The homological-algebra tag has no wiki summary.

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### Hochschild cohomology of de Rham algebra

I am interested in the relation between the Hochschild cohomology of the de Rham algebra on a manifold and the de Rham cohomology.
As we know, we can always take the de Rham algebra as a D.G.A. or ...

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### A map from the coinvariants of the dual to the dual of the invariants for a G-module

Suppose $G$ is a group and $X$ is a $\mathbb{Z}[G]$-module. Recall that the augmentation ideal $I \subset \mathbb{Z}[G]$ is generated by elements of the form $g - 1$ for $g \in G$, the coinvariants ...

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### What is the relation between the Lie bracket on $TX$ as commutator and that coming from the Atiyah class?

Let X be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction of the global existence ...

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### Homotopy equivalence of curved A_\infty algebra

I am quite curious:
What is the precise definition of "homotopy equivalence" or "isomorphism" of two curved $A_\infty$ algebra $A$ and $B$?
What is the condition to set for the morphism $f:A ...

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### The normalised cochain complex, totalisation and cosimplicial simplicial $R$-modules

Short Version
Given a cosimplicial space $X_\bullet$, what is the relationship between (co)chains on the totalisation of $X_\bullet$ and the totalisation of the cosimplicial chain complex obtained by ...

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### Balanced dualizing complexes according to A. Yekutieli

I am reading A. Yekutieli's original article on dualizing complexes for noncommutative algebras and I found a problem I cannot solve.
First, some background. We start with a field $k$ and a ...

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### The Hochschild cohomology of a variety “with coefficient” in a vector bundle

This question is related to one of my previous question Do we have the following isomorphism for $\mathcal{Ext}$?
Let $X$ be a smooth variety (over $\mathbb{C}$) and $\Delta: X \rightarrow X \times ...

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### Serre duality for compactly supported sheaves

Given a smooth quasi-projective variety $X$ over $\mathbb{C}$ and bounded complexes of vector bundles $(P,d)$ and $(P',d')$ with compactly supported cohomology. It is well-known that such complexes ...

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### Do we have the following isomorphism for $\mathcal{Ext}$?

Let $X$ be a smooth variety (over $\mathbb{C}$) and $\Delta: X \rightarrow X \times X$ be the diagonal embedding and $p_1: X\times X\rightarrow X, ~p_2: X\times X\rightarrow X$ be the projections to ...

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### Dualizing Complexes

Let $R$ be a dualizing complex of a Noetherian graded algebra $A$ (not necessary commutative). For any $M\in D_c^b(A)$, there is a natural morphism
$$
\theta: R\Gamma_m(M) \to ...

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### cofree modules and dual

1, Why do people pay special attention to Q/Z in the definition of cofree modules instead of ordinary abelian groups?
2, Over a PID, is every injective module cofree? Just like the relationship ...

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### A question on some computation of group cohomologies

Let $G=H\times J$, where $H\cong J\cong C_2$ (cyclic group of order 2). Let $M \cong \mathbb{Z}$ be a $G$-module via "trivial $H$-action and negation $J$-action". My question is "What are the group ...

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### Sheaf cohomology with compact supports (and Verdier duality?)

Consider a manifold and a complex where cochains are sections of vector bundles and coboundary maps are differential operators, which are locally exact except in lowest degree (think de Rham complex). ...

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### Künneth formula for Ext groups

Setup: Let $X,Y$ be quasi-compact quasi-separated schemes defined over a field $k$. If necessary, you can also assume that $X,Y$ are noetherian, but I don't want to assume that $X,Y$ have the ...

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### is every finitely n-presented (S^{-1})R-module a localization of a finitely n-presented R-module?

Let S be a multiplicative set in a ring R. We can see that every finitely generated $(S^{-1})R$-module is a localization of a finitely generated R-module.
Then, more generally, is every finitely ...

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### Tilting objects and Koszul algebras

Let $X$ be a smooth variety over a field, and $T\in D^b(Coh(X))$ be a tilting object, i.e. (1) $Ext^i(T,T)=0$ for all $ i\neq 0$; (2)Tilting algebra $A:=End(T)$ has finite global dimension; ...

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### Koszul (Exterior/Symmetric) duality for a 1-dim vector space

The simplest example of Koszul duality (see introduction of http://www.ams.org/journals/jams/1996-9-02/S0894-0347-96-00192-0/)
Let $V = \mathbb{C}x$ be a $1$ dimensional vector space. Then the ...

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### Quasi-equivalences of DG categories

There are several definitions of a quasi-equivalence $\newcommand{\T}{\mathscr{T}}F : \T \to \T'$ of DG categories in the literature, e.g.
(i) the induced functor $H^0(F) : H^0(\T) ...

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### example of Local cohomology

Let $S=k[x_1,...,x_n]$ be a polynomial ring over field $k$ with maximal ideal $m=(x_1,...,x_n)$. I wanna make a $3$-dimensional $S$-module $M$ such that $H^0_m(M)=H^1_m(M)=0$ and $H^2_m(M)\neq 0$ be ...

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### Is Hochschild cohomology finitely generated?

I am sorry if this is a naive question.
Let $k$ be a field, and let $A$ be a finitely generated commutative $k$-algebra.
Let $M$ be a finite $A$-module.
Consider the Hochschild cohomologies of $A$ ...

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### Epimorphisms $\mathbb{Z}^{\mathbb{N}} \to \mathbb{Z}^{\mathbb{N}}$ are split

Consider the additive group of integer sequences $\mathbb{Z}^{\mathbb{N}}$. Why does every epimorphism of groups $\mathbb{Z}^{\mathbb{N}} \to \mathbb{Z}^{\mathbb{N}}$ split? $(\star)$
Actually this ...

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### Computing multiplication in the Ext (for a simple example)

This is a follow-up to Computing Ext in Exterior algebra (related to Koszul duality) .
Let $V = \mathbb{C} x$, $A = \Lambda^{\bullet}(V) = A_0 \oplus A_1$ is graded (with $A_0 =
\mathbb{C}, A_1 = ...

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### “as close to being semisimple as it can possibly be.”

I had originally asked this question on math stack exchange but I think maybe it's more appropriate to ask it here.
In the paper of Beilinson, Ginzburg and Soergel entitled "Koszul Duality ...

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### Is it possible to take “limits up to homotopy”?

Before I begin, an apology: it's been a while since I've done much analysis, so I might be misusing (or just missing) terminology.
I have a chain complex $(V_\bullet,\partial)$ of topological vector ...

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### Is the functor $mod(Tw\mathcal{C})\rightarrow mod(\mathcal{C})$ cohomologically full and faithful?

Let $\mathcal{C}$ be a $c$-unital $A_\infty$-category. If $\mathcal{A}$ is a $c$-unital and triangulated $A_\infty$-category, then there is a $c$-unital $A_\infty$- functor
$$Tw: ...

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### Does abstract nonsense of model categories determine the “nonlinear” morphisms of $L_\infty$ algebras?

Recall that a Lie algebra is a module for the operad $\mathrm{Lie}$, which is freely generated by a binary antisymmetric operation $\beta$ modulo an equation that is quadratic in $\beta$. There is a ...

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### The 2-group of extensions

Let $A,B$ objects of an abelian category. Then we can define the abelian group $\mathrm{Ext}^1(A,B)$ as the set of isomorphism classes of extensions $0 \to B \to E \to A \to 0$, endowed with the Baer ...

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### Realization of a formal duality isomorphism via integration

This is a more precise version of my previous question. Let $X$ be a smooth variety of dimension $n$ over $\mathbb{C}$ and $Z$ a proper sub-scheme. We denote by $\tilde{X}$ the formal completion of ...

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### When are MCM ideals principal?

Suppose $(R,\mathfrak m, k)$ is a $d$-dimensional Cohen-Macaulay local ring with canonical module $\omega_R$ and $d>1$. Suppose $I\subset R$ is an ideal which is MCM (=maximal Cohen-Macaulay, ...

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### Grothendieck trace on local cohomology?

Let R be an augmented regular local ring over a field $k$ with maximal ideal m. There is the Grothendieck residue symbol:
$$Res: H^n_m(\Omega^n) \to k$$
If $k=\mathbb{C}$ and $R$ is affine space, ...

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### The global dimension of fields

In the absence of the Axiom of Choice, it is not necessarily true that all vector spaces over a field have bases.
What are the possible global dimensions of fields in a model of ZF in which AoC ...

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### finiteness of local cohomology

Well-known Theorem:
Let $a$ be an ideal of the noetherian ring $R$ and let $M$ be a finitely generated
$R$-module. Let $i \in \Bbb N_0$ be such that $H^j_a
(M)$ is finitely generated for all $j < ...

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### Rejects and injectives

Let $A$ be any ring and consider modules on the left. For $M$ $A$-module, the trace $Tr(M,A)$ is a two-sided ideal of $A$. If $A$ is a unitary ring then:
$Tr(P,A)P=P$, for $P$ projective;
...

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### Cisinski-Deglise model structure on chain complexes

Let $\newcommand{\A}{\mathscr{A}}\A$ be a grothendieck abelian category. In their paper "Local and stable homological algebra in grothendieck abelian categories", Cisinski and Déglise define the ...

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### A Isomorphism between the extension group and cohomology group of Lie algebras [closed]

Within the book An introduction to homological algebra by Weibel, I am trying to prove the following isomorphism, but I am not sure this is true. But I really want to know how to prove or disprove ...

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### Is this a description of the $\aleph_1$-localizing subcategory generated by a compact generator?

This should be obvious but I'm not seeing it:
The $\mathfrak T$ be a triangulated category with coproducts and with a compact generator $A$ (that is, the functor $\mathfrak T(A,\_)$ preserves ...

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### Does the forgetful G-Groups to Groups (nonabelian) have a left adjoint?

Motivation
For a fixed group $G$, consider the category of $G$-groups, whose objects consist of (not necessarily abelian) groups $A$ together with an action $G\to \mathrm{Aut}(A)$. Now we can ...

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### Quasi-isomorphisms in exact categories

I am trying to understand quasi-isomorphisms in an exact category as defined via the mapping cylinder. I would like to know whether these form a category of weak equivalences in the sense of ...

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### Decomposition of a quotient module

Let $R=k[v,x,y,z]/I$, with $I=\langle v^2,z^2,xy,vx+xz,vy+yz,vx+y^2,vy-x^2\rangle$,and let
$f:R^2 \rightarrow R^2$ denote the map given by the matrix
$$M=\begin{pmatrix}
v & y \\
x & z
...

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### Computing Ext in Exterior algebra (related to Koszul duality)

Let $V = \mathbb{C}^n$, $A = \Lambda^{\bullet}(\mathbb{C}^n)$ is a graded algebra (with $A_0 =
\mathbb{C}, A_1 = V$, etc).
Consider $A_0$ as a left $A$-module, how do we compute the graded ring ...

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### When an exact embedding of abelian categories induces a full embedding of their derived categories?

Let $F:A\to A'$ be a (full) exact embedding of abelian categories. When $D(F):D(A)\to D(A')$ (or its bounded version) is a full embedding also?
I would be interested in any necessary or sufficient ...

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### When is Ext*(M,N) finitely generated as a Ext*(M,M) module?

Let A be a finite dimensional algebra over a field k and M,N a finitely generated A-module.
Im searching for examples where the module $ Ext^{o} (M,N) $ is a finitely generated $ Ext^{o}(M,M) $ ...

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### Non-degeneracy of cup products on Tate-cohomology groups

I am working on a paper of R.P Langlands called "Representations of abelian algebraic groups", available here: http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/pdf/AbelianAlg-ps.pdf
Now on page ...

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### Properties of quotient categories.

I asked this on math.stackexchange.com, but didn't get any answer.
Let $\mathcal{A}$ be an abelian category and $\mathcal{C}$ a localizing subcategory in the sense of Gabriel. (A Serre/thick/dense ...

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### Is there an $A$ such that $B$ injective iff 1st Ext functor vanishes?

In the category of $\mathbb{Z}$-modules, there exists a module $A$---for instance $\bigoplus_{k=2}^\infty \mathbb{Z}/k\mathbb{Z}$---such that a $\mathbb{Z}$-module $B$ is injective iff ...

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### When does Ext^2 vanish in a category of group representations.

Let $G$ be a linear algebraic group over field $k$ of characteristic zero. It is well known that the category of finite dimensional $k$--linear representations of $G$ is abelian, and that it is ...

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### Model structure on the category of chain complexes in an abelian category gives rise to the derived category

Let $\mathcal A$ be an abelian category with enough projectives. Consider a morphism $f \colon X^\bullet \longrightarrow Y^\bullet$ in the category $C(\cal A)$ of chain complexes in $\mathcal A$.
Is ...

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### Natural Isomorphism of $S(V[1])$ and $(\bigwedge V)[n]$

Let $V:=\oplus_{j\in\mathbb{Z}}V_j$ be a graded $\mathbb{F}$-vector space over
the field $\mathbb{F}$. The graded tensor product of graded vector spaces is given
by
$V \otimes W:= \oplus_{j\in ...

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### Why are the left exact functors from an abelian category to abelian groups cocomplete and have a injective generator?

Let $\mathcal{C}$ be an abelian category, $\mathcal{Ab}$ the category of abelian groups and $Lex(\mathcal{C}, \mathcal{B})$ the category of left exact functors between abelian categories.
What is the ...

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### Why every complex of injectives is homotopically injective (provided that, the injective dimension is finite)?

Let $\scr A$ be an abelian category with exact products and a cogenerator (e.g. $\scr A$ is a category of modules). Let ${\mathbf K}(\scr A)$ be the homotopy category of cochain complexes over $\scr ...