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Say I have an ind scheme $X = \cup_i X_i$ over a field $k$. I have its tangent bundle $\hom_k(k[\epsilon], X)$ which I can think of as ind scheme via $\cup_i \hom_k(k[\epsilon],X_i)$. The problem is even if $X$ is smooth it might be the case that most of the $X_i$ are not smooth. I believe this happens at least for polynomial loop groups $G[z^\pm]$. In this case the sheaf of differentials is not locally free. This seems to be an obstruction to constructing the canonical sheaf inductively.

Additionally if each $X_i$ is infinite dimensional, which happens for the formal loop group, then it seems like top exterior power doesn't make much sense. And finally if you looked at say $\mathbb{P}^\infty := \cup_n \mathbb{P}^n$ then it also seems unclear what a canonical sheaf should be. If it were a line bundle it could be described as a line bundle $L_n$ on each $\mathbb{P}^n$ which are compatible under pull backs. But then each $L_n$ would have the same degree $d$. But the canonical line bundles $O(-n-1)$ have a different degree on each $\mathbb{P}^n$!

So is there any sense in asking for something like a canonical sheaf or dualizing sheaf for smooth ind schemes?

UPDATE: Brian Conrad shared the following with me:

If $f:X \to Y$ is a map between finite type schemes over a field (or one can be much more general...) then for a relative dualizing complex $\omega_Y$ on $Y$ we have that $f^!(\omega_Y)$ is a relative dualizing complex on $X$ (for suitable functor $f^!$ at derived category level). In other words, one does have "compatibility" for relative dualizing complexes, but with respect to the appropriate "derived pullback" operation $f^!$. Also, the property of being a dualizing sheaf is insensitive to tensoring by a line bundle of shifting a complex, so for example every line bundle on a smooth scheme over a field is a "dualizing complex". (One has to think about duality and "canonical sheaf" in a much broader sense than Serre duality over a field in order to define "relative dualizing object" in a derived category.)

The upshot is that one has to work in derived categories (and so develop a suitable formalism of direct/inverse limits in derived categories) to make a good theory of duality on ind-schemes.

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So is there any sense is in asking for something like a canonical sheaf or dualizing sheaf for smooth ind schemes?

UPDATE: Brian Conrad shared the following with me:

If $f:X \to Y$ is a map between finite type schemes over a field (or one can be much more general...) then for a dualizing complex $\omega_Y$ on $Y$ we have that $f^!(\omega_Y)$ is a dualizing complex on $X$ (for suitable functor $f^!$ at derived category level). In other words, one does have "compatibility" for dualizing complexes, but with respect to the appropriate "derived pullback" operation $f^!$. Also, the property of being a dualizing sheaf is insensitive to tensoring by a line bundle of shifting a complex, so for example every line bundle on a smooth scheme over a field is a "dualizing complex". (One has to think about duality and "canonical sheaf" in a much broader sense than Serre duality over a field in order to define "dualizing object" in a derived category.)

The upshot is that one has to work in derived categories (and so develop a suitable formalism of direct/inverse limits in derived categories) to make a good theory of duality on ind-schemes.

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# Do smooth ind schemes have Dualizing sheafs?

Say I have an ind scheme $X = \cup_i X_i$ over a field $k$. I have its tangent bundle $\hom_k(k[\epsilon], X)$ which I can think of as ind scheme via $\cup_i \hom_k(k[\epsilon],X_i)$. The problem is even if $X$ is smooth it might be the case that most of the $X_i$ are not smooth. I believe this happens at least for polynomial loop groups $G[z^\pm]$. In this case the sheaf of differentials is not locally free. This seems to be an obstruction to constructing the canonical sheaf inductively.

Additionally if each $X_i$ is infinite dimensional, which happens for the formal loop group, then it seems like top exterior power doesn't make much sense. And finally if you looked at say $\mathbb{P}^\infty := \cup_n \mathbb{P}^n$ then it also seems unclear what a canonical sheaf should be. If it were a line bundle it could be described as a line bundle $L_n$ on each $\mathbb{P}^n$ which are compatible under pull backs. But then each $L_n$ would have the same degree $d$. But the canonical line bundles $O(-n-1)$ have a different degree on each $\mathbb{P}^n$!

So is there any sense is asking for something like a canonical sheaf or dualizing sheaf for smooth ind schemes?