Gluing Techniques in Triangular Geometry

Balmer, Paul; Favi, Giordano

doi:10.1093/qmath/ham002

Cited by 26 publications

(39 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For instance, in [2, Theorem 2.11], we proved that if an object x 2 has a disconnected support, ie supp.x/ D Y 1 [ Y 2 with Y 1 and Y 2 closed and disjoint, then the object is decomposable accordingly: x ' x 1˚x2 with supp.x i / D Y i . This neat result has applications, like the gluing technique of Balmer-Favi [6] and its representation theoretic incarnations in Balmer-Benson-Carlson [5] and in our paper [4]. Another application, this time to algebraic K -theory of schemes, can be found in our paper [3].…”

Section: Introductionmentioning

confidence: 54%

Spectra, spectra, spectra – Tensor triangular spectra versus Zariski spectra of endomorphism rings

Balmer

2010

Algebr. Geom. Topol.

109

176

View full text Add to dashboard Cite

We construct a natural continuous map from the triangular spectrum of a tensor triangulated category to the algebraic Zariski spectrum of the endomorphism ring of its unit object. We also consider graded and twisted versions of this construction. We prove that these maps are quite often surjective but far from injective in general. For instance, the stable homotopy category of finite spectra has a triangular spectrum much bigger than the Zariski spectrum of Z. We also give a first discussion of the spectrum in two new examples, namely equivariant KK -theory and stable A 1 -homotopy theory.18E30; 14F05, 19K35, 20C20, 55P42, 55U35 IntroductionAlgebraic geometers, stable topologists, finite group representation theorists, motivic theorists, noncommutative geometers and many other mathematicians have triangulated categories in common: The derived category of sheaves of modules over a scheme, the stable homotopy category of topological spectra, the derived category or the stable category of representations of a finite group or finite group scheme, the various motivic derived categories, Morel and Voevodsky's stable A 1 -homotopy category and equivariant KK -theory or E -theory of C -algebras are famous examples. In several cases, a tensor structure is also available and is especially well-behaved on the triangulated subcategory of compact objects. In the above examples, this leads us to focus on perfect complexes, finite topological spectra, finite dimensional representations, geometric motives, etc.This profusion of examples motivates the study of tensor triangulated categories per se. Emphasizing the geometric aspects of this unified theory leads us to a subject called tensor triangular geometry, to which the present paper belongs.To explain these ideas, let us denote by one of our tensor triangulated categories (say, of compact objects), by˝W ! the symmetric tensor x˝y D y˝x and by 1 2 the˝-unit: 1˝x D x . We began our geometric study in [1] with the definition of a topological space Spc./ called the spectrum of (see also Definition 1.3 below). We call it the triangular spectrum here, to avoid confusion with other meanings of the word "spectrum". This fundamental space Spc./ is the canvas on which to draw tensor triangular geometry. For instance, every object x 2 has a support, supp.x/ Spc./, which is a closed subset behaving nicely with respect to exact triangles and tensor product. In all applications though, the crucial anchor point is the computation of the triangular spectrum Spc./ in the first place. Without this knowledge, abstract results of tensor triangular geometry are difficult to translate into concrete terms. It is therefore a major challenge to compute the spectrum Spc./ in as many examples as possible, or at least to provide some information about that space when the full determination of Spc./ lies beyond reach of the community's current forces.Actually, by our paper [1, Theorem 4.10, Theorem 5.2] and Buan, Krause and Solberg [9, Corollary 5.2], we know that the information about contained in the ...

show abstract

Section: Introductionmentioning

confidence: 54%

Spectra, spectra, spectra – Tensor triangular spectra versus Zariski spectra of endomorphism rings

Balmer

2010

Algebr. Geom. Topol.

109

176

View full text Add to dashboard Cite

show abstract

“…Proof of Lemma 7.2. As we are dealing with honest morphisms of complexes we may first reduce to open subsets on which F is a sum of line bundles L i (note that two morphisms in D.X/ are not necessarily equal if they are equal when restricted to all open sets of an affine covering, see for example [5]). We then reduce to the case of codimension d D 1, by multiplicativity of Koszul complexes:…”

Section: Regular Embeddingsmentioning

confidence: 99%

Push-forwards for Witt groups of schemes

Calmès¹,

Hornbostel²

2011

Comment. Math. Helv.

View full text Add to dashboard Cite

Abstract. Using suitable closed symmetric monoidal structures on derived categories of schemes, as well as adjunctions of the type .Lf ; Rf / and .Rf ; f Š / (i.e. Grothendieck duality theory), we define push-forwards for coherent Witt groups along proper morphisms between separated noetherian schemes. We also establish fundamental theorems for these push-forwards (e.g. base change and projection formula) and provide some computations.

show abstract

“…(K) is the subgroup of locally trivial objects and finally, whereȞ 1 (Spc(K), G m ) denotes the firstČech cohomology group of Spc(K) with coefficients in the presheaf of units (see Construction 1.12 and Remark 2.5). The main result of the paper, Theorem 3.9, says that for K in characteristic p, the map α is an isomorphism after inverting p. Its inverse is obtained via the gluing construction of [7]. Putting things together we get the general result :…”

Section: Moreover This Isomorphism Maps Omentioning

confidence: 98%

“…It relies on the concept of gluing, which is very standard in algebraic geometry and which inspired the gluing construction of Balmer-Favi [7] in tensor triangular geometry.…”

Section: Moreover This Isomorphism Maps Omentioning

confidence: 99%

See 1 more Smart Citation

Picard groups in triangular geometry and applications to modular representation theory

Balmer

2010

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

Gluing Techniques in Triangular Geometry

Abstract: We discuss gluing of objects and gluing of morphisms in triangulated categories. We illustrate the results by producing, among other things, a Mayer-Vietoris exact sequence involving Picard groups.

Cited by 26 publications

References 11 publications

Spectra, spectra, spectra – Tensor triangular spectra versus Zariski spectra of endomorphism rings

Spectra, spectra, spectra – Tensor triangular spectra versus Zariski spectra of endomorphism rings

Push-forwards for Witt groups of schemes

Picard groups in triangular geometry and applications to modular representation theory

Contact Info

Product

Resources

About