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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 $f^!$. (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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What definition of "smooth" are you using? I usually see "formally smooth and locally of finite presentation", but that might be overly restrictive for you. –  S. Carnahan Feb 6 '12 at 4:31
I mean algebraic smoothness. That is $\varprojlim S_n^q(m_n/m_n^2) \to \varprojlim m^q_n/m^{q+1}_n$ is an isomorphism for all $q\ge 0$. Locally of finite presentation I think is too restrictive because the main example I'm interested in is the loop group. –  solbap Feb 6 '12 at 16:41
I'm a little bit confused, how can the structure sheaf be a dualizing sheaf on any smooth variety? –  Yosemite Sam Feb 10 '12 at 15:23
@Y.Sam from Brian `The answer is that "dualizing complex" is intrinsic to the scheme (as is needed to relate local and global duality on reasonable schemes); it is a different notion than "relative dualizing complex"; Yosemite Sam is thinking about relative dualizing complex/sheaf.' –  solbap Feb 12 '12 at 1:35
Y. Sam comment was a response to `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".' Originally, Brian did not want mention relative dualizing complex but only indicate that one has to work in the derived category. But to avoid confusion the above sentenced was removed and the comments were changed to be about relative dualizing complex. –  solbap Feb 12 '12 at 1:46

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