Formal Bott–Thurston Cocycle and Part of a Formal Riemann–Roch Theorem

Abstract

The Bott–Thurston cocycle is a $2$-cocycle on the group of orientation-preserving diffeomorphisms of the circle. We introduce and study a formal analog of the Bott–Thurston cocycle. The formal Bott–Thurston cocycle is a $2$-cocycle on the group of continuous $A$-automorphisms of the algebra $A((t))$ of Laurent series over a commutative ring $A$ with values in the group $A^*$ of invertible elements of $A$. We prove that the central extension given by the formal Bott–Thurston cocycle is equivalent to the 12-fold Baer sum of the determinantal central extension when $A$ is a $\mathbb Q$-algebra. As a consequence of this result we prove a part of a new formal Riemann–Roch theorem. This Riemann–Roch theorem is applied to a ringed space on a separated scheme $S$ over $\mathbb Q$, where the structure sheaf of the ringed space is locally on $S$ isomorphic to the sheaf $\mathcal O_S((t))$ and the transition automorphisms are continuous. Locally on $S$ this ringed space corresponds to the punctured formal neighborhood of a section of a smooth morphism to $U$ of relative dimension $1$, where $U \subset S$ is an open subset.

Appendix A. Lie algebra-valued functors constructed from group functors

In this appendix we collected some statements on how to construct a Lie algebra-valued functor from a group functor with some conditions and how to construct a $2$-cocycle on the Lie algebra-valued functor from a corresponding $2$-cocycle on a group functor.

Above in the paper, we applied these statements to group functors represented by ind-schemes.

We fix a commutative ring $R$. All functors considered below will be from the category of commutative $R$-algebras. By $A$ we denote an arbitrary commutative $R$-algebra.

In Subsections A.1 and A.2 below we mainly follow, but with some additions, the work [12, Exp. II] by M. Demazure.

Appendix A.1. Tangent space functors

Let $G$ be a (covariant) functor. Let $x\in G(R)$. Then the tangent space functor $TG_x$ of $G$ at $x$ is defined as

$$TG_x(A) = \rho^{-1}(x),$$

where $\rho \colon\, G(A[\varepsilon]/\varepsilon^2)\to G(A)$ is the natural map and we consider the image of $x$ in $G(A)$.

For any free $A$-module $V$ of finite rank, we consider the commutative $A$-algebra $I_V$ which is isomorphic to $A\oplus V$ as an $A$-module and in which the multiplication is defined by the relation $A=A\cdot 1$ and the fact that the second direct summand $V$ is an ideal of $I_V$ with $V^2=(0)$. For any $x\in G(R)$ we define

$$G(I_V)_x = \varrho^{-1}(x),$$

where $\varrho \colon\, G(I_V)\to G(A)$ is the natural map. Note that $G(I_A)_x = TG_x(A)$.

We suppose that $G$ satisfies the following condition for any free $A$-modules $V_1$ and $V_2$ of finite ranks:

$$ G(I_{V_1 \oplus V_2})_x \xrightarrow{\sim} G(I_{V_1})_x\times G(I_{V_2})_x,$$

(A.1)

where the map is induced by the natural homomorphisms $I_{V_1 \oplus V_2}\to I_{V_i}$. In this case $TG_x$ has a natural structure of an ${\mathbb A} ^1_R$-module, where ${\mathbb A} ^1_R(A) = A$ (see the appendix to Lecture 4 in [30] in the case when $A =R$ is a field, and see [12, Exp. II, Definition 3.5, Proposition 3.6] in the general case, which has the same form). For any fixed $\xi_1,\xi_2\in G(I_A)_x$ and any $\alpha,\beta\in A$, the result $\alpha \xi_1 + \beta \xi_2$ comes from the following diagram:

$$G(I_A)_x\times G(I_A)_x \xleftarrow{\sim} G (I_{A \oplus A})_x \xrightarrow{\langle\alpha,\beta\rangle} G(I_A)_x,$$

where the map $\langle \alpha, \beta \rangle$ is induced by the $A$-module homomorphism $(\gamma, \delta) \mapsto (\alpha \gamma + \beta \delta)$ from $A \oplus A$ to $A$. The image of the element $\xi_1\times\xi_2$ is equal, by definition, to $\alpha \xi_1 + \beta \xi_2$.

Remark A.1.

By an ind-scheme we mean an ind-object of the category of schemes such that all transition maps in the ind-object are closed embeddings of schemes. If a functor $G$ is represented by an ind-scheme over $R$, then condition (A.1) is satisfied, since it is satisfied for schemes. Moreover, in this case $TG_x$ is the inductive limit of tangent space functors to the corresponding schemes, and the ${\mathbb A} ^1_R$-module structure on $TG_x$ induced by this inductive limit coincides with the ${\mathbb A} ^1_R$-module structure described above.

Now we will consider only group functors and we suppose that they satisfy condition (A.1).

Note that for a group functor $G$, it suffices to check condition (A.1) only for $x=e$, where $e\in G(R)$ is the identity element. Moreover, the group structure on $G$ induces a group structure on $TG_e$, and this group structure coincides with the group structure coming from the structure of an ${\mathbb A} ^1_R$-module defined above (see [12, Exp. II, Corollary 3 to Proposition 3.9]).

Now according to [12, Exp. II, Definition 4.6], we say that a group functor $G$ is good if additionally to condition (A.1) the functor $G$ satisfies the condition

$$ TG_e(A) \otimes_A I_V \xrightarrow{\sim} TG_e (I_V)$$

(A.2)

for any commutative $R$-algebra $A$ and any free $A$-module $V$ of finite rank. The map is induced by the natural embedding $A \hookrightarrow I_V$ and the $I_V$-module structure on $TG_e (I_V)$.

Remark A.2.

If a group functor $G$ is represented by a group ind-scheme, then $G$ is good, because a functor represented by a scheme satisfies condition (A.1) and condition (A.2), which can be formulated for any functors satisfying condition (A.1).

Appendix A.2. Lie bracket

For a good group functor $G$ we introduce the notation $\operatorname{Lie} G = TG_e$. Now we consider only good group functors. For good group functors it is possible to define the Lie bracket on $\operatorname{Lie} G$ in the same way as it is done before Proposition 4.8 in [12, Exp. II]. We have the following commutative diagram with exact columns and rows:

Here $\operatorname{Lie} G(A) \subset G(A[\varepsilon_1 \varepsilon_2]/(\varepsilon_1^2 \varepsilon_2^2))$ in the upper left corner is canonically isomorphic to the group $\operatorname{Ker} \alpha \cap \operatorname{Ker} \beta$.

Now we take an element $f_1\in \operatorname{Lie} G(A) \subset G (A[\varepsilon_1]/(\varepsilon_1^2))$ in the lower left corner of the diagram and an element $f_2\in \operatorname{Lie} G(A) \subset G (A[\varepsilon_2]/(\varepsilon_2^2))$ in the upper right corner of the diagram. Using the fact that all columns and rows of the diagram have canonical group splittings, we find that $f_1, f_2\in G(A[\varepsilon_1,\varepsilon_2]/(\varepsilon_1^2, \varepsilon_2^2))$. By definition, the Lie bracket

$$ [f_1, f_2] = f_1 f_2 f_1^{-1} f_2^{-1}\in G (A[\varepsilon_1,\varepsilon_2]/(\varepsilon_1^2, \varepsilon_2^2))$$

(A.3)

is the image of the element of $\operatorname{Lie} G(A)$ in the upper left corner under the embedding induced by the embedding of rings $A [\varepsilon_1 \varepsilon_2]/(\varepsilon_1^2 \varepsilon_2^2) \hookrightarrow A [\varepsilon_1,\varepsilon_2]/(\varepsilon_1^2, \varepsilon_2^2)$.

The described bracket

$$[ \kern1pt\cdot\kern1pt,\cdot\kern1pt ] \colon\, \, \operatorname{Lie} G(A)\times \operatorname{Lie} G(A)\to \operatorname{Lie} G(A)$$

is functorial with respect to $A$, $A$-bilinear, and satisfies the Jacobi identity (see formulas (5) and (6) after Lemma 5.11 in [28, §5] for commutators in a group).

By construction, we have $[f_1, f_2] + [f_2, f_1]= 0$ for any $f_1, f_2\in \operatorname{Lie} G(A)$.

By an ind-affine ind-scheme we mean an ind-scheme $M= \mathop{\mathopen{{}^{\backprime\backprime}}\varinjlim_{i\in I}^{\prime\prime}} \operatorname{Spec} A_i$. By ${\mathcal O} (M)= \varprojlim_{i\in I} A_i$ we denote the corresponding topological algebra of regular functions on $M$ (with the topology of projective limit, taking the discrete topology on each $A_i$).

If $G$ is represented by an ind-affine ind-scheme, then $[f,f]=0$ for any $f\in \operatorname{Lie} G(A)$, because in this case the $A$-algebra $\operatorname{Lie} G(A)$ is isomorphic to the Lie $A$-algebra of all continuous left-translation-invariant $A$-derivations of the topological $A$-algebra ${\mathcal O} (G_A)$ of regular functions on the ind-scheme $G_A$. (See [12, Exp. II, Proposition 3.13, Theorem 4.1.4, and §4.11] for the case of right-translation-invariant $A$-derivations; the $A$-algebra $\operatorname{Lie} G(A)$ is anti-isomorphic to the Lie $A$-algebra of such continuous derivations.)

Thus, in the case when $G$ is represented by an ind-affine ind-scheme, the bracket $[ \kern1pt\cdot\kern1pt,\cdot\kern1pt ]$ on $\operatorname{Lie} G$ defines the Lie $A$-algebra structure on the $A$-module $\operatorname{Lie} G(A)$ for any commutative $R$-algebra $A$.

Remark A.3.

It is also easy to see that $[f_1, f_2]$ does not depend on the choice of the liftings of $f_1$ and $f_2$ to $G (A[\varepsilon_1,\varepsilon_2]/(\varepsilon_1^2, \varepsilon_2^2))$.

Appendix A.3. $2$-cocycles

We consider a central extension of group functors

$$1\to F\to \widetilde{G}{} \xrightarrow{\pi\,} G\to 1$$

such that there is a section $\sigma$ of $\pi$, but we do not require that $\sigma$ should be a group section. We suppose that $F$ and $G$ are represented by group ind-affine ind-schemes. Then $\widetilde{G}{}$ is also represented by a group ind-affine ind-scheme (here we use the fact that there is a section $\sigma$). The section $\sigma$ gives a $2$-cocycle $\Lambda$ on $G$ with values in $F$.

We obtain the corresponding central extension of Lie algebra-valued functors

$$0\to \operatorname{Lie} F\to \operatorname{Lie} \widetilde{G}{} \to \operatorname{Lie} G\to 0,$$

and $\sigma$ induces a section $\operatorname{Lie} G\to \operatorname{Lie} \widetilde{G}{}$. In the usual way (similarly to the case of central extensions of group functors in the proof of Proposition 2.2), this section gives a Lie algebra $2$-cocycle

$$\operatorname{Lie} \Lambda \colon\, \, \operatorname{Lie} G\times \operatorname{Lie} G\to \operatorname{Lie} F,$$

which can be explicitly obtained from the cocycle $\Lambda$ in the following way.

Let $h_1$ and $h_2$ be from $\operatorname{Lie} G(A)$, where $A$ is any commutative $R$-algebra. We consider the elements $h_i$ ($i=1,2$) as elements of $\operatorname{Lie} G (A[\varepsilon_1,\varepsilon_2]/(\varepsilon_1^2, \varepsilon_2^2))$ through the embeddings

$$h_i\in \operatorname{Lie} G(A) \subset G(A[\varepsilon_i]/(\varepsilon_i^2)) \subset G (A[\varepsilon_1,\varepsilon_2]/(\varepsilon_1^2, \varepsilon_2^2)).$$

Then

$$\Upsilon (h_1,h_2) = \Lambda (h_1,h_2) \cdot \Lambda (h_2, h_1)^{-1}$$

is an element of the Lie $A$-algebra $\operatorname{Lie} F(A) \subset F (A[\varepsilon_1 \varepsilon_2]/(\varepsilon_1^2 \varepsilon_2^2)) \subset F(A[\varepsilon_1,\varepsilon_2]/(\varepsilon_1^2,\varepsilon_2^2))$. This follows immediately from the diagram in Subsection A.2 written for the functor $F$ and from the identity $\Lambda(1, y)= \Lambda(y,1)= \Lambda(1,1)$, which follows from the cocycle identity for $\Lambda$.

The proof of the following proposition is similar to the proof of Proposition 2.6.1 from [23] given in the case of a central extension of Lie groups. In our case we have to work over the ring $A[\varepsilon_1,\varepsilon_2]/(\varepsilon_1^2, \varepsilon_2^2)$. Moreover, analogs of one-parameter subgroups are obtained from the functors over the subrings $A[\varepsilon_i]/(\varepsilon_i^2)$, and an analog of the mixed second-order partial derivative is the functor over the subring $A[\varepsilon_1 \varepsilon_2]/(\varepsilon_1^2 \varepsilon_2^2)$.

Proposition A.1.

We have $\, \operatorname{Lie} \Lambda (h_1,h_2) = \Upsilon (h_2, h_2)$ for any $\,h_1$ and $\,h_2$ from the Lie $\,A$-algebra $\, \operatorname{Lie} G(A)$.