(Co)chain complexes, abelian Categories, (pre)sheaves, (co)homology in various (possibly highly generalized) settings, spectra, derived functors, resolutions, spectral sequences, homotopy categories. Chain complexes in an abelian category form the heart of homological algebra.

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3
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122 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 ...
-2
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0answers
73 views

Singular cohomology groups [on hold]

I have a question concerning inclusion of singular cohomology groups with different coefficients. To be more precise, we know that the inclusion $i:\mathbb{Z}\hookrightarrow\mathbb{R}$ naturally ...
6
votes
1answer
268 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
0answers
193 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
1answer
104 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
1answer
180 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
1answer
528 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
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0answers
154 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 ...
3
votes
0answers
56 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
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0answers
73 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
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3answers
464 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
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0answers
132 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
1answer
104 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
1answer
55 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 ...
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0answers
20 views

Is there any description for 1-Yoneda-Extension of $A$-moduls in the spirits of factor systems?

The 1-Yoneda-Extension group of two abelian groups ($\mathbb{Z}$-modules) in the category of abelian groups can be described explicitly by symmetric factor systems(modulo coboundaries). Now my ...
1
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0answers
151 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
0answers
122 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 ...
2
votes
2answers
164 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
1answer
285 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
0answers
138 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
1answer
389 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
1answer
126 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 ...
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0answers
88 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 ...
1
vote
1answer
126 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
1answer
88 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
1answer
79 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, ...
13
votes
0answers
567 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
0answers
57 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
0answers
151 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
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0answers
112 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
1answer
64 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
1answer
230 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
1answer
155 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
1answer
190 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 ...
8
votes
0answers
192 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
1answer
38 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
0answers
114 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 ...
0
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0answers
33 views

Exactness in the zeroth term and (relative) vector field

Recently I've asked the following question: Contraction with a vector field and pullback bundle When I already know how $i_X$ should be understood (please see the discussion; recall that we are ...
1
vote
1answer
168 views

Dimension of Ext modules [closed]

Let $(R,m)$ be a noetherian local ring, and $M$ and $N$ be two finitely generated $R$-module. Then is it true that $\dim \text{Ext}^k(M,N)\leq \dim M-k$? If not does the reversed inequality hold?
3
votes
1answer
175 views

When is the cohomology of a space Ext(k,k)?

If $X$ is aspherical, we know that $H^*(X,k) = \text{Ext}_R(k,k)$, with $R = k\pi_1$. For non-aspherical spaces, do we ever have $H^*(X,k) = \text{Ext}_R(k,k)$ for some ring $R$? Obviously we need ...
1
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0answers
38 views

Antisymmetrization of the Hochschild cocycle

Let $A$ be a commutative (unital, complex) algebra and let $\varphi$ be a $n+1$-linear functional on $A$ (we will call it cochain). Define ...
3
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1answer
116 views

LES for relative cohomology via sheaves

I was unable to find a suitable answer for the following question: Once one learns that singular cohomology is the same as cohomology with coefficients in locally constant sheaf, it is natural to try ...
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0answers
274 views

End of the Ext functors

Let $R$ be a ring, and consider the hom functor $\hom\colon Mod(R)^\text{op}\times Mod(R)\to Mod(R)$; the end of $\hom$ is well-known to be the set of endomorphisms (endonatural transformations) of ...
0
votes
1answer
133 views

$\inf\{i\in \mathbb N \cup \{0\}\cup\infty\mid Ext^i_R(R/I,R)\neq 0\}=0 ?$

Let $R := k[x_1, \cdots, x_n, \cdots]/(x_1^1, \cdots , x^n_n, \cdots),$ where $k$ is a field. Set $I:=(x_1, \cdots, x_n, \cdots)$. the questions are: Is $\inf\{i\in \mathbb N \cup \{0\}\cup ...
3
votes
1answer
217 views

A question on the cohomology of elliptic curves over local fields

Let $K$ be a number field,$\nu$ a nonarchimedian prime of $K$, $K_{\nu} $ the completion of $K $ at $\nu $ with maximal unramified extension $K_{\nu}^{unr} $. Let $E $ be an elliptic curve defined ...
1
vote
0answers
81 views

Global dimension of graded Lie algebra

The rational global dimension of a graded algebra $A=(A_k)_{k\geq 0}$, with $A_0=\mathbb Q$, denoted here ${\rm gl}\dim A$ is defined to be the greatest integer $k$ (or $\infty$) such that ${\rm ...
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0answers
47 views

Minimal free resolution of sum of ideals

Let $S$ be the polynomial ring in $n$ variables, and let $I_1$ and $I_2$ be ideals in $S$. What can be said about the $\mathbb{Z}$-graded minimal free resolution of $I_1+ I_2$ in terms of the ...
2
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1answer
145 views

Decide two indices of Ext functor

This question is from the proof of Theorem 11.34 in the book: Twenty-four Hours of Local Cohomology. Let $R$ and $S$ be CM local ring and $R\to S$ a local homomorphism such that $S$ is a finite ...
0
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0answers
127 views

Is it a correct description of the bounded above derived category of coherent sheaves?

Let $X$ be a (Noetherian) scheme. Let $D^{-}_{\text{coh}}(X)$ be the derived category of complexes of $\mathcal{O}_X$-modules with bounded above and coherent cohomologies. Do we have the following ...
6
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45 views

A “lower-central” filtration of Steenrod algebra?

$\renewcommand{\Atwo}{\mathcal{A}_2}$ So, a lot of good work has been accomplished by filtering the Steenrod algebras $\mathcal{A}_p$ in powers of the Augmentation ideal; For reasons partly ...