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exponential exact sequence

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topos theory

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In higher category theory

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Contents

Idea

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

0ker(exp)𝔾 aexp(i())𝔾 m0 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(i())\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 11 or to 1/2π1/2 \pi.

Hence more explicitly over the complex numbers this is

02πexp(i()) ×0, 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 ×={0}\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 XX, where in terms of the structure sheaf 𝒪 X\mathcal{O}_{X} it reads

0Lconst()𝒪 Xexp(i())𝒪 X ×0. 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(,𝔾 m)H n+1(,) 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.

References

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)

Last revised on February 19, 2018 at 08:34:05. See the history of this page for a list of all contributions to it.