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The generalization in (∞,1)-category theory of the notion of group of units in ordinary category theory.
Let be an A-∞ ring spectrum.
For the underlying A-∞ space and the ordinary ring of connected components, write for its group of units.
Then the ∞-group of units
of is the (∞,1)-pullback in
There is slight refinement of the above definition, which essentially adds one 0-th “grading” homotopy group to and thereby makes the -group of units of E-∞ rings be canonically augmented over the sphere spectrum (Sagave 11).
There is a functor
given by …
This is (Sagave 11, def. 3.14 in view of example 3.8, Sagave-Schlichtkrull 11, above theorem 1.8). See also (Sagave 11, section 1.4) for comments on how this yields an -version of -grading on an abelian group.
In fact this grading extends form the group of units to the full -ring (Sagave-Schlichtkrull 11, theorem 1.7- 1.8).
Here the existence of the map is (Sagave 11, lemma 2.12 + lemma 3.16). The fact that the resulting sequence is a homotopy fiber sequence is (Sagave 11, prop. 4.1).
Using this, there is now a modified delooping of the ordinary -group of units:
Write for the homotopy cofiber of to yield
(Sagave 11, prop. 4.3)
Adjointness to -group -ring
This is (ABGHR 08, theorem 2.1/3.2, remark 3.4).
Also the augmented -group of units functor of def. 2 is a homotopy right adjoint. (Sagave 11, theorem 1.7).
The homotopy groups of are
Cohomology and logarithm
Given an E-∞ ring, then write for its -group of units regarded as a connective spectrum. For the homotopy type of a topological space, then the cohomology represented by in degree 0 is the ordinary group of units in the cohomology ring of :
In positive degree the canonical map of pointed homotopy types is in fact an isomorphism on all homotopy groups
On cohomology elements this map
is logarithm-like, in that it sends .
But there is not a homomorphism of spectra of this form. This only exists after K(n)-localization, where it is called then the logarithmic cohomology operation, see there for more.
Relation to Picard -group and Brauer -group
Given an E-∞ ring , the looping of the Brauer -group is the Picard ∞-group (Szymik 11, theorem 5.7).
The looping of that is the ∞-group of units (Sagave 11, theorem 1.2).
Snaith’s theorem and the units of K-theory and complex cobordism
Snaith's theorem asserts that
the K-theory spectrum for complex K-theory is the ∞-group ∞-ring of the circle 2-group localized away from the Bott element :
the periodic complex cobordism spectrum is the ∞-group ∞-ring of the classifying space for stable complex vector bundles (the classifying space for topological K-theory) localized away from the Bott element :
Units of topological modular forms
Analysis of the -group of units of tmf is in (Ando-Hopkins-Rezk 10, from section 12 on).
Inclusion of circle -bundles into higher chromatic cohomology
By Snaith’s theorem above there is a canonical map
that sends circle bundles to cocycles in topological K-theory.
At the next level there is a canonical map
that sends circle 2-bundles to tmf. See at tmf – Inclusion of circle 2-bundles.
Write for the ∞-group of units of the (a) Morava K-theory spectrum.
(Sati-Westerland 11, theorem 1)
A notion of spectrum of units of an -ring was originally described in
- Peter May, ring spaces and ring spectra Lecture Notes in Mathematics, Vol. 577. Springer-Verlag, Berlin, 1977. With contributions by Frank Quinn, Nigel Ray, and Jørgen Tornehave.
One explicit model was given in
A general abstract discussion in stable (∞,1)-category theory is in
Remarks alluding to this are also on p. 20 of
Theorem 3.2 there is proven using classical results which are collected in
- Peter May, What precisely are -ring spaces and -ring spectra?, Geometry and Topology Monographs 16 (2009) 215–282 (pdf)
A survey of the situation in (∞,1)-category theory is also in section 3.1 of
A construction in terms of a model structure on spectra is in
- John Lind, Diagram spaces, diagram spectra, and spectra of units (arXiv:0908.1092)
A refinement of the construction of -groups of units to augmented ∞-groups over the sphere spectrum, such as to distinguish of a periodic E-∞ ring from its connective cover, is in
based on (Schlichtkrull 04). See also
Steffen Sagave, Christian Schlichtkrull, Diagram spaces and symmetric spectra, Advances in Mathematics, Volume 231, Issues 3–4, October–November 2012, Pages 2116–2193 (arXiv:1103.2764)
Markus Szymik, Brauer spaces for commutative rings and structured ring spectra (arXiv:1110.2956)
Andrew Baker, Birgit Richter, Markus Szymik, Brauer groups for commutative -algebras, J. Pure Appl. Algebra 216 (2012) 2361–2376 (arXiv:1005.5370)
The -group of units of Morava K-theory is discussed in
The cohomology with coefficients in and the corresponding logarithmic cohomology operations are discussed in
The group of units of tmf is analyzed from section 12 on in