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This is basically a question of functoriality for base change of CM morphisms.

EDIT: $\text{ }$ Brian Conrad sent me an email explaining the that this is indeed true, and follows from his book. I'll explain this in my case a little later. First I will state the question.

Suppose that $f : X \to V$ is an equidimensional (dimension $d$) finite type (reduced, if it helps) Cohen-Macaulay morphism (flat with Cohen-Macaulay fibers). I'm also happy to assume that $V$ is integral, excellent and has a dualizing complex. Additionally suppose that we have $f' : X' \to V$ another equidimensional (dimension $d$) finite-type (reduced) Cohen-Macaulay morphism that factors through $f$ as below.

$$f' : X' \xrightarrow{\phi} X \to V.$$

Further suppose that $\phi$ is finite (although the question could be asked more generally for proper $\phi$, I'll phrase it for finite $\phi$). If it helps at any point, please feel free to assume that $f$ and $f'$ are Gorenstein morphisms.

Recall that $\omega_{f}[d] = f^! \mathcal{O}_V$ and that by Brian Conrad's book [LINK: Google books][1] we know that both $\omega_f$ and $\omega_{f'}$ are compatible with base change.

I'd like to conclude that the following natural map is also compatible with base change:

$$\phi_* \omega_{f'} \cong R \mathcal{H}om_{O_X}(\phi_* O_{X'}, \omega_f ) \cong \mathcal{H}om_{O_X}(\phi_* O_{X'}, \omega_f ) \to \omega_f$$.
The map can be interpreted as evaluation at 1.

In other words, I'd like to know that the trace map of $\phi$ is compatible with base change. Furthermore, it would be even good enough to prove this in the $f, f'$ Gorenstein morphisms case.

One way to do this would be as follows. If $g : T \to V$ is any other morphism and $f_T : X_T \to T$ and $f_T': X_T' \to T$ are the base changes and $g_X : X_T \to X$ is the projection, is it true that the natural map (denoted [*] below) $$g_X^* \phi_* \omega_{f'} \cong g_X^* \mathcal{H}om_{O_X}(\phi_* O_{X'}, \omega_f ) \to \mathcal{H}om_{O_{X_T}}(\phi_* O_{X_T'}, \omega_{f_T} ) \cong (\phi_T)_* \omega_{f_T'}$$ between abstractly isomorphic sheaves is an isomorphism?

Edit: Brian Conrad pointed out to me that this is already a special case of what is in his Theorem 3.6.1 (assuming I understand everything right). I'll try to write down an a precise answer tomorrow.

This is basically a question of functoriality for base change of CM morphisms.

EDIT: $\text{ }$ Brian Conrad sent me an email explaining the that this is indeed true, and follows from his book. I'll explain this in my case a little later. First I will state the question.

Suppose that $f : X \to V$ is an equidimensional (dimension $d$) finite type (reduced, if it helps) Cohen-Macaulay morphism (flat with Cohen-Macaulay fibers). I'm also happy to assume that $V$ is integral, excellent and has a dualizing complex. Additionally suppose that we have $f' : X' \to V$ another equidimensional (dimension $d$) finite-type (reduced) Cohen-Macaulay morphism that factors through $f$ as below.

$$f' : X' \xrightarrow{\phi} X \to V.$$

Further suppose that $\phi$ is finite (although the question could be asked more generally for proper $\phi$, I'll phrase it for finite $\phi$). If it helps at any point, please feel free to assume that $f$ and $f'$ are Gorenstein morphisms.

Recall that $\omega_{f}[d] = f^! \mathcal{O}_V$ and that by Brian Conrad's book [LINK: Google books][1] we know that both $\omega_f$ and $\omega_{f'}$ are compatible with base change.

I'd like to conclude that the following natural map is also compatible with base change:

$$\phi_* \omega_{f'} \cong R \mathcal{H}om_{O_X}(\phi_* O_{X'}, \omega_f ) \cong \mathcal{H}om_{O_X}(\phi_* O_{X'}, \omega_f ) \to \omega_f$$.
The map can be interpreted as evaluation at 1.

In other words, I'd like to know that the trace map of $\phi$ is compatible with base change. Furthermore, it would be even good enough to prove this in the $f, f'$ Gorenstein morphisms case.

One way to do this would be as follows. If $g : T \to V$ is any other morphism and $f_T : X_T \to T$ and $f_T': X_T' \to T$ are the base changes and $g_X : X_T \to X$ is the projection, is it true that the natural map (denoted [*] below) $$g_X^* \phi_* \omega_{f'} \cong g_X^* \mathcal{H}om_{O_X}(\phi_* O_{X'}, \omega_f ) \to \mathcal{H}om_{O_{X_T}}(\phi_* O_{X_T'}, \omega_{f_T} ) \cong (\phi_T)_* \omega_{f_T'}$$ between abstractly isomorphic sheaves is an isomorphism?

Edit: Brian Conrad pointed out to me that this is already a special case of what is in his Theorem 3.6.1 (assuming I understand everything right). Essentially the point is that in my case everything is finite type, which makes it all much easier.

This is basically a question of functoriality for base change of CM morphisms.

EDIT: Brian Conrad sent me an email explaining the that this is indeed true, and follows from his book. I'll explain this after I state the question.

Suppose that $f : X \to V$ is an equidimensional (dimension $d$) finite type (reduced, if it helps) Cohen-Macaulay morphism (flat with Cohen-Macaulay fibers). I'm also happy to assume that $V$ is integral, excellent and has a dualizing complex. Additionally suppose that we have $f' : X' \to V$ another equidimensional (dimension $d$) finite-type (reduced) Cohen-Macaulay morphism that factors through $f$ as below.

$$f' : X' \xrightarrow{\phi} X \to V.$$

Further suppose that $\phi$ is finite (although the question could be asked more generally for proper $\phi$, I'll phrase it for finite $\phi$). If it helps at any point, please feel free to assume that $f$ and $f'$ are Gorenstein morphisms.

Recall that $\omega_{f}[d] = f^! \mathcal{O}_V$ and that by Brian Conrad's book [LINK: Google books][1] we know that both $\omega_f$ and $\omega_{f'}$ are compatible with base change.

I'd like to conclude that the following natural map is also compatible with base change:

$$\phi_* \omega_{f'} \cong R \mathcal{H}om_{O_X}(\phi_* O_{X'}, \omega_f ) \cong \mathcal{H}om_{O_X}(\phi_* O_{X'}, \omega_f ) \to \omega_f$$.
The map can be interpreted as evaluation at 1.

In other words, I'd like to know that the trace map of $\phi$ is compatible with base change. Furthermore, it would be even good enough to prove this in the $f, f'$ Gorenstein morphisms case.

One way to do this would be as follows. If $g : T \to V$ is any other morphism and $f_T : X_T \to T$ and $f_T': X_T' \to T$ are the base changes and $g_X : X_T \to X$ is the projection, is it true that the natural map (denoted [*] below) $$g_X^* \phi_* \omega_{f'} \cong g_X^* \mathcal{H}om_{O_X}(\phi_* O_{X'}, \omega_f ) \to \mathcal{H}om_{O_{X_T}}(\phi_* O_{X_T'}, \omega_{f_T} ) \cong (\phi_T)_* \omega_{f_T'}$$ between abstractly isomorphic sheaves is an isomorphism?

Edit: Brian Conrad pointed out to me that this is already a special case of what is in his Theorem 3.6.1 (assuming I understand everything right). I'll try to write down an a precise answer tomorrow.

This is basically a question of functoriality for base change of CM morphisms.

EDIT: $\text{ }$ Brian Conrad sent me an email explaining the that this is indeed true, and follows from his book. I'll explain this in my case a little later. First I will state the question.

Suppose that $f : X \to V$ is an equidimensional (dimension $d$) finite type (reduced, if it helps) Cohen-Macaulay morphism (flat with Cohen-Macaulay fibers). I'm also happy to assume that $V$ is integral, excellent and has a dualizing complex. Additionally suppose that we have $f' : X' \to V$ another equidimensional (dimension $d$) finite-type (reduced) Cohen-Macaulay morphism that factors through $f$ as below.

$$f' : X' \xrightarrow{\phi} X \to V.$$

Further suppose that $\phi$ is finite (although the question could be asked more generally for proper $\phi$, I'll phrase it for finite $\phi$). If it helps at any point, please feel free to assume that $f$ and $f'$ are Gorenstein morphisms.

Recall that $\omega_{f}[d] = f^! \mathcal{O}_V$ and that by Brian Conrad's book [LINK: Google books][1] we know that both $\omega_f$ and $\omega_{f'}$ are compatible with base change.

I'd like to conclude that the following natural map is also compatible with base change:

$$\phi_* \omega_{f'} \cong R \mathcal{H}om_{O_X}(\phi_* O_{X'}, \omega_f ) \cong \mathcal{H}om_{O_X}(\phi_* O_{X'}, \omega_f ) \to \omega_f$$.
The map can be interpreted as evaluation at 1.

In other words, I'd like to know that the trace map of $\phi$ is compatible with base change. Furthermore, it would be even good enough to prove this in the $f, f'$ Gorenstein morphisms case.

One way to do this would be as follows. If $g : T \to V$ is any other morphism and $f_T : X_T \to T$ and $f_T': X_T' \to T$ are the base changes and $g_X : X_T \to X$ is the projection, is it true that the natural map (denoted [*] below) $$g_X^* \phi_* \omega_{f'} \cong g_X^* \mathcal{H}om_{O_X}(\phi_* O_{X'}, \omega_f ) \to \mathcal{H}om_{O_{X_T}}(\phi_* O_{X_T'}, \omega_{f_T} ) \cong (\phi_T)_* \omega_{f_T'}$$ between abstractly isomorphic sheaves is an isomorphism?

Edit: Brian Conrad pointed out to me that this is already a special case of what is in his Theorem 3.6.1 (assuming I understand everything right). I'll try to write down an a precise answer tomorrow.

This is basically a question of functoriality for base change of CM morphisms.

EDIT: Brian Conrad sent me an email explaining the that this is indeed true, and follows from his book. I'll explain this after I state the question.

Suppose that $f : X \to V$ is an equidimensional (dimension $d$) finite type (reduced, if it helps) Cohen-Macaulay morphism (flat with Cohen-Macaulay fibers). I'm also happy to assume that $V$ is integral, excellent and has a dualizing complex. Additionally suppose that we have $f' : X' \to V$ another equidimensional (dimension $d$) finite-type (reduced) Cohen-Macaulay morphism that factors through $f$ as below.

$$f' : X' \xrightarrow{\phi} X \to V.$$

Further suppose that $\phi$ is finite (although the question could be asked more generally for proper $\phi$, I'll phrase it for finite $\phi$). If it helps at any point, please feel free to assume that $f$ and $f'$ are Gorenstein morphisms.

Recall that $\omega_{f}[d] = f^! \mathcal{O}_V$ and that by Brian Conrad's book [LINK: Google books][1] we know that both $\omega_f$ and $\omega_{f'}$ are compatible with base change.

I'd like to conclude that the following natural map is also compatible with base change:

$$\phi_* \omega_{f'} \cong R \mathcal{H}om_{O_X}(\phi_* O_{X'}, \omega_f ) \cong \mathcal{H}om_{O_X}(\phi_* O_{X'}, \omega_f ) \to \omega_f$$.
The map can be interpreted as evaluation at 1.

In other words, I'd like to know that the trace map of $\phi$ is compatible with base change. Furthermore, it would be even good enough to prove this in the $f, f'$ Gorenstein morphisms case.

One way to do this would be as follows. If $g : T \to V$ is any other morphism and $f_T : X_T \to T$ and $f_T': X_T' \to T$ are the base changes and $g_X : X_T \to X$ is the projection, is it true that the natural map (denoted [*] below) $$g_X^* \phi_* \omega_{f'} \cong g_X^* \mathcal{H}om_{O_X}(\phi_* O_{X'}, \omega_f ) \to \mathcal{H}om_{O_{X_T}}(\phi_* O_{X_T'}, \omega_{f_T} ) \cong (\phi_T)_* \omega_{f_T'}$$ between abstractly isomorphic sheaves is an isomorphism?

Edit: Brian Conrad pointed out to me that this is already a special case of what is in his Theorem 3.6.1 (assuming I understand everything right). I'll try to write down an a precise answer shortly.

This is basically a question of functoriality for base change of CM morphisms.

EDIT: Brian Conrad sent me an email explaining the that this is indeed true, and follows from his book. I'll explain this after I state the question.

Suppose that $f : X \to V$ is an equidimensional (dimension $d$) finite type (reduced, if it helps) Cohen-Macaulay morphism (flat with Cohen-Macaulay fibers). I'm also happy to assume that $V$ is integral, excellent and has a dualizing complex. Additionally suppose that we have $f' : X' \to V$ another equidimensional (dimension $d$) finite-type (reduced) Cohen-Macaulay morphism that factors through $f$ as below.

$$f' : X' \xrightarrow{\phi} X \to V.$$

Further suppose that $\phi$ is finite (although the question could be asked more generally for proper $\phi$, I'll phrase it for finite $\phi$). If it helps at any point, please feel free to assume that $f$ and $f'$ are Gorenstein morphisms.

Recall that $\omega_{f}[d] = f^! \mathcal{O}_V$ and that by Brian Conrad's book [LINK: Google books][1] we know that both $\omega_f$ and $\omega_{f'}$ are compatible with base change.

I'd like to conclude that the following natural map is also compatible with base change:

$$\phi_* \omega_{f'} \cong R \mathcal{H}om_{O_X}(\phi_* O_{X'}, \omega_f ) \cong \mathcal{H}om_{O_X}(\phi_* O_{X'}, \omega_f ) \to \omega_f$$.
The map can be interpreted as evaluation at 1.

In other words, I'd like to know that the trace map of $\phi$ is compatible with base change. Furthermore, it would be even good enough to prove this in the $f, f'$ Gorenstein morphisms case.

One way to do this would be as follows. If $g : T \to V$ is any other morphism and $f_T : X_T \to T$ and $f_T': X_T' \to T$ are the base changes and $g_X : X_T \to X$ is the projection, is it true that the natural map (denoted [*] below) $$g_X^* \phi_* \omega_{f'} \cong g_X^* \mathcal{H}om_{O_X}(\phi_* O_{X'}, \omega_f ) \to \mathcal{H}om_{O_{X_T}}(\phi_* O_{X_T'}, \omega_{f_T} ) \cong (\phi_T)_* \omega_{f_T'}$$ between abstractly isomorphic sheaves is an isomorphism?

Edit: Brian Conrad pointed out to me that this is already a special case of what is in his Theorem 3.6.1 (assuming I understand everything right). I'll try to write down an a precise answer tomorrow.