# 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 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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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

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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.

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It's interesting that this is also in a higher category level. I wander if section's 4.2 and 4.3 in Lurie's DAG XIV are relevant as well. – Reimundo Heluani Nov 26 '14 at 11:25
@ReimundoHeluani Those sections of DAG XIV are quite similar to Gaitsgory's treatment, but developed in a different direction, in slightly different language and with more precise proofs. I think any relationships between dualizing complexes and determinant gerbes may be better illuminated by Sho Saito's recent work relating K-theory delooping with Beilinson's locally compact object construction (e.g., arxiv.org/abs/1405.0923), but I don't know how to state anything precise. – S. Carnahan Nov 27 '14 at 1:10