Costratification and actions of tensor-triangulated categories

Null hypersurfaces are a mathematical consequence of the Lorentzian signature of general relativity; singularities in mathematical models usually indicate where the interesting physics takes place. This book discusses what happens when you combine these ideas.Right from the preface, this is a no-nonsense book. There are two principalapproaches to singular shells, one distributional and the other 'cut andpaste'; both are treated in detail. A working knowledge of GR is assumed,including familiarity with null tetrads, differential forms, and 3 + 1decompositions. Despite my own reasonably extensive, closely relatedknowledge, there was material unfamiliar to me already in chapter 3, althoughI was reunited with some old friends in later chapters.The exposition is crisp, with a minimum of transition from chapter to chapter.In fact, my main criticism is that there is no clear statement of theorganization of the book, nor is there an index. Everything is here, and thestory is compelling if you know what to look for, although it is less easy tofollow the story if you are not already familiar with it.But this is really a book for experts, and the authors certainly qualify,having played a significant role in developing and extending the results theydescribe. It is also entirely appropriate that the book is dedicated toWerner Israel, who pioneered the thin-shell approach to (non-null) singularsurfaces and later championed the use of similar methods for analysing nullshells.After an introductory chapter on impulsive signals, the authors show how theBianchi identities can be used to classify spacetimes with singular nullhypersurfaces. This approach, due to the authors, generalizes the frameworkoriginally proposed by Penrose [1]. While astrophysical applicationsare discussed only briefly, the authors point out that detailed physicalcharacteristics of signals from isolated sources can be determined in thismanner. In particular, they describe the behaviour of test particles in such aspacetime, in an initial attempt to outline a framework for the detection ofimpulsive gravitational waves.Subsequent chapters describe the singular null hypersurfaces obtained byboosting isolated gravitational sources, building on the work of Aichelburgand Sexl [2], and by colliding impulsive waves, building onthe work of Khan and Penrose [3]. In between, the special caseof spherical symmetry is considered, both with and without collisions. Thereis also a short chapter discussing the effect of replacing GR by alternativetheories of gravity, and an appendix which briefly summarizes the non-nullcase.The references are reasonably complete, from Synge and Penrose to the recentwork of the authors. However, there are a few relatively minor errors andomissions. For instance, the results in chapter 3 about shells of matter inboth Schwarzschild and Reissner–Nordström geometries are presented without reference or derivation. And I was disappointed to see that my own work with 't Hooft [4] on the horizon shift due to the impulsive wave of amassless particle at the horizon of a Schwarzschild black hole—a directgeneralization of the work by Aichelburg and Sexl—is not mentioned.But none of these minor complaints detracts from my appreciation of having acomplete discussion of singular null hypersurfaces all in one place. Thethree fundamental papers [1–3] whichstarted this area of research all appeared at essentially the same time, 35years ago; it is high time there was a unified presentation of the entirefield. This book fills that need admirably, and could serve as the core of agraduate seminar for students having already taken a course in generalrelativity, or as a reference. My copy will have a treasured place in mylibrary. ReferencesPenrose R 1972 The geometry of impulsive gravitational waves General Relativity: Papers in Honour of J L Syngeed L Ó Raifeartaigh (Oxford: Clarendon) pp 101–30Aichelburg P C and Sexl R U 1971 On the gravitational field of a massless particle Gen. Rel. Grav. 2 303–12 Khan K A and Penrose R 1971 Scattering of two impulsive gravitational plane waves Nature 229 185–6 Dray T and 't Hooft G 1985 The gravitational shock wave of a massless particle Nucl. Phys. B 253 173–88

Colocalizing subcategories of singularity categories

Local cohomology and quasicoherent sheaves

In this chapter we define the general notion of cohomology of a sheaf of abelian groups on a topological space, and then study in detail the cohomology of coherent and quasi-coherent sheaves on a noetherian scheme.

We study the notion of pure-direct-injective objects in categories of sheaves by introducing the class of pure-direct-injective sheaves with respect to the geometrical purity. We investigate whether pure-direct-injective objects are preserved or reflected by the three functors associated with an open subset: the restriction, the extension by zero, and the direct image. We also exhibit a few examples for the pure-direct-injective sheaves, which enable us to distinguish pure-injective sheaves from the pure-direct-injective sheaves. Finally, we show that any coherent sheaf is pure-injective in the subcategory of coherent sheaves over a Noetherian scheme [Formula: see text]. Also, over [Formula: see text], where [Formula: see text] is an Artinian principal ideal ring, every quasi-coherent sheaf is pure-direct-injective in the subcategory of quasi-coherent sheaves.

This note proposes a new method to complete a triangulated category, which is based on the notion of a Cauchy sequence. We apply this to categories of perfect complexes. It is shown that the bounded derived category of finitely presented modules over a right coherent ring is the completion of the category of perfect complexes. The result extends to non-affine noetherian schemes and gives rise to a direct construction of the singularity category. The parallel theory of completion for abelian categories is compatible with the completion of derived categories. There are three appendices. The first one by Tobias Barthel discusses the completion of perfect complexes for ring spectra. The second one by Tobias Barthel and Henning Krause refines for a separated noetherian scheme the description of the bounded derived category of coherent sheaves as a completion. The final appendix by Bernhard Keller introduces the concept of a morphic enhancement for triangulated categories and provides a foundation for completing a triangulated category.

Intersection sheaves are usually defined for a projective flat surjective morphism of Noetherian schemes of relative dimension d and for d+1 invertible sheaves on the ambient scheme. In this article, the construction is generalized to the case of the equi-dimensional projective surjective morphisms to normal separated Noetherian schemes. Applications to the studies on family of effective algebraic cycles and on polarized endomorphisms are also given.

The derived category of quasi-coherent sheaves and axiomatic stable homotopy

Morita theory for non–commutative noetherian schemes

This paper aims at studying the homotopy category of cotorsion flat left modules $${{\mathbb {K}}({\mathrm{CotF}}\text {-}R)}$$ over a ring R. We prove that if R is right coherent, then the homotopy category $${\mathbb {K}}(\mathrm{dg}\text {-}\mathrm{CotF}\text {-}R)$$ of dg-cotorsion complexes of flat R-modules is compactly generated. This uses firstly the existence of cotorsion flat preenvelopes over such rings and, secondly, the existence of a complete cotorsion pair $$({{\mathbb {K}}_{\mathrm{p}}({\mathrm{Flat}}\text {-}R)}, {\mathbb {K}}(\mathrm{dg}\text {-}\mathrm{CotF}\text {-}R))$$ in the homotopy category $${{\mathbb {K}}({\mathrm{Flat}}\text {-}R)}$$ of complexes of flat R-modules, for arbitrary R. In the setting of quasi coherent sheaves over a Noetherian scheme, this cotorsion pair was discovered in the literature. However, we use a more elementary argument that gives this cotorsion pair for arbitrary R. Next we deal with cotorsion flat resolutions of complexes and define and study the notion of cotorsion flat dimension for complexes of flat R-modules. We also obtain an equivalence $${\mathbb {K}}(\mathrm{dg}\text {-}\mathrm{CotF}\text {-}R)\approx {{\mathbb {K}}({\mathrm{Proj}}\text {-}R)}$$ of triangulated categories where $${{\mathbb {K}}({\mathrm{Proj}}\text {-}R)}$$ is the homotopy category of projective R-modules. Combined with the aforementioned result, this recovers a result from Neeman, asserting the compact generation of $${{\mathbb {K}}({\mathrm{Proj}}\text {-}R)}$$ over right coherent R. Also we get the unbounded derived category $${\mathbb {D}} (R)$$ of R as a Verdier quotient of $${\mathbb {K}}(\mathrm{dg}\text {-}\mathrm{CotF}\text {-}R)$$ .

Relative tensor triangular Chow groups for coherent algebras

Totally acyclic complexes over noetherian schemes

We prove that the derived category of a Grothendieck abelian category has a unique dg enhancement. Under some additional assumptions, we show that the same result holds true for its subcategory of compact objects. As a consequence, we deduce that the unbounded derived category of quasi-coherent sheaves on an algebraic stack and the category of perfect complexes on a noetherian concentrated algebraic stack with quasi-finite affine diagonal and enough perfect coherent sheaves have a unique dg enhancement. In particular, the category of perfect complexes on a noetherian scheme with enough locally free sheaves has a unique dg enhancement.

For a semiseparated noetherian scheme, we show that the category of cotorsion Gorenstein flat quasi-coherent sheaves is Frobenius and a natural non-affine analogue of the category of Gorenstein projective modules over a noetherian ring. We show that this coheres perfectly with the work of Murfet and Salarian that identifies the pure derived category of F -totally acy- clic complexes of flat quasi-coherent sheaves as the natural non-affine analogue of the homotopy category of totally acyclic complexes of projective modules.

Let $(\mathcal C,\otimes,1)$ be an abelian symmetric monoidal category satisfying certain conditions and let $X$ be a scheme over $(\mathcal C,\otimes,1)$ in the sense of Toen and Vaquie. In this paper we show that when $X$ is quasi-compact and semi-separated, any quasi-coherent sheaf on $X$ may be expressed as a directed colimit of its finitely generated quasi-coherent submodules. Thereafter, we introduce a notion of objects in $(\mathcal C,\otimes,1)$ that satisfy several properties similar to those of fields in usual commutative algebra. Finally we show that the points of a Noetherian, quasi-compact and semi-separated scheme $X$ over such a field object $K$ in $(\mathcal C,\otimes,1)$ can be recovered from certain kinds of functors between categories of quasi-coherent sheaves. The latter is a partial generalization of some recent results of Brandenburg and Chirvasitu.