nLab infinity-group of units

Idea

The generalization in (∞,1)-category theory of the notion of group of units in ordinary category theory.

Definition

Unaugmented definition

Definition

Let $A$ be an A-∞ ring spectrum.

For $\Omega^\infty A$ the underlying A-∞ space and $\pi_0 \Omega^\infty A$ the ordinary ring of connected components, write $(\pi_0 \Omega^\infty A)^\times$ for its group of units.

Then the ∞-group of units

$$A^\times \coloneqq GL_1(A)$$

of $A$ is the (∞,1)-pullback $GL_1(A)$ in

(1)$$\array{ GL_1(A) &\to& \Omega^\infty A \\ \downarrow &(pb)& \downarrow \\ (\pi_0 \Omega^\infty A)^\times &\to& \pi_0 \Omega^\infty A } \,.$$

Remark

In terms of derived algebraic geometry one has that

$$GL_1(A) \simeq \mathbb{G}_m(A) = Hom(Spec A, \mathbb{G}_m)$$

is the mapping space from $Spec A$ into the multiplicative group. This point of view is adopted for instance in (Lurie, p. 20).

Augmented definition

There is slight refinement of the above definition, which essentially adds one 0-th “grading” homotopy group to $B gl_1(E)$ and thereby makes the $\infty$-group of units of E-∞ rings be canonically augmented over the sphere spectrum (Sagave 11).

Definition

There is a functor

$$gl_1^J \colon CRing_\infty \to AbGrp_\infty/\mathbb{S} \,,$$

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 $\infty$-version of $\mathbb{Z}$-grading on an abelian group.

In fact this grading extends form the group of units to the full $\infty$-ring (Sagave-Schlichtkrull 11, theorem 1.7- 1.8).

Proposition

For $E$ an E-∞ ring, there is a homotopy fiber sequence of abelian ∞-groups

$$gl_1(E) \to gl_1^J(E) \to \mathbb{S} \,,$$

where on the left we have the ordinary $\infty$-group of units of def. and on the right we have the sphere spectrum, regarded (being a connective spectrum) as an abelian ∞-group.

Here the existence of the map $gl_1(E) \to gl_1^J(E)$ 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 $\infty$-group of units:

Definition

Write $bgl_1^\ast(E)$ for the homotopy cofiber of $gl_1^J(E) \to \mathbb{S}$ to yield

$$gl_1(E) \to gl_1^J(E) \to \mathbb{S} \to bgl_1^\ast(E) \,.$$

(Sagave 11, prop. 4.3)

Remark

It ought to be true that the non-connective delooping $bgl_1^\ast(E)$ sits inside the full Picard ∞-group of $E Mod$. (Sagave 11, remark 4.11). (Apparently it’s the full inclusion on those degree-0 twists which are grading twists, i.e. on the elements $(-)\wedge\Sigma^n E$.)

See also at twisted cohomology – by R-module bundles.

Properties

Adjointness to $\infty$-group $\infty$-ring

Unaugmented case

Definition

Write

$$gl_1 \; \colon \; CRing_\infty \to AbGrp_\infty$$

for the (∞,1)-functor which sends a commutative ∞-ring to its ∞-group of units.

Theorem

The ∞-group of units (∞,1)-functor of def. is a right-adjoint (∞,1)-functor

$$CRing_\infty \stackrel{\overset{\mathbb{S}[-]}{\leftarrow}}{\underset{gl_1}{\to}} AbGrp_\infty \,.$$

This is (ABGHR 08, theorem 2.1/3.2, remark 3.4).

Remark

The left adjoint

$$\mathbb{S}[-] \colon AbGrp_\infty \to CRing_\infty$$

is a higher analog of forming the group ring of an ordinary abelian group over the integers

$$\mathbb{Z}[-] \colon Ab \to CRing \,,$$

which is indeed left adjoint to forming the ordinary group of units of a ring.

We might call $\mathbb{S}[A]$ the ∞-group ∞-ring of $A$ over the sphere spectrum.

Augmented case

Also the augmented $\infty$-group of units functor of def. is a homotopy right adjoint. (Sagave 11, theorem 1.7).

Homotopy groups

The homotopy groups of $GL_1(E)$ are

$$\pi_n(GL_1(E)) = \left\{ \array{ \pi_0(E)^\times & |\, n = 0 \\ \pi_n(E) & | \, n \geq 1 } \right.$$

Cohomology and logarithm

Given $E$ an E-∞ ring, then write $gl_1(E)$ for its $\infty$-group of units regarded as a connective spectrum. For $X$ the homotopy type of a topological space, then the cohomology represented by $gl_1(E)$ in degree 0 is the ordinary group of units in the cohomology ring of $E$:

$$H^0(X, gl_1(E)) \simeq (E^0(X))^\times \,.$$

In positive degree the canonical map of pointed homotopy types $GL_1(E) = \Omega^\infty gl_1(E) \to \Omega^\infty E$ is in fact an isomorphism on all homotopy groups

$$\pi_{\bullet \geq 1} GL_1(E) \simeq \pi_{\bullet \geq 1} \Omega^\infty E \,.$$

On cohomology elements this map

$$\pi_q(gl_1(E)) \simeq \tilde H^0(S^q, gl_1(E)) \simeq (1+ \tilde R^0(S^q))^\times \subset (R^0(S^q))^\times$$

is logarithm-like, in that it sends $1 + x \mapsto x$.

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.

(Rezk 06)

Relation to Picard $\infty$-group and Brauer $\infty$-group

Given an E-∞ ring $E$, the looping of the Brauer $\infty$-group is the Picard ∞-group (Szymik 11, theorem 5.7).

$$\Omega Br(E) \simeq Pic(E).$$

The looping of that is the ∞-group of units (Sagave 11, theorem 1.2).

$$\Omega^2 Br(E) \simeq \Omega Pic(E) \simeq GL_1(E) \,.$$

Examples

Snaith’s theorem and the units of K-theory and complex cobordism

Snaith's theorem asserts that

1. the K-theory spectrum for complex K-theory is the ∞-group ∞-ring of the circle 2-group localized away from the Bott element $\beta$:

   $$KU \simeq (\mathbb{S}[B U(1)])[\beta^{-1}] \,;$$

2. 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 $\beta$:

   $$MU \simeq (\mathbb{S}[B U])[\beta^{-1}] \,.$$

Units of topological modular forms

Analysis of the $\infty$-group of units of tmf is in (Ando-Hopkins-Rezk 10, from section 12 on).

Inclusion of circle $n$-bundles into higher chromatic cohomology

By Snaith’s theorem above there is a canonical map

$$B U(1) \to \mathbb{S}[B U(1)] \to KU$$

that sends circle bundles to cocycles in topological K-theory.

At the next level there is a canonical map

$$B^2 U(1) \to \mathbb{S}[B^2 U(1)] \to tmf$$

that sends circle 2-bundles to tmf. See at tmf – Inclusion of circle 2-bundles.

Write $gl_1(K(n))$ for the ∞-group of units of the (a) Morava K-theory spectrum.

Proposition

For $p = 2$ and all $n \in \mathbb{N}$, there is an equivalence

$$Maps(B^{n+1}U(1), B gl_1(K(n))) \simeq \mathbb{Z}/(2)$$

between the mapping space from the classifying space for circle (n+1)-bundles to the delooping of the ∞-group of units of $K(n)$.

(Sati-Westerland 11, theorem 1)

Remark

By the discussion at (∞,1)-vector bundle this means that for each such map there is a type of twist of Morava K-theory (at $p = 2$).

Related concepts

• (∞,1)-vector bundle

References

A notion of spectrum of units of an $E_\infty$-ring was originally described in

• Peter May, p. 54 of: $E_\infty$ ring spaces and $E_\infty$ ring spectra, Lecture Notes in Mathematics, Vol. 577. Springer-Verlag, Berlin, 1977. With contributions by Frank Quinn, Nigel Ray, and Jørgen Tornehave (pdf)

One explicit model was given in

• Christian Schlichtkrull, Units of ring spectra and their traces in algebraic K-theory, Geom. Topol. 8 (2004) 645-673 (arXiv:math/0405079, euclid:gt/1513883412)

Further discussion in

• Peter May, Johann Sigurdsson, Section 22.2 of: Parametrized Homotopy Theory, Mathematical Surveys and Monographs, vol. 132, AMS 2006 (ISBN:978-0-8218-3922-5, arXiv:math/0411656, pdf)

A general abstract discussion in stable (∞,1)-category theory is in

• Charles Rezk, section II of The units of a ring spectrum and a logarithmic cohomology operation, J. Amer. Math. Soc. 19 (2006), 969-1014 (arXiv:math/0407022)

• Matthew Ando, Andrew Blumberg, David Gepner, Michael Hopkins, Charles Rezk, Units of ring spectra and Thom spectra [arXiv:0810.4535]

• Matthew Ando, Andrew Blumberg, David Gepner, Michael Hopkins, Charles Rezk, Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory, Journal of Topology 7 4 (2014) [arXiv:1403.4320, doi:10.1112/jtopol/jtu009]

• Jacob Lurie, construction 3.9.4 of Elliptic Cohomology I: Spectral Abelian Varieties (pdf)

Remarks alluding to this are also on p. 20 of

• Jacob Lurie, A Survey of Elliptic Cohomology

Theorem 3.2 there is proven using classical results which are collected in

• Peter May, What precisely are $E_\infty$-ring spaces and $E_\infty$-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

• Jacob Lurie, A Survey of Elliptic Cohomology

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 (using $\mathcal{I}$-spaces) of the construction of $\infty$-groups of units to augmented ∞-groups over the sphere spectrum, such as to distinguish $gl_1$ of a periodic E-∞ ring from its connective cover:

• Steffen Sagave, Spectra of units for periodic ring spectra, Algebr. Geom. Topol. 16 (2016) 1203-1251 (arXiv:1111.6731)

based on (Schlichtkrull 04). See also

• Steffen Sagave, Christian Schlichtkrull, Diagram spaces and symmetric spectra, Advances in Mathematics 231 3-4 (2012) 2116-2193 [arXiv:1103.2764, doi:10.1016/j.aim.2012.07.013]

• Markus Szymik, Brauer spaces for commutative rings and structured ring spectra (arXiv:1110.2956)

• Andrew Baker, Birgit Richter, Markus Szymik, Brauer groups for commutative $\mathbb{S}$-algebras, J. Pure Appl. Algebra 216 (2012) 2361–2376 (arXiv:1005.5370)

The $\infty$-group of units of Morava K-theory is discussed in

• Hisham Sati, Craig Westerland, Twisted Morava K-theory and E-theory (arXiv:1109.3867)

The cohomology with coefficients in $gl_1(E)$ and the corresponding logarithmic cohomology operations are discussed in

• Charles Rezk, The units of a ring spectrum and a logarithmic cohomology operation, J. Amer. Math. Soc. 19 (2006), 969-1014 (arXiv:math/0407022)

The group of units of tmf is analyzed from section 12 on in

• Matthew Ando, Mike Hopkins, Charles Rezk, Multiplicative orientations of KO-theory and the spectrum of topological modular forms, 2010 (pdf)