symmetric monoidal (∞,1)-category of spectra
In the context of complex analytic geometry, the term “exponential exact sequence” typically referes to the short exact sequence
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
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
In this form the sequence is then also called the exponential sheaf sequence.
The connecting homomorphisms of the long exact sequence in cohomology induces by the exponential exact sequence
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 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:
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)
Last revised on February 19, 2018 at 08:34:05. See the history of this page for a list of all contributions to it.