# nLab exponential exact sequence

Contents

topos theory

## Theorems

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

In the context of complex analytic geometry, the term “exponential exact sequence” typically referes to the short exact sequence

$0 \to ker(\exp) \longrightarrow \mathbb{G}_{a} \stackrel{\exp(\tfrac{i}{\hbar}(-))}{\longrightarrow} \mathbb{G}_m \to 0$

given by the exponential map $\exp(\tfrac{i}{\hbar}(-))$ from the additive group to the multiplicative group. Here $\hbar$ is any element of $\mathbb{R}^\times$ (“Planck's constant”) but is traditionally set either to $1$ or to $1/2 \pi$.

Hence more explicitly over the complex numbers this is

$0 \to \hbar2\pi\mathbb{Z} \longrightarrow \mathbb{C} \stackrel{\exp(\tfrac{i}{\hbar}(-))}{\longrightarrow} \mathbb{C}^\times \to 0 \,,$

where $\mathbb{C}$ denotes the complex numbers as the additive abelian group and where $\mathbb{C}^\times = \mathbb{C} - \{0\}$ is the group of units of the ring structure of the complex numbers.

Often this is considered and displayed relative to a complex analytic space $X$, where in terms of the structure sheaf $\mathcal{O}_{X}$ it reads

$0 \to Lconst(\mathbb{Z}) \longrightarrow \mathcal{O}_X \stackrel{\exp(\tfrac{i}{\hbar}(-))}{\longrightarrow} \mathcal{O}_X^\times \to 0 \,.$

In this form the sequence is then also called the exponential sheaf sequence.

## Properties

### Long sequence in cohomology

The connecting homomorphisms of the long exact sequence in cohomology induces by the exponential exact sequence

$H^n(-,\mathbb{G}_m) \longrightarrow H^{n+1}(-,\mathbb{Z})$

encode the canonical characteristic classes of line n-bundles.

and so on.

### In logarithmic geometry

In algebraic geometry there is no exponential sequence, the closest analogs being the Kummer sequence and the Artin-Schreier sequence. But in logarithmic geometry there is again a kind of exponential sequence (e.g. Ogus 01, chapter IV, remark 1.1.7, Brylinski 94, page 15). Compare also the sequences in Kato-Nakayama 99, section 1.4.

Discussion in real analytic geometry:

• Johannes Huisman, The exponential sequence in real algebraic geometry and Harnack’s Inequality for proper reduced real schemes, Communications in Algebra, Volume 30, Issue 10, 2002 (pdf)

Discussion in logarithmic geometry

• Jean-Luc Brylinski, Holomorphic gerbes and the Beilinson regulator, Astérisque 226 (1994): 145-174 (pdf)

• Arthur Ogus, Lectures on logarithmic algebraic geometry, TeXed notes, 2001, pdf

• Kazuya Kato, Chikara Nakayama, Log Betti cohomology, log étale cohomology, and log de Rham cohomology of log schemes over $\mathbb{C}$ Kodai Math. J.

Volume 22, Number 2 (1999), 161-186. (ProjectEuclid)