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Let me first recall a known fact. Suppose $X$ is a complex algebraic variety, $\mathscr{L}$ is a line bundle on $X$ and $L^\times$ is the total space of $\mathscr{L}$ with the zero section removed. Write $\pi:L^\times\longrightarrow X$ for the projection. Then $\mathscr{A}=\pi_*\mathscr{O}_{L^\times}$ is naturally a $\mathbb{Z}$-graded quasicoherent sheaf of commutative $\mathscr{O}_X$-algebras, and $\mathscr{L}$ can be recovered from $\mathscr{A}$ (because we can recover $L^\times$ together with the map $\pi$ and the action of the multiplicative group $\mathbb{G}_m$).

My question is whether this picture can be categorified along the following lines.

On the one hand, instead of line bundles let us look at $\mathbb{G}_m$-gerbes on $X$.

On the other hand, instead of $\mathscr{A}$ let us suppose (for instance) that we have a $\mathbb{Z}$-graded symmetric monoidal $\mathbb{C}$-linear abelian category $\mathscr{M}=\bigoplus_{d\in\mathbb{Z}}\mathscr{M}^d$, so that $\mathscr{M}^0$ is the category of coherent sheaves on $X$ with the usual tensor product.

Is there a setup of this sort where one gets a suitable equivalence between $\mathbb{G}_m$-gerbes on one side and a special class of $\mathbb{Z}$-graded symmetric monoidal categories on the other side? (It is fine with me if the description of the other side needs to be modified.)

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The answer should be yes: the datum of a line bundle is the datum of a morphism from $X$ to the stack $\mathbf{B}\mathbb{G}_m$, together with the datum of the defining representation of $\mathbf{B}\mathbb{G}_m$ in $\mathbb{K}$-vector spaces, i.e.., $\mathbb{G}_m\to \mathrm{Aut}_{\mathbb{K}}(\mathbb{K})$. Going to gerbes means considering the 2-stack $\mathbf{B}^2\mathbb{G}_m$; this should have a natural defining 2-representation in 2-vector spaces, givin the kind of correspondence you're after. I guess details can be found in David Ben-Zvi work on geometric function theory. –  domenico fiorenza Jan 1 '11 at 9:45
Could you give me a reference to the relevant work of Ben-Zvi? (I'm not trying to be lazy, but he does have many papers, and since you mentioned him, I thought you may have a specific paper in mind.) –  senti_today Jan 1 '11 at 22:48
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2 Answers

up vote 7 down vote accepted

This question can be approached abstractly through the general Tannakian formalism, as laid out e.g. here, or very concretely by hand. You construct maps in both directions. To a $G_m$ gerbe assign its tensor category $QC(Gerbe)$ of sheaves (I'll speak of quasicoherent sheaves out of force of habit - presumably you can work just with coherent). This is a commutative algebra over $QC(X)$, which locally is isomorphic to graded sheaves on $X$ (i.e. to $QC(X)\otimes Rep G_m$). Conversely to such a category assign its spectrum, the stack which to any ring attaches the groupoid of tensor functors from your category to modules. This carries a map to $X$ which is a $G_m$ gerbe.

Put another way, given a sheaf of tensor categories over $X$ (or commutative algebra over $QC(X)$) which is locally isomorphic to $Rep G$, you consider the stack (sheaf of groupoids) of isomorphisms of this sheaf of categories with $QC(X)\otimes Rep G$ [EDIT: Better and more Tannakian to say, the stack of fiber functors to $QC(X)$]. This is a $G$ gerbe (ie 's locally of the form $X\times BG$). This is of course just the usual Tannakian reconstruction as in Deligne, except that we have the base $X$ be a scheme (or geometric stack) instead of the spectrum of a field.

Of course you could also ask to give a more global characterization of such $Z$-graded commutative algebras over $QC(X)$. I think it's equivalent to characterize the module category given by sheaves of degree 1, aka twisted sheaves on the corresponding gerbe. This is your version of a categorified line bundle -- it's a module category locally isomorphic to sheaves. Presumably it can be characterized as an invertible module category -- one for which there exists (or maybe for which you specify - I'm conveniently pretending that everything has been taking place one level of categoricity down, which is fine if you only care about say a class in $H^2(X,G_m)$) an inverse with respect to tensor product of module categories over $QC(X)$. Then the above argument proves that such categorified line bundles (via Spec of the $Z$-graded $QC(X)$-algebra they generate) are equivalent to $G_m$-gerbes. [Edit:] You'll also need to make sure such invertible modules are locally trivial, ie again given by the same cohomology group. You might also want to think of things through a third perspective on this story after the gerby/Tannakian ones, namely that of Azumaya algebras. You want to know that your invertible module category has a generator as an $O_X$-linear category, and thus is equivalent to the category oif modules over a sheaf of algebras, namely the endomorphisms of this generator. And then appeal to a classification of these algebras up to Morita equivalence by the same cohomology group.

[By the way a very interesting recent paper about the derived version of this story is here.]

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Thanks so much David! This does look exactly like the sort of thing I was looking for. –  senti_today Jan 2 '11 at 4:35
And I take it, you will not confirm the rumor I heard on some website (I think it may have been MathOverflow) that the answer to my question is contained in one of your works? :) –  senti_today Jan 2 '11 at 4:41
I enjoy such rumors, and am certainly very fond of Tannakian pictures, but I'm afraid there's nothing directly relevant in this case.. –  David Ben-Zvi Jan 2 '11 at 15:53
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When $X={\rm spec}\; k$, the spectrum of a field, your question seems to be related to (non neutral) Tannaka duality : gerbes over a field are characterized by they categories of representations (see Deligne, P. Catégories tannakiennes. The Grothendieck Festschrift, Vol. II). This is not specific to $\mathbb G_m$, of course. Over an arbitrary basis, you are looking for a categorical interpretation of the cohomological Brauer group $H^2(X, \mathbb G_m)$, I don't know of a description along the lines you suggest. One could also understand your question in this way: it is possible to understand this $H^2$ as a $H^1$, arguing that "a $\mathbb G_m$-gerbe is a form of $B\mathbb G_m$", and this seems indeed feasible, see Breen, L Tannakian categories. Motives (Seattle, WA, 1991). To classify these forms, one has then to describe the automorphism group of $B\mathbb G_m$. This can be seen as categorical analogue of the set-theoretic fact that invertible sheaves are forms of $\mathcal O_X$, and as such are classified by $H^1(X,\mathbb G_m)$, where $\mathbb G _m$ is the automorphism group of $\mathcal O_X$.

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