A semiorthogonal decomposition for Brauer–Severi schemes

Bernardara, Marcello

doi:10.1002/mana.200610826

Cited by 49 publications

(59 citation statements)

References 9 publications

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“…Let

K / k

be a field extension such that

X_{K} ≃ {double-struckP}_{K}^{n}

. Thanks to Beilinson , we have the following semiorthogonal decomposition

{sans-serifD}^{normalb} false(P_{K}^{n} false) = 〈O_{{double-struckP}_{K}^{n}}, O_{{double-struckP}_{K}^{n}} (1), \dots, O_{{double-struckP}_{K}^{n}} (n)〉 .

On the other hand,

{sans-serifD}^{normalb} false(X false)

admits the following semiorthogonal decomposition [, Corollary 4.7]

{sans-serifD}^{normalb} false(X false) = false⟨ D^{b} (k), D^{b} (k, A), \dots, D^{b} (k, A^{n}) false⟩ .

We can see the semiorthogonal decomposition as the base‐change of using descent of vector bundles as follows. The exceptional line bundle

O_{{double-struckP}_{K}^{n}}

clearly descends to

X

, and hence generates an admissible subcategory of

{sans-serifD}^{normalb} false(X false)

equivalent to

{sans-serifD}^{normalb} false(k false)

.…”

Section: Descent For Semiorthogonal Decompositionsmentioning

confidence: 99%

Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields

Auel

Bernardara

2018

Proc. London Math. Soc.

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We study the birational properties of geometrically rational surfaces from a derived categorical perspective. In particular, we give a criterion for the rationality of a del Pezzo surface S over an arbitrary field, namely, that its derived category decomposes into zero‐dimensional components. When S has degree at least 5 we construct explicit semiorthogonal decompositions by subcategories of modules over semisimple algebras arising as endomorphism algebras of vector bundles and we show how to retrieve information about the index of S from Brauer classes and Chern classes associated to these vector bundles.

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“…Let

K / k

be a field extension such that

X_{K} ≃ {double-struckP}_{K}^{n}

. Thanks to Beilinson , we have the following semiorthogonal decomposition

{sans-serifD}^{normalb} false(P_{K}^{n} false) = 〈O_{{double-struckP}_{K}^{n}}, O_{{double-struckP}_{K}^{n}} (1), \dots, O_{{double-struckP}_{K}^{n}} (n)〉 .

On the other hand,

{sans-serifD}^{normalb} false(X false)

admits the following semiorthogonal decomposition [, Corollary 4.7]

{sans-serifD}^{normalb} false(X false) = false⟨ D^{b} (k), D^{b} (k, A), \dots, D^{b} (k, A^{n}) false⟩ .

We can see the semiorthogonal decomposition as the base‐change of using descent of vector bundles as follows. The exceptional line bundle

O_{{double-struckP}_{K}^{n}}

clearly descends to

X

, and hence generates an admissible subcategory of

{sans-serifD}^{normalb} false(X false)

equivalent to

{sans-serifD}^{normalb} false(k false)

.…”

Section: Descent For Semiorthogonal Decompositionsmentioning

confidence: 99%

Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields

Auel

Bernardara

2018

Proc. London Math. Soc.

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“…As pointed out by the referee, the combination of [33, Lemma 5.1] with Bernardara's semi-orthogonal decomposition [4] leads to a generalization of the above motivic decomposition 3.9 to every Severi-Brauer variety X → S over a smooth projective k-scheme S. As pointed out by the referee, the combination of [33, Lemma 5.1] with Bernardara's semi-orthogonal decomposition [4] leads to a generalization of the above motivic decomposition 3.9 to every Severi-Brauer variety X → S over a smooth projective k-scheme S.…”

Section: Severi-brauer Varietiesmentioning

confidence: 99%

Noncommutative motives of Azumaya algebras

Tabuada

Bergh

2014

J. Inst. Math. Jussieu

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Let k be a base commutative ring, R a commutative ring of coefficients, X a quasi-compact quasi-separated k-scheme, A a sheaf of Azumaya algebras over X of rank r, and Hmo(R) the category of noncommutative motives with R-coefficients. Assume that 1/r belongs to R. Under this assumption, we prove that the noncommutative motives with R-coefficients of X and A are isomorphic. As an application, we show that all the R-linear additive invariants of X and A are exactly the same. Examples include (nonconnective) algebraic K-theory, cyclic homology (and all its variants), topological Hochschild homology, etc. Making use of these isomorphisms, we then computer the R-linear additive invariants of differential operators in positive characteristic, of cubic fourfolds containing a plane, of Severi-Brauer varieties, of Clifford algebras, of quadrics, and of finite dimensional k-algebras of finite global dimension. Along the way we establish two results of independent interest. The first one asserts that every element of the Grothendieck group of X which has rank r becomes invertible in the R-linearized Grothendieck group, and the second one that every additive invariant of finite dimensional algebras of finite global dimension is unaffected under nilpotent extensions.Comment: 22 pages; revised versio

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“…Its compatibility with the classical base-change mechanism is proved in §7. 3. As an application, we obtain new tools for the study of motivic decompositions and Schur/Kimura finiteness.…”

Section: Theorem 18 (I)mentioning

confidence: 99%

Noncommutative Artin motives

Marcolli

Tabuada

2013

Sel. Math. New Ser.

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Abstract. In this article we introduce the category of noncommutative Artin motives as well as the category of noncommutative mixed Artin motives. In the pure world, we start by proving that the classical category AM(k) Q of Artin motives (over a base field k) can be characterized as the largest category inside Chow motives which fully-embeds into noncommutative Chow motives. Making use of a refined bridge between pure motives and noncommutative pure motives we then show that the image of this full embedding, which we call the category NAM(k) Q of noncommutative Artin motives, is invariant under the different equivalence relations and modification of the symmetry isomorphism constraints. As an application, we recover the absolute Galois group Gal(k/k) from the Tannakian formalism applied to NAM(k) Q . Then, we develop the base-change formalism in the world of noncommutative pure motives. As an application, we obtain new tools for the study of motivic decompositions and Schur/Kimura finiteness. Making use of this theory of base-change we then construct a short exact sequence relating Gal(k/k) with the noncommutative motivic Galois groups of k and k. Finally, we describe a precise relationship between this short exact sequence and the one constructed by Deligne-Milne. In the mixed world, we introduce the triangulated category NMAM(k) Q of noncommutative mixed Artin motives and construct a faithful functor from the classical category MAM(k) Q of mixed Artin motives to it. When k is a finite field this functor is an equivalence. On the other hand, when k is of characteristic zero NMAM(k) Q is much richer than MAM(k) Q since its higher Ext-groups encode all the (rationalized) higher algebraic K-theory of finité etale k-schemes. In the appendix we establish a general result about short exact sequences of Galois groups which is of independent interest. As an application, we obtain a new proof of Deligne-Milne's short exact sequence.

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A semiorthogonal decomposition for Brauer–Severi schemes

Cited by 49 publications

References 9 publications

Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields

Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields

Noncommutative motives of Azumaya algebras

Noncommutative Artin motives

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