**4**

votes

**2**answers

187 views

### Vanishing natural transformation and strong generator

Let $X$ be a smooth projective variety (over the field of complex numbers). Let $T$ be strong generator of $D^b(X)$ : this means that every object in $D^b(X)$ can be obtained in a given finite number ...

**3**

votes

**1**answer

117 views

### Classifying space for homology endomorphisms supported on a graph?

Let $X$ be a reasonable topological space (say one that has the homotopy type of a finite CW complex) and consider a subset $\Gamma$ of $X \times X$ so that the projection $p:\Gamma \to X$ onto the ...

**0**

votes

**1**answer

163 views

### Can you detect homological dimensions from homology?

Suppose you are given a bounded chain complex $M$ over a commutative ring $R$.
Is there a clear relation between homological dimensions of $M$ and homological dimensions of its cohomologies?
For ...

**1**

vote

**0**answers

64 views

### On (universal) additive functors making a given complex contractible: examples?

Let $M=(M^i)$ be a (cohomological) complex of objects of some additive category $A$ (I am mostly interested in "short" complexes; yet one may also consider an unbounded $M$). I am interested in those ...

**1**

vote

**0**answers

62 views

### Concerning the $SBI$-sequence for dihedral homology (Loday, Cyclic Homology, 5.2)

I was wondering about the signs of the $SBI$-sequence ("Connes' periodicity exact sequence") in equations $(5.2.7.2)$ and $(5.2.7.3)$ of 'Cyclic Homology' by Jean-Louis Loday. Why is the sequence ...

**2**

votes

**0**answers

100 views

### Unital $A_{\infty}$-algebra?

I'm not familiar with the subject and i don't know if my question make sense.
In Homological mirror symmetry and torus fibrations (http://arxiv.org/abs/math/0011041) a structure of a non-unital ...

**1**

vote

**2**answers

101 views

### Showing that the stable module category of a ring $R$ restricted to maximal Cohen-Macaulay objects is trivial if $\text{gldim } R < \infty$

(In the following, a (not necessarily commutative) ring $R$ is Gorenstein if it has finite injective dimension as a module over itself on either side, and a finitely generated (right) $R$-module is ...

**1**

vote

**1**answer

89 views

### Independent set of relations in an algebra [closed]

Let $k⟨X⟩$ be a free associative algebra generated by a set $X$ over a field $k$. Let $S$ be a set of $k$-algebra relations. Then what does it mean by the set of relations are independent ?

**2**

votes

**0**answers

128 views

### Existence of universal extension between two modules?

I need a reference for the following fact:
Let $R$ be (for simplicity) an algebra over the field $k$, let $A, B$ be $R$-modules. Let $E = Ext^1_R(B, A)$. There is a natural isomorphism between ...

**5**

votes

**1**answer

485 views

### reference for “Topological algebra of Grothendieck”

I would like to have some references for Grothendieck's theory of "Topological algebra": a synthesis of homotopical and homological algebra,
with special emphasis on topoi.

**2**

votes

**1**answer

124 views

### Projectivity of torsion-free modules over integral group rings

Let $G$ be a torsion-free group and assume that the integral group ring $\mathbb{Z}G$ is torsion-free as well. Let $M$ be a torsion-free, finitely generated module over $\mathbb{Z}G$.
If we assume ...

**4**

votes

**1**answer

164 views

### locally noetherian categories and the category of quasi-coherent sheaves over a noetherian scheme

It is known that a ring $R$ is noetherian if and only if direct sums of injective $R$-modules
are injective if and only if every injective $R$-module is a direct sum of indecomposable injective ...

**4**

votes

**0**answers

165 views

### Hochschild Cohomology of the Quantum Torus

I would like some advice on how to compute directly, or by a higher powered method the Hochschild Cohomology groups of the quantum torus using the stated complex I have found. I think there are ...

**5**

votes

**1**answer

360 views

### soft: Reference/ Suggested Read: Homological Algebraic techniques in PDEs

I was reading this article on wikipiedia and was interested by the apparent link between Homological Algebra and PDEs. What is an accessible reference which showcases the link between these topics? ...

**7**

votes

**1**answer

202 views

### Frobenius $A_{\infty}$-bialgebras?

Recall that a finite dimensional associative algebra $A$ over a field $k$ is called a symmetric Frobenius algebra (sometimes called "closed" Frobenius algebra) if its equipped with a symmetric non ...

**4**

votes

**0**answers

158 views

### Generation of cohomology of graded algebras

Let $A$ be an unital, associative, graded algebra over a base ring $k$. I'm happy to assume that $k$ is a field if need be, and will insist that $A$ free and of finite rank in each degree (locally ...

**6**

votes

**1**answer

297 views

### Connected CW complex, isomorphism?

Let $\pi$ be a group and let $K(\pi, 1)$ be a connected CW complex such that $\pi_1(K(\pi, 1)) = \pi$ and $\pi_q(K(\pi, 1)) = 0$ for $q \neq 1$. My question is, are $H_*(K(\pi, 1);A)$ and ...

**1**

vote

**0**answers

211 views

### Image of an $F$-acyclic resolution homotopic to a projective resolution?

This is a crosspost of this MSE question according to the recommendation in the comments. I know this question is elementary, but I'm hoping the author of these notes could provide a more detailed ...

**1**

vote

**1**answer

126 views

### When the restriction of derived equivalence to a summand is a derived equivalence as well

I have a question about the equivalence of derived categories. Let $\mathcal{A} = \mathcal{A}'\oplus \mathcal{A}''$ and $\mathcal{B} = \mathcal{B}' \oplus \mathcal{B}''$ are direct sum of abelian ...

**6**

votes

**1**answer

223 views

### When the restriction of a derived functor to a subcategory is the derived functor of the restriction

Let $\mathcal{D},\mathcal{E}$ be abelian categories and $\mathcal{C}$ be a Serre subcategory of $\mathcal{D}$.
Let $D(\mathcal{C}), \, D(\mathcal{D})$ denote the derived categories of ...

**5**

votes

**1**answer

564 views

### How to compute this $\mathrm{Ext}^1$?

Let $A$ be a regular local $\mathbb{C}$-algebra of dimension $2$, such as the localization of $\mathbb{C}[x,y]$ at $(x,y)$, and let $\nu=(\nu_1\geq\nu_2\geq\cdots\geq\nu_{\ell}\geq0)$, ...

**3**

votes

**0**answers

183 views

### Ring epimorphisms, and epimorphism in the category of small preadditive cats

This question is related to this other question I have asked some time ago. Let $R$ and $S$ be two rings and let $\phi:R\to S$ be a ring homomorphism.
It is well-known that $\phi$ is an epimorphism ...

**4**

votes

**0**answers

69 views

### Homological dimension of Joseph quotients

Let $\mathfrak g$ be a simple Lie algebra over $\mathbb C$ not isomorphic to $sl(n)$.
Let $\mathcal O$ be the minimal nilpotent orbit in $\mathfrak g^*$. Joseph proved that there exists unique ...

**2**

votes

**0**answers

76 views

### Counting chain maps

I initially asked this question over at math stack exchange, you can find it here. I haven't really gotten any traction and I'm beginning to wonder if maybe its a harder question than I originally ...

**6**

votes

**3**answers

520 views

### Homological vs. cohomological dimension of a group/space

I have several related questions regarding homological vs. cohomological dimension of a space/group (this is not a duplicate of this).
The standard definition of the cohomological dimension $cd(X)$ ...

**1**

vote

**0**answers

146 views

### How do I check if a sequence of R-modules is exact?

Let R be a ring. For example, take $R=k[x_1,\ldots,x_n]$ or, if possible, $R = \Bbb{Z}[x_1,\ldots,x_n]$.
Consider a sequence of free R-modules
$$R^a \stackrel{f}\to R^b \stackrel{g}\to R^c$$
where ...

**0**

votes

**1**answer

117 views

### Spliting of short exact exact sequences of partially ordered groups

Consider a short exact sequence of partially ordered groups
$$0 \longrightarrow H \stackrel{\alpha}{\longrightarrow} G \stackrel{\beta} {\longrightarrow} G/H \longrightarrow 0 ,$$ where $H$ is a ...

**2**

votes

**1**answer

67 views

### Is the class of acyclic complexes deconstructible?

Let $\mathcal{C}$ be a category, then a class $\mathcal{A}\subseteq \mathcal{C}$ is deconstructible if there is a set $\mathcal{S}\subseteq\mathcal{C}$ such that $\mathcal{A}$ consists of ...

**4**

votes

**0**answers

191 views

### Can triangulated categories be “approximated by countable subcategories” (that are triangulated but not full!)?

For a given (finite) set of (objects and) morphisms $f_i$ in a triangulated category $C$ I am interested in a (non-full!) triangulated subcategory $C'\subset C$ of "small size" that would contain ...

**3**

votes

**0**answers

126 views

### When there exists some “cone” of a morphism of (ind-representable) cohomological functors?

I am interested in cohomological functors from a certain small triangulated category $C$ to abelian groups.
The question is: given a tranformation $F\to G$ of two functors of this sort, is it ...

**3**

votes

**2**answers

211 views

### A conservative, non faithful functor between triangulated categories

Suppose that we have:
1) triangulated categories $C,D$, each equipped with a $t$-structure.
2) triangulated functor $F: C \to D$ which is $t$-exact.
3) $F$ reflects isomorphisms, i.e. is ...

**0**

votes

**1**answer

301 views

### On a claim of Zagier on extending a map to cocycle

Zagier, in his paper 'Some Surprising Consequences of the Cohomology of SL$_2(\bf{ Z})$' (link, p. 6), studies the action of $\Gamma=PSL_2(\bf Z)$ on a vector space $V$, denoting the action by $v\ |\ ...

**0**

votes

**0**answers

162 views

### Conditions for splitting of short exact sequence?

Are there conditions under which the short exact sequence
$$0\rightarrow E (K)/mE (K)\rightarrow H^1_{Sel}(K,E_m)\rightarrow \Sha(E|K)_m\rightarrow 0$$
splits? I assume $K $ to be a number field and ...

**8**

votes

**1**answer

412 views

### Deformations of Ext rings

Let $k$ be a base ring and $k[x]$ the ring of polynomials in an indeterminate $x$ over $k$. Consider a (not necessarily commutative) algebra $A$ over $k[x]$ and two $A$-modules $M$ and $N$. Then for ...

**5**

votes

**1**answer

140 views

### Left orthogonals to compact objects in triangulated categories: existence and “control”?

Let $C$ be a compactly generated triangulated category. Can it contain a non-zero object $M$ such that there are no non-zero morphisms FROM $M$ into compact objects? I would be grateful for any ...

**0**

votes

**0**answers

114 views

### If the direct sum of cyclic modules is cyclic, what happens to nontrivial extensions?

Let $R$ be any ring with unit over some field $k$ and let $M_1$ and $M_2$ be cyclic left $R$-modules with $dim_k(Ext^1_R(M_2,M_1))\geq 1$.
Assume $M_1\oplus M_2$ is a cyclic left $R$-module. Given ...

**2**

votes

**1**answer

144 views

### Ext groups in the equivariant derived category

I apologize in advance that this question is probably too basic for MO, but I reckoned I would not get an answer on Math.Stackexchange.
I am starting to learn about perverse sheaves, the ...

**4**

votes

**1**answer

120 views

### Extension-closed subcategories of triangulated categories as “almost exact” categories

Did anybody study those subcategories of triangulated categories that are closed with respect to "extensions" (in the sense of distinguished triangles; in particular, any such $B$ is additive)? If we ...

**3**

votes

**1**answer

91 views

### Can you test flatness on $FP_3$-modules?

Let $A$ be a ring, and let $M$ be a right $A$-module. Then $M$ is flat if and only if for each left $A$-module $N$ we have that $Tor^1_A(M,N) = 0$. Becasuse $Tor$ commutes with filtered direct limits, ...

**15**

votes

**0**answers

712 views

### Derived functors - homotopical vs homological approach

This question is a crosspost of the second part of this MSE question.
In my first course in homological algebra, derived functors were defined in terms of universal $\delta$-functors. In the text ...

**4**

votes

**0**answers

60 views

### Minimal rank of a permutation resolution of a $G$-lattice

Let $G$ be a finite group.
By a $G$-lattice I mean a finitely generated free abelian group $L$ with an action of $G$.
One says that $L$ is a permutation lattice if $L$ has a $\mathbb{Z}$-basis ...

**3**

votes

**0**answers

181 views

### Is there a Hochschild-Serre spectral sequence for unramified cohomology?

Similar to the Hochschild-Serre spectral sequence for etale cohomology ($H^p(G, H^q_{et}(X_L, \mathcal F|_{X_L})) \Rightarrow H^{p+q}_{et}(X, \mathcal F)$ for a Galois field extension $L/k$ with ...

**4**

votes

**0**answers

145 views

### Reference for Grothendieck's duality and Cousin, Dualizing and Residual complexes

I am a graduate student currently reading Hartshorne's Residues and Duality. In order to reach the construction of the right adjoint $f^!$ of $Rf_*$ for some special types of maps of locally ...

**2**

votes

**1**answer

74 views

### Morphisms $P \to M$ in the derived category of a dg-category, if $P$ is h-projective

Let $\mathbf A$ be a dg-category. Denote by $\mathsf{C}_{\mathrm{dg}}(\mathbf A)$ the dg-category of right $\mathbf A$-modules, and by $\mathsf{C}(\mathbf A) = Z^0(\mathsf{C}_{\mathrm{dg}}(\mathbf ...

**4**

votes

**1**answer

294 views

### On various “extension closures” and “orthogonals” in triangulated categories

A vague form of my question is the following one: for a class of objects $D$ of a triangulated category $C$ we consider the class $E$ of objects that satisfy $Mor_{C}(d,e)=\{0\}\ \forall d\in D$; ...

**4**

votes

**1**answer

198 views

### Different ways of having infinite global dimension

Is there any ring $R$ of infinite global dimension such that any $R$-module is a retract (i.e. direct summand) of some $\oplus_{i\in I}M_i$ where each $M_i$ has finite projective dimension?
I ask ...

**3**

votes

**1**answer

223 views

### Model structure on non-negative differential graded algebras with homological grading

I was wondering if there exists a model structure on the category of non-negative differential graded algebras with homological grading. To be more precise: Let $Ch_{k}$ the model category of ...

**13**

votes

**1**answer

387 views

### Homotopy transfer in the opposite direction

Let $X\rightleftarrows Y\circlearrowleft$ be a strong deformation retraction of chain complexes (a.k.a. contraction), i.e. $X\rightarrow Y\rightarrow X$ is the identity, $Y\rightarrow Y$ is a homotopy ...

**0**

votes

**1**answer

42 views

### Subgroups with some closed property

Assume $A$ and $B$ are infinite abelian groups, $B$ is a subgroup of $A$. Is it true that if every homomorphism from $B$ to $\mathbb Z$ can be extended to a homomorphism from $A$ to $\mathbb Z$, then ...

**5**

votes

**0**answers

133 views

### Matrix factorizations as a dg-category?

Matrix factorizations (in the graded case) give a triangulated category. I would imagine that there should be an underlying dg-category. Is there such a definition, and if so, where can I find it in ...