Questions tagged [derived-categories]

For questions about the derived categories of various abelian categories and questions regarding the derived category construction itself.

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47 views

Heart of a bounded $t$-structure on the derived category of coherent sheaves

Let $X$ be an elliptic curve and $D(X)$ the bounded derived category of $Coh(X)$, coherent sheaves on $X$. If $(D^{\leq 0}, D^{>0})$ is a bounded $t$-structure, then can we already say that the ...
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K-equivalence => isomorphism of Chow motives?

An old conjecture of Bondal-Orlov-Kawamata predicts that K-equivalent varieties are D-equivalent, see Kawamata's paper for definitions. In particular this applies to birational Calabi-Yau varieties. ...
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85 views

The derived $\infty$-category of sheaves on a site is closed symmetric monoidal

Let $X$ be a quasicompact semiseparated scheme. I am trying to recover the (closed) symmetric monoidal structure on $\mathcal{D}(\mathrm{QCoh}(X))$, the derived $\infty$-category of quasicoherent ...
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1answer
122 views

Derived Category of strictly simplicial algebraic space vs. systems of objects in the derived categories

Let $X_{\bullet}^+$ be a strictly simplicial algebraic space and for a morphism $\delta:[m]\to[n]$ in $\Delta^+$, let $\delta:X_n\to X_m$ also denote the associated map (by abuse of notation). Then ...
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What is equivariant chains on a representation sphere?

For a finite group $G$ and a finite-dimensional real representation $V$ of $G$, denote by $S^V$ the one-point compactification of $V$, with basepoint at infinity. What is the reduced chain complex $...
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1answer
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Rickard's strengthening of Broué's abelian defect group conjecture and the lifting of some equivalences up to splendid derived equivalences

Let $G$ be a finite group. Let $p$ be a prime dividing $|G|$. Let $K:=\overline{\mathbb{F}_p}$. Let $b$ be a $p$-block of $G$ with abelian defect group $D$. Let $H:=N_G(D)$. Let $c$ be the Brauer ...
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1answer
189 views

Conflicting definitions of RHom

I am trying to understand the bifunctor $R\operatorname{Hom} : D(\mathcal{A}) ^{op} \times D(\mathcal{A}) \to D(\operatorname{Ab})$ (I am also interested in the total right derived functor of the ...
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1answer
146 views

2-limits of triangulated categories

Let $\mathcal{D}_{i}$ be a family of triangulated categories, labelled by a countable poset $I$ with a lowest element. Suppose further that for $i\leq j$, we have exact functors $F_{i,j}: \mathcal{D}_{...
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1answer
98 views

Serre functor on the category $Perf(A)$, $A$ - k-algebra

Consider a finite-dimensional $k$-algebra $A$ of finite global dimension. Then it is known that the Serre functor on $D^b(mod-A)$ exists and is given by the Nakayama functor. The proof goes something ...
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123 views

3x3 lemma in triangulated categories

I am currently reading Le Stum's Rigid Cohomology and have encountered the following passage (proof of Proposition 5.2.16): The deduction made here seems to be purely "triangulated category-...
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147 views

Which derived categories of coherent sheaves are equivalent (or “$t$-related”) to derived categories of rings?

As far as I understand, it was Beilinson who proved that the bounded derived category of coherent sheaves $D^b(\mathbb{P}^n)$ is equivalent to the bounded derived category of a certain (non-...
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61 views

Is the secondary Euler characteristic a categorical trace?

Context: The ordinary Euler characteristic of a complex (satisfying appropriate finiteness conditions so that all cohomology groups are finite-dimensional over some field ''k'', say, and only finitely ...
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1answer
782 views

Computations in condensed mathematics, page 32-34

I started reading the Lectures on Condensed Mathematics. I am looking at the material at page 32-34. I have three fundamental computation questions: At the last line of pg 32 - it seems to imply that ...
4
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1answer
162 views

Questions about $\text{Perf}(A)$ of dg algebra $A$

[ALEXEY ELAGIN AND VALERY A. LUNTS, p.4.] Recall that triangulated category $\text{Perf}(A)$ is defined as the full thick triangulated subcategory of $D(A)$ generated by the dg $A$-module $A$. [...
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100 views

Perverse sheaves and maximal genus Gopakumar-Vafa invariants

Let $f: X \to Y$ be a proper morphism between complex varieties (the varieties as well as the map may be non-smooth) and let $\phi \in \text{Perv}(X)$ be a perverse sheaf on $X$. Given this data, it ...
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502 views

What is the status of a result of Kontsevich and Rosenberg?

In their influential paper Noncommutative Smooth Spaces (https://arxiv.org/abs/math/9812158), Kontsevich and Rosenberg define the notion of a noncommutative projective space. In Section 3.3 they ...
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57 views

$L^r_M = i_* \circ \hat{L}^{r-1}_M \circ i^*$ by the projection formula and the Poincare duality

This is a question arising when I am reading M. A. A. de Cataldo, L. Migliorini - The Hard Lefschetz Theorem and the topology of semismall maps, Ann. sci. École Norm. Sup., Serie 4 35 (2002) 759-772. ...
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1answer
138 views

Cohomology of normal bundle and tangent bundle on Gushel-Mukai threefold

Let $X$ be a smooth general ordinary Gushel-Mukai threefold. There is an embedding $X\rightarrow\mathrm{Gr}(2,5):=G$. Consider the normal bundle $\mathcal{N}_{X|G}$, how to compute cohomology of this ...
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243 views

Is there a bestiary of “derived 2-vector spaces”?

The appendix "A bestiary of 2–vector spaces" of Bartlett, Douglas, Schommer-Pries, Vicary, "Modular categories as representations of the 3-dimensional bordism 2-category" analyzes ...
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47 views

Naive pushforward of D-modules and Gauss--Manin connection

Suppose that $f\colon X\to Y$ is a morphism of smooth quasi-projective varieties over a field of characteristic $0$. We then have a naive pullback functor $f^\circ:=\mathcal D_{X\to Y}\otimes_{f^{-1}\...
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Is there a direct way to show Fano surface of lines and conics on the pairs of Fano threefolds isomorphic?

I am considering the following setting: Let $(Y_d, X_{4d+2})$ be the pair of degree $d$ and index 2 Fano threefold $Y_d$ and degree $4d+2$ index 1 Fano threefold and both of them are Picard number 1. ...
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138 views

Smoothness of a variety implies homological smoothness of DbCoh

I have been told that $D^bCoh(X)$ is homologically smooth if $X$ is a smooth variety, and I am trying to construct a proof. My background is not in algebra, so I apologize for elementary questions. It ...
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1answer
400 views

Embedding of a derived category into another derived category

I am considering the following two cases: Assume that there is an embedding: $D^b(\mathcal{A})\xrightarrow{\Phi} D^b(\mathbb{P}^2)$and the homological dimension of $\mathcal{A}$ is equal to $1$($\...
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196 views

When is the cotangent complex perfect?

Let $X\rightarrow S$ be a proper flat morphism of schemes. When is the cotangent complex $L_{X/S}$ perfect ? It is well known, that for local complete intersections the cotangent complex is perfect, ...
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131 views

2 K3s and cubic fourfolds containing a plane

Two K3 surfaces show up when talking about cubic fourfolds containing a plane. Let $P\subset X\subset \mathbb{P}^5$ be the plane inside the cubic. Since $P$ is cut out by 3 linear equations then $X$ ...
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222 views

Are two versions of Kuznetsov components equivalent?

Let $X$ be a degree $10$, genus $6$ index $1$ and Picard number $1$ smooth Fano threefold. There are two versions of Semi-orthogonal decompositions. The First version is $$D^b(X)=\langle\mathrm{Ku}(X)...
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When is an algebra derived indecomposable?

Call a finite dimensional (acyclic) quiver $K$-algebra A derived indecomposable in case $A$ is not derived equivalent to an algebra of the form $B \otimes_K C$. For example when the number of simples ...
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1answer
181 views

Does formation of the derived $\infty$-category preserve pushouts?

Let $B\leftarrow A\to C$ be a diagram of commutative rings, and let $\mathcal{D}(A)$ be the derived $\infty$-category of $A$-modules (as in Lurie's "Higher Algebra"). Then is there an ...
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73 views

Cone of a morphism of complexes that are concentrated in degree $0$ and $1$

Let $R$ be a ring and $f:A\to A'$ and $g:B\to B'$ be morphisms of $R$-modules. Let $h:C_{\bullet}\to C_{\bullet}'$ be a morphism of $R$-module complexes fitting in a morphism of distinguished ...
3
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1answer
104 views

Smallness condition for augmented algebras

I'm not sure this question is research level question. Sorry in advance. Hypothesis $k$ is a commutative ring. $A$ is an augmented $k$-algebra. $A^e$ is defined as the $k$-algebra $A\otimes_{k}A^{op}$...
2
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2answers
460 views

On the link between homology and homotopy

In the last semester I learned homological algebra and higher category theory/homotopy theory. But I am kind of confused when I try to really understand the link between the two subjects (this is ...
2
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0answers
75 views

Cone of morphism in families

I am working in derived category $D^b(X)$ of coherent sheaves on a smooth projective varitey. Let $E,F$ be two sheaves on $X$, with $\mathrm{R}Hom(E,F)=k\oplus k[-1]$, I consider the following ...
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1answer
500 views

$H^{p,q}(X)$ versus $H^{q}(X, \bigwedge^p TX)$

Let $X$ be a Kahler manifold. To $X$ one can associate the cohomology groups $H^{p,q}(X)$, and $H^{(0,q)}(X, \bigwedge^p TX)$ with $TX$ being the holomorphic tangent bundle of $X$. Is there a general ...
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203 views

Variety over $\mathbb{F}_p$ that does not embed into flat scheme over $\mathbb{Z}/p^2\mathbb{Z}$

Let $X\to\mathrm{Spec}\:\mathbb{F}_p$ be a smooth proper morphism. Is there a closed immersion $X\to Y$ where $Y$ is flat of finite type over $\mathbb{Z}/p^2\mathbb{Z}$? As mentioned in the comments ...
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97 views

Computing the derived $I$ invariants $D(G\operatorname{Rep})\to D((G/I)\operatorname{Rep})$

In Scholze's paper The Langlands–Kottwitz approach for the modular curve (published version) about $\operatorname{GL}_2$ when he wants to find the relation between the semisimple trace and the usual ...
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101 views

detecting a semi-free module from its bar-resolution

Let $A$ be a DG-algebra over a field (say $k$). A DG-module $M$ over $A$ is said to be semi-free if it admits an exhaustive filtration $0 = M_0 \subset M_1 \subset \ldots \subset M_p = M$ such that ...
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63 views

Compute Cech cohomology with two open sets

Let $(X,\mathcal{O}_X)$ be a topological ringed space and $(U,V)$ be an open covering of $X$ (i.e. $U$ and $V$ are two open subsets of $X$ such that $U\cup V=X$). Let $\mathcal{F}$ be a sheaf of $\...
4
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0answers
55 views

Morphism of distinguished triangles where one of the arrows is a quasi-isomorphism

Let $R$ be any ring and let $A\to B\to C\to [1]$ and $A'\to B'\to C'\to [1]$ be distinguished triangles of complexes of $R$-modules. Let $f:A\to A'$, $g:B\to B'$ and $h:C\to C'$ be morphism of ...
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114 views

Commutative group stacks and Galois cohomology

"Classically", if we consider an abelian variety $A$ over some number field $k$, we get a $Gal(\bar{k}/k)$-module $A(\bar{k})$, or equivalently a sheaf of abelian groups on the étale site $\...
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Examples of “non equivalent” algebras that are derived equivalent?

One can define different equivalence relations between algebras depending on what one want to study, but also these definitions may have their own life and not result as one expected at first. My ...
3
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1answer
277 views

“Universal coefficent theorem” for pro-étale cohomology

In algebraic topology, for any space with finite homology type, the universal coefficient theorem states that for any abelian group $G$, we have $$H^n(X,G)\cong \left( H^n(X,\mathbb{Z})\otimes G\right)...
4
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1answer
149 views

Does localization at quasi-isomorphisms imply homotopy invariance?

Usually, the derived category of some abelian category $A$ (I'm happy already with $A$-mod) is defined first taking chain complexes up to homotopy, and then localize at quasi-isomorphisms. My question ...
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102 views

Understanding a step in proof of how the localization of an additive category by a subclass of morphism satisfies Ore is also additive

I started to study localization in additive and triangulated categories via a subclass of morphism which satisfies the Ore conditions by my own. Right now, I'm studying how for an additive category $...
4
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1answer
196 views

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 ...
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97 views

Constructing functorial homotopies in derived infinity-category

I'm interested in the following problem : let $\mathcal{C}$ be an $\infty$-category and $\mathcal{D}:=D_\infty(\mathbb{Z})$ the derived $\infty$-category of abelian groups. Consider functors $A, B, C ,...
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129 views

Isomorphic objects in the derived category

Let $\mathcal{A}$ be an abelian category, $D(\mathcal{A})$ be its derived category and $X,Y$ be complexes with morphisms in $\mathcal{A}$. I am trying to understand what does it mean to say that $X$ ...
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1answer
302 views

References for quivers and derived categories of coherent sheaves for a string theory student

I'm a student mostly from physics knowledge hoping to learn about the math involved the string theory paper Topological Quiver Matrix Models and Quantum Foam. Context: The topological string theory ...
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1answer
248 views

Is the derived category of local systems equivalent to the derived category of sheaves of vector spaces with local system cohomology?

Let $k$ be a field and $X$ a topological space. Write $\mathrm{Sh}(X)$ for the category of sheaves of vector spaces on $X$, and $\mathrm{Loc}(X)$ for the subcategory of local systems of finite ...
11
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2answers
379 views

The relation between t-structures and derived category

Let $\mathcal{D}$ be a triangulated category and a $t$-structure $(\mathcal{D}^{\leq 0},\mathcal{D}^{\geq 0})$ on $\mathcal{D}$. The heart of the $t$-structure, $\mathcal{A}=\mathcal{D}^{\leq 0} \cap ...
4
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0answers
48 views

Reflexive vs. pseudo-coherent abelian groups

Recall that a module M over some ring R is pseudo-coherent if it admits a resolution whose terms are finitely generated projective R-modules. Such a module is reflexive when regarded as an object in ...

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