Colliot-Thélène–Skorobogatov: Brauer groups of schemes

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corollary_3_7_3: For a regular integral scheme X with generic point Spec(F) and a prime l different from the residual characteristics of X, Br(X){l} injects into Br(F){l} with image the kernel of the residues at all codimension-one points of X.

theorem_3_3_2: For a quasi-compact separated scheme X with an ample invertible sheaf, for example a quasi-projective scheme over an affine scheme, the map from the Brauer–Azumaya group of X to the torsion subgroup of Br(X) is an isomorphism.

theorem_3_5_4: For a geometrically locally factorial integral scheme X, for example a regular one, with generic point Spec(F), the map Br(X) to Br(F) is injective, and so is the restriction Br(X) to Br(U) for every non-empty open subset U of X.

theorem_3_7_1: For a regular integral scheme X, a dense open U and a prime l different from the residual characteristics of X, Br(X){l} injects into Br(U){l} with image the kernel of the residues along the codimension-one components of the complement: those of its regular locus in Theorem 3.7.1, all irreducible divisors in X minus U in Theorem 3.7.2.

theorem_3_7_6: For a noetherian, regular, integral scheme X with function field F, the subgroup Br(X) of Br(F) is the intersection of Br(O_{X,x}) over the points x of codimension one, by Česnavičius's purity theorem 3.7.5.

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The copy read for this card is a 29-page excerpt printed pp. 71--99 (PDF p. n is printed p. 70+n): the end of Chapter 2 (pp. 71--74) and Chapter 3 from p. 75 to Proposition 3.8.2 on p. 99. Its pagination is not that of the published chapter: Crossref's record puts the chapter on pp. 71--99, while here Chapter 3 begins on p. 75 and continues past p. 99. The page numbers below and on the result pages are this copy's. No notice is printed on its pages; the publisher's chapter page (DOI 10.1007/978-3-030-74248-5_3, read 2026-10-02) shows "© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG" behind a paywall and names no Open Access or Creative Commons license, every other right reserved.

Jean-Louis Colliot-Thélène and Alexei N. Skorobogatov, "Brauer groups of schemes," in The Brauer–Grothendieck Group, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 71-99, 2021. https://doi.org/10.1007/978-3-030-74248-5_3

Overview

Content and organizing questions. The chapter compares two extensions of the Brauer group of a field to a scheme: the Brauer–Azumaya group $\operatorname{Br}_{\mathrm{Az}}(X)$, formed from Morita-equivalence classes of Azumaya algebras, and the Brauer–Grothendieck group $\operatorname{Br}(X)=H^2_{\acute et}(X,\mathbb G_m)$ (Definition 3.2.1, printed p. 76). Its principal questions are when these groups agree, how Brauer classes behave under localization and passage to the generic point, and how the unramified subgroup of a function-field Brauer group is detected by divisorial residues.

Theorem 3.1.1 (printed p. 76) gives the fibrewise-central-simple, endomorphism, and étale-local matrix-algebra characterizations of an Azumaya algebra. Theorem 3.3.1 (pp. 79–80), using the central extension

$$1\to\mathbb G_m\to\operatorname{GL}_n\to\operatorname{PGL}_n\to1 \tag{3.4}$$

identifies degree-$n$ Azumaya algebras with $\operatorname{PGL}_n$-torsors, constructs the natural injection $\operatorname{Br}_{\mathrm{Az}}(X)\hookrightarrow\operatorname{Br}(X)$, and places its degree-$n$ image in $\operatorname{Br}(X)[n]$. The main comparison result is Gabber’s Theorem 3.3.2 (p. 80): if $X$ is quasi-compact and separated and has an ample invertible sheaf—for example, if it is quasi-projective over an affine scheme—then

$$\operatorname{Br}_{\mathrm{Az}}(X)\simeq\operatorname{Br}(X)_{\mathrm{tors}}.$$

The separatedness hypothesis is necessary: the text cites non-separated normal complex varieties with torsion Brauer classes outside the image of $\operatorname{Br}_{\mathrm{Az}}(X)$ (p. 80).

Methods for the comparison theorem. The chapter sketches de Jong’s proof rather than proving every imported ingredient. Proposition 3.3.3 (p. 80) associates to an Azumaya algebra its $\mathbb G_m$-gerbe of local splittings. Proposition 3.3.4 (p. 81) characterizes gerbes arising in this way: a $\mathbb G_m$-gerbe $\mathcal X\to X$ comes from an Azumaya algebra exactly when it carries a finite locally free, positive-rank, $1$-twisted sheaf $\mathcal M$, in which case $A=\pi_*\mathcal End(\mathcal M)$. Lemma 2.6.1 of the preceding chapter (p. 72) identifies such sheaves with Čech-style $\alpha$-twisted sheaves. For torsion $\alpha$, the proof constructs coherent twisted sheaves and repeatedly raises the codimension of their non-flat locus through the induction $(H_c)$ (printed pp. 82–85). The kernel construction in Step 2 removes codimension-$c$ components, while Step 3 obtains a sufficiently general global map by a high-codimension avoidance argument and Rumely’s cited local-to-global principle. The affine case of Gabber’s theorem is explicitly imported rather than reproved (p. 82).

Cohomological tools and local behavior. For a prime $\ell$ invertible on $X$, the Kummer sequence yields the exact sequences

$$0\to\operatorname{Pic}(X)/\ell^n\to H^2_{\acute et}(X,\mu_{\ell^n})\to\operatorname{Br}(X)[\ell^n]\to0 \tag{3.2}$$

and (3.3) on printed p. 77. Theorem 3.2.2 (p. 78) gives the Mayer–Vietoris sequence for an open cover. Propositions 3.2.3 and 3.2.4 (pp. 78–79) describe passage to $X_{\mathrm{red}}$ for noetherian $X$, with isomorphism in the affine or dimension-at-most-one cases, surjectivity in dimension at most two, and, for $n$ invertible on $X$, injectivity on $\operatorname{Br}(X)/n$, surjectivity on $n$-torsion and, over a field of characteristic $0$, surjectivity on torsion.

Every Brauer class becomes zero on an étale cover by Lemma 3.4.1 (p. 86). Azumaya’s Theorem 3.4.2 (p. 86) proves $\operatorname{Br}(R)\simeq\operatorname{Br}(k)$ for a henselian local ring with residue field $k$, hence vanishing for strictly henselian local rings. Corollaries 3.4.3 and 3.4.4 (p. 86) give invariance under completion and an étale-neighborhood trivialization criterion at a rational point.

Generic points, residues, and purity. For a geometrically locally factorial integral scheme, the divisor sequence (3.6) leads to torsion of $H^n_{\acute et}(X,\mathbb G_m)$ for $n\ge2$ (Lemma 3.5.2, p. 88) and to the residue sequence (3.7) (Lemma 3.5.3, p. 88). Theorem 3.5.4 (pp. 88–89) proves that $\operatorname{Br}(X)\to\operatorname{Br}(F)$ is injective for such $X$, in particular for regular integral noetherian schemes. Theorem 3.5.5 (p. 89) gives injectivity of $\operatorname{Br}(X)\to\operatorname{Br}(U)$ for a separated noetherian $X$ and an open $U$ containing every generic and every singular point of $X$.

For regular one-dimensional schemes, Proposition 3.6.1 (pp. 89–91) gives long exact residue sequences: for the $\ell$-primary parts, $\ell$ invertible on $X$, in general, and for the full groups with $\mathbb Q/\mathbb Z$ coefficients when the residue fields at closed points are perfect. The residue is identified with the Witt residue. Theorem 3.6.2, equation (3.10) (p. 91), gives the split sequence

$$0\to\operatorname{Br}(k)\to\operatorname{Br}(K)\to H^1(k,\mathbb Q/\mathbb Z)\to0$$

for a henselian discretely valued field with perfect residue field. Theorem 3.6.4 (pp. 92–93) proves surjectivity of the prime-to-characteristic residue map for a semilocal Dedekind domain.

For a regular integral scheme, Theorems 3.7.1 and 3.7.2, equations (3.11)–(3.12) (pp. 93–94), express prime-to-residual-characteristic purity: for a prime $\ell$ different from the residual characteristics, an $\ell$-primary Brauer class on a dense open extends precisely when its codimension-one residues vanish. Corollary 3.7.3, equation (3.13) (p. 94), gives the corresponding function-field sequence. Its proof uses cohomology with supports, the Kummer sequence, and Gabber’s cited absolute-purity theorem; equation (3.16) (p. 95) is the finite-coefficient form. Theorem 3.7.4 (p. 96) gives the pullback formula for residues, including divisor multiplicities. The stronger Theorem 3.7.5 (p. 96), attributed to Česnavičius, says that deleting a codimension-at-least-two subset from a regular integral scheme does not change its full Brauer group; its proof is not reproduced. Consequently, Theorem 3.7.6 (pp. 96–97) identifies, for noetherian, regular, integral $X$,

$$\operatorname{Br}(X)=\bigcap_{x\in X^{(1)}}\operatorname{Br}(\mathcal O_{X,x})\subset\operatorname{Br}(F),$$

and Propositions 3.7.7–3.7.9 (p. 97) recast this valuation-theoretically and obtain birational invariance for regular proper models.

Finally, Section 3.8 constructs restriction and corestriction for finite locally free morphisms; their composite is multiplication by the rank. Proposition 3.8.1 (pp. 98–99) proves compatibility of corestriction with base change. Thus the chapter is a structural survey and proof account for scheme-theoretic Brauer groups, emphasizing gerbes, twisted sheaves, étale cohomology, purity, and valuation-theoretic detection rather than explicit computation in a particular Diophantine family.

Relation to E940

This source bears on Problem 940.

Write

$$\mathcal P_r=\{m\ge1: v_p(m)=0\text{ or }v_p(m)\ge r\text{ for every prime }p\}$$

and

$$\Sigma_r=\bigcup_{0\le j\le r}\left\{a_1+\cdots+a_j:a_i\in\mathcal P_r\right\}.$$

For $r\ge3$, E940 asks whether infinitely many integers lie outside $\Sigma_r$, and whether

$$\lim_{B\to\infty}\frac{\#(\Sigma_r\cap[1,B])}{B}=0.$$

The paper contains no theorem about $\mathcal P_r$, $\Sigma_r$, additive representations, or natural density, so its relation to E940 is weak and infrastructural.

A possible translation would begin with a separately constructed representation family $V\to\mathbb A^1_N$, whose fibre $V_N$ parametrizes decompositions of $N$ into at most $r$ $r$-powerful summands. The chapter’s scheme $X$ would then be such a regular model $V$, with function field $F=k(V)$. For a proposed torsion class $\alpha\in\operatorname{Br}(V)$, equation (3.1) (printed p. 77) supplies pullback and specialization; Theorem 3.3.2 (p. 80) realizes $\alpha$ by an Azumaya algebra under its hypotheses (for example, $V$ quasi-projective over an affine scheme). Corollary 3.7.3, equation (3.13) (p. 94), could certify that the $\ell$-primary part of a generic class, for $\ell$ different from the residual characteristics, extends over $V$ by checking all codimension-one residues, and Theorem 3.7.6 (pp. 96–97) reduces extension of any class to the local rings at codimension-one points. Theorem 3.7.4 (p. 96) would control those residues after a change of variables or a ramified base change, and Proposition 3.8.1 (pp. 98–99) would control restriction and corestriction under finite locally free base change.

These tools could therefore organize cohomological data on an algebraic family already known to encode the representations. They do not construct such a family: the condition $v_p(m)=0$ or $v_p(m)\ge r$ simultaneously over all primes is not represented in the chapter by a finite-type scheme. More decisively, none of the cited results bounds $\#(\Sigma_r\cap[1,B])$, supplies a sieve or additive estimate, or converts fibrewise cohomological information into a density statement. The paper proves neither density zero nor even nonrepresentability for any positive-density set of integers, for $r=3$ or for $r\ge4$. It is relevant only if an approach to E940 first produces a regular parameter space and Brauer classes whose specialization can be tied—by additional arguments absent here—to integral representations.

Read status: claims checked for Theorems 3.3.2, 3.5.4, 3.7.1, 3.7.2, 3.7.5 and 3.7.6, Corollary 3.7.3, Lemmas 3.5.2 and 3.5.3 and Propositions 3.7.7--3.7.9, read clause by clause on the page images of the copy; the proofs of Theorems 3.5.4, 3.7.1 and 3.7.6 were followed and the sketch of Theorem 3.3.2 read for structure. Gabber's affine case, absolute purity and Theorem 3.7.5 are cited, not proved, in the chapter. Nothing here is independently reviewed. Result pages: Theorem 3.3.2, Theorem 3.5.4, Theorems 3.7.1 and 3.7.2, Corollary 3.7.3 and Theorem 3.7.6.

Bears on. #940: the chapter proves nothing about $r$-powerful numbers, sums of them or density, and decides neither question of the problem.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.