Do smooth ind schemes have Dualizing sheafs?

Question

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.

Answer 1

$\require{AMScd}$

I think the issue here is that the right notion of top exterior power in infinite dimensions is given by a gerbe instead of a line. Specially in the context of loop spaces as you are interested, the right notion of determinant bundle is given by the determinantal gerbe of Kapranov and Vasserot [1]. I'll only sketch the idea here. The notion of determinant for finite dimensional vector spaces is beautifully abstracted and specially adapted for this point of view by Knudsen and Mumford [2]. A nice explanation of Kapranov's ideas and much more on extensions of groups by groupoids is found in Osipov-Zhu [3].

The key point is that the tangent space to these formally smooth ind-schemes is a locally compact vector space that wants to look like $k((t))$. These are called Tate vector spaces. $Tate_0$ spaces are finite dimensional vector spaces and $Tate_{n+1}$ spaces are vector spaces that can be written as projective limit of directed limits of $Tate_n$ spaces (the limits take place in a fixed category defined by Kato and Beilinson). The stereotypical example of a $Tate_1$ space is $k((t))$ and a $Tate_2$ space is $k((t))((s))$ and so on. We need one more definition, that of a lattice in a $Tate_{n+1}$ space $V$, these corresponds to subspaces $V' \subset V$ that are projective limits of spaces in $Tate_n$. So a typical lattice in $V=k((t))$ is $V' = k[[t]]$.

In $Tate_0$ we have the usual notion of determinant: it is a functor from $Tate_0$ to the Picard groupoid $Pic^\mathbb{Z}$ of $\mathbb{Z}$ graded lines: $$ V \mapsto \det V := \wedge^{n} V[-n], \qquad \mathrm{dim } V = n$$ For each injective homomorphism $V' \hookrightarrow V$ we have $$\det(V') \otimes \det (V/V') \simeq \det (V)$$ For any diagram \begin{CD} 0 @>>> {U'} @>>> U @>>> {U/U'} @>>> 0 \\ @. @VVV @VVV @VVV @. \\ 0 @>>> {V'} @>>> V @>>> {V/V'} @>>> 0 \end{CD} The following diagram is commutative: \begin{CD} \det(U') \otimes \det(U/U') @>>> \det U \\ @VVV @VVV \\ \det(V') \otimes \det(V/V') @>>> \det V \end{CD} And there's a larger diagram for a quotient of three short exact sequences as above.

On $Tate_1$ the situation is subtler. A graded determinantal theory on $V$ is a rule that to each lattice $V' \subset V$ it assigns a graded line $\Delta(V') \in Pic^\mathbb{Z}$ and for any lattice $V'' \supset V'$ we have isomorphisms $$\Delta(V') \otimes \det(V''/V') \simeq \Delta(V'')$$ with the natural compatibility condition when $V''' \supset V''$. \begin{CD}\Delta(V') \otimes \det (V''/V') \otimes \det (V'''/V'') @> >> \Delta(V'') \otimes \det(V'''/V'') \\ @VVV @VVV \\ \Delta(V') \otimes \det(V'''/V') @> >> \Delta(V''') \end{CD}

The set of graded determinantal theories on $V$ is a category (the notion of morphisms is straightforward) and moreover it is a $Pic^\mathbb{Z}$-torsor since for each graded line $l[n]$ and determinantal theory $\Delta$ we can define $\Delta'(V') := l[n] \otimes \Delta(V')$.

So now if we sheafify things, and we look at a sheaf $\mathcal{V}$ of $Tate_1$ spaces over a space $X$, its top exterior power is the category of graded determinantal theories on $\mathcal{V}$. This is a torsor over $Pic^\mathbb{Z}_X$ and forgetting the grading this is the same thing as a $\mathbb{G}_m$-gerbe.

In [1] Kapranov and Vasserot work this out in detail when $X$ is a formally smooth ind-scheme and $\mathcal{V}$ is its tangent bundle, so they construct the determinantal gerbe $\mathcal{D}et (TX)$ and I think this is the closest you would get to a notion of (anti)canonical bundle on these spaces. Duality is now a different story that for the most part has to be written down.

[1] Formal loops IV: chiral differential operators. http://arxiv.org/abs/math/0612371
[2] The projectivity of the moduli space of stable curves I: preliminaries on "det" and "div" Math. Scand. 39 (1976), 19-55 http://www.mscand.dk/article/viewFile/12001/10017
[3] A categorical proof of the Parshin reciprocity laws on algebraic surfaces http://arxiv.org/abs/1002.4848

Answer 2

By judicious use of Kan extension, one can define a functorial !-pullback, and hence a dualizing complex, for a broad class of geometric objects, including arbitrary DG prestacks (Gaitsgory section 10.1), and in particular smooth ind-schemes as you ask.

This is used in Theorem 10.1.1 of Gaitsgory-Rozenblyum, where tensoring with the dualizing complex yields an equivalence between quasicoherent sheaves and ind-coherent sheaves in the case of formally smooth DG ind-schemes that are weakly $\aleph_0$ and locally almost of finite type. In this case, the dualizing complex is an ind-coherent sheaf that presumably looks something like a shifted line bundle, but I couldn't extract that information directly from the paper. Serre duality for DG ind-schemes locally almost of finite type also appears as Corollary 2.6.2 of the same paper, but its statement doesn't seem to use the dualizing complex directly.