nLab exponential exact sequence

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.

• For $n=1$ this is the first Chern class in complex analytic geometry defined on the Picard group of holomorphic line bundles;

• for $n = 2$ this is the complex analytic-analog of the Dixmier-Douady class of holomorphic 2-line bundles, defined on the bigger Brauer group;

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.

Related concepts

• Kummer exact sequence

• Artin-Schreier exact sequence

• Kummer-Artin-Schreier-Witt exact sequence

• Bockstein homomorphism

References

• Wikipedia, Exponential sheaf sequence

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)