**Example 6.1.4.** The preceding example generalizes to a quasi-compact quasi-separated scheme $X$. Let $D(\mathcal{O}_X)$ be the derived category of $\mathcal{O}_X$-modules, endowed with its closed symmetric monoidal structure via the derived tensor product. Let $K$ be an object of $D(\mathcal{O}_X)$. Then the following conditions are equivalent (see for instance [Stack, Ex. 0FPC, Lem. 0FPD, Prop. 09M1]):

(i) $K$ is strongly dualizable in $D(\mathcal{O}_X)$;

(ii) $K$ is a perfect complex of $\mathcal{O}_X$-modules;

(iii) $K$ is compact.

**Definition 6.1.5.** Consider the above setting and let $M$ be a strongly dualizable object with dual $(M^{\vee}, \pi, \delta)$. We define the trace of an endomorphism $f: M \to M$ as the following element of the ring $\mathrm{End}_c(1)$:

$$ \mathrm{tr}_M(f) : 1 \xrightarrow{f'} M \otimes M^{\vee} \xrightarrow{\pi} 1 $$

where $f'$ is obtained by adjunction from $f$. This defines a map:

$$ \mathrm{tr}_M : \mathrm{End}_c(M) \to \mathrm{End}_c(1). $$

*Remark 6.1.6 (exercise).* One can derive the following formulas for the traces just defined:

$$ \bullet \mathrm{tr}_M(f \circ g) = \mathrm{tr}_M(g \circ f). $$

$$ \bullet \mathrm{tr}_{M \otimes N}(f \otimes g) = \mathrm{tr}_M(f) \otimes \mathrm{tr}_N(g). $$

$$ \bullet \mathrm{tr}_M(\lambda.f) = \lambda.\mathrm{tr}_M(f), \lambda \in \mathrm{End}_c(1). $$

**Example 6.1.7.** Consider the setting of Example 6.1.3. Obviously, $1 = A$ and $\mathrm{End}_A(A) = A$, as a ring.

Given a strongly dualizable $A$-module $M$, the trace map $\mathrm{Tr}_M : \mathrm{End}_A(M) \to A$ defined above coincides with the classical notion in number theory. In particular, when $M$ admits a (global) $A$-basis $(f_1, \dots, f_n)$, through the induced isomorphism $\mathrm{End}_A(M) \simeq \mathcal{M}_n(A)$, the map $\mathrm{Tr}_M$ is the usual trace map of matrices.

**6.1.8.** Consider again the abstract situation of a symmetric monoidal category $(\mathcal{C}, \otimes, 1)$, and a strongly dualizable object $M$ of $\mathcal{C}$ with dual $(M^{\vee}, \pi, \delta)$.

We remark that the trace map $\mathrm{tr}_M$ is induced by an internal trace map:

$$ \underline{\mathrm{tr}}_M : \underline{\mathrm{Hom}}(M, M) \simeq M^{\vee} \otimes M \simeq M \otimes M^{\vee} \xrightarrow{\pi} 1. $$

This means that $\mathrm{tr}_M = \underline{\mathrm{Hom}}(\mathrm{tr}_M, 1)$.

Assume now that $M$ admits a product map $\mu: M \otimes M \to M$ (for example, $M$ is a monoid, [Mac71, VII.3]). Then one gets a $\mu$-trace morphism:

$$ \mathrm{Tr}_M^{\mu}: M \xrightarrow{\mu'} \underline{\mathrm{Hom}}(M, M) \xrightarrow{\mathrm{tr}_M} 1 $$

As a particular case, one gets back the following classical definition from algebra:

**Definition 6.1.9.** Let $A$ be a ring and $B$ be a commutative $A$-algebra which is projective and finitely generated as an $A$-module.
Then $B$ is a strongly dualizable $A$-module and we define the trace morphism

$$ \mathrm{Tr}_{B/A} : B \to A $$

as the $A$-linear map associated above with respect to the multiplication map $B \otimes_A B \to B$.

Concretely, the trace of an element $b \in B$ is the trace of the endomorphism $\gamma_b$ such that $\gamma_b(x) = b \cdot x$. It can be computed locally by choosing bases of the $A$-module $B$ and using the trace of matrices. The local definitions then glue using faithfully flat descent.

Let us recall the following classical result.

**Proposition 6.1.10.** Let $B/A$ be a finitely generated projective ring extension. Then the following conditions are equivalent:

(i) $B/A$ is étale.

(ii) For every prime ideal $\mathfrak{q}$ in $B$, $\mathfrak{p}$ being its inverse image in $A$, $L = B/\mathfrak{q}$, $K = A/\mathfrak{p}$, one has: $\mathrm{Tr}_{L/K} \neq 0$.

(iii) The bilinear form $B \otimes_A B \to A$, $x \otimes y \mapsto \mathrm{Tr}_{B/A}(xy)$ is non-degenerate — i.e. induces by adjunction an isomorphism $B \to B^\vee$ of $A$-modules.

In particular, a finite field extension $L/K$ is separable if and only if $\mathrm{Tr}_{L/K} \neq 0$.

Therefore, the above notion of trace map is inadequate for inseparable field extensions, since it yields the zero map. This justifies the use of a finer duality theory, which was introduced by Grothendieck. We recall the abstract setting to end up this section.

**Definition 6.1.11.** Let $(C, \otimes, \mathbb{1})$ be a closed symmetric monoidal category. Let $K$ be an object of $C$, and write $\mathrm{D}_K(M) = \mathrm{Hom}(M, K)$. The evaluation map $M \otimes \mathrm{Hom}(M, K)$ induces by adjunction a canonical map $\omega_M : M \to \mathrm{D}_K \circ \mathrm{D}_K(M)$.

One says that $K$ is *dualizing* if the natural transformation $\omega : \mathrm{Id}_C \to \mathrm{D}_K \circ \mathrm{D}_K$ is an isomorphism.

In words, $\mathrm{D}_K(M)$ is called the weak $K$-dual of $M$, and the definition asks that any object $M$ is isomorphic to its double weak $K$-dual, by the canonical map $\omega_M$.

**Remark 6.1.12.** (1) In the original definition of a dualizing complex ([Har66, Definition p. 258]), one had additional assumptions (finite injective dimension and lower boundedness). One has progressively dismissed this kind of assumptions, in order to extend Grothendieck’s theory to other context (torsion étale sheaves [ILO14, VII, 6.1.1], constructible pro-étale sheaves [BS15, 6.7.20], $D$-modules, motivic homotopy [Ayo07, 2.3.73] and motivic complexes [CD19, 4.4.24]).

(2) The seminal definition of Grothendieck has been extended in several directions. We refer the reader to [BD13] for further developments.
**6.1.13.** Consider a dualizing object $K$ of $\mathcal{C}$ as in the above definition. Then one has the following basic properties:

(1) The map $1 \to \operatorname{Hom}(K, K)$, deduced from $\operatorname{Id}_K$ by adjunction, is an isomorphism.

(2) For any object $M, N$ in $\mathcal{C}$, one has an isomorphism:

$$D_K(M \otimes D_K(N)) \simeq \operatorname{Hom}(M, N).$$

(3) An object $K'$ of $\mathcal{C}$ is dualizing if and only if there exists a $\otimes$-invertible object $L$ such that $K' = K \otimes L$.

Moreover, in this case, one has $L \simeq D_{K'}(K) = \operatorname{Hom}(K, K')$.

(4) If $M$ is a strongly dualizable object in $\mathcal{C}$ with dual $M^{\vee}$, then $D_K(M) \simeq M^{\vee} \otimes K$.

(5) The dualizing object $K$ is strongly dualizable if and only if it is invertible.

We give the arguments for completeness:

(1) use the isomorphism $1 \simeq D_K(D_K(1))$ and $\operatorname{Hom}(1, -) \simeq \operatorname{Id}_\mathcal{C}$ (as a right adjoint to the functor $(1 \otimes -)$).

(2) Use the sequence of isomorphisms:

$$D_K(M \otimes D_K(N)) = \operatorname{Hom}(M \otimes D_K(N), K) \simeq \operatorname{Hom}(M, D_K D_K(N)) \simeq \operatorname{Hom}(M, N).$$

(3) $\Leftarrow$: use $\operatorname{Hom}(M, N \otimes L) \simeq \operatorname{Hom}(M \otimes L^{-1}, N) \simeq \operatorname{Hom}(M, N) \otimes L$.

$\Rightarrow$: one reduces to proving that $M \mapsto M \otimes D_K(K')$ is an equivalence. It suffices to apply the equivalence $D_K$, point (2) to reduce to the fact that $D_{K'}$ is an equivalence.

(4) Follows from definitions.

(5) Follows from point (4).

**Example 6.1.14.** In the category of locally compact abelian groups, the unit circle $T = \mathbb{R}/\mathbb{Z}$ is a dualizing object. We will see more examples in the next section.

## 6.2. Grothendieck differential trace map and duality.

**6.2.1.** Let $f : X \to S$ be a morphism of quasi-compact and quasi-separated schemes. We have an adjoint pair:

$$\mathbf{L}f^* : \mathrm{D}(\mathcal{O}_S) \hookrightarrow \mathrm{D}(\mathcal{O}_X) : \mathbf{R}f_*.$$

We say that a complex $K$ of $\mathrm{D}(\mathcal{O}_X)$ is quasi-coherent if its cohomology sheaves are quasi-coherent. We let $\mathrm{D}_{\mathrm{qc}}(X)$ be the full sub-category of $\mathrm{D}(\mathcal{O}_X)$ made of quasi-coherent complexes.<fnref n="51" /> Both functors $\mathbf{L}f^*$ and $\mathbf{R}f_*$ preserves quasi-coherent complexes (see [Stack, Lemmas 08DW, 08D5]). The following theorem is one of the essential part of Grothendieck’s duality theory.

<footnote n="51">⁵¹For Noetherian schemes, this category is equivalent to the derived category of the abelian category of quasi-coherent sheaves: see [Stack, Prop. 09T4].</footnote>
**Theorem 6.2.2.** Consider the above assumptions.

(1) The functor $\mathbf{R}f_* : \mathrm{D}_{\mathrm{qc}}(X) \to \mathrm{D}_{\mathrm{qc}}(S)$ admits a right adjoint. When $f$ is in addition proper, we denote this right adjoint by $f^! : \mathrm{D}_{\mathrm{qc}}(S) \to \mathrm{D}_{\mathrm{qc}}(X)$.

(2) If $f$ is proper smoothable lci with canonical sheaf $\omega_{X/S}$ and relative dimension $d$ (see Definition 4.1.5), there exists a canonical isomorphism:

$$ \mathfrak{p}_f : \omega_{X/S}[d] \to f^!(\mathcal{O}_S) $$

with the notation of the preceding point. It is called the purity isomorphism associated to $f$.

*Proof.* The first statement is Neeman's theorem (see [Nee96], [Stack, 48.3.1]).

We could not find an appropriate reference for point (2). However, it follows from the results of [Har66], with some complements brought by many years of improvement. Let us summarize the arguments from the literature. We now erase the symbols $\mathbf{L}$ and $\mathbf{R}$ for readability.

First, we fix a factorization of $f$ as $X \xrightarrow{i} P \xrightarrow{p} S$ such that $i$ is a regular closed immersion and $p$ is a smooth morphism. We will consider the functors $i^!$ and $p^!$ restricted to $\mathrm{D}_{\mathrm{qc}}^+(X)$, which is legitimate thanks to [Stack, Lemma 0A9I].

According to [Stack, Lemma 0A76], there exists a canonical isomorphism (uniquely characterized by the adjoint property) of functors $i^! \simeq i^b$ where $i^b : \mathrm{D}_{\mathrm{qc}}^+(P) \to \mathrm{D}_{\mathrm{qc}}^+(X)$ is the functor defined in [Har66, III. §6]. As $i$ is a regular closed immersion, there exists a canonical isomorphism of functors by [Har66, III. Cor. 7.3]:

$$ \mathfrak{p}_i : \omega_{X/P}[-m] = \omega_{X/P}[-m] \otimes i^*(\mathcal{O}_P) \simeq i^b(\mathcal{O}_P) \simeq i^!(\mathcal{O}_P) $$

where $m$ is the codimension of $i$.

As $p$ is proper and smooth, there exists a canonical isomorphism of functors, as defined in [Nee20, 4.1.6]:

$$ \mathfrak{p}_p : \omega_{P/S}[n] = (\omega_{P/S}[n] \otimes p^*(\mathcal{O}_S)) \simeq p^!(\mathcal{O}_S) $$

where $n$ is the dimension of $p$. In particular, the complex $p^!(\mathcal{O}_S)$ is perfect.

We now build the desired map as the following composition:

$$ \begin{aligned} \omega_{X/S}[d] &\simeq \omega_{X/P}[-m] \otimes i^*(\omega_{P/S}[n]) \xrightarrow{\mathfrak{p}_i \otimes i^*(\mathfrak{p}_p)} i^!(\mathcal{O}_P) \otimes i^*(p^!(\mathcal{O}_S)) \\ &\simeq i^!(\mathcal{O}_P \otimes p^!(\mathcal{O}_S)) \simeq f^!(\mathcal{O}_S) \end{aligned} $$

where the isomorphism $(*)$ exists as $i$ is lci ([Stack, Lemma 0A9T]).

To justify the word “canonical”, one needs to prove that the above isomorphism does not depend on the choice of the factorization. The steps for this fact are well-known. The main points may be found in [Har66, §III]: 2.2, 6.2, 6.4, 8.1 (see also the proof of Th. 3.3.2 [DJK21]). □

**Remark 6.2.3.** In fact, the purity isomorphism can be generalized in the coherent context as follows. For any bounded quasi-coherent complex $K$, one defines an
isomorphism by the following composite maps:

$$f'(K) \xrightarrow{(*)} f'(K) \otimes f'(\mathcal{O}_S) \xrightarrow{\mathfrak{p}_f} f'(K) \otimes \omega_{X/S}[d],$$

where the isomorphism (*) follows from [Har66, III, 8.8].

Note this is specific to the coherent case. The analogue isomorphism does not hold in other six functors formalism such as the étale $\ell$-adic or motivic one, unless further restrictions to $f$ are assumed (e.g., $f$ is smooth, or a nil-immersion).

**Definition 6.2.4.** Assuming $f$ is proper smoothable lci of relative dimension $d$, we will denote by

$$\mathrm{Tr}_{f}^{\omega}: \mathbf{R}f_{*}(\omega_{X/S})[d] \to \mathcal{O}_{S}$$

the map in $D_{qc}(S)$ obtained by adjunction from $\mathfrak{p}_f$ and call it the differential trace map associated with $f$.

When $f$ is finite lci, the source and target of the map $\mathrm{Tr}_{f}^{\omega}$ are concentrated in degree 0. Therefore, it gives, by taking cohomology in degree 0, a canonical morphism of coherent $\mathcal{O}_S$-modules, and we will consider it as such. Taking global sections, we will also consider the induced trace map:

$$\mathrm{Tr}_{X/S}^{\omega}: \Gamma(X, \omega_{X/S}) \to \Gamma(S, \mathcal{O}_S).$$

Finally, if $X/S$ is the spectrum of a finite lci ring extension $B/A$ the above map will be denoted by:

$$\mathrm{Tr}_{B/A}^{\omega}: \omega_{B/A} \to A.$$

*Remark 6.2.5.* The above definition clarifies several properties of the differential trace map. It is functorial with respect to flat base change in $X$, and compatible with disjoint sums in $X$.

This means in particular that if we have an isomorphism of finite lci $A$-algebras:

$$\Theta: B \xrightarrow{\sim} \prod_{i \in I} B_i,$$

the following diagram is commutative:

<figure><img src="image_4.png" /></figure>

*Remark 6.2.6.* The compatibility with composition of the trace map is more involved. Consider a factorization $X \xrightarrow{g} Y \xrightarrow{h} S$ of $f$ by proper smoothable and lci morphisms, of respective dimensions $n$ and $m$. First recall that there exists a canonical isomorphism (see 4.1.7):

$$\psi: \omega_{X/S} \simeq \omega_{X/Y} \otimes (f^*\omega_{Y/S}).$$
The compatibility with composition of the differential trace map is expressed by the following commutative diagram (again we discard the symbols **R** and **L** for readability):

<figure><img src="image_5.png" /></figure>

The second vertical map is obtained by the so-called projection formula, which holds here either because $g$ is proper or even simply as $\omega_{Y/S}$ is an invertible sheaf. This statement follows from [Har66, III, 10.5] (see also [Con00, Th. 3.4.1]).

In the affine case, $X = \text{Spec}(C)$, $Y = \text{Spec}(B)$, $S = \text{Spec}(A)$, $f$, $g$ and $h$ being finite, the diagram takes the following simpler form:

<figure><img src="image_6.png" /></figure>

**6.2.7.** *A particular case of duality.* For any quasi-coherent complex $K$ over $X$, and any proper smoothable lci morphism $f: X \to S$, the adjunction property of the pair $(\mathbf{R}f_*, f')$ gives an isomorphism:

$$ \operatorname{Hom}_{\mathcal{D}(\mathcal{O}_X)}(K, \omega_{X/S}[d]) \xrightarrow{\sim} \operatorname{Hom}_{\mathcal{D}(\mathcal{O}_S)}(\mathbf{R}f_*(K), \mathcal{O}_S), \\ (u : K \to \omega_{X/S}[d]) \mapsto (\operatorname{Tr}^\omega_{X/S} \circ \mathbf{R}f_*(u)). $$

In the case of a finite lci ring extension $B/A$, and for a $B$-module $M$, this boils down to an isomorphism:

$$ \operatorname{Hom}_B(M, \omega_{B/A}) \xrightarrow{\sim} \operatorname{Hom}_A(M, A), \\ (u : M \to \omega_{B/A}) \mapsto (\operatorname{Tr}^\omega_{B/A} \circ u). $$

Taking $M = B$, we get an isomorphism between $A$-linear forms on $B$ and elements of $\omega_{B/A}$:

$$ \omega_{B/A} \xrightarrow{\sim} \operatorname{Hom}_A(B, A), \\ w \mapsto (\psi_w : \lambda \mapsto \operatorname{Tr}^\omega_{B/A}(\lambda \cdot w)). $$

**Example 6.2.8.** We end-up this section with a few classical examples of duality in the case of coherent sheaves.

(1) A concrete case of duality is obtained when $S$ is the spectrum of any field $k$, $X$ a proper smoothable lci $k$-scheme. In that case, the first isomorphism of 6.2.7
applied to $K[n]$ where $K$ is a bounded complex with coherent cohomology, gives
an isomorphism of $k$-vector spaces:

$$
\mathrm{Ext}_{\mathcal{O}_X}^{d-n}(K, \omega_{X/k}) \xrightarrow{\sim} \mathrm{H}^n(X, K)^*
$$

The trace map $\mathrm{Tr}_{X/k}^{\omega}$ induces what I will call the *Gysin map* associated with $f$:

$$
f_! : H^d(X, \omega_{X/k}) \to k
$$

and the above duality isomorphism is induced by the *Poincaré duality* (perfect) pairing:

$$
\operatorname{Ext}_{\mathcal{O}_X}^{d-n}(K, \omega_{X/k}) \otimes \operatorname{H}^n(X, K) \to k \\
(x, y) \mapsto f_!(x.y).
$$

(2) In the case of a proper smoothable lci morphism $f : X \to S$, one can interpret Grothendieck duality, for $K = \mathcal{O}_X$, by saying that $\mathbf{R}f_*(\mathcal{O}_X)$ is strongly dualizable (Definition 6.1.2, Example 6.1.4) with dual given by $\mathbf{R}f_*(\omega_{X/S})[d]$. One of the pairings coming from this duality is a relative version of the Poincaré duality pairing:

$$
\mathbf{R}f_*(\mathcal{O}_X) \otimes \mathbf{R}f_*(\omega_{X/S})[d] \to \mathbf{R}f_*(\omega_{X/S})[d] \xrightarrow{\mathrm{Tr}_{X/S}^{\omega}} \mathcal{O}_S
$$

where the first map comes from the fact $\mathbf{R}f_*$ is weakly monoidal (as the right adjoint of a monoidal functor).

(3) Of course, the theory can be considerably generalized - but we will only need the case of finite field extensions! Indeed, Grothendieck's main objective was to obtain duality for any proper morphism $f : X \to \mathrm{Spec}(k)$. He achieved this by constructing a dualizing complex $K_X = \omega_{X/k}$, which is no longer an invertible sheaf in general (except if $X$ is *Gorenstein*, see [Har66, V, 9.3]). We refer the reader to [Har66, Con00] or [LH09, Chap. 1].

6.3. Grothendieck and Scheja-Storch Residues.

6.3.1. We consider a commutative diagram of schemes:

<figure><img src="image_7.png" /></figure>

such that $f$ is finite lci, $p$ is smooth of relative dimension $n$, and $i$ is a closed immersion with ideal sheaf $\mathcal{I} \subset \mathcal{O}_p$. The hypothesis imply that $i$ is regular of codimension $n$.

Recall that we can associate to the above commutative diagram a canonical
isomorphism (see paragraph 4.1.7):

$$
\Theta : \omega_{X/S} \simeq \omega_{X/P} \otimes_{\mathcal{O}_X} i^* \omega_{P/S} \simeq (\Lambda^n(\mathcal{I}/\mathcal{I}^2))^\vee \otimes_{\mathcal{O}_P} \Omega_{P/S}^n.
$$
**Definition 6.3.2.** Consider a global differential $n$-form $w \in \Gamma(P, \Omega_{P/S}^n)$ and a global regular parametrization $(f_1, \dots, f_n)$ of $\mathcal{I}$. We define the Grothendieck residue (symbol) of $w$ at $(f_1, \dots, f_n)$ as the element of $\Gamma(S, \mathcal{O}_S)$:

$$ \mathrm{Res}_{P/S} \left[ \begin{matrix} w \\ f_1 \dots f_n \end{matrix} \right] = \mathrm{Tr}_{X/S}^{\omega} \left( (\bar{f}_1 \wedge \dots \wedge \bar{f}_n)^* \otimes i^*(w) \right) $$

where we have used the differential trace map of Definition 6.2.4 and we have considered the element $(\bar{f}_1 \wedge \dots \wedge \bar{f}_n)^* \otimes i^*(w)$ as an element of $\omega_{B/A}$ via the isomorphism $\Theta$.

This definition agrees with that of [Har66, III, §9] and that of [Con00, Appendix A, (A.1.4)].$^{52}$

**6.3.3.** We now explain a method of Scheja and Storch to compute the above residue, and therefore the differential trace map. Our reference is [Kun08, §8].

We will restrict to the affine case. Let $B$ be a finite projective $A$-algebra: in other words, $B$ is strongly dualizable as an $A$-module, see Example 6.1.3.

In what follows, we will use a set of indeterminates $\underline{t} = (t_1, \dots, t_n)$, and put $A[\underline{t}] = A[t_1, \dots, t_n]$ for brevity. We assume that $B$ is a complete intersection $A$-algebra: there exist elements $\alpha = (\alpha_1, \dots, \alpha_n) \in B^n$ which generate $B$ as an $A$-algebra and the kernel of the surjective map

$$ A[\underline{t}] \to B, t_i \mapsto \alpha_i $$

admits a regular parametrization $I = (f_1, \dots, f_n)$ for polynomials $f_i \in A[\underline{t}]$. We say that $f = (f_1, \dots, f_n)$ is a *presentation* of the lci $A$-algebra $B$.

Putting $S = \mathrm{Spec}(A)$, $X = \mathrm{Spec}(B)$, $P = A_S^n = \mathrm{Spec}(A[\underline{t}])$, we are therefore in the conditions of paragraph 6.3.1. Consider the composite map$^{53}$:

$$ \sigma : B[\underline{t}] \to B \otimes_A B \xrightarrow{\mu} B $$

where $\mu$ is the multiplication map, and the first arrow is the natural surjection coming from the identification $B \otimes_A B = B \otimes_A A[\underline{t}]/I = B[\underline{t}]/I$. We consider the ideals:

$$ J = \mathrm{Ker}(\mu) \subset B \otimes_A B, $$

$$ K = \mathrm{Ker}(\sigma) \subset B[\underline{t}]. $$

$$ (\bar{f}_1 \wedge \dots \wedge \bar{f}_n)^* = \bar{f}_n^* \wedge \dots \wedge \bar{f}_1^* = (-1)^{n(n-1)/2} \bar{f}_1^* \wedge \dots \wedge \bar{f}_n^* $$

<footnote>$^{52}$The sign in the latter can be explained as:</footnote>

<footnote>$^{53}$Geometrically, this map corresponds to the graph $\gamma_i : X \to X \times_S P$ of the closed immersion $i : X \to P$. As $i$ is regular and $P/S$ is smooth, $\gamma_i$ is regular. Algebraically, it is just the map evaluating $t_i$ at $\alpha_i$.</footnote>
Moreover, K admits a regular parametrization, $K = (t_1 - \alpha_1, \dots, t_n - \alpha_n)$ and one obtains the identification $J = K/I$ as obviously $I \subset K$ as ideals of $B[t]$. Therefore, there exist polynomials $c_{ij} \in B[t]$ such that

$$ \forall i \in [1, n], f_i = \sum_{j=1}^{n} c_{ij} (t_j - \alpha_j). $$

Finally, the element

$$ \Delta_f = \det (c_{ij}(\alpha)_{1 \le i, j \le n}) \in B \otimes_A B $$

is independent of the chosen polynomials $c_{ij}$ (see [Kun08, Lemma 4.10]).

**Definition 6.3.4.** Consider the above notation: $B/A$ is a finite projective complete intersection and $f = (f_1, \dots, f_n) \in A[t]^n$ is a fixed presentation of $B/A$. Then the element $\Delta_f \in B \otimes_A B$ is called the *Bézoutian* associated with the presentation $f$ of $B/A$.

*Remark 6.3.5.* This definition is of course an extension of the classical Bézoutian (or rather the determinant of the Bézout matrix) arising from Euler and Bézout elimination theory, which corresponds to the case where $A = k$ is a field and $n = 2$.

**6.3.6.** Consider again the setting of 6.3.3. We put $B^* = \operatorname{Hom}_A(B, A)$, which is the (canonical) dual of the strongly dualizable $A$-module $B$ (see Definition 6.1.2 and what follows).$^{54}$ As $B$ is strongly dualizable, the canonical map:

$$ \Phi : B \otimes_A B \to \operatorname{Hom}_A(B^*, B), \\ b \otimes b' \mapsto (\varphi \mapsto \varphi(b).b') $$

is an isomorphism. The following lemma is now a formality (see [Kun08, 8.13]; beware to translate the notation: $I$ (resp. $\omega_{B/A}$) in loc. cit. is what we denote by $J$ (resp. $B^*$) here.)

**Lemma 6.3.7.** *Consider the above notation. Recall that $J = \operatorname{Ker}(B \otimes_A B \xrightarrow{\mu} B)$, seen as an ideal of $B \otimes_A B$. Then $\Phi$ induces an isomorphism:*

$$ \operatorname{Ann}(J) \xrightarrow{\sim} \operatorname{Hom}_A(B^*, B). $$

With this lemma in hand, we see that there exists a unique $A$-linear map $\tau_f : B \to A$, equivalently $\tau_f \in B^*$ such that:

$$ (6.3.7.a) \qquad \Phi(\Delta_f)(\tau_f) = 1_B $$

**Definition 6.3.8.** Consider the above notation, as in Definition 6.3.4. We call the $A$-linear map $\tau_f : B \to A$ the *Scheja-Storch trace map* associated with the presentation $f$ of $B/A$.

<footnote>$^{54}$Following the usage, we identify the set of morphisms $\operatorname{Hom}_A$ with the internal Hom-functor in the category of $A$-modules.</footnote>
**Example 6.3.9.** We consider the *monogenic* case:

$$B = A[\alpha] = A[t]/(f)$$

where $f$ is a monic polynomial in one variable $t$:

$$f(t) = a_0 + \cdots + a_{n-1} t^{n-1} + t^n.$$

Thus $B$ is a free $A$-module with basis $1, \alpha, \dots, \alpha^{n-1}$. Then one can compute $\Delta_f$ explicitly and one finds that

$$\tau_f = (\alpha^{n-1})^*, \alpha^i \mapsto \delta_{n-1}^i.$$

*Remark 6.3.10.* In the above example, the map $\tau_f$ does depend on the chosen generator $\alpha$ of $B/A$, or more explicitly on the chosen presentation of $B/A$. Therefore, it is sometimes customary to put:

$$\tau_{B/A}^{\alpha} = \tau_f.$$

In view of [Tat52, §1, (2)], corresponding to the case where $B/A$ is an inseparable extension field, the map $\tau_f$ is sometimes called the *Tate trace map* (cf. [Kun08]).

**Theorem 6.3.11.** Consider the assumption of the above definition.

Then the $A$-linear map $\tau_f: B \to A$ is not $B$-torsion, and in fact is a $B$-basis of $\text{Hom}_A(B, A)$.

In other words, the symmetric bilinear form

$$\varphi_f : B \otimes_A B \to A, b \otimes b' \mapsto \tau_f(bb')$$

is non-degenerate: the associated map

$$B \to B^* = \text{Hom}_A(B, A), b \mapsto b \cdot \tau_f = \varphi_f(b, -)$$

is an isomorphism.

*Proof.* Using the definitions of 6.3.3, one obtains that $\text{Ann}(J)$ is a principal ideal generated by the Bézoutian $\Delta_f$; a result attributed to Wiebe, see [Kun08, Cor. 4.12]. According to the previous lemma, $\text{Ann}(J) \simeq \text{Hom}_A(B^*, B)$ is also an invertible $B$-module. So $\Delta_f$ is a $B$-basis of the $B$-module $\text{Ann}(J)$. Relation (6.3.7.a) then implies that $\tau_f$ is a $B$-basis of $B^*$ as expected. The other assertions are formal consequences of this fact. $\square$

We are now ready to state the link between the concrete construction of Scheja and Storch and the theory of Grothendieck residue symbols (Definition 6.3.2).

**Proposition 6.3.12.** Recall the situation of the previous theorem and definition:

* $B$ is a complete intersection, finite and projective $A$-algebra

* $f$ is a presentation of $B/A$: $f = (f_1, \dots, f_n)$ is regular sequence of elements of $R = A[t_1, \dots, t_n]$, $I = (f_1, \dots, f_n)$ and $B = R/I$.
Recall that we have a canonical isomorphism (see Example 4.1.9)

$$ \Theta : \omega_{B/A} \simeq \Lambda^n(I/I^2)^{\vee} \otimes_R \Omega_{R/A}^n. $$

Then, for any $\lambda \in R$, with image $\bar{\lambda}$ in $B = R/I$, we get:

$$ \mathrm{Res}_{R/A} \begin{bmatrix} \lambda \cdot dt_1 \wedge \dots \wedge dt_n \\ f_1 \cdots f_n \end{bmatrix} = \tau_f(\bar{\lambda}). $$

In other words, if we let $w = (\bar{f}_1 \wedge \dots \wedge \bar{f}_n)^* \otimes i^*(dt_1 \wedge \dots \wedge dt_n)$ seen as an element of $\omega_{B/A}$ via the isomorphism $\Theta$, for any $b \in B$, one gets:

$$ (6.3.12.a) \qquad \mathrm{Tr}_{B/A}^{\omega}(b.w) = \tau_f(b). $$

Or equivalently, with the notation of 6.2.7: $\psi_w = \tau_f$.

*Proof.* In the case where $A = k$ is a field (the only case we will need!), this is [Kun08, Prop. 8.32]. In general, one can reduce to this case by base change: we need to compare two trace maps associated with $f: X = \mathrm{Spec}(B) \to \mathrm{Spec}(A) = S$ which is finite and syntomic. Both traces $\mathrm{Tr}_{B/A}^{\omega}$ and $\tau_f$ are compatible under arbitrary base change (as syntomic finite morphisms are stable under base change) so that we can reduce to residue fields of $S$.

Alternatively, both definition of residues, via respectively Grothendieck and Scheja-Storch methods, satisfy the properties (R1)-(R10) of [Har66, III, §9] (see respectively [Con00, Appendix A] and [HL79, Hop83]). This uniquely characterizes the residue symbol. $\square$

In view of Example 6.3.9, we deduce the following comparison of the Grothendieck differential trace map and the Tate trace map (Remark 6.3.10).

**Corollary 6.3.13.** Suppose $B/A$ is a monogenic extension ring, of the form $B = A[t]/I$ where $I = (f)$ for a monic polynomial $f \in A[t]$. We identify $\omega_{B/A}$ with the $B$-module $(I/I^2)^* \otimes_{A[t]} \Omega_{A[t]/A}$ (via the isomorphism $\Theta$ of Example 4.1.9).

Then for any $b \in B$, one gets:

$$ (6.3.13.a) \qquad \mathrm{Tr}_{B/A}^{\omega}(b.\bar{f}^* \otimes dt) = \tau_{B/A}^{\alpha}(b) $$

with the notation of Remark 6.3.10.

**Corollary 6.3.14.** Let $B$ be a finite étale $A$-algebra. Then $\omega_{B/A} = B$ and the following diagram commutes:

<figure><img src="image_8.png" /></figure>

where $\mathrm{Tr}_{B/A}$ is the “usual” trace map (Definition 6.1.9).
Proof. This is asserted without proof in [Har66, Remark p. 187]. As both trace maps are compatible with composition, the proof reduces to the case where $B/A$ is monogenic, $B = A[\alpha] = A[t]/(f)$, $f$ being a monic polynomial in one variable $t$ such that $f'(\alpha) \in B^\times$. Note that under the identification

$$ \Theta : B = \omega_{B/A} \simeq (I/I^2)^* \otimes \Omega_{A[t]/A}^1 $$

one has $\Theta^{-1}(\bar{f} \otimes dt) = f'(\alpha)^{-1}$ (as explained in Example 4.1.10, separable case). Example 6.3.9 shows that $\tau_f = (\alpha^{n-1})^*$ where $n$ is the degree of $f$ in $t$.

Therefore, the relation of the corollary follows from the previous corollary and the “Euler formula” (see for instance [NS03, Prop. 1]):

$$ \mathrm{Tr}_{B/A} (f'(\alpha)^{-1}\lambda) = (\alpha^{n-1})^*(\lambda). $$

<figure><img src="image_9.png" /></figure>

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