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1 answer
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Left exact functor $F$ preserves quasi-isomorphism between $F$-acyclics

In this math overflow page, the poster gives a proof of the statement "an additive left exact functor $F$ preserves quasi-isomorphisms between $F$-acyclic objects." I'm having trouble ...
Reinder van der Weide's user avatar
4 votes
1 answer
227 views

Literature Request: The derived category is Krull-Schmidt

I am looking for literature where it is proven that the derived category of bounded complexes over a finite-dimensional algebra is Krull-Schmidt. I found this question Literature request: $K^b(\text{...
Sebastian Pozo's user avatar
2 votes
0 answers
117 views

Tilting complexes arising from homotopy equivalences

Let $k$ be a field and let $A$ and $B$ be finite-dimensional selfinjective $k$-algebras. Suppose we have an isomorphism of homotopy categories $F: K^b(A-mod) \cong K^b(B-mod)$ that descends to a ...
Sam K's user avatar
  • 175
4 votes
2 answers
285 views

Does $\mathbf R\text{Hom}_R(k, -)\otimes_R^{\mathbf L} k$ commute with co-products?

Let $(R, \mathfrak m, k)$ be a commutative Noetherian local ring. Then, is it true that $\mathbf R\text{Hom}_R(k, -)\otimes_R^{\mathbf L} k$ commutes with arbitrary co-products?
uno's user avatar
  • 412
11 votes
1 answer
513 views

When does derived tensor product commute with arbitrary products?

Let $R$ be a commutative Noetherian ring. Let $M$ be an $R$-module. It is well-known that $M$ is finitely generated if and only if the functor $M\otimes_R (-)$ preserves arbitrary products (for ...
uno's user avatar
  • 412
4 votes
0 answers
86 views

Lifting maps on the spectral sequence of a double complex to the derived category

Question The differentials on the $(r+1)$th page of a spectral sequence are maps on the cohomologies of the complexes on $r$th page. So, between two adjacent complexes $K^\bullet,L^\bullet$ on the $r$...
Joseph Sullivan's user avatar
3 votes
0 answers
120 views

Derived tensor by perfect complex preserves exact triangle in singularity category?

Let $R$ be a commutative Noetherian ring. Let $\operatorname{D}_{sg}(R)$ be the singularity category of $R$, i.e., the Verdier localization of $D_b(\text{mod } R)$ by the thick subcategory of perfect ...
Snake Eyes's user avatar
1 vote
0 answers
126 views

full strong exceptional collection

I am wondering whether, if a triangulated category $\mathcal{D}$ has a full strong exceptional collection (infinite), it is triangle-equivalent to the bounded derived category of finitely generated ...
Paulo Rossi's user avatar
3 votes
0 answers
107 views

Dimension of hom spaces between indecomposable modules

Undergraduate-Level Background Let $A$ be an Artin algebra over an algebraically closed field $k$, and let $C = Rep(A)$ denotes the category of $k$-linear, $k$-finite dimensional representations of $A$...
Student's user avatar
  • 5,230
6 votes
2 answers
442 views

Existence of functorial (K-)flat resolutions?

I am wondering about the following: Suppose $X$ is a reasonable scheme or stack with the resolution property. (So, all quasi-coherent sheaves admit a surjection from a flat sheaf.) Then I believe ...
C00's user avatar
  • 91
4 votes
0 answers
238 views

Derived functors from localization vs animation

I got a bit confused with the derived functors getting from the localization and the animation. More specifically, let $\mathcal{A}$ be an abelian category generated by compact projective objects $\...
Johnny's user avatar
  • 255
6 votes
0 answers
126 views

Explicit proof that $\mathbb{k}[x]/(x^n)$ is not derived discrete

In the question Explicit proof that algebra is derived wild it was asked whether there are examples of algebras $A$ where it is possible to show explicitly that $A$ is derived wild by finding an ...
Jannik Pitt's user avatar
  • 1,474
4 votes
1 answer
267 views

A particular morphism being zero in the singularity category

Let $R$ be a commutative Noetherian ring and $D^b(R)$ be the bounded derived category of finitely generated $R$-modules. Let $D_{sg}(R)$ be the singularity category, which is the Verdier localization $...
strat's user avatar
  • 361
3 votes
1 answer
129 views

Thick subcategory containment in bounded derived category vs. singularity category

Let $R$ be a commutative Noetherian ring, and $D^b(\operatorname{mod } R)$ the bounded derived category of the abelian category of finitely generated $R$-modules. Let me abbreviate this as $D^b(R)$. ...
Alex's user avatar
  • 480
3 votes
1 answer
385 views

Concrete examples of derived categories

What examples of abelian categories $\mathcal{A}$ are there such that the derived category $\mathcal{D}(\mathcal{A})$ can be described concretely? For example, is there a concrete way of describing $\...
Jannik Pitt's user avatar
  • 1,474
4 votes
1 answer
335 views

Gluing objects of derived category of sheaves

Let $X$ be a locally compact topological space (may be assumed to be a stratified space with finite stratification). Let $\{U_i\}$ be an open finite covering. Assume that over each $U_i$ we are given ...
asv's user avatar
  • 21.8k
2 votes
0 answers
51 views

When can GKZ setup encompass HMS?

Are there any instances when the Landau-Ginzburg superpotential describing the mirror of a smooth projective Fano variety $X_\Sigma$ is encompassed by a GKZ hypergeometric system? In some sense I am ...
locally trivial's user avatar
6 votes
1 answer
233 views

Comparing stabilization of stable category modulo injectives and a Verdier localization

Let $\mathcal A$ be an abelian category with enough injectives. Let $\mathcal I$ be the collection of injective objects. Let $\mathcal A/\mathcal I$ be the quotient category whose objects are same as ...
Snake Eyes's user avatar
1 vote
0 answers
158 views

When is a functor of chain complexes triangulated?

Let $\textsf{A}, \textsf{B}$ be abelian categories. Let $F: \operatorname{Ch}(\textsf{A}) \to \operatorname{Ch}(\textsf{B})$ be an additive functor of chain complexes. If $F$ preserves chain ...
Jannik Pitt's user avatar
  • 1,474
2 votes
1 answer
242 views

Derived category of local systems of finite type on a $K(\pi,1)$ space: an explicit counterexample

Let $X$ be a nice enough topological space. I am mostly interested in smooth complex algebraic varieties. One may ask whether the bounded derived category of the category $\mathrm{Loc}(X)$ of local ...
Sergey Guminov's user avatar
3 votes
0 answers
106 views

Multiplication map by a ring element on an object vs. all its suspensions in singularity category

Let $R$ be a commutative Noetherian ring, consider the bounded derived category of finitely generated $R$-modules $D^b(R)$ and consider the singularity category $D_{sg}(R):=D^b(R)/D^{perf}(R)$. Let $r\...
uno's user avatar
  • 412
5 votes
0 answers
660 views

Hypercohomology spectral sequence from the derived category point of view

Let $F\colon \mathsf{A}\to\mathsf{B}$ be an additive functor between abelian categories and let $M$ be a complex on $\mathsf{A}$. There's a "hypercohomology spectral sequence" $$E_1^{i,j}=\...
Gabriel's user avatar
  • 711
2 votes
0 answers
133 views

Formulation of cap product in group-equivariant sheaf cohomology + applications?

Originally asked on Math SE but it was suggested I move it here. Suppose one has a distinguished cocycle in the group-equivariant sheaf cohomology $\Phi \in H^n(X, G, \mathcal{F})$ for a "nice&...
xion3582's user avatar
  • 101
2 votes
0 answers
154 views

Non-triviality of a morphism

Let $X$ be a smooth Gushel–Mukai fourfold and $Y$ a smooth hyperplane section, which is a Gushel–Mukai threefold. I consider semi-orthogonal decomposition of $X$ and $Y$: $$D^b(X)=\langle\mathcal{O}_X(...
user41650's user avatar
  • 1,982
3 votes
0 answers
160 views

Does a functor preserving injectives also preserve K-injective complexes?

Let $F:A\to B$ be an exact functor of Grothendieck abelian categories. If $F$ preserves injective objects, then does the exact functor $F:K(A)\to K(B)$ preserves K-injective complexes? For example, ...
Doug Liu's user avatar
  • 615
2 votes
0 answers
136 views

dg-Künneth formula for qcqs schemes

Let $X$ and $Y$ be qcqs schemes over a field $k$ (or I am happy to assume any nice condition up to smooth (quasi-)projective varieties if that makes the folloiwng question true). Let us define $Perf(X)...
P. Usada's user avatar
  • 256
-1 votes
1 answer
177 views

When morphism of complexes is homotopic to 0?

Let $f\colon A^\bullet\to I^\bullet$ be a morphism of bounded below complexes in an abelian category. Assume all $I^i$ are injective objects. Assume also that $f$ induces the zero map on cohomology. ...
asv's user avatar
  • 21.8k
4 votes
1 answer
230 views

Decompose an unbounded (cochain) complex in the homotopy category

Let $\mathcal{A}$ be an abelian category, it is known that any complex $A^{\bullet}$ admits a distinguished triangle $$B^{\bullet}\rightarrow A^{\bullet}\rightarrow C^{\bullet}\rightarrow B^{\bullet}[...
user avatar
3 votes
1 answer
187 views

Explicit proof that algebra is derived wild

Following the terminology of Drozd, Yuriy A., Derived tame and derived wild algebras, Algebra Discrete Math. 2004, No. 1, 57-74 (2004). ZBL1067.16028. let $A$ and $R$ be algebras over a field $k$. A ...
Jacob FG's user avatar
  • 497
1 vote
0 answers
111 views

Kunneth formula for hypercohomology

Let $A_{\bullet}$ and $B_{\bullet}$ be two bounded complexes of sheaves over a variety $X$. Is there a Kunneth-like formula for the hypercohomology of the tensor product $A_{\bullet}\otimes B_{\bullet}...
S.D.'s user avatar
  • 494
2 votes
0 answers
119 views

dg-natural transformation between FM functors and Hom between kernels

The question is related to Morphism between Fourier-Mukai functors implies the morphism between kernels? Consider a complex smooth projective variety $X$ and the bounded derived category $D^b(X)$, it ...
user avatar
6 votes
1 answer
478 views

Unbounded acyclic resolutions

Let $\mathscr A$ be a Grothendieck abelian category. Then every object in $\operatorname{Ch}(\mathscr A)$ is quasi-isomorphic to a $K$-injective object [Stacks, Tag 079P]. In particular, for any left ...
R. van Dobben de Bruyn's user avatar
3 votes
1 answer
147 views

What do we know about a sheaf $M$ if we know its derived fibers $\mathsf{L}x^* M$, for $x\in X(k)$?

Let $X$ be a scheme over a field $k$. (Feel free to assume that $X$ is an algebraic variety, if needed.) Also, let $M^\bullet$ be a complex in the derived category of quasi-coherent sheaves $\mathsf{D}...
Gabriel's user avatar
  • 711
2 votes
0 answers
310 views

Invariants of objects in $\operatorname{Ch}(\mathrm{Ab})$ up to chain homotopy

$\newcommand\Ab{\mathrm{Ab}}\newcommand\ab{\mathrm{ab}}\DeclareMathOperator\Ch{Ch}\DeclareMathOperator\Kom{Kom}\newcommand\ho{\mathrm{ho}}$Let $\Ab$ be the category of finitely generated abelian ...
Student's user avatar
  • 5,230
0 votes
0 answers
281 views

What can be said about the derived functor of a composition between unbounded derived categories?

Let $\mathcal A, \mathcal B,\mathcal C$ be abelian categories and let $F:\mathcal A \to \mathcal B,G: \mathcal B \to \mathcal C$ be left exact functors such that $RF:D(\mathcal A) \to D(\mathcal B), ...
Lukas Heger's user avatar
6 votes
1 answer
526 views

How does the Balmer spectrum fail to describe the algebraic geometry of categories of non-compact objects?

In trying to understand the higher algebraic geometry of the stable homotopy category, one thing I've come across repeatedly is the claim that one should only consider the Balmer spectrum of a tt-...
Doron Grossman-Naples's user avatar
3 votes
2 answers
376 views

Is there an elementary reason that this colocalisation map of complexes is a quasi-isomorphism?

A fact about triangulated categories is that (exact) localisation functors and so-called colocalisation functors come in pairs, making an exact localisation triangle. I've tried to come up with less ...
Justin Bloom's user avatar
1 vote
1 answer
149 views

Finitely generated module, which is a virtually small complex, embeds into a module of finite projective dimension?

Let $R$ be a commutative Noetherian ring, and let $\text{mod } R$ denote the abelian category of finitely generated $R$-module. Consider the bounded derived category $D^b(\text{mod } R) $ which is a ...
feder's user avatar
  • 73
6 votes
1 answer
397 views

Vanishing of higher limits

Let $I$ be a directed set and let $X_I$ be a corresponding inverse system of, say, (complex) vector spaces or abelian groups (in my case in general not finite-dimensional, resp. not finitely generated)...
AlexE's user avatar
  • 2,998
0 votes
0 answers
306 views

Distinguished triangles as generalizations of short exact sequences

If you'll have patience with me, I understand that this is not the first time that a variation of this question is being asked on MathOverflow, but alas, I am unable to truly make sense of those ...
StormyTeacup's user avatar
7 votes
1 answer
241 views

On the derived categories of coherent sheaves on a quadric

I am reading On the derived categories of coherent sheaves on some homogeneous spaces - Kapranov for the proof for quadrics. Consider a vector space $V=\mathbb{C}^{n+2}$ with the standard quadratic ...
user avatar
8 votes
2 answers
897 views

Derived functors out of an unbounded derived $\infty$-category

Let $\mathcal A$ be an abelian category. In this lecture, Thomas Nikolaus Defines the unbounded derived category $\mathcal D(\mathcal A)$ as $\mathcal K(\mathcal A)[W^{-1}]$, where $\mathcal K(\...
Tomo's user avatar
  • 1,217
2 votes
1 answer
239 views

How to complete $i^*i_*\mathcal{F}\to \mathcal{F}$ into an exact triangle for a smooth divisor $i: X\hookrightarrow Y$?

Let $Y$ be a smooth algebraic variety and $i: X\hookrightarrow Y$ be a smooth divisor. We consider the derived functors $i^*: D^b_{coh}(Y)\to D^b_{coh}(X)$ and $i_*: D^b_{coh}(X)\to D^b_{coh}(Y)$. By ...
Zhaoting Wei's user avatar
  • 9,019
2 votes
0 answers
244 views

Do we have a left adjoint of $i^*$ for a closed immersion $i$?

Let $i: X\hookrightarrow Y$ be a closed immersion of varieties. We have the derived pullback functor $i^*: D^b_{coh}(Y)\to D^b_{coh}(X)$. My questions is: can we construct a left adjoint of $i^*$ in ...
Zhaoting Wei's user avatar
  • 9,019
3 votes
1 answer
360 views

Derived Hom without injectives nor projectives

I am stuck with the following farce on derived Homs. I have an abelian category $A$ and I showed that, given any two objects $X$ and $Y$ of $A$, the group of $1$fold extensions $\operatorname{Ext}^1_{...
Stabilo's user avatar
  • 1,479
6 votes
2 answers
506 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 ...
Leo Herr's user avatar
  • 1,084
8 votes
2 answers
406 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, ...
Piotr Pstrągowski's user avatar
2 votes
0 answers
546 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 $\...
Sam K's user avatar
  • 175
7 votes
1 answer
441 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 ...
Marius Nielsen's user avatar
1 vote
0 answers
156 views

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 ...
Flavius Aetius's user avatar

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