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

We prove in this paper that for a quasi-compact and semi-separated (nonnecessarily noetherian) scheme X, the derived category of quasi-coherent sheaves over X, D(Aqc(X)), is a stable homotopy category in the sense of Hovey, Palmieri and Strickland, answering a question posed by Strickland. Moreover we show that it is unital and algebraic. We also prove that for a noetherian semi-separated formal scheme X, its derived category of sheaves of modules with quasi-coherent torsion homologies Dqct(X) is a stable homotopy category. It is algebraic but if the formal scheme is not a usual scheme, it is not unital, therefore its abstract nature differs essentially from that of the derived category Dqc(X) (which is equivalent to D(Aqc(X))) in the case of a usual scheme.     

Cartesian Closed Topological Categories and Tensor Products

The projective tensor product in a category of topological R-modules (where R is a topological ring) can be defined in Top, the category of topological spaces, by the same universal property used to define the tensor product of R-modules in Set. In this article, we extend this definition to an arbitrary topological category X and study how the Cartesian closedness of X is related to the monoidal closedness of the category of R-module objects in X. Mathematics Subject Classifications (2000) 18D15, 18D35, 18A40.     

Unifying two results of Orlov on singularity categories

Let \mathbbX\mathbb{X} be a separated Noetherian scheme of finite Krull dimension which has enough locally free sheaves of finite rank and let U í \mathbbXU\subseteq \mathbb{X} be an open subscheme. We prove that the singularity category of U is triangle equivalent to the Verdier quotient triangulated category of the singularity category of \mathbbX\mathbb{X} with respect to the thick triangulated subcategory generated by sheaves supported in the complement of U. The result unifies two results of Orlov. We also prove a noncommutative version of this result.     

On the Existence of a Compact Generator on the Derived Category of a Noetherian Formal Scheme

In this paper, we prove that for a noetherian formal scheme \mathfrak X\mathfrak X, its derived category of sheaves of modules with quasi-coherent torsion homologies D_qct(\mathfrak X)\boldsymbol{\mathsf{D}}_\mathsf{qct}(\mathfrak X) is generated by a single compact object. In an Appendix we prove that the category of compact objects in D_qct(\mathfrak X)\boldsymbol{\mathsf{D}}_\mathsf{qct}(\mathfrak X) is skeletally small.     

Triangulated functors,homological functors,tilts, and lifts

 Let T be a triangulated category and let X be an object of T. This paper studies the questions: Does there exist a triangulated functor G : D(ℤ)  T with G(ℤ)≌X? Does there exist a triangulated functor H : T  D(ℤ) with h⁰ ⊚ H ⋍ Hom_T (X, −)? To what extent are G and H unique? One spin off is a proof that the homotopy category of spectra is not the stable category of any Frobenius category with set indexed coproducts. Received: 8 March 2002 / Revised version: 18 October 2002 Published online: 14 February 2003 Mathematics Subject Classification (2000): 18E30, 55U35     

Naïve noncommutative blowups at zero-dimensional schemes

In an earlier paper [D.S. Keeler, D. Rogalski, J.T. Stafford, Naïve noncommutative blowing up, Duke Math. J. 126 (2005) 491–546, MR 2120116], we defined and investigated the properties of the naïve blowup of an integral projective scheme X at a single closed point. In this paper we extend those results to the case when one naïvely blows up X at any suitably generic zero-dimensional subscheme Z. The resulting algebra A has a number of curious properties; for example it is noetherian but never strongly noetherian and the point modules are never parametrized by a projective scheme. This is despite the fact that the category of torsion modules in qgr-A is equivalent to the category of torsion coherent sheaves over X. These results are used in the companion paper [D. Rogalski, J.T. Stafford, A class of noncommutative projective surfaces, in press] to prove that a large class of noncommutative surfaces can be written as naïve blowups.     

A Serre-Swan Theorem for Bundles of Topological Modules

Let R be a unital topological ring whose set of invertible elements is open and inversion is continuous, and let X be a compact Hausdorff space admitting continuous R-valued partitions of unity. Considering bundles over X of fibre type a projective finitely generated R-module, we prove a Serre-Swan type theorem: namely, the category of these bundles is equivalent to the category of projective finitely generated modules over the ring of continuous R-valued functions on X.     

Projective limits of MV-spaces

An MV-space is a topological space X such that there exists an MV-algebra A whose prime spectrum Spec A is homeomorphic to X. The characterization of the MV-spaces is an important open problem.We shall prove that any projective limit of MV-spaces in the category of spectral spaces is an MV-space. In this way, we obtain new classes of MV-spaces related to some preservation properties of the Belluce functor.     

On the derived category of a regular toric scheme

Let X be a quasi-compact scheme, equipped with an open covering by affine schemes U _σ = Spec A ^σ . A quasi-coherent sheaf on X gives rise, by taking sections over the U _σ , to a diagram of modules over the coordinate rings A ^σ , indexed by the intersection poset Σ of the covering. If X is a regular toric scheme over an arbitrary commutative ring, we prove that the unbounded derived category of quasi-coherent sheaves on X can be obtained from a category of Σ^op-diagrams of chain complexes of modules by inverting maps which induce homology isomorphisms on hyper-derived inverse limits. Moreover, we show that there is a finite set of weak generators, one for each cone in the fan Σ. The approach taken uses the machinery of Bousfield–Hirschhorn colocalisation of model categories. The first step is to characterise colocal objects; these turn out to be homotopy sheaves in the sense that chain complexes over different open sets U _σ agree on intersections up to quasi-isomorphism. In a second step it is shown that the homotopy category of homotopy sheaves is equivalent to the derived category of X.     

Noetherianity of the space of irreducible representations

LetR be an associative ring with identity. We study an elementary generalization of the classical Zariski topology, applied to the set of isomorphism classes of simple leftR-modules (or, more generally, simple objects in a complete abelian category). Under this topology the points are closed, and whenR is left noetherian the corresponding topological space is noetherian. IfR is commutative (or PI, or FBN) the corresponding topological space is naturally homeomorphic to the maximal spectrum, equipped with the Zariski topology. WhenR is the first Weyl algebra (in characteristic zero) we obtain a one-dimensional irreducible noetherian topological space. Comparisons with topologies induced from those on A. L. Rosenberg’s spectra are briefly noted. The author’s research was supported in part by NSF grants DMS-9970413 and DMS-0196236.     

A Gabriel Theorem for coherent twisted sheaves

The aim of this work is to give a generalization of Gabriel’s Theorem on coherent sheaves to coherent twisted sheaves on noetherian schemes. We start by showing that we can recover a noetherian scheme X from the category Coh(X, α) of coherent α-twisted sheaves over X, where α lies in the cohomological Brauer group of X. This follows from the bijective correspondence between closed subsets of X and Serre subcategories of finite type of Coh(X, α). Moreover, any equivalence between Coh(X, α) and Coh(Y, β), where X and Y are noetherian schemes, and , β Br ′(Y) induces an isomorphism between X and Y.     

Desingularization of quasi-excellent schemes in characteristic zero

Grothendieck proved in EGA IV that if any integral scheme of finite type over a locally noetherian scheme X admits a desingularization, then X is quasi-excellent, and conjectured that the converse is probably true. We prove this conjecture for noetherian schemes of characteristic zero. Namely, starting with the resolution of singularities for algebraic varieties of characteristic zero, we prove the resolution of singularities for noetherian quasi-excellent Q-schemes.     

Function Spaces and Continuous Algebraic Pairings for Varieties

Given a quasi-projective complex variety X and a projective variety Y, one may endow the set of morphisms, Mor(X, Y), from X to Y with the natural structure of a topological space. We introduce a convenient technique (namely, the notion of a functor on the category of 'smooth curves') for studying these function complexes and for forming continuous pairings of such. Building on this technique, we establish several results, including (1) the existence of cap and join product pairings in topological cycle theory; (2) the agreement of cup product and intersection product for topological cycle theory; (3) the agreement of the motivic cohomology cup product with morphic cohomology cup product; and (4) the Whitney sum formula for the Chern classes in morphic cohomology of vector bundles.     

Recovering Quivers from Derived Quiver Representations

We compute Balmer's prime spectrum for the derived category of quiver representations for a finite ordered quiver with the vertex-wise tensor product and show that it does not recover the quiver. We then associate an algebra to every k-linear triangulated tensor category and show that the path algebra can be recovered in this way.     

Curve counting via stable pairs in the derived category

For a nonsingular projective 3-fold X, we define integer invariants virtually enumerating pairs (C,D) where C⊂X is an embedded curve and D⊂C is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of X. The resulting invariants are conjecturally equivalent, after universal transformations, to both the Gromov-Witten and DT theories of X. For Calabi-Yau 3-folds, the latter equivalence should be viewed as a wall-crossing formula in the derived category. Several calculations of the new invariants are carried out. In the Fano case, the local contributions of nonsingular embedded curves are found. In the local toric Calabi-Yau case, a completely new form of the topological vertex is described.     

Scalar Extensions of Triangulated Categories

Given a triangulated category ${{\mathcal T}}$ over a field K and a field extension L/K, we investigate how one can construct a triangulated category ${{\mathcal T}}_L$ over L. Our approach produces the derived category of the base change scheme X _L if ${{\mathcal T}}$ is the bounded derived category of a smooth projective variety over K and the field extension is finite and Galois. We also investigate how the dimension of a triangulated category behaves under scalar extensions.     

General Heart Construction on a Triangulated Category (I): Unifying <Emphasis Type="Italic">t</Emphasis>-Structures and Cluster Tilting Subcategories

In the paper of Keller and Reiten, it was shown that the quotient of a triangulated category (with some conditions) by a cluster tilting subcategory becomes an abelian category. After that, Koenig and Zhu showed in detail, how the abelian structure is given on this quotient category, in a more abstract setting. On the other hand, as is well known since 1980s, the heart of any t-structure is abelian. We unify these two constructions by using the notion of a cotorsion pair. To any cotorsion pair in a triangulated category, we can naturally associate an abelian category, which gives back each of the above two abelian categories, when the cotorsion pair comes from a cluster tilting subcategory, or a t-structure, respectively.     

Zero-divisor graph of C(X)

Summary In this article the zero-divisor graph Γ(C(X)) of the ring C(X) is studied. We associate the ring properties of C(X), the graph properties of Γ(C(X)) and the topological properties of X. Cycles in Γ(C(X)) are investigated and an algebraic and a topological characterization is given for the graph Γ(C(X)) to be triangulated or hypertriangulated. We have shown that the clique number of Γ(C(X)), the cellularity of X and the Goldie dimension of C(X) coincide. It turns out that the dominating number of Γ(C(X)) is between the density and the weight of X. Finally we have shown that Γ(C(X)) is not triangulated and the set of centers of Γ(C(X)) is a dominating set if and only if the set of isolated points of X is dense in X if and only if the socle of C(X) is an essential ideal.