ \begin{align*} f_! : \operatorname{H}^*(X, \Lambda) \to \operatorname{H}^{*-2d}(Y, \Lambda(-d)). \end{align*} $$

Remark 2. Claim (iii) in Theorem A contains in particular the statement that for any closed immersion $i : X \to Y$ between regular schemes X and Y, there is an isomorphism $i^!(\Lambda )(d)[2d] \simeq \Lambda $ in $\operatorname {\mathbf {D}}(X_{\mathrm {\acute {e}t}}, \Lambda )$. This is Grothendieck’s absolute purity conjecture, proven by Gabber (see [Reference GrothendieckSGA5, Exposé I, 3.1.4], [Reference FujiwaraAzu], [Reference Illusie, Laszlo and OrgogozoILO, Exposé XVI, Thm. 3.1.1]). However, the proof of (iii) uses this as input; that is, we do not provide a new proof of absolute purity.

1 Deformation to the normal stack

The main ingredient is deformation to the normal stack, a variant of deformation to the normal cone that makes sense not just for closed immersions.

Given an lci morphism $f : X \to Y$ of schemes, the normal stack $N_{X/Y}$ is the ‘total space’ of the $(-1)$-shifted cotangent complex $L_{X/Y}[-1]$. To make sense of this, recall that the total space construction $\mathcal {E} \mapsto \mathbf {V}_X(\mathcal {E}) = \operatorname {Spec}_X(\operatorname {\mathrm {\operatorname {Sym}}}_{\mathcal {O}_X}(\mathcal {E}))$ defines an equivalence between finite locally free sheaves and vector bundles over X. This extends to an equivalence between perfect complexes of Tor-amplitude $[0,1]$ and vector bundle stacks over X, so that we can write

$$ \begin{align*} N_{X/Y} := \mathbf{V}_X(L_{X/Y}[-1]). \end{align*} $$

See [Reference KhanKh, §1.3], [Reference Behrend and FantechiBF, §2], [Reference Artin, Grothendieck and VerdierSGA4, Exposé XVIII, §1.4]. In [Reference Behrend and FantechiBF] this is called the ‘intrinsic normal cone’ or ‘intrinsic normal sheaf’ (they agree for lci morphisms).

If f is a closed immersion, then $L_{X/Y}[-1]$ is just the conormal sheaf in degree zero so $N_{X/Y}$ is just the normal bundle. In general, $L_{X/Y}[-1]$ will typically have nonzero cohomology in degree $1$, which is why $N_{X/Y}$ will only exist as an algebraic stack. For example, if f is smooth, then $L_{X/Y}[-1]$ is the cotangent sheaf in degree $-1$, so $N_{X/Y}$ is the classifying stack $BT_{X/Y}$ of the tangent bundle (viewed as a group scheme over X under addition). If there is a global factorisation of f through a regular immersion $i : X \hookrightarrow M$ and a smooth morphism $p : M \to Y$, then $N_{X/Y}$ is isomorphic to the stack quotient

$$ \begin{align*} N_{X/Y} \simeq [N_{X/M}/i^*T_{M/Y}] \end{align*} $$

where $N_{X/M}$ is the normal bundle of i and $T_{M/Y}$ is the relative tangent bundle of p. No choices are involved in the definitions of $L_{X/Y}$ and $N_{X/Y}$; that is, they are intrinsic to f.

Deformation to the normal stack is an $\mathbf {A}^1$-family of algebraic stacks which deforms $f : X \to Y$ to the zero section $0 : X \to N_{X/Y}$.

Theorem 2. Let $f : X \to Y$ be an lci morphism. Then there exists a commutative diagram of algebraic stacks

(1.1)

where each square is Cartesian and Tor-independent.

Proof. See [Reference KreschKr, §5.1] and [Reference ManolacheMa, Thm. 2.31]. At the referee’s request we include the more ‘intrinsic’ construction using derived algebraic geometry mentioned in [Reference KhanKh, §1.4] (a more general and detailed version of the following argument will appear in [Reference Hekking, Khan and RydhHKR]).

Denote by $D_{X/Y} \to Y \times \mathbf {A}^1$ the derived Weil restriction of $f : X \to Y$ along $0 : Y \to Y \times \mathbf {A}^1$. Thus, $D_{X/Y}$ is a derived stack such that for a derived scheme T over $Y\times \mathbf {A}^1$, the T-points of $D_{X/Y}$ are given by

(1.2)$$ \begin{align} \operatorname{\mathrm{\operatorname{Hom}}}_{Y\times{\mathbf{A}^1}}(T, D_{X/Y}) \simeq \operatorname{\mathrm{\operatorname{Hom}}}_{Y}(T \operatorname*{\mathrm{\times}}^{\mathbf{R}}_{\mathbf{A}^1} 0, X) \end{align} $$

where $T \operatorname *{\mathrm {\times }}^{\mathbf {R}}_{\mathbf {A}^1} 0$ is the derived fibre over $0$. In particular, for every derived scheme $T_0$ over Y, we have natural isomorphisms

$$ \begin{align*} \operatorname{\mathrm{\operatorname{Hom}}}_{Y}(T_0, D_{X/Y} \operatorname*{\mathrm{\times}}^{\mathbf{R}}_{\mathbf{A}^1} 0) \simeq \operatorname{\mathrm{\operatorname{Hom}}}_{Y\times{\mathbf{A}^1}}(T_0, D_{X/Y})\\ \simeq \operatorname{\mathrm{\operatorname{Hom}}}_{Y}(T_0 \operatorname*{\mathrm{\times}}^{\mathbf{R}}_{\mathbf{A}^1} 0, X) \simeq \operatorname{\mathrm{\operatorname{Hom}}}_{Y}(T_0, N_{X/Y}) \end{align*} $$

where $T_0$ is regarded over $Y \times \mathbf {A}^1$ by composing with $0 : Y \to Y \times \mathbf {A}^1$ and the last isomorphism comes from the identification

$$ \begin{align*} T \operatorname*{\mathrm{\times}}^{\mathbf{R}}_{\mathbf{A}^1} 0 = T \operatorname*{\mathrm{\times}}_{0} 0 \operatorname*{\mathrm{\times}}^{\mathbf{R}}_{\mathbf{A}^1} 0 \simeq \operatorname{Spec}_T(\mathcal{O}_T \oplus \mathcal{O}_T[1])\\[-17pt] \end{align*} $$

with the trivial square-zero extension (in the derived sense) over T and the universal property of the cotangent complex in derived algebraic geometry. By the Yoneda lemma, it follows that $N_{X/Y}$ is the derived fibre of $D_{X/Y} \to \mathbf {A}^1$ over $0$. Similarly, the fibre over $\mathbf {G}_m$ is $Y \times \mathbf {G}_m$ since

$$ \begin{align*} \operatorname{\mathrm{\operatorname{Hom}}}_{Y\times\mathbf{G}_m}(T_\eta, D_{X/Y} \operatorname*{\mathrm{\times}}_{\mathbf{A}^1} \mathbf{G}_m) \simeq \operatorname{\mathrm{\operatorname{Hom}}}_{Y\times{\mathbf{A}^1}}(T_\eta, D_{X/Y})\\ \simeq \operatorname{\mathrm{\operatorname{Hom}}}_{Y}(T_\eta \operatorname*{\mathrm{\times}}_{\mathbf{G}_m} \mathbf{G}_m \operatorname*{\mathrm{\times}}_{\mathbf{A}^1} 0, X) \simeq \operatorname{\mathrm{\operatorname{Hom}}}_{Y}(\varnothing, X)\\[-17pt] \end{align*} $$

is naturally isomorphic to $\operatorname {\mathrm {\operatorname {Hom}}}_{Y\times \mathbf {G}_m}(T_\eta , Y\times \mathbf {G}_m) \simeq \{\ast \}$ for all $T_\eta $ over $Y\times \mathbf {G}_m$.

Through (1.2) we get a canonical morphism $X \times \mathbf {A}^1 \to D_{X/Y}$ corresponding to $\mathrm {id}_X \in \operatorname {\mathrm {\operatorname {Hom}}}_Y(X,X)$, which factors $f\times \mathrm {id} : X \times \mathbf {A}^1 \to Y \times \mathbf {A}^1$. The commutativity of the two upper squares in (1.1) is witnessed by two isomorphisms in the mapping $\infty $-groupoids

$$ \begin{align*} \operatorname{\mathrm{\operatorname{Hom}}}_{Y\times{\mathbf{A}^1}}(X, D_{X/Y}) &\simeq \operatorname{\mathrm{\operatorname{Hom}}}_Y(X\times 0\operatorname*{\mathrm{\times}}^{\mathbf{R}}_{\mathbf{A}^1} 0, X),\\ \operatorname{\mathrm{\operatorname{Hom}}}_{Y\times{\mathbf{A}^1}}(X\times\mathbf{G}_m, D_{X/Y}) &\simeq \operatorname{\mathrm{\operatorname{Hom}}}_Y(X\times \mathbf{G}_m\operatorname*{\mathrm{\times}}_{\mathbf{A}^1} 0, X) \simeq \{\ast\}.\\[-17pt] \end{align*} $$

Both squares are homotopy Cartesian since the lower two squares and both vertical composite rectangles are.

So far we have constructed the diagram (1.1) in the $\infty $-category of derived stacks. To show that $D_{X/Y}$ is algebraic, we can appeal to either of two algebraicity results for derived Weil restrictions. The first is [Reference Halpern-Leistner and PreygelHP, Thm. 5.1.1], which is stated for mapping stacks but applies in view of the formula for derived Weil restriction in [HP, Proposition 5.1.14]. Alternatively, in our lci situation there is a more general and easier result in [Reference Hekking, Khan and RydhHKR]. Briefly, the question of algebraicity is local, so using the local structure of lci morphisms ([Reference Khan and RydhKRy, Prop. 2.3.14]) and the fact that derived Weil restriction commutes with fibred products, it boils down to the case where $X = \mathbf {V}_Y(\mathcal {E})$ is a vector bundle over Y, whose derived Weil restriction along $0 : Y \to Y\times \mathbf {A}^1$ is the vector bundle stack

$$ \begin{align*} \mathbf{V}_{Y\times{\mathbf{A}^1}}\big(0_*(\mathcal{E}^\vee)^\vee\big).\\[-17pt] \end{align*} $$

This argument also shows that $D_{X/Y}$ is in fact a classical algebraic stack. Thus, (1.1) is a diagram in the ordinary category of algebraic stacks and homotopy Cartesianness of the squares translates to Cartesianness and Tor-independence.

2 Borel–Moore homology of stacks

Using the extension of the six operations to algebraic stacks defined in [Reference Liu and ZhengLZ],Footnote 6 we can define Borel–Moore homology of an algebraic stack $\mathcal {X}$ (locally of finite type over some base S) again by the formula

$$ \begin{align*} \operatorname{H}_k(\mathcal{X}/S, \Lambda) = \operatorname{H}^{-k}(\mathcal{X}, p^!(\Lambda)) \\[-17pt]\end{align*} $$

where $p : \mathcal {X} \to S$ is the structural morphism. Equivalently, these are the homology groups of the complexes

$$ \begin{align*} \operatorname{\mathbf{R}\Gamma}(\mathcal{X}/S, \Lambda) &:= \operatorname{\mathbf{R}\Gamma}(\mathcal{X}, p^!\Lambda)\\&\simeq \mathbf{R}\!\!\varprojlim_{(T,t)} \operatorname{\mathbf{R}\Gamma}(T, t^* p^!\Lambda) \simeq \mathbf{R}\!\!\varprojlim_{(T,t)} \operatorname{\mathbf{R}\Gamma}(T, (p\circ t)^!\Lambda)(-d_t)[-2d_t], \end{align*} $$

where the homotopy limits are over pairs $(T,t)$ where T is a scheme and $t : T \to \mathcal {X}$ is a smooth morphism of relative dimension $d_t$. (By Zariski descent, T can also be taken affine.)

It is straightforward to deduce that the localisation exact triangle extends to stacks.

Proposition 1 Localisation

If $i : \mathcal {Z} \to \mathcal {X}$ is a closed immersion with open complement $j : \mathcal {U} \to \mathcal {X}$, then we have an exact triangle

$$ \begin{align*} \operatorname{\mathbf{R}\Gamma}(\mathcal{Z}/S, \Lambda) \xrightarrow{i_*} \operatorname{\mathbf{R}\Gamma}(\mathcal{X}/S, \Lambda) \xrightarrow{j^!} \operatorname{\mathbf{R}\Gamma}(\mathcal{U}/S, \Lambda), \end{align*} $$

whence a long exact sequence

$$ \begin{align*} \cdots \to \operatorname{H}_k(\mathcal{Z}/S, \Lambda) \xrightarrow{i_*} \operatorname{H}_k(\mathcal{X}/S, \Lambda) \xrightarrow{j^!} \operatorname{H}_k(\mathcal{U}/S, \Lambda) \xrightarrow{\partial} \operatorname{H}_{k-1}(\mathcal{Z}/S, \Lambda) \to \cdots. \end{align*} $$

For example, consider the closed/open pair $(\hat {i},\hat {j})$ from (1.1). The boundary map gives rise to a specialisation map

(2.1)$$ \begin{align} &\mathrm{sp}_{X/Y} : \operatorname{\mathbf{R}\Gamma}(Y/Y, \Lambda) \xrightarrow{\mathrm{incl}} \operatorname{\mathbf{R}\Gamma}(Y/Y, \Lambda) \oplus \operatorname{\mathbf{R}\Gamma}(Y/Y, \Lambda)(1)[1]\nonumber\\ &\quad\simeq \operatorname{\mathbf{R}\Gamma}(Y\times\mathbf{G}_m/Y, \Lambda)[-1] \xrightarrow{\partial} \operatorname{\mathbf{R}\Gamma}(N_{X/Y}/Y, \Lambda). \end{align} $$

We will also need homotopy invariance for vector bundle stacks:

Proposition 2. Let $\mathcal {E}$ be the total space of a perfect complex of Tor-amplitude $[0,1]$ over X, say of virtual rank r. Then there is a canonical isomorphism in the derived category of $\Lambda $-modules

$$ \begin{align*} \operatorname{\mathbf{R}\Gamma}(\mathcal{E}/Y, \Lambda) \simeq \operatorname{\mathbf{R}\Gamma}(X/Y, \Lambda)(r)[2r]. \end{align*} $$

In particular,

$$ \begin{align*} \operatorname{H}_k(\mathcal{E}/Y, \Lambda) \simeq \operatorname{H}_{k-2r}(X/Y, \Lambda)(r) \end{align*} $$

for all $k\in \mathbf {Z}$.

Proof. Since the projection $\pi : \mathcal {E} \to X$ is smooth of relative dimension r, we have the Poincaré duality isomorphism

$$ \begin{align*} \operatorname{\mathbf{R}\Gamma}(\mathcal{E}, \pi^!f^!\Lambda) \simeq \operatorname{\mathbf{R}\Gamma}(\mathcal{E}, f^!\Lambda)(r)[2r], \end{align*} $$

where there is an implicit $\pi ^*$ on the right-hand side. This is the homotopy limit over $(T,t)$ of the Poincaré duality isomorphisms

$$ \begin{align*} \operatorname{\mathbf{R}\Gamma}(T, (\pi \circ t)^!f^!\Lambda)(-d_t)[-2d_t] \simeq \operatorname{\mathbf{R}\Gamma}(T, f^!\Lambda)(r)[2r] \end{align*} $$

for the smooth morphism $\pi \circ t : T \to \mathcal {X} \to Y$ of relative dimension $d_t+r$.Footnote 7

Secondly, there is a canonical map

$$ \begin{align*} \pi^* : \operatorname{\mathbf{R}\Gamma}(X, f^!\Lambda) \to \operatorname{\mathbf{R}\Gamma}(\mathcal{E}, f^!\Lambda) \end{align*} $$

which (as a consequence of étale descent) can be described as the homotopy limit of the maps $\pi _U^*$, where $\pi _U : \mathcal {E} \operatorname *{\mathrm {\times }}_Y U \to U$, taken over smooth morphisms $U \to Y$ with U affine. Therefore, the claim is local on Y and we may assume that the perfect complex defining $\mathcal {E}$ admits a global resolution, so that $\mathcal {E}$ is globally the stack quotient $[E^1/E^0]$ of a vector bundle morphism $E^0 \to E^1$. In this case, $\pi ^*$ factors through isomorphisms

$$ \begin{align*} \operatorname{\mathbf{R}\Gamma}(X, f^!\Lambda) \to \operatorname{\mathbf{R}\Gamma}(E^1, f^!\Lambda) \gets \operatorname{\mathbf{R}\Gamma}(\mathcal{E}, f^!\Lambda) \end{align*} $$

by homotopy invariance for the vector bundle $E^1 \to X$ (follows by descent from the case of trivial bundles, see [Reference Artin, Grothendieck and VerdierSGA4, Exposé XV, Cor. 2.2]) and for the $E^0$-torsor $E^1 \twoheadrightarrow \mathcal {E}$ (can be checked after base change to affines, over which vector bundle torsors are split).

3 Construction of the trace

We return to the situation of an lci morphism $f : X \to Y$, say of relative virtual dimension d. The $(-1)$-shifted cotangent complex $L_{X/Y}[-1]$ is perfect of Tor-amplitude $[0, 1]$ (of virtual rank $-d$), so Proposition 2 yields a canonical isomorphism

$$ \begin{align*} \operatorname{\mathbf{R}\Gamma}(N_{X/Y}/Y, \Lambda) \simeq \operatorname{\mathbf{R}\Gamma}(X/Y, \Lambda)(-d)[-2d]. \end{align*} $$

Combining this with the specialisation map (2.1) produces now a canonical map

$$ \begin{align*} \operatorname{\mathbf{R}\Gamma}(Y/Y, \Lambda) \xrightarrow{\mathrm{sp}_{X/Y}} \operatorname{\mathbf{R}\Gamma}(N_{X/Y}/Y, \Lambda) \simeq \operatorname{\mathbf{R}\Gamma}(X/Y, \Lambda)(-d)[-2d]. \end{align*} $$

In particular, the image of the unit $1 \in \operatorname {\mathbf {R}\Gamma }(Y/Y, \Lambda )$ gives rise to a canonical element (a relative fundamental class)

(3.1)$$ \begin{align} [X/Y] \in \operatorname{\mathbf{R}\Gamma}(X/Y, \Lambda)(-d)[-2d]. \end{align} $$

Our Gysin morphism is then the corresponding morphism

(3.2)$$ \begin{align} \mathrm{gys}_f : \Lambda(d)[2d] \to f^!\Lambda \end{align} $$

in $\operatorname {\mathbf {D}}(X_{\mathrm {\acute {e}t}}, \Lambda )$ and the trace morphism $\mathrm {tr}_f : f_! \Lambda (d)[2d] \to \Lambda $ is its transpose.

It will also be useful to note that these can be refined to natural transformations

(3.3)$$ \begin{align} &\mathrm{gys}_f : f^*(d)[2d] \to f^! \end{align} $$
(3.4)$$ \begin{align} &\mathrm{tr}_f : f_!f^*(d)[2d] \to \mathrm{id}. \end{align} $$

For example, $\mathrm {tr}_f$ is the composite

$$ \begin{align*} f_!f^*(-)(d)[2d] \simeq (-) \otimes f_!\Lambda(d)[2d] \xrightarrow{\mathrm{id} \otimes \mathrm{tr}_f} (-) \otimes \Lambda = \mathrm{id} \end{align*} $$

where the isomorphism is the projection formula. Note that when (3.2) is invertible, (3.3) will also be invertible on dualisable objects in $\operatorname {\mathbf {D}}(Y_{\mathrm {\acute {e}t}}, \Lambda )$ (but not necessarily on arbitrary ones).

4 Proofs of the asserted properties

We begin by noting that, in case f is a closed immersion, our construction of the Gysin morphism obviously coincides with that of [Reference Déglise, Jin and KhanDJK, §3.2], which itself agrees with Gabber’s construction [Reference Illusie, Laszlo and OrgogozoILO, Exposé XVI, §2.3] by [Reference Déglise, Jin and KhanDJK, Paragraph 4.4.3]. The base change and functoriality properties are proven exactly as in the case of closed immersions, using respectively Tor-independent base change of the deformation space $D_{X/Y}$ (see [Reference KhanKh, Thm. 1.3(ii)]) and the double deformation space associated to lci morphisms $X \to Y \to Z$,

$$ \begin{align*} D_{X/Y/Z} := D_{D_{X/Z} \operatorname*{\mathrm{\times}}_Z Y/D_{X/Z}}, \end{align*} $$

the deformation to the normal stack of the morphism $D_{X/Z} \operatorname *{\mathrm {\times }}_Z Y \to D_{X/Z}$. See the proof of [Reference Déglise, Jin and KhanDJK, Thm. 3.2.21].

Let us show that if f is smooth of relative dimension d, then $\mathrm {gys}_f$ is the Poincaré duality isomorphism $f^!(\Lambda ) \simeq \Lambda (d)[2d]$. Form the Cartesian square

The diagonal morphism $\Delta : X \to X\operatorname *{\mathrm {\times }}_Y X$ is lci of relative virtual dimension $-d$ and the natural transformation $\mathrm {tr}_\Delta : \Delta _!\Delta ^*(-d)[-2d] \to \mathrm {id}$ (3.4) gives rise to

$$ \begin{align*} \eta_f : \mathrm{id} = {\mathrm{pr}}_{2,!}\Delta_!\Delta^*{\mathrm{pr}}_1^* \xrightarrow{\mathrm{tr}_\Delta} {\mathrm{pr}}_{2,!}{\mathrm{pr}}_1^*(d)[2d] \simeq f^*f_!(d)[2d]. \end{align*} $$

We claim that $\eta _f$ and $\mathrm {tr}_f$ form the unit and counit of an adjunction $(f_!, f^*(d)[2d])$. Indeed, it is easy to check that both composites

are identity by using the functoriality of the trace for the composite ${\mathrm {pr}}_1\circ \Delta $ (respectively for the composite ${\mathrm {pr}}_2\circ \Delta $) and by base change for the trace of f. This argument shows not only that $\mathrm {gys}_f$ is an isomorphism but also that it agrees with the Poincaré duality isomorphisms of [Reference Artin, Grothendieck and VerdierSGA4, Exposé XVIII, Thm. 3.2.5] and [Reference Liu and ZhengLZ, Thm. 0.1.4] or, equivalently, that $\mathrm {tr}_f$ agrees with the trace of [Reference Artin, Grothendieck and VerdierSGA4, Exposé XVIII, Thm. 2.9] or [Reference Liu and ZhengLZ, Thm. 0.1.4]; indeed, both are counits for the same adjunction.

It remains to show that if X and Y are regular (in which case $f : X \to Y$ is automatically lci), then $\mathrm {gys}_f$ gives the isomorphism $f^!\Lambda \simeq \Lambda (d)[2d]$ asserted in Theorem A(iii). But invertibility of $\mathrm {gys}_f$ can be checked after inverse image along a Zariski cover, and by functoriality we have for any open immersion $j : U \hookrightarrow X$ a commutative diagram

where $\mathrm {gys}_j$ is invertible. Thus, we may localise on X and choose a global factorisation through a closed immersion $i : X \hookrightarrow X'$ and a smooth morphism $p : X' \to Y$. By functoriality of Gysin morphisms again and the fact that $\mathrm {gys}_p$ is an isomorphism by above, we reduce to the case of a closed immersion between regular schemes (note that $X'$ is still regular). Finally, since $\mathrm {gys}_f$ agrees with Gabber’s construction in this case, the claim now follows from absolute purity [Reference Illusie, Laszlo and OrgogozoILO, Exposé XVI, Thm. 3.1.1].

5 Remarks

Using the formalism of [Reference Liu and ZhengLZ], our construction of the traces $\mathrm {tr}_f$ immediately extends to the case where the schemes X and Y are algebraic stacks. Absolute Poincaré duality also extends to regular algebraic stacks with the same proof.

We can also allow X and Y to be derived (schemes or stacks) and $f : X \to Y$ to be any quasi-smooth morphism. Indeed, an lci morphism is precisely a quasi-smooth morphism whose source and target happen to be classical (underived). The construction of Theorem A goes through mutatis mutandis, since the deformation space $D_{X/Y}$ exists in that setting (see [Reference KhanKh, §1.4]): it is simply the Weil restriction of X along $0 : Y \hookrightarrow Y \times \mathbf {A}^1$ (in the derived sense). For a quasi-smooth morphism, the trace is a kind of categorification of Kontsevich’s virtual fundamental class (cf. (3.1)) and gives rise, for example, to the Gromov–Witten theory of smooth projective varieties in arbitrary characteristic. On the other hand, absolute Poincaré duality does not hold for derived schemes whose classical truncations are not regular.

Finally, the construction can be refined from étale cohomology to motivic cohomology. For this one can use the limit-extended motivic cohomology of algebraic stacks defined in [Reference Khan and RaviKRa, §12] as a substitute for [Reference Liu and ZhengLZ]. Note that the trace formalism for flat maps (as developed in [Reference Artin, Grothendieck and VerdierSGA4, Exposé XVIII, Thm. 2.9]) has recently been extended to motivic cohomology by Abe (see [Reference AbeAbe]). Note that in this setting our proof of absolute Poincaré duality goes through only in equicharacteristic, since absolute purity in motivic cohomology is open in general (see [Reference Déglise, Fasel, Jin and KhanDFJK, Thm. C.1] for the equicharacteristic case).

In the setting of rational and étale motivic cohomology, the results of this article appeared in a somewhat different form in the preprint [Reference KhanKh]. The present article is an attempt to give a short and self-contained account without using the language of motives or derived algebraic geometry.

In a future paper, I will explain how to use deformation to the normal stack to generalise Verdier’s specialisation functor [Reference VerdierVe]. This will be combined with a derived version of Laumon’s homogeneous Fourier transform [Reference LaumonLa2] to give an analogue of microlocalisation in the sense of Kashiwara–Schapira [Reference Kashiwara and SchapiraKS] for singular schemes.

Acknowledgments

I thank Luc Illusie, Joël Riou and the anonymous referee for helpful comments on previous versions. These results were first announced during a conference at the Euler International Mathematical Institute in St. Petersburg, Russia, in 2019; I would like to thank the organisers for the invitation to speak there. These results were obtained while I was partially supported by the SFB 1085 ‘Higher Invariants’, Universität Regensburg, and by MOST 110-2115-M-001-016-MY3.

Conflict of Interest

None.

Footnotes

1 Classically, the $!$-operations were constructed for compactifiable morphisms between quasi-compact, quasi-separated schemes (see [Reference Artin, Grothendieck and VerdierSGA4, Exposé XVII]), which by Nagata is the class of separated morphisms of finite type. In [Reference Liu and ZhengLZ], they were extended to locally of finite type morphisms.

2 If S is regular and $f : X \to S$ is separated of finite type, then $K_X$ is a dualising complex on X by [Reference Illusie, Laszlo and OrgogozoILO, Exposé XVII, Thm. 0.2]. Professor Illusie has informed me that this definition of ‘Borel–Moore homology’, as cohomology with coefficients in $K_X$, is in fact due to Grothendieck. See also [Reference LaumonLa1, §2].

3 Throughout the note, all functors are implicitly derived (wherever necessary).

4 Surprisingly, this does not appear in [Reference Berthelot, Grothendieck and IllusieSGA6] or other classical references. In fact, it seems to be a common misconception that this requires Noetherianness or that it relies on Quillen’s conjecture.

5 If S is a field and X is quasi-projective, then this is the image of the fundamental class in the Chow group $\operatorname {A}_d(X)$ by the cycle class map.

6 If we work over a base satisfying some strong hypotheses, which hold, for example, for spectra of finite or separably closed fields, then we can also use the formalism of [Reference Laszlo and OlssonLO]; cf. [Reference Liu and ZhengLZ, §6.5].

7 Note that to form this homotopy limit, we need the Poincaré duality isomorphism for schemes to be functorial in a homotopy coherent sense; however, this coherence comes for free using a standard t-structure argument; see [Reference Liu and ZhengLZ, Thm. 6.2.9, Rem. 4.1.10].

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