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Tensor Categorical Foundations of Algebraic Geometry

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Tensor Categorical Foundations of Algebraic Geometry Tensor categorical foundations of algebraic geometry Martin Brandenburg { 2014 { Abstract Tannaka duality and its extensions by Lurie, Sch¨appi et al. reveal that many schemes as well as algebraic stacks may be identified with their tensor categories of quasi-coherent sheaves. In this thesis we study constructions of cocomplete tensor categories (resp. cocontinuous tensor functors) which usually correspond to constructions of schemes (resp. their morphisms) in the case of quasi-coherent sheaves. This means to globalize the usual local-global algebraic geometry. For this we first have to develop basic commutative algebra in an arbitrary cocom- plete tensor category. We then discuss tensor categorical globalizations of affine morphisms, projective morphisms, immersions, classical projective embeddings (Segre, Pl¨ucker, Veronese), blow-ups, fiber products, classifying stacks and finally tangent bundles. It turns out that the universal properties of several moduli spaces or stacks translate to the corresponding tensor categories. arXiv:1410.1716v1 [math.AG] 7 Oct 2014 This is a slightly expanded version of the author's PhD thesis. Contents 1 Introduction 1 1.1 Background . .1 1.2 Results . .3 1.3 Acknowledgements . 13 2 Preliminaries 14 2.1 Category theory . 14 2.2 Algebraic geometry . 17 2.3 Local Presentability . 21 2.4 Density and Adams stacks . 22 2.5 Extension result . 27 3 Introduction to cocomplete tensor categories 36 3.1 Definitions and examples . 36 3.2 Categorification . 43 3.3 Element notation . 46 3.4 Adjunction between stacks and cocomplete tensor categories . 49 4 Commutative algebra in a cocomplete tensor category 53 4.1 Algebras and modules . 53 4.2 Ideals and affine schemes . 58 4.3 Symtrivial objects . 63 4.4 Symmetric and exterior powers . 65 4.5 Derivations . 74 4.6 Flat objects . 81 4.7 Dualizable objects . 83 4.8 Invertible objects . 86 4.9 Locally free objects . 90 4.10 Descent theory . 95 4.11 Cohomology . 101 5 Constructions with cocomplete tensor categories 102 5.1 Basic free constructions . 102 5.1.1 Free tensor categories . 102 5.1.2 Cocompletions . 103 5.1.3 Indization . 108 5.1.4 Limits and colimits . 109 5.2 Global schemes and stacks . 111 5.3 Module categories over algebras . 114 5.4 Gradings . 119 5.5 Representations . 124 5.6 Local functors . 129 5.7 Tensoriality . 134 5.7.1 The case of qc qs schemes . 134 5.7.2 The case of algebraic stacks . 137 5.8 Localization . 140 5.8.1 General theory . 140 5.8.2 Examples . 149 5.8.3 Localization at sections . 152 5.8.4 Ideals . 154 5.9 Idempotents . 157 5.10 Projective tensor categories . 160 5.10.1 Definition and comparison to schemes . 160 5.10.2 Blow-ups . 166 5.10.3 The Segre embedding . 168 5.10.4 The Veronese embedding . 172 5.10.5 The Pl¨ucker embedding . 174 5.11 Products of schemes . 177 5.11.1 Tensorial base change . 178 5.11.2 The case of qc qs schemes over a field . 180 5.12 Tangent tensor categories . 187 6 Monoidal monads and their modules 206 6.1 Overview . 206 6.2 Reflexive coequalizers . 208 6.3 Monoidal monads . 212 6.4 Tensor product of modules . 218 6.5 Module categories . 228 Bibliography 235 \According to Peter Freyd: 'Perhaps the purpose of categorical al- gebra is to show that which is trivial is trivially trivial.' That was written early on, in 1966. I prefer an update of that quote: \Perhaps the purpose of categorical algebra is to show that which is formal is formally formal". It is by now abundantly clear that mathematics can be formal without being trivial. Categorical algebra allows one to articulate analogies and to perceive unexpected relationships between concepts in different branches of mathematics." Peter May, [May01] Chapter 1 Introduction 1.1 Background It is a common principle of mathematics to study geometric objects by consid- ering categories of algebraic objects acting on them. For example in algebraic geometry one tries to understand a scheme X via its category of quasi-coherent sheaves Qcoh(X). Do we really understand X by looking at Qcoh(X)? This has been answered in 1962 by Gabriel ([Gab62]) who proved his famous Reconstruction Theorem: A noetherian scheme X can be reconstructed from Qcoh(X). The idea is to associate to an abelian category A a ringed space Spec(A), its spectrum, and then prove X ∼= SpecQcoh(X). Later this paved the way for the vision of noncommutative algebraic geometry in which abelian categories are regarded as noncommutative schemes ([Ros95], [AZ94]). In 1998 A. L. Rosenberg used a different spectrum construction to give a short proof of the Reconstruction Theorem for arbitrary schemes ([Ros96], [Ros98]), then in 2004 another proof for quasi-separated schemes ([Ros04]). The proof has been corrected and simplified by Gabber ([Bra13]). In 2013 Antieau obtained a generalization of the Reconstruction Theorem to twisted quasi-coherent sheaves on quasi-compact quasi-separated schemes ([Ant13]). With a completely new approach Calabrese and Groechenig have proven a Reconstruction Theorem for quasi-compact separated algebraic spaces ([CG13]). Although a smooth variety X cannot be reconstructed from its derived category of quasi-coherent sheaves Db(Qcoh(X)), Bondal and Orlov have shown in 2001 that this works at least if the canonical bundle of X is ample or anti-ample ([BO01]). However, Qcoh(X) is actually a tensor category, so it is reasonable to use this additional structure. In 2002 Balmer has reconstructed a noetherian scheme from its tensor triangulated category of perfect complexes ([Bal02]). This lead to the development of tensor triangular geometry ([Bal10]). Balmer's result has been generalized to quasi-compact quasi-separated schemes by Buan, Krause and Solberg ([BKS07]). 1 Chapter 1. Introduction What about reconstruction for algebraic stacks? Classical Tannaka duality gives an affirmative answer in the case of classifying stacks since it reconstructs an affine group scheme from its tensor category of representations ([Del90]). More generally, Lurie has reconstructed an arbitrary geometric stack from its tensor category of quasi-coherent sheaves ([Lur04]). In the case of algebraic stacks the tensor structure becomes essential, since for example the discrete scheme with two points and the classifying stack of a group of order 2 have the same category of quasi-coherent sheaves { but not the same tensor category of quasi-coherent sheaves. Obviously very much has been done in order to recover the object of interest X from Qcoh(X). However, it seems that little has been done towards the reconstruction of morphisms of schemes, which is certainly useful for recovering properties of X (such as smoothness) from Qcoh(X). One might ask which functors Qcoh(X) ! Qcoh(Y ) are induced by morphisms Y ! X via the pullback construction. A minimal requirement is that these functors preserve colimits and tensor prod- ucts (up to specified isomorphisms). Therefore, let us call a scheme (or stack) X tensorial if colimit-preserving tensor functors Qcoh(X) ! Qcoh(Y ) are induced by morphisms Y ! X. Then for example affine and projective schemes are ten- sorial ([Bra11]). More generally, in 2012 Chirvasitu and the author showed that any quasi-compact quasi-separated scheme is tensorial ([BC14]). In the same year, Sch¨appiproved that Adams stacks (i.e. geometric stacks with enough lo- cally free sheaves) are tensorial ([Sch12a]). He even gave a classification of those tensor categories which are equivalent to Qcoh(X) for some Adams stack X ([Sch12a],[Sch13]). It seems to be an open problem if every algebraic stack (satisfying mild finiteness conditions) is tensorial and to characterize those tensor categories which are equivalent to Qcoh(X) for some algebraic stack X. This thesis aims to expand the relationship between algebraic geometry and cocomplete tensor categories. We will find constructions with cocomplete ten- sor categories which correspond to constructions with schemes or stacks un- der Qcoh(−). This leads to a theory which might be called tensor categori- cal algebraic geometry or 2-algebraic geometry which has been already intro- duced by Chirvasitu and Johnson-Freyd ([CJF13]). In some sense this is like Rosenberg's noncommutative algebraic geometry with additional tensor products (which makes it commutative) or Balmer's tensor triangular geometry without triangulations. The precise results will be explained in the next section. 2 1.2. Results 1.2 Results Tensoriality A cocomplete tensor category is a symmetric monoidal category with all colimits which commute with the tensor product in each variable (section 3.1). These constitute a 2-category Catc⊗. The Hom-categories of cocontinuous tensor func- tors are denoted by Homc⊗(−; −). We often restrict to linear categories over some fixed commutative ring. We have the following adjunction (section 3.4) of 2-categories: Spec u op Stack > Cat 5 c⊗ Qcoh op Here, the 2-functor Qcoh : Stack ! Catc⊗ associates to a stack in the fpqc topol- ogy its cocomplete tensor category of quasi-coherent modules and to a morphism ∗ op f its pullback functor f . Its right adjoint Spec : Catc⊗ ! Stack is defined by Spec(C)(X) := Homc⊗ C; Qcoh(X) : We call the fixed points of this adjunction tensorial stacks resp. stacky tensor categories. Hence, we get an equivalence of 2-categories ftensorial stacksg ' fstacky tensor categoriesgop: Specifically, a stack X is tensorial if X ' SpecQcoh(X); i.e. for every scheme Y the pullback functor ∗ Hom(Y; X) ! Homc⊗ Qcoh(X); Qcoh(Y ) ; f 7! f is an equivalence of categories. The following results have already appeared, but shorter proofs will be byprod- ucts of our theory (section 5.7): Theorem 1.1. 1. [Lur04] If X is geometric, then ∗ Hom(Y; X) ! Homc⊗=∼ Qcoh(X); Qcoh(Y ) ; f 7! f is fully faithful.
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