Cyclic Group Brauer Construction

• Cyclic Group Brauer Construction is a method that converts cyclic symmetries into Brauer classes using 2-cocycles and crossed-product techniques.
• It is applied across algebra, geometry, topology, and arithmetic to yield explicit Azumaya algebras and compute Brauer groups in moduli spaces and cyclic covers.
• The approach leverages cyclic algebras, Galois descent, and twisting mechanisms to produce Brauer classes with orders matching the group actions, offering insights for lifting and deformation theories.

Across several settings, Brauer classes are constructed from cyclic group actions by crossed products, cyclic algebras, and Galois descent. The common pattern is the passage from a cyclic symmetry—usually a group $C_n$ or $\mathbb Z/p$—to a class in a Brauer group, often through a $2$-cocycle, a factor set, or a universal projective bundle. In algebraic geometry this yields explicit computations such as $${\rm Br}({\mathcal P}{\mathcal M}^\alpha_s)\cong \mathbb Z/m\mathbb Z$$; in homotopy theory it produces topological Azumaya algebras and $K(1)$-local cyclic generators; and in arithmetic settings it controls relative Brauer groups, cyclic covers, and mixed-characteristic lifting phenomena (Baker et al., 2010).

1. Crossed-product mechanism for cyclic Brauer classes

A basic cyclic-group Brauer construction starts with a faithful $C_n$-Galois extension $F/E$ of commutative $S$-algebras in the sense of Rognes and a normalized $2$-cocycle

$u\in Z^2(C_n;\pi_0(F)^\times).$

Baker–Richter–Szymik form the free rank-$\mathbb Z/p$0 $\mathbb Z/p$1-module

$\mathbb Z/p$2

with twisted multiplication

$\mathbb Z/p$3

Associativity is exactly the cocycle condition, and after base change one has

$\mathbb Z/p$4

This places the resulting algebra in the relative Brauer group, and the isomorphism class is controlled by the cohomology class $\mathbb Z/p$5. Under mild hypotheses, including discrete Galois behavior on $\mathbb Z/p$6 or $\mathbb Z/p$7-local Lubin–Tate type, the cyclic construction yields a bona-fide topological Azumaya algebra over $\mathbb Z/p$8 (Baker et al., 2010).

This factor-set viewpoint persists in the purely algebraic cases. In characteristic $\mathbb Z/p$9, mixed characteristic, and degree-$2$0 settings, the cyclic algebra is often presented by generators and relations rather than by the direct-sum crossed-product basis, but the governing datum remains a cyclic action together with a class in $2$1 or an equivalent factor set (Saltman, 2022).

2. Cyclic Brauer groups on moduli of parabolic bundles

For a smooth projective complex curve $2$2 of genus $2$3, fixed rank $2$4 (with $2$5 if $2$6), degree $2$7, determinant line bundle $2$8, and parabolic data $2$9 at points $${\rm Br}({\mathcal P}{\mathcal M}^\alpha_s)\cong \mathbb Z/m\mathbb Z$$0, the moduli space $${\rm Br}({\mathcal P}{\mathcal M}^\alpha_s)\cong \mathbb Z/m\mathbb Z$$1 of $${\rm Br}({\mathcal P}{\mathcal M}^\alpha_s)\cong \mathbb Z/m\mathbb Z$$2-stable parabolic bundles is a smooth quasi-projective variety of dimension

$${\rm Br}({\mathcal P}{\mathcal M}^\alpha_s)\cong \mathbb Z/m\mathbb Z$$3

Biswas–Dey define

$${\rm Br}({\mathcal P}{\mathcal M}^\alpha_s)\cong \mathbb Z/m\mathbb Z$$4

and prove

$${\rm Br}({\mathcal P}{\mathcal M}^\alpha_s)\cong \mathbb Z/m\mathbb Z$$5

The generator is the Brauer class of the universal projective bundle restricted from $${\rm Br}({\mathcal P}{\mathcal M}^\alpha_s)\cong \mathbb Z/m\mathbb Z$$6 to $${\rm Br}({\mathcal P}{\mathcal M}^\alpha_s)\cong \mathbb Z/m\mathbb Z$$7, and a universal vector bundle exists on $${\rm Br}({\mathcal P}{\mathcal M}^\alpha_s)\cong \mathbb Z/m\mathbb Z$$8 if and only if $${\rm Br}({\mathcal P}{\mathcal M}^\alpha_s)\cong \mathbb Z/m\mathbb Z$$9 (Biswas et al., 2010).

The proof uses a reduction to sufficiently small parabolic weights. One obtains a forgetful morphism

$K(1)$0

where $K(1)$1 is the moduli of stable rank-$K(1)$2 bundles with fixed determinant and the fiber is the generalized flag variety

$K(1)$3

Thaddeus’s wall-crossing implies that $K(1)$4 and the small-weight space $K(1)$5 differ in codimension at least $K(1)$6, so their Brauer groups agree. Applying the Leray spectral sequence for $K(1)$7 gives the exact segment

$K(1)$8

The map $K(1)$9 is then computed explicitly: if $C_n$0 is the class of the projective bundle induced from the universal projective bundle on $C_n$1, then

$C_n$2

Since $C_n$3, exactness gives $C_n$4, hence the same for $C_n$5 (Biswas et al., 2010).

3. Cyclic covers of surfaces and geometric realizations

For a $C_n$6-cyclic cover of the projective plane

$C_n$7

with $C_n$8 smooth of degree divisible by $C_n$9, the Galois group is

$F/E$0

generated by

$F/E$1

Ingalls–Obus–Ozman–Viray study unramified $F/E$2-torsion Brauer classes on $F/E$3 fixed by this cyclic action. If $F/E$4 is a reference line meeting $F/E$5 transversely and $F/E$6, there is a natural injection

$F/E$7

and an exact description of the $F/E$8-fixed subgroup of $F/E$9 in terms of divisor classes on $S$0. In particular,

$S$1

and $S$2 (Ingalls et al., 2013).

When $S$3, the same paper gives a second construction by Clifford algebras. From a line bundle $S$4 whose class is $S$5-torsion modulo $S$6, Catanese’s theorem provides a symmetric resolution with matrix $S$7. This determines a quadratic form and hence a sheaf of Clifford algebras. On $S$8,

$S$9

and this is an Azumaya algebra in $2$0; when the branch degree is even, it pulls back to an Azumaya algebra on $2$1. If $2$2, the cyclic-algebra construction and the Clifford-algebra construction give the same class: $2$3 Thus, in the double-cover case, every $2$4-torsion Brauer class on $2$5 admits a globally defined Azumaya representative via either description (Ingalls et al., 2013).

4. Relative Brauer groups of cyclic twists of genus-one curves

Haile–Han–Wadsworth study curves

$2$6

by realizing $2$7 as a cyclic twist of the elliptic curve

$2$8

The relevant $2$9-module is

$u\in Z^2(C_n;\pi_0(F)^\times).$0

and $u\in Z^2(C_n;\pi_0(F)^\times).$1 is a principal homogeneous space under $u\in Z^2(C_n;\pi_0(F)^\times).$2 whose class in $u\in Z^2(C_n;\pi_0(F)^\times).$3 lies in the image of $u\in Z^2(C_n;\pi_0(F)^\times).$4. The resulting cohomological description yields an explicit isomorphism

$u\in Z^2(C_n;\pi_0(F)^\times).$5

This identifies relative Brauer classes with cyclic-twist data coming from rational points on the Jacobian (Haile et al., 2010).

In the diagonalizable case $u\in Z^2(C_n;\pi_0(F)^\times).$6, one sets

$u\in Z^2(C_n;\pi_0(F)^\times).$7

with $u\in Z^2(C_n;\pi_0(F)^\times).$8. For a point $u\in Z^2(C_n;\pi_0(F)^\times).$9, the connecting homomorphism is computed using

$\mathbb Z/p$00

and every relative Brauer class is represented by the degree-$\mathbb Z/p$01 cyclic algebra

$\mathbb Z/p$02

Equivalently, the map

$\mathbb Z/p$03

is surjective with kernel $\mathbb Z/p$04. In this diagonalizable situation, every class in $\mathbb Z/p$05 is of cyclic form (Haile et al., 2010).

In the non-diagonalizable case, the paper replaces the cyclic-algebra presentation over $\mathbb Z/p$06 by cup products. One still obtains degree-$\mathbb Z/p$07 algebras representing the relative Brauer classes, but “not as cyclic algebras.” After passage to the quadratic field $\mathbb Z/p$08, these classes become ordinary $\mathbb Z/p$09-Kummer algebras, and Proposition 6.4 explains their relation to cyclic descriptions after further twisting (Haile et al., 2010).

5. Characteristic $\mathbb Z/p$10, mixed characteristic, and lifting

Saltman develops a cyclic-group Brauer construction in both characteristic $\mathbb Z/p$11 and mixed characteristic $\mathbb Z/p$12. If $\mathbb Z/p$13 is a commutative ring with $\mathbb Z/p$14, the Artin–Schreier extension

$\mathbb Z/p$15

is Galois with group $\mathbb Z/p$16 generated by $\mathbb Z/p$17. The associated “differential crossed-product” algebra

$\mathbb Z/p$18

is generated by $\mathbb Z/p$19 subject to

$\mathbb Z/p$20

It is Azumaya of degree $\mathbb Z/p$21, splits over $\mathbb Z/p$22, and every $\mathbb Z/p$23-torsion Brauer class of $\mathbb Z/p$24 is represented by some $\mathbb Z/p$25 (Saltman, 2022).

For mixed characteristic, with $\mathbb Z/p$26, a primitive $\mathbb Z/p$27th root of unity $\mathbb Z/p$28, and $\mathbb Z/p$29, the cyclic degree-$\mathbb Z/p$30 extension is defined by

$\mathbb Z/p$31

chosen so that modulo $\mathbb Z/p$32 it becomes $\mathbb Z/p$33, and the Galois action is

$\mathbb Z/p$34

The corresponding algebra

$\mathbb Z/p$35

has relations

$\mathbb Z/p$36

and is Azumaya of degree $\mathbb Z/p$37 whenever $\mathbb Z/p$38. The same paper extends this to cyclic degree $\mathbb Z/p$39 extensions and exponent $\mathbb Z/p$40 Brauer classes via twisted group algebras

$\mathbb Z/p$41

and formulates lifting theorems: every cyclic Galois $\mathbb Z/p$42-extension of degree $\mathbb Z/p$43 lifts to a cyclic Galois $\mathbb Z/p$44-extension of degree $\mathbb Z/p$45, and every Brauer class over $\mathbb Z/p$46 of exponent $\mathbb Z/p$47 lifts to a class over $\mathbb Z/p$48 of the same order (Saltman, 2022).

These constructions retain the factor-set description. The algebra is a crossed product for a cyclic group $\mathbb Z/p$49 of order $\mathbb Z/p$50, with factor set $\mathbb Z/p$51 satisfying the cocycle relation

$\mathbb Z/p$52

This shows that the mixed-characteristic theory is not a separate formalism but a deformation of the cyclic crossed-product picture (Saltman, 2022).

6. Topological and $\mathbb Z/p$53-local cyclic algebras, and related terminology

In chromatic height $\mathbb Z/p$54, the cyclic-group Brauer construction becomes explicit. Let $\mathbb Z/p$55 be the $\mathbb Z/p$56-local Morava $\mathbb Z/p$57-theory at an odd prime $\mathbb Z/p$58, so that $\mathbb Z/p$59 acts on $\mathbb Z/p$60, and choose a primitive $\mathbb Z/p$61st root of unity

$\mathbb Z/p$62

The cyclic algebra

$\mathbb Z/p$63

is the free rank-$\mathbb Z/p$64 $\mathbb Z/p$65-module with basis $\mathbb Z/p$66 and relations

$\mathbb Z/p$67

Using Galois descent, one identifies

$\mathbb Z/p$68

and the class $\mathbb Z/p$69 has exact order $\mathbb Z/p$70 and generates the group (Mor, 2023).

A nearby but broader topological statement is supplied by Baker–Richter–Szymik: cyclic Azumaya algebras over commutative $\mathbb Z/p$71-algebras are classified by $\mathbb Z/p$72, and for $\mathbb Z/p$73, $\mathbb Z/p$74, and $\mathbb Z/p$75 prime, the standard cocycle in $\mathbb Z/p$76 produces an Azumaya $\mathbb Z/p$77-algebra of topological degree $\mathbb Z/p$78. For $\mathbb Z/p$79, one recovers the classical quaternion algebra in spectra (Baker et al., 2010).

The literature also uses “Brauer” in distinct cyclic-group contexts. In block theory, a block of $\mathbb Z/p$80 with cyclic defect group has a Brauer tree algebra structure: if $\mathbb Z/p$81 has nontrivial cyclic defect group $\mathbb Z/p$82, then

$\mathbb Z/p$83

and when $\mathbb Z/p$84, the Brauer tree is of star type with multiplicity $\mathbb Z/p$85 (Flores et al., 2021). Craven likewise determines Brauer trees for unipotent blocks with cyclic defect group in the non-crystallographic objects $\mathbb Z/p$86, $\mathbb Z/p$87, and $\mathbb Z/p$88, using perverse equivalences and Deligne–Lusztig theory (Craven, 2015). A different usage again appears in the cyclic group–Brauer algebra $\mathbb Z/p$89, where strands in Brauer diagrams are oriented and labeled by powers of a generator $\mathbb Z/p$90 of $\mathbb Z/p$91, and multiplication is defined by stacking diagrams and evaluating closed loops by the trace (Geetha et al., 2012). These constructions involve cyclic groups and Brauer-theoretic terminology, but they concern Brauer tree algebras or diagram algebras rather than cyclic classes in a Brauer group.

The recurring mathematical content is therefore specific: a cyclic group acts, a cocycle or projective obstruction is extracted, and the resulting datum produces either an Azumaya algebra or an explicit cyclic Brauer group. In the sources considered here, that mechanism appears in moduli spaces, cyclic covers, relative Brauer groups of genus-one curves, structured ring spectra, and mixed-characteristic lifting, with the cyclicity visible either in the algebra presentation, in the order of the Brauer group, or in the Galois symmetry that generates the class.