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First of all, dualizing sheaves are unfortunately not treated in EGA. The treatment in Hartshorne has some limitations. Perhaps some of them are related to your questions.

For pointers to more recent and complete treatments of duality, I suggest you to look at the MO question "Serre duality in families".

Let me start by your second question. The defining property of $\omega_X$, namely $$ H^n(X, \mathcal{F})^* \cong \operatorname{Hom}_{\mathcal{O}_X}(\mathcal{F},\omega_X) $$ expresses the fact that the functor $H^n(X, -)^*$ is representable. This means that there is a representing pair $(\omega_X, \int_X)$ with $\int_X \colon H^n(X, \omega_X) \to k$ a canonical isomorphism that, as you explain, indices the previously displayed isomorphism. This is what you denote "$t$" in your post. Notice that $\int_{X}$ is what corresponds to $\operatorname{id_{\omega_X}}$ in the isomorphism $$ H^n(X, \omega_X)^* \cong \operatorname{Hom}_{\mathcal{O}_X}(\omega_X,\omega_X) $$

The fact that the representing object of a functor is unique up to unique isomorphism means that there is no choice for it, once you have a concrete description of $\omega_X$ it forces a unique description of $\int_X$.

How to get such a description? On the projective space $\mathbb{P}^n_k$ one gets a characterization of $\Omega_{\mathbb{P}^n_k|k}$ as $\mathcal{O}_{\mathbb{P}^n_k}(-n-1)$ and from this characterization a canonical isomorphism: $$ \int_{\mathbb{P}^n_k} \colon H^n(\mathbb{P}^n_k, \Omega^n_{\mathbb{P}^n_k|k}) \longrightarrow k $$ Therefore $\omega_{\mathbb{P}^n_k|k} = \Omega^n_{\mathbb{P}^n_k|k}$. Once you get this description you extend it to other projective varieties and with ba little more work to proper varieties over $k$. This is explained under the assumption that $k$ is perfect in J. Lipman's blue book, Dualizing sheaves, differentials and residues on algebraic varieties, Astérisque No. 117 (1984).

Now for your fist question. The pairing $$ \operatorname{Hom}_{\mathcal{O}_X}(\mathcal{F},\omega_X) \times H^n(X, \mathcal{F}) \to k $$ that assigns to a linear map $\varphi \colon \mathcal{F} \to \omega_X$ and a cohomology class $\alpha \in H^n(X, \mathcal{F})$ the element $\int_{X} (\alpha \circ \varphi)$ where "$\circ$" denotes Yoneda composition. So the fact that $H^n(X, \omega_X)$ is 1-dimensional means, essentially, that the integral is unique up to "rescaling", so any time you take a multiple of $\alpha$ you are basically recalling it. In a perhaps more abstract point of view one may interpret $\alpha \colon \mathcal{O}_X \to \mathcal{F}[n]$ (in the derived category), therefore $\alpha \circ \varphi \colon \mathcal{O}_X \to \omega_X[n]$ is just a scalar multiple of the "volume form": the element in $H^n(X, \omega_X)$ whose image by $\int_{X}$ is $1 \in K$.

The fact that you are dealing with canonical maps suggests that one should avoid identifying a space with its dual, unless there is canonical choice of the isomorphism. This is crucial in this theory.

What is fascinating to me is that this story makes sense in any characteristic, and that there is an interesting counterpoint between the abstract aspects (dualizing sheaves, representable functors) and the more concrete ones in the sense of computations with cohomology classes, traces and differentials.

First of all, dualizing sheaves are unfortunately not treated in EGA. The treatment in Hartshorne has some limitations. Perhaps some of them are related to your questions.

For pointers to more recent and complete treatments of duality, I suggest you to look at the MO question "Serre duality in families".

Let me start by your second question. The defining property of $\omega_X$, namely $$ H^n(X, \mathcal{F})^* \cong \operatorname{Hom}_{\mathcal{O}_X}(\mathcal{F},\omega_X) $$ expresses the fact that the functor $H^n(X, -)^*$ is representable. This means that there is a representing pair $(\omega_X, \int_X)$ with $\int_X \colon H^n(X, \omega_X) \to k$ a canonical isomorphism that, as you explain, indices the previously displayed isomorphism. This is what you denote "$t$" in your post. Notice that $\int_{X}$ is what corresponds to $\operatorname{id_{\omega_X}}$ in the isomorphism $$ H^n(X, \omega_X)^* \cong \operatorname{Hom}_{\mathcal{O}_X}(\omega_X,\omega_X) $$

The fact that the representing object of a functor is unique up to unique isomorphism means that there is no choice for it, once you have a concrete description of $\omega_X$ it forces a unique description of $\int_X$.

How to get such a description? On the projective space $\mathbb{P}^n_k$ one gets a characterization of $\Omega_{\mathbb{P}^n_k|k}$ as $\mathcal{O}_{\mathbb{P}^n_k}(-n-1)$ and from this characterization a canonical isomorphism: $$ \int_{\mathbb{P}^n_k} \colon H^n(\mathbb{P}^n_k, \Omega^n_{\mathbb{P}^n_k|k}) \longrightarrow k $$ Therefore $\omega_{\mathbb{P}^n_k|k} = \Omega^n_{\mathbb{P}^n_k|k}$. Once you get this description you extend it to other projective varieties and with a little more work to proper varieties over $k$. This is explained under the assumption that $k$ is perfect in J. Lipman's blue book, Dualizing sheaves, differentials and residues on algebraic varieties, Astérisque No. 117 (1984).

Now for your fist question. The pairing $$ \operatorname{Hom}_{\mathcal{O}_X}(\mathcal{F},\omega_X) \times H^n(X, \mathcal{F}) \to k $$ that assigns to a linear map $\varphi \colon \mathcal{F} \to \omega_X$ and a cohomology class $\alpha \in H^n(X, \mathcal{F})$ the element $\int_{X} (\alpha \circ \varphi)$ where "$\circ$" denotes Yoneda composition. So the fact that $H^n(X, \omega_X)$ is 1-dimensional means, essentially, that the integral is unique up to "rescaling", so any time you take a multiple of $\alpha$ you are basically recalling it. In a perhaps more abstract point of view one may interpret $\alpha \colon \mathcal{O}_X \to \mathcal{F}[n]$ (in the derived category), therefore $\alpha \circ \varphi \colon \mathcal{O}_X \to \omega_X[n]$ is just a scalar multiple of the "volume form": the element in $H^n(X, \omega_X)$ whose image by $\int_{X}$ is $1 \in K$.

The fact that you are dealing with canonical maps suggests that one should avoid identifying a space with its dual, unless there is canonical choice of the isomorphism. This is crucial in this theory.

What is fascinating to me is that this story makes sense in any characteristic, and that there is an interesting counterpoint between the abstract aspects (dualizing sheaves, representable functors) and the more concrete ones in the sense of computations with cohomology classes, traces and differentials.

First of all, dualizing sheaves are not unfortunately treated in EGA. The treatment in Hartshorne has some limitations. Perhaps some of them are related to your questions.

For pointers to more recent and complete treatments of duality, I suggest you to look at the MO question "Serre duality in families".

Let me start by your second question. The defining property of $\omega_X$, namely $$ H^n(X, \mathcal{F})^* \cong \operatorname{Hom}_{\mathcal{O}_X}(\mathcal{F},\omega_X) $$ expresses the fact that the functor $H^n(X, -)^*$ is representable. This means that there is a representing pair $(\omega_X, \int_X)$ with $\int_X \colon H^n(X, \omega_X) \to k$ a canonical isomorphism that, as you explain, indices the previously displayed isomorphism. This is what you denote "$t$" in your post. Notice that $\int_{X}$ is what corresponds to $\operatorname{id_{\omega_X}}$ in the isomorphism $$ H^n(X, \omega_X)^* \cong \operatorname{Hom}_{\mathcal{O}_X}(\omega_X,\omega_X) $$

The fact that the representing object of a functor is unique up to unique isomorphism means that there is no choice for it, once you have a concrete description of $\omega_X$ it forces a unique description of $\int_X$.

How to get such a description? On the projective space $\mathbb{P}^n_k$ one gets a characterization of $\Omega_{\mathbb{P}^n_k|k}$ as $\mathcal{O}_{\mathbb{P}^n_k}(-n-1)$ and from this characterization a canonical isomorphism: $$ \int_{\mathbb{P}^n_k} \colon H^n(X, \Omega^n_{\mathbb{P}^n_k|k}) \longrightarrow k $$ Therefore $\omega_{\mathbb{P}^n_k|k} = \Omega^n_{\mathbb{P}^n_k|k}$. Once you get this description you extend it to other projective varieties and with ba little more work to proper varieties over $k$. This is explained under the assumption that $k$ is perfect in J. Lipman's blue book, Dualizing sheaves, differentials and residues on algebraic varieties, Astérisque No. 117 (1984).

Now for your fist question. The pairing $$ \operatorname{Hom}_{\mathcal{O}_X}(\mathcal{F},\omega_X) \times H^n(X, \mathcal{F}) \to k $$ that assigns to a linear map $\varphi \colon \mathcal{F} \to \omega_X$ and a cohomology class $\alpha \in H^n(X, \mathcal{F})$ the element $\int_{\mathbb{P}^n_k} (\alpha \circ \varphi)$ where "$\circ$" denotes Yoneda composition. So the fact that $H^n(X, \omega_X)$ is 1-dimensional means, essentially, that the integral is unique up to "rescaling", so any time you take a multiple of $\alpha$ you are basically recalling it. In a perhaps more abstract point of view one may interpret $\alpha \colon \mathcal{O}_X \to \mathcal{F}[n]$ (in the derived category), therefore $\alpha \circ \varphi \colon \mathcal{O}_X \to \omega_X[n]$ is a scalar multiple of the "volume form": the element in $H^n(X, \omega_X)$ whose image by $\int_{X}$ is $1 \in K$.

The fact that you are dealing with canonical maps suggests that one should avoid identifying a space with its dual, unless there is canonical choice of the isomorphism. This is crucial in this theory.

What is fascinating to me is that this story makes sense in any characteristic, and that there is an interesting counterpoint between the abstract aspects (dualizing sheaves, representable functors) and the more concrete ones in the sense of computations with cohomology classes, traces and differentials.

First of all, dualizing sheaves are unfortunately not treated in EGA. The treatment in Hartshorne has some limitations. Perhaps some of them are related to your questions.

For pointers to more recent and complete treatments of duality, I suggest you to look at the MO question "Serre duality in families".

Let me start by your second question. The defining property of $\omega_X$, namely $$ H^n(X, \mathcal{F})^* \cong \operatorname{Hom}_{\mathcal{O}_X}(\mathcal{F},\omega_X) $$ expresses the fact that the functor $H^n(X, -)^*$ is representable. This means that there is a representing pair $(\omega_X, \int_X)$ with $\int_X \colon H^n(X, \omega_X) \to k$ a canonical isomorphism that, as you explain, indices the previously displayed isomorphism. This is what you denote "$t$" in your post. Notice that $\int_{X}$ is what corresponds to $\operatorname{id_{\omega_X}}$ in the isomorphism $$ H^n(X, \omega_X)^* \cong \operatorname{Hom}_{\mathcal{O}_X}(\omega_X,\omega_X) $$

The fact that the representing object of a functor is unique up to unique isomorphism means that there is no choice for it, once you have a concrete description of $\omega_X$ it forces a unique description of $\int_X$.

How to get such a description? On the projective space $\mathbb{P}^n_k$ one gets a characterization of $\Omega_{\mathbb{P}^n_k|k}$ as $\mathcal{O}_{\mathbb{P}^n_k}(-n-1)$ and from this characterization a canonical isomorphism: $$ \int_{\mathbb{P}^n_k} \colon H^n(\mathbb{P}^n_k, \Omega^n_{\mathbb{P}^n_k|k}) \longrightarrow k $$ Therefore $\omega_{\mathbb{P}^n_k|k} = \Omega^n_{\mathbb{P}^n_k|k}$. Once you get this description you extend it to other projective varieties and with ba little more work to proper varieties over $k$. This is explained under the assumption that $k$ is perfect in J. Lipman's blue book, Dualizing sheaves, differentials and residues on algebraic varieties, Astérisque No. 117 (1984).

Now for your fist question. The pairing $$ \operatorname{Hom}_{\mathcal{O}_X}(\mathcal{F},\omega_X) \times H^n(X, \mathcal{F}) \to k $$ that assigns to a linear map $\varphi \colon \mathcal{F} \to \omega_X$ and a cohomology class $\alpha \in H^n(X, \mathcal{F})$ the element $\int_{X} (\alpha \circ \varphi)$ where "$\circ$" denotes Yoneda composition. So the fact that $H^n(X, \omega_X)$ is 1-dimensional means, essentially, that the integral is unique up to "rescaling", so any time you take a multiple of $\alpha$ you are basically recalling it. In a perhaps more abstract point of view one may interpret $\alpha \colon \mathcal{O}_X \to \mathcal{F}[n]$ (in the derived category), therefore $\alpha \circ \varphi \colon \mathcal{O}_X \to \omega_X[n]$ is just a scalar multiple of the "volume form": the element in $H^n(X, \omega_X)$ whose image by $\int_{X}$ is $1 \in K$.

The fact that you are dealing with canonical maps suggests that one should avoid identifying a space with its dual, unless there is canonical choice of the isomorphism. This is crucial in this theory.

What is fascinating to me is that this story makes sense in any characteristic, and that there is an interesting counterpoint between the abstract aspects (dualizing sheaves, representable functors) and the more concrete ones in the sense of computations with cohomology classes, traces and differentials.

First of all, dualizing sheaves are not unfortunately treated in EGA. The treatment in Hartshorne has some limitations. Perhaps some of them are related to your questions.

For pointers to more recent and complete treatments of duality, I suggest you to look at the MO question "Serre duality in families".

Let me start by your second question. The defining property of $\omega_X$, namely $$ H^n(X, \mathcal{F})^* \cong \operatorname{Hom}_{\mathcal{O}_X}(\mathcal{F},\omega_X) $$ expresses the fact that the functor $H^n(X, -)^*$ is representable. This means that there is a representing pair $(\omega_X, \int_X)$ with $\int_X \colon H^n(X, \omega_X) \to k$ a canonical isomorphism that, as you explain, indices the previously displayed isomorphism. This is what you denote "$t$" in your post. Notice that $\int_{X}$ is what corresponds to $\operatorname{id_{\omega_X}}$ in the isomorphism $$ H^n(X, \omega_X)^* \cong \operatorname{Hom}_{\mathcal{O}_X}(\omega_X,\omega_X) $$

The fact that the representing object of a functor is unique up to unique isomorphism means that there is no choice for it, once you have a concrete description of $\omega_X$ it forces a unique description of $\int_X$.

How to get such a description? On the projective space $\mathbb{P}^n_k$ one gets a characterization of $\Omega_{\mathbb{P}^n_k|k}$ as $\mathcal{O}_{\mathbb{P}^n_k}(-n-1)$ and from this characterization a canonical isomorphism: $$ \int_{\mathbb{P}^n_k} \colon H^n(X, \Omega^n_{\mathbb{P}^n_k|k}) \longrightarrow k $$ Therefore $\omega_{\mathbb{P}^n_k|k} = \Omega^n_{\mathbb{P}^n_k|k}$. Once you get this description you extend it to other projective varieties and with ba little more work to proper varieties over $k$. This is explained under the assumption that $k$ is perfect in J: Lipman's blue book, Dualizing sheaves, differentials and residues on algebraic varieties, Astérisque No. 117 (1984).

Now for your fist question. The pairing $$ \operatorname{Hom}_{\mathcal{O}_X}(\mathcal{F},\omega_X) \times H^n(X, \mathcal{F}) \to k $$ that assigns to a linear map $\varphi \colon \mathcal{F} \to \omega_X$ and a cohomology class $\alpha \in H^n(X, \mathcal{F})$ the element $\int_{\mathbb{P}^n_k} (\alpha \circ \varphi)$ where "$\circ$" denotes Yoneda composition. So the fact that $H^n(X, \omega_X)$ is 1-dimensional means, essentially, that the integral is unique up to "rescaling", so any time you take a multiple of $\alpha$ you are basically recalling it. In a perhaps more abstract point of view one may interpret $\alpha \colon \mathcal{O}_X \to \mathcal{F}[n]$ (in the derived category), therefore $\alpha \circ \varphi \colon \mathcal{O}_X \to \omega_X[n]$ is a scalar multiple of the "volume form": the element in $H^n(X, \omega_X)$ whose image by $\int_{X}$ is $1 \in K$.

The fact that you are dealing with canonical maps suggests that one should avoid identifying a space with its dual, unless there is canonical choice of the isomorphism. This is crucial in this theory.

What is fascinating to me is that this story makes sense in any characteristic, and that there is an interesting counterpoint between the abstract aspects (dualizing sheaves, representable functors) and the more concrete ones in the sense of computations with cohomology classes, traces and differentials.

First of all, dualizing sheaves are not unfortunately treated in EGA. The treatment in Hartshorne has some limitations. Perhaps some of them are related to your questions.

For pointers to more recent and complete treatments of duality, I suggest you to look at the MO question "Serre duality in families".

Let me start by your second question. The defining property of $\omega_X$, namely $$ H^n(X, \mathcal{F})^* \cong \operatorname{Hom}_{\mathcal{O}_X}(\mathcal{F},\omega_X) $$ expresses the fact that the functor $H^n(X, -)^*$ is representable. This means that there is a representing pair $(\omega_X, \int_X)$ with $\int_X \colon H^n(X, \omega_X) \to k$ a canonical isomorphism that, as you explain, indices the previously displayed isomorphism. This is what you denote "$t$" in your post. Notice that $\int_{X}$ is what corresponds to $\operatorname{id_{\omega_X}}$ in the isomorphism $$ H^n(X, \omega_X)^* \cong \operatorname{Hom}_{\mathcal{O}_X}(\omega_X,\omega_X) $$

The fact that the representing object of a functor is unique up to unique isomorphism means that there is no choice for it, once you have a concrete description of $\omega_X$ it forces a unique description of $\int_X$.

How to get such a description? On the projective space $\mathbb{P}^n_k$ one gets a characterization of $\Omega_{\mathbb{P}^n_k|k}$ as $\mathcal{O}_{\mathbb{P}^n_k}(-n-1)$ and from this characterization a canonical isomorphism: $$ \int_{\mathbb{P}^n_k} \colon H^n(X, \Omega^n_{\mathbb{P}^n_k|k}) \longrightarrow k $$ Therefore $\omega_{\mathbb{P}^n_k|k} = \Omega^n_{\mathbb{P}^n_k|k}$. Once you get this description you extend it to other projective varieties and with ba little more work to proper varieties over $k$. This is explained under the assumption that $k$ is perfect in J. Lipman's blue book, Dualizing sheaves, differentials and residues on algebraic varieties, Astérisque No. 117 (1984).

Now for your fist question. The pairing $$ \operatorname{Hom}_{\mathcal{O}_X}(\mathcal{F},\omega_X) \times H^n(X, \mathcal{F}) \to k $$ that assigns to a linear map $\varphi \colon \mathcal{F} \to \omega_X$ and a cohomology class $\alpha \in H^n(X, \mathcal{F})$ the element $\int_{\mathbb{P}^n_k} (\alpha \circ \varphi)$ where "$\circ$" denotes Yoneda composition. So the fact that $H^n(X, \omega_X)$ is 1-dimensional means, essentially, that the integral is unique up to "rescaling", so any time you take a multiple of $\alpha$ you are basically recalling it. In a perhaps more abstract point of view one may interpret $\alpha \colon \mathcal{O}_X \to \mathcal{F}[n]$ (in the derived category), therefore $\alpha \circ \varphi \colon \mathcal{O}_X \to \omega_X[n]$ is a scalar multiple of the "volume form": the element in $H^n(X, \omega_X)$ whose image by $\int_{X}$ is $1 \in K$.

The fact that you are dealing with canonical maps suggests that one should avoid identifying a space with its dual, unless there is canonical choice of the isomorphism. This is crucial in this theory.

What is fascinating to me is that this story makes sense in any characteristic, and that there is an interesting counterpoint between the abstract aspects (dualizing sheaves, representable functors) and the more concrete ones in the sense of computations with cohomology classes, traces and differentials.