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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Fourier transform for constructible sheaves on spheres

Let $S_1 = S_2 = S^d$ be two copies of the $d$-dimensional sphere. Let $p_i : S_1 \times S_2 \to S_i$ be the projection, $j : U \to S_1 \times S_2$ the inclusion of the complement of the diagonal and $...
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Projectivization in the derived category of coherent sheaves

Let $X$ be a compact Kahler manifold. There exists a notion of projectivization of holomorphic vector bundles and coherent sheaves over $X$. Does that concept extend to objects in the derived category ...
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Understanding an involution of the category of perverse sheaves on $\mathbb{C}$

It is well-known (for example: chapter 2 in [GGM] A. Galligo, M. Granger, P. Maisonobe. D-modules et faisceaux pervers dont le support singulier est un croisement normal. Ann. Inst. Fourier Grenoble, ...
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6 votes
2 answers
282 views

Distinguished triangle of dualizing complexes and/or determinants?

Q1 : If $X \to Y \to Z$ are maps of schemes, is there a relation such as $$\omega_{X/Z} \overset{?}{=} \omega_{Y/Z}|_X \overset{L}{\otimes} \omega_{X/Y}$$ between their dualizing complexes? Or maybe ...
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6 votes
2 answers
227 views

Relative and absolute Ext groups

Given a homomorphism of rings $S \rightarrow R$, for a pair of $R$-modules $M, N$ the machinery of relative homological algebra defines relative $Ext$-groups $Ext_{R, S}(M, N)$. These can be defined, ...
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2 votes
0 answers
80 views

Existence of quasi-isomorphisms between complexes with same homology

Consider an abelian category $\mathcal{A}$ (or more specifically, $R$-Mod). Suppose $C_1$ and $C_2$ are chain complexes with componentwise isomorphic homology. What conditions must be imposed upon $\...
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7 votes
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278 views

Derived functor of functor tensor product

Suppose $\mathcal{A}$ is a Grothendieck abelian category with enough projectives, then $\mathcal{A}$ is tensored and cotensored over $\mathrm{Ab}$ with $\mathbb{Z}^{\oplus S}\otimes X\cong \bigoplus_S ...
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$\operatorname{Ext}$-group in the category of modules versus in the subcategory of finitely generated ones

I am trying to refine my understanding of derived categories. Let $\text{Mod}_R$ and $\text{Mod}^f_R$ be respectively the categories of modules and finitely generated modules over a Notherian ring $R$ ...
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2 votes
0 answers
86 views

A question about t-structures in derived category

Let R be a ring and $_{R}P$ be a projective module, my question is whether $P^{\perp_{>0}}:=\{X\in D(R)|Hom(P,X[i])=0, i>0\}$ is an aisle i.e. if $(P^{\perp_{>0}}, (P^{\perp_{>0}})^{\perp}...
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5 votes
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233 views

Appropriate notion of derived category over condensed set

If we have a compact Hausdorff space $S$, then my understanding is that the appropriate notion of the derived category of sheaves of condensed abelian groups is to consider the derived category $D_{\...
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Does Grothendieck duality hold without taking RHom?

I asked this very basic question about Grothendieck Duality on the Stack--exchange some time ago, without any replies. I'm therefore asking the question here to test my luck. Let $f:X\to Y$ be a ...
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Derived category of dg modules vs. graded modules over a formal dg-algebra

Let $R = \oplus_{i\geq0} R_i$, $R_0 = k$ ($k$ a field) be a positively-graded commutative noetherian algebra, regarded as an augmented dg-algebra with zero differential. Depending on one's interest, ...
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4 votes
1 answer
174 views

Which complexes of coherent sheaves can be presented as countable homotopy limits of perfect complexes?

Let $X$ be a noetherian scheme (actually, I need the case where $X$ is proper over an affine scheme), $C$ is an object of the derived category $D_{coh}(X)$ of coherent sheaves on $X$. Under which ...
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3 votes
1 answer
211 views

Confusion about the (Grothendieck–Poincaré) double dual of reflexive differentials vs usual differentials on a normal Cohen–Macaulay scheme

$\DeclareMathOperator\Hom{Hom}$Let $\mathcal{A}$ be an abelian category, my question is about the case when $\mathcal{A}$ is the category of quasi-coherent sheaves on a scheme $X$. There is a fully ...
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Pseudo-coherent complexes over sheaves of non-commutative rings

I am posing a question on derived categories to which I was not able to find an answer anywhere in the literature. I would appreciate any answer, hint or suggestion. Assume that $\mathcal{R}_X$ is a ...
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2 votes
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137 views

In which generality we have $K$-flat resolutions?

Let $\mathsf{A}$ be an abelian monoidal symmetric category (i.e., an abelian tensor category in the sense of Milne / Deligne). I wonder what more I should impose on $\mathsf{A}$ in order to have $K$-...
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1 vote
1 answer
132 views

Inducing an equivalence of $G$-equivariant categories

Suppose we have an equivalence of triangulated categories $\Phi : \mathcal{A} \to \mathcal{B}$. Let $G$ be a finite group. Are there any methods/conditions for specifying when one has an induced ...
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Computing the cotangent complex of morphisms of perfect complexes

In Lurie's Spectral Algebraic Geometry the cotangent complex of $\textbf{Perf}$ is computed as $ \Sigma^{-1}( \mathscr{F} \otimes \mathscr{F}^\vee)$ for some universal $\mathscr{F} \in \text{Qcoh}(\...
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Dimension of Hilbert scheme of curves on Gushel-Mukai varieties

I have several questions on Hilbert scheme of Gushel-Mukai varieties. Let $X$ be a Gushel-Mukai fourfold and let $\mathcal{H}_3$ be Hilbert scheme of twisted cubics. I was wondering what is the ...
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semi-orthogonal decomposition of Fano fourfold associated to threefold

Let $Y$ be Gushel-Mukai threefold and $X$ a Gushel-Mukai fourfold containing $Y$ as its hyperplane section, the semi-orthogonal decomposition of $X$ and $Y$ are both known. Also, for cubic fourfold ...
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2 votes
1 answer
143 views

Semi-orthogonal decompositions over singular schemes

Where can I find any more or less explicit semi-orthogonal decompositions of derived categories of perfect complexes or of bounded derived categories for singular schemes that are proper over a ring R?...
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Adjunctions of residue categories of Gushel-Mukai threefolds and Gushel-Mukai fourfolds

Let $X$ be an ordinary Gushel-Mukai fourfold and $Y$ its hyperplane section, which is a Gushel-Mukai threefold. I consider semi-orthogonal decompositions of $X$ and $Y$: $D^b(X)=\langle\mathcal{K}u(X),...
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4 votes
1 answer
198 views

When is the birational Torelli problem for CY threefolds true?

I am aware from Borisov, Căldăraru, Perry and Ottem, Rennemo that what is known as the birational Torelli problem is false in general for Calabi-Yau threefolds, but I would like to know if there are ...
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If a natural transform is an equivalence, under which circumstances is the induced derived natural transform also an equivalence?

More specifically, let A and B be two abelian categories. Suppose $F:A\to B$ is a left exact functor, $G:A\to A$ and $H:B\to B$ are two right exact functors such that $F\circ G=H\circ F$. With which ...
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2 votes
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Normal bundle of a Fano threefold as Brill-Noether loci

Let $X$ be a degree 12 or degree 16 index one prime Fano threefold. In the paper of Mukai https://arxiv.org/pdf/math/0304303.pdf page 500, Theorem 4 and Theorem 5. He said $X_{12}$ has two ambient ...
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3 votes
0 answers
125 views

Bounded derived categories of which smooth projectives possess bounded t-structures whose hearts are categories of modules?

I am interested in $P$ that is smooth and proper over a field and such that the derived category of coherent sheaves $D^b(P)$ possesses a $t$-structure whose heart is the category of finitely ...
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4 votes
1 answer
327 views

Why two definitions of localization of categories coincide?

Let $\mathsf{C}$ be a category and $S$ be a collection of morphisms in $\mathsf{C}$. In this generality, we can construct the localization $S^{-1}\mathsf{C}$ by posing its objects to be the same as ...
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1 vote
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Hochschild cohomology of a sheaf of associative algebras

Assume that $X$ is a complex manifold. Let $\delta: X\to X\times X$ be the diagonal map. Assume that $\mathcal{A}_X$ is a $\mathbb C_X$-algebra and $\mathcal{M}_X$ is a left $\mathcal{A}_X\otimes_{\...
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2 votes
0 answers
143 views

Conics on Gushel-Mukai fourfold

Let $X$ be a very general Gushel-Mukai fourfold, let $\mathcal{U}$ be the tautological sub-bundle and $\mathcal{Q}$ be the tautological quotient bundle. Let $C\subset X$ be a $\rho$-conic, then $\...
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1 vote
0 answers
99 views

Does the Mukai's lemma hold for non-algebraic $K3$ surfaces?

In Huybrechts' book Fourier-Mukai Transforms in Algebraic Geometry I found the following result due to Mukai (Page 232, Lemma 10.6) Let $X$ and $Y$ be two $K3$ surfaces. Then the Mukai vector of any ...
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0 votes
0 answers
121 views

How to construct functorialy objects in a derived category

Let $F$ be a left-exact functor from an abelian category $A$ to an abelian category $B$. Let $A'$ be a subcategory of $A$ (not necessarily abelian) and $B'$ a subcategory of $B$ which is abelian such ...
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4 votes
1 answer
311 views

Perverse sheaves on the complex affine line

Show that a perverse sheaf on $\mathbb{A}^1(\mathbb{C})$ (the complex plane with the analytic topology) is a bounded complex $A$ of sheaves of $\mathbb{Q}$-vector spaces with constructible cohomology ...
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2 votes
0 answers
82 views

Minus sign in rotated triangles in Triangulated categories

Let $T$ be a triangulated category and $$ X \xrightarrow{u} Y \xrightarrow{v} Z \xrightarrow{w} X[1]$$ an exact triangle (or distinguished triangle). TR 2 implies that then the two rotated triangles $$...
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4 votes
0 answers
182 views

Formality of a category of constructible sheaves

Let $X= S^1 \wedge S^1$ be a wedge of circles. Then $X$ admits a natural stratification $\mathcal{S}$ as a union of two disjoint open intervals $I_1, I_2$ and a point $\{*\}$. Let $D_{\mathcal{S}}(X)$ ...
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3 votes
0 answers
61 views

Derived category supported in a Serre subcategory of a locally noetherian category

This is a cross-post from math.stackexchange at https://math.stackexchange.com/questions/4251692/derived-category-supported-in-a-serre-subcategory-of-a-locally-noetherian-catego, since I didn't get ...
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0 votes
0 answers
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The topological Grothendieck group of a mixed category

I have recently become interested in the notion of mixed categories, as well as the topological Grothendieck group of their derived categories. I am still very new to the field. For that, I would like ...
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2 votes
0 answers
124 views

Product and coproduct in derived category

I'm sure this is either a standard result or false, but I don't have enough experience with the derived category to decide either way. I have tried looking in Kashiwara-Schapira's Sheaves on Manifolds ...
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7 votes
1 answer
289 views

$\infty$-local systems

Let $X$ be a "nice" topological space, $R$ a ring. I believe that there is an equivalence of $\infty$-categories betweeen: the full subcategory of $D(X,R)$ (derived category of sheaves of $...
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3 votes
0 answers
151 views

A Nakayama type of claim for countably generated modules on complex affine varieties

Let $U\subset \mathbb{A}^n_{\mathbb{C}}$ be any Zariski open affine subvarity. Let $M$ be an $\mathcal{O}(U)$-module. Suppose $M$ satisfies $M\overset{L}{\otimes}\mathbb{C}_{\mathfrak{M}}\cong 0$ for ...
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7 votes
0 answers
288 views

What's the definition of a microlocal sheaf?

I'm slowly becoming familiar with what microsupport of a sheaf is, but none of the references I've seen give a definition of what a microlocal sheaf should be in general. In this paper of ...
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5 votes
0 answers
136 views

Extension groups in quotient categories

Let $\mathcal{A}$ be an abelian category and let $\mathcal{B}$ be a Serre subcategory of $\mathcal{A}$. We can form the quotient category $\mathcal{A}/\mathcal{B}$, and the canonical functor $Q:\...
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3 votes
0 answers
68 views

Coxeter polynomials of graphs

Let $Q$ be a finite connected and directed graph with $n$ points. Assume $Q$ is acyclic as a directed graph. Let $C=C_Q$ be the Cartan matrix of $Q$, that is the matrix with entries $c_{i,j}$ being ...
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4 votes
1 answer
378 views

Proof of derived tensor-hom adjunction

This is a cross-post from math.stackexchange, since I didn't get any answers there. As far as I know, for $R,S,V,W$ rings and $M$ an $(R,W)$-bimodule, $N$ an $(R,S)$-bimodule and $L$ an $(S,V)$-...
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8 votes
1 answer
225 views

Chain complexes split in the derived category over rings of global dimension 1

Let $R$ be a ring of global dimension $1$. Then I have seen the claim (in a paper, and in this MO post When do chain complexes decompose as a direct sum?) that any chain complex over $R$ is equivalent ...
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12 votes
1 answer
491 views

Unbounded resolutions for Grothendieck abelian categories

Consider the following result: Theorem 1: Let $\mathsf{A}$ be a Grothendieck abelian category. Then every complex in $\mathsf{C}(\mathsf{A})$ has a $K$-injective resolution. As far as I know, the ...
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2 votes
0 answers
54 views

Edge morphism of a particular spectral sequence

I am not sure if this question is too elementary for MathOverflow and should strictly belong to Math StackExchange but I shall try my luck. Please feel free to close it in this case and I will migrate ...
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4 votes
1 answer
230 views

Why do we need the axiom MS3 for localizing categories?

Background: Let $\mathsf{C}$ be a category and $S$ be a collection of morphisms (let's suppose that $S$ has all the identities and is closed under multiplication just to simplify a bit). We can ...
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3 votes
0 answers
134 views

How do the shift and truncation functors affect hypercohomology groups?

For $k$ a number field and $X$ a smooth geometrically integral variety, we denote by $\pi:X \rightarrow \mathrm{Spec} \, k$ the canonical morphism. For the etale sheaf $\mathbb{G}_{m,X}$, we have a ...
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3 votes
0 answers
133 views

Obtaining an exact sequence of Galois modules via derived functors

This question has two parts, the first part will be to obtain the desired exact sequence while the second will be to study it in the corresponding derived category and try to obtain it from there. Let ...
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3 votes
0 answers
115 views

How to distinguish the singularities on moduli space?

Let me start with concrete examples. Let $X$ be a smooth special Gushel-Mukai threefold and $\mathcal{C}(X)$ be its honest Fano surface of conics, it has two irreducible components $\mathcal{C}(X)=\...
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