Related papers: Derived categories of absolutely flat rings
We offer a solution to the long-standing problem of group completing within the context of rig categories (also known as bimonoidal categories). Given a rig category R we construct a natural additive group completion R' that retains the…
In this article we prove that all the inclusions between the 'classical' and naturally defined full triangulated subcategories of a weakly approximable triangulated category are intrinsic (in one case under a technical condition). This…
A totally umbilical submanifold in pseudo-Riemannian manifolds is a fundamental notion, which is characterized by the condition that the second fundamental form is proportional to the metric. It is also a generalization of the notion of a…
If $X$ is a quasi-compact and quasi-separated scheme, the category $Qcoh(X)$ of quasi-coherent sheaves on $X$ is locally finitely presented. Therefore categorical flat quasi-coherent sheaves naturally arise. But there is also the standard…
Fix a noetherian scheme S. For any flat map f: X->Y of separated essentially-finite-type perfect S-schemes we define a canonical derived-category map c(f):\H(X)->f^!\H(Y), the fundamental class of f, where \H(Z) is the (pre-)Hochschild…
We formulate a general abstract criterion for verifying the local-to-global principle for a rigidly-compactly generated tensor triangulated category. Our approach is based upon an inductive construction using dimension functions. Using our…
We continue our study of the local theory for quasiperiodic cocycles in $\mathbb{T} ^{d} \times G$, where $G=SU(2)$, over a rotation satisfying a Diophantine condition and satisfying a closeness-to-constants condition, by proving a…
It is proved that a map $\varphi\colon R\to S$ of commutative noetherian rings that is essentially of finite type and flat is locally complete intersection if and only $S$ is proxy small as a bimodule. This means that the thick subcategory…
We classify all non-degenerate skew-hermitian forms defined over certain local rings, not necessarily commutative, and study some of the fundamental properties of the associated unitary groups, including their orders when the ring in…
This paper establishes the homological and geometric foundations of non-commutative n-ary Gamma-semirings, unifying two previously distinct directions in Gamma-algebra: the derived Gamma-geometry developed for the commutative ternary case…
We show that every thick subcategory of the singularity category of a complete intersection ring is self dual. We also prove the analogous statement for thick subcategories of the bounded derived category and give applications to the…
We discuss dualisable objects in minimal subcategories of compactly generated tensor triangulated categories, paying special attention to the derived category of a commutative noetherian ring. A cohomological criterion for detecting these…
New homotopy invariant finiteness conditions on modules over commutative rings are introduced, and their properties are studied systematically. A number of finiteness results for classical homological invariants like flat dimension,…
We introduce and study relatively divisible and relatively flat objects in exact categories in the sense of Quillen. For every relative cotorsion pair $(\mathcal{A},\mathcal{B})$ in an exact category $\mathcal{C}$, $\mathcal{A}$ coincides…
Let $X$ be a quasi-compact and quasi-separated scheme. There are two fundamental and pervasive facts about the unbounded derived category of $X$: (1) $\mathsf{D}_{\mathrm{qc}}(X)$ is compactly generated by perfect complexes and (2) if $X$…
Principal affine open subsets in affine schemes are an important tool in the foundations of algebraic geometry. Given a commutative ring $R$, $\,R$-modules built from the rings of functions on principal affine open subschemes in…
We analyse a class of quantum field theory models illustrating some of the possibilities that have emerged in the general study of the short distance properties of superselection sectors, performed in a previous paper (together with R.…
Gillam proved that the category of locally ringed spaces admits a fully faithful embedding into a certain category, which has a right adjoint that maps some simple objects to the spectra of rings. In this paper, we use condensed mathematics…
We generalize the classical semiregularity theorem of Buchweitz and Flenner to the setting of noncommutative algebraic geometry, with group actions. This applies in particular to twisted derived categories, in which case it answers a…
We study smooth maps that arise in derived algebraic geometry. Given a map $A \to B$ between non-positive commutative noetherian DG-rings which is of flat dimension $0$, we show that it is smooth in the sense of To\"{e}n-Vezzosi if and only…