When does the canonical tt-structure restrict to perfect complexes?

I am interested in non-Noetherian(!) rings such that the canonical t-structure on D(R) (the derived category of left R-modules) restricts to perfect complexes i.e. to the subcategory of complexes of finitely generated projective R-modules; i.e., for a complex C of this sort the complex …0→Coker(C−1→C0)→C1→C2→… should be quasi-isomorphic to a perfect complex. This is equivalent … Read more

Relative completeness of a relative cocompletion of a subcategory

I’m going to use the language from Lack and Rosicky’s Notions of Lawvere theory, but I won’t be touching on actual enriched category theory. Suppose I have a category C with a class of limits and colimits, Φ,Ψ respectively, that commute with each other. Now suppose I have a subcategory C′ that is closed to … Read more

Continuity property for Čech cohomology

Suppose we have an inverse system of compact Hausdorff spaces {Xi,φij}i∈I and that each space has a presheaf Γi assigned to it in such a way that Γi(φij(U))=Γj(U) whenever i≤j. Then X:=lim←Xi has a presheaf Γ defined on it by Γ(U):=Γi(φi(U)) where φi:X→Xi is the map from X as an inverse limit; this is well-defined … Read more

Canonical presentation of pro-modules over pro-rings

Let $A = (\dotsc \twoheadrightarrow A_2 \twoheadrightarrow A_1 \twoheadrightarrow A_0)$ be a (commutative) pro-ring with surjective transition maps. Consider the category $\mathcal{M} := \varprojlim_i \,\mathsf{Mod}(A_i)$: Objects are families of right $A_i$-modules $M=(M_i)$ together with isomorphisms $M_{i+1} \otimes_{A_{i+1}} A_i \cong M_i$. We let $\widehat{M} := \varprojlim_i M_i$. For each $j$, there is a natural epimorphism of … Read more

Chow group over function field and algebraic equivalence

It is known that for smooth projective varieties X,Y over k=ˉk, CHd(Xk(Y))=lim→U⊂Y openCHd(X×kU) I was wondering whether there was such an equality with algebraic equivalence (instead of rational equivalence). Answer It is false: see Remark 2.3.9 2) here. (Appeared in AKT 1 (2016), 379-440.) AttributionSource : Link , Question Author : user100915 , Answer Author : … Read more

Dual of colimit in Ban1\text{Ban}_1

I learned in J. Castillo’s Hitchhiker guide to categorical Banach space theory that, by a theorem of Semadeni and Zidenberg, limits and colimits exist in the category Ban1 of Banach spaces and contractive maps. I am hoping now that the Banach space dual of a projective limit is the inductive limit of the corresponding duals; … Read more

Do coproducts of infinity-groupoids commute with pullbacks?

As stated in this question, coproducts commute with pullbacks in the category of sets. Let $Grpd_{\infty}$ denote the $\infty$-category of $\infty$-groupoids. Do coproducts commute with pullbacks in $Grpd_{\infty}$? Answer Yes, they do, if you mean what is meant in the question you linked, but it is misleading to describe this property as “coproducts commute with … Read more

Algorithmically deciding existence of finite limits in a category

Given Σ a consistent finite first order theory in vocabulary L, one can consider the category of its models M(Σ), its objects are the models of Σ and arrows are exactly L-structure homomorphisms. After fixing a finite index category J we define a decision problem P(J)={Σ a finite theory∣∀D∈Func(J,M(Σ))limD exists}. I want to ask if it’s known … Read more

Computation on homotopy colimit cocomplete triangulated categories

I have a couple of questions about dealing with homotopy (co)limits cocomplete triangulated categories. Question I:The first one concerns a comment by Peter Arndt in this discussion about derived categories regarding posibibility to calculate the homotopy colimit when working with nice enough category. Peter wrote: I also find this a very enlightening view point, but … Read more

Relation between dual of nuclear space $(\substack{\text{lim} \\ \leftarrow i} H_i)’$ and $\substack{\text{colim} \\ i \rightarrow } H_i$

Let $\substack{\text{lim} \\ \leftarrow i} H_i$ be a nuclear space, considered as the limit of the codirected diagram $$… \to H_2 \to H_1 \to H_0,$$ with $f_{ji}:H_i \to H_j$ being the trace class operators in the diagram. I’m hoping that $$(\substack{\text{lim} \\ \leftarrow i} H_i)’ \cong \substack{\text{colim} \\ i \rightarrow} H_i$$ holds as topological vector … Read more