All Questions
Tagged with homological-algebra reference-request
271 questions
3
votes
1
answer
129
views
Schur multiplier of finite-dimensional simple Lie algebras in positive characteristic
The Schur multipliers of finite simple groups are known and easily accessible:
https://en.wikipedia.org/wiki/List_of_finite_simple_groups
Moreover, as a consequence of the second Whitehead's Lemma, if ...
- 630
3
votes
1
answer
517
views
Künneth spectral sequence for cohomology of chain complexes of $R$-modules
Let $R$ be a unital ring. Let $\mathbf{A}_\bullet$ and $\mathbf{C}_\bullet$ be positive chain complexes of $R$-modules. If $\mathbf{A}_\bullet$ consists of flat $R$-modules then there is homology ...
- 855
3
votes
0
answers
117
views
Twisting an L_{\infty} module quasi-isomorphism with a sufficiently small Maurer-Cartan element
I was wondering if someone could help me understand this result, or point me towards a reference. Suppose that $M$ and $N$ are $L_{\infty}$ modules over a dgla $L$. Suppose that $\phi: M \rightarrow N$...
- 295
3
votes
1
answer
177
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Quiver and relations for ADE singularities in dimension one
Let $A$ be an ADE-hypersurface singularity in dimension one.
For example in Dynkin type $A_n$, A is given by $K[[x,y]]/(x^2+y^{n+1})$.
Then $A$ is CM-finite and let $M$ be the direct sum of all ...
- 26.5k
5
votes
0
answers
87
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Reference request: Étale base change of differential-graded algebras
I've asked this question on Math.StackExchange, but didn't receive an answer, so I'd like to try my luck here.
I'm looking for a reference for the following fact, which I've recently stumbled upon:
...
- 171
4
votes
2
answers
299
views
Relation of the first Hochschild cohomology and the outer automorphism group
Let $R$ be a ring.
Qeustion: Is it true that the first Hochschild cohomology of $R$ is zero if and only if the outer automorphism group of $R$ is finite?
(It is not true, by the two answers. Is it ...
- 26.5k
2
votes
0
answers
257
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Taking the homology of a chain complex, seen as a symmetric monoidal functor
I've found, in quite some places online (e.g. it's the last example in the wikipedia page about monoidal functors), a statement similar to this:
Homology can be seen as a symmetric, monoidal functor $...
- 613
3
votes
2
answers
337
views
Quasi-isomorphism preserves group hypercohomology
I am looking for a reference for the assertion in the title.
In more detail, let $\Gamma=\{1,\gamma\}$ be a group of order 2.
Let $A$ be a $\Gamma$-module (an abelian group on which $\Gamma$ acts).
...
- 14.1k
3
votes
1
answer
591
views
Finitely generated modules over Noetherian local ring that become isomorphic after faithfully flat base change
Let $(R,\mathfrak m)$ be a Noetherian local ring. Let $S$ be a Noetherian ring which is a faithfully flat $R$-algebra. If $M,N$ are finitely generated $R$-modules such that $M\otimes_R S \cong N \...
- 31
6
votes
1
answer
244
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What is a Serre-smooth algebra?
Let $A$ be an $R$-algebra.
In the book "Noncommutative Geometry and Cayley-smooth Orders" by Le Bruyn one can find the notion of "Serre-smooth" in the introduction.
But no formal ...
- 26.5k
3
votes
0
answers
261
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On the Hochschild cohomology of the minimal model of an $A_\infty$ algebra
Suppose $(A, (\mu_k))$ is a (curved) $A_\infty$ algebra, and let $(\tilde A, (\tilde\mu_k))$ be its minimal model. Now, we have two Hochschild cohomology rings $HH^*(A)$ and $HH^*(\tilde A)$. (It may ...
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1
vote
0
answers
175
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Cochains with multilinear differentials
Let $G$ be a group and let $M$ be a $G$-module. We denote by $(C^*(M,G),d)$ the complex of inhomogeneous cochains, i.e. $C^n(G,M)=M^{G^n}$.
We say that a cochain $a\in C^n(G,M)$ is multilinear if it ...
2
votes
0
answers
536
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A long exact sequence involving the mapping cone (Exercise 1.5.9 in Weibel)
Let $f:B\to C$ be a morphism of chain complexes. Exercise 1.5.9 from Weibel's "An introduction to homological algebra", page 24, states that the natural maps $\ker f[-1]\xrightarrow\alpha{\...
1
vote
0
answers
92
views
Group structure extension
Let $G$ be a finite group and $X$ a finite $G$-set. Let $H$ be the set-theoretical cartesian product of $G$ and $X$.
Is there an homological theory controlling all possible group structure on $H$ (...
- 2,384
2
votes
0
answers
91
views
When does a stable endomorphism ring have injective dimension at most one?
tLet $A$ be a Frobenius algebra (we can assume that $A$ is given by quiver and relations) and let $M$ be a basic $A$-module without projective direct summands (we can assume we know the decomposition ...
- 26.5k
3
votes
0
answers
277
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Dimension three spectral sequences
If you have two triangulated functors $F,G$, you can compute the cohomology of $FG(-)$ in terms of the cohomology of $G(-)$ and the cohomology of $F(-)$, in terms of a ``Grothendieck spectral sequence'...
- 5,701
5
votes
1
answer
425
views
Can we construct a filtered chain complex from a spectral sequence?
Suppose $\{(E_r,d_r)\}$ for $r>0$ forms a spectral sequence over a field $\mathbb{F}$, i.e. for any $r$, $(E_r,d_r)$ is a chain complex over $\mathbb{F}$ and $E_{r+1}=H(E_r,d_r)$. For simplicity, ...
- 673
2
votes
0
answers
195
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Small abelian categories and module categories - preservation of injective and projective objects
A soft question on small abelian categories:
https://en.wikipedia.org/wiki/Grothendieck%27s_T%C3%B4hoku_paper
Wikipedia: "The article "Sur quelques points d'algèbre homologique" by ...
user122276
3
votes
1
answer
263
views
Cohomology of commutative monoid acting on module
I have a some naive questions about how to define the cohomology of a commutative monoid.
One way to express the cohomology of a group $G$ with coefficients in a module $A$ is as $\text{Ext}^i_{\...
- 2,044
4
votes
1
answer
158
views
Linearity of covariant and contravariant $Ext^1$ functors defined via short exact sequences
Let $R$ be a Commutative ring. Let $M,X,Y$ be $R$-modules. Let $f: X \to Y$ be an $R$-linear map.
Then, given an exact sequence $\eta: 0\to X \to Z_{\eta} \to M \to 0$ in $Ext^1(M,X)$, the pushout of $...
- 642
4
votes
0
answers
158
views
Postnikov square explicitly on a simplicial complex
$\DeclareMathOperator\Z{\mathbb{Z}}$
Following Wikipedia, a Postnikov square is a certain cohomology operation from a first cohomology group $H^1$ to a third cohomology group $H^3$, introduced by ...
- 10.5k
7
votes
0
answers
244
views
Operad-free proofs of rectification of homotopy ($A_\infty/L_\infty$) algebras?
If (say) $L$ is an $L_\infty$-algebra, then it is known that under certain conditions there exists a quasi-isomorphic $L_\infty$-algebra $L'$ which is a differential graded Lie algebra: all ternary ...
- 784
4
votes
1
answer
104
views
Which Auslander algebras satisfy $Ext_B^1(D(B),B)=0$?
Let $B$ be the Auslander algebra of a representation-finite algebra $A$.
Question: When do we have $Ext_B^1(D(B),B)=0$? Can this be expressed in terms of nice properties of $A$?
This is for example ...
- 26.5k
10
votes
1
answer
307
views
Rings where all indecomposable projective modules are finitely generated
Let $X$ be the class of (unital, associative and not necessarily commutative) rings $R$ where every indecomposable projective $R$-module is finitely generated.
Question 1: Is there a nice equivalent ...
- 26.5k
0
votes
0
answers
108
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Connecting homomorphism and Baer sum in an abelian category
I would like to prove that the connecting homomorphism $\delta \colon \mathrm{Hom}_{\mathcal{A}}(N,M_3) \to \mathrm{Ext}_{\mathcal{A}}(N,M_1)$ from part (2) of Lemma 12.6.4 of the Stacks Project is ...
- 237
1
vote
0
answers
178
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Zeroth cohomology of tensor product of complexes concentrated in nonpositive degrees
This is probably an easy problem, but I can't find any reference.
Let $V$ and $W$ be cochain complexes over some commutative ring, and assume that they have both cohomologies concentrated in ...
- 2,079
5
votes
0
answers
142
views
A practical way to check whether a module is periodic
A module $M$ over a finite dimensional selfinjective algebra $A$ over a field $K$ is called periodic if $M \cong \Omega^n(M)$ for some $n \geq 1$. We assume here that $M$ is simple and that A is a ...
- 26.5k
3
votes
0
answers
180
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$k$-invariants of $KO$ and $ko$ and differentials in the AHSS spectral sequence
Let $KO$ and $ko$ denote real $K$-theory and connective real $K$-theory. It appears to be a well done result that the $k$-invariants can be used to determine the early differentials in the Atiyah-...
- 855
8
votes
1
answer
362
views
Reference request: Origins of differential homological algebra
Differential homological algebra in its initial formulation is due to Eilenberg and Moore, who published the homological version of the Eilenberg–Moore spectral sequence in 1965 (and the cohomological ...
- 2,995
6
votes
1
answer
362
views
Comparing Hochschild (co)homology for algebras and coalgebras
Given a field $k$, an associative $k$-algebra $A$, and an $A$-bimodule $M$, one can define as the Hochschild homology and cohomology as the homology of the complexes
$$M\otimes A^{\otimes n}$$
and
$$\...
- 518
2
votes
0
answers
64
views
contracting homotopies of Koszul resolution of $\mathbb{C}[x_1, \ldots, x_n]$ and $\mathbb{C}_{q}[x_1, \ldots, x_n]$
Let $A : = \mathbb{C}[x_1, \ldots, x_n],$ $A_q : =\mathbb{C}_q[x_1, \ldots, x_n] = \mathbb{C} \langle x_1, \ldots, x_n \rangle / (x_ix_j = q x_jx_i)$.
By Koszul resolution I mean
$$\ldots \to A \...
- 161
1
vote
1
answer
360
views
Computing Ext groups in a functor stable $\infty$-category
Let $I$ be a small category and $\mathcal{D}=D^b_\infty(\mathbb{Z})$ the bounded derived $\infty$-category of abelian groups. Consider the $\infty$-category $\mathcal{C}:=\mathrm{Fun}(I,\mathcal{D})$. ...
- 617
6
votes
1
answer
186
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Endomorphism ring of trivial source modules for abelian p-groups
Bernhard Böhmler (who is also on MO) and myself had the following idea:
Let $G$ be a finite group and $k$ a field of characteristic $p$ (algebraically closed when it is needed) such that $p$ divides ...
- 26.5k
4
votes
0
answers
75
views
Lie algebra "semi" coinvariants
In the process of my research, I've come across the need to understand the following construction:
Let $\mathfrak{g}$ be a (finite-dimensional) complex Lie algebra, $\beta\in \mathfrak{g}^*$ a Lie ...
- 3,079
3
votes
0
answers
48
views
Questions on piecewise hereditary algebras
Let $A$ be a finite dimensional quiver algebra over a field $k$ that is quasi-tilted and representation-finite (this implies that $A$ is a tilted algebra). Assume that the Coxeter polynomial of $A$ is ...
- 26.5k
3
votes
0
answers
69
views
On Ext-duals of injective modules for commutative rings
Let $R$ be a commutative noetherian ring and $I=E(R/p)$ the injective hull of the module $R/p$ for a prime ideal $p$.
Question: Is there a (more) explicit description of the $R$-modules $Ext_R^i(I,R)$...
- 26.5k
10
votes
1
answer
396
views
Generalising the union-closed sets conjecture from lattice to a larger class of posets
(edit: I decided to simplify the question and only pose it for bounded posets first)
The Union-closed sets conjecture is equivalent for lattices P to:
There exists a join-irreducible element $a$ with ...
- 26.5k
5
votes
0
answers
76
views
Reference on two numbers associated to a module of finite homological dimension
Let $A$ be a finite dimensional algebra over a field $K$ with a module $M$ which has finite projective dimension and finite injective dimension.
Let $n \geq 1$.
Let $(P_i)$ be a minimal projective ...
- 26.5k
8
votes
2
answers
2k
views
Original proof of Hilbert's syzygy theorem
Does anyone know an English reference for the original proof of Hilbert's syzygy theorem? The three proofs that I know use either:
the theory of projective dimension and change of rings (plus a step ...
- 14.7k
4
votes
1
answer
290
views
When does $\operatorname{Ext}_C^1(M,N_i)=0$ imply $\operatorname{Ext}_C^1\left(M,\lim\limits_\longleftarrow N_i\right)=0$?
Let $C$ be an abelian category. Suppose that $(N_i)_{i\in I}$ is an inverse system of objects in $C$. Under which conditions does the hypothesis that $$\operatorname{Ext}_C^1(M,N_i)=0\quad\forall i\...
- 143
1
vote
1
answer
872
views
Homotopy equivalences and Mapping Cones
Edit: Version 2:
Suppose that $A,B,C$ are chain complexes and $f: A \rightarrow B$ is a chain map. Suppose that there is a homotopy equivalence
$$ \text{Cone}(f: A \rightarrow B) \simeq C.$$
The chain ...
- 29
1
vote
0
answers
132
views
On the dual version of an isomorphism of spectral sequence term (from Cartan and Eilenberg)
I'm trying to take spectral sequences as a black box for application in commutative algebra and I admit that I haven't really gone through (or understand) all the proofs of all the isomorphisms ...
- 642
2
votes
0
answers
55
views
Free resolutions of universal enveloping algebras for simple, finite dimensional Lie algebras
I'm currently studying Anick's resolution on the context of universal enveloping algebras for certain Lie algebras, namely some of the smallest cases: $A_1,A_2,A_3,B_2,G_2$, and so on.
What are some ...
5
votes
0
answers
132
views
On a reference for computing global spectrum of $A_n$-curve singularities, by H.Dao and E.Faber
This question is about chasing down a reference in a paper relating to non-commutative crepant resolutions and Cohen-Macaulay representation theory.
Allow me to first give a minor introduction.
Let $(...
- 51
4
votes
1
answer
259
views
Induced map in K-theory by a "trivial" bimodule
Let $R$ be a ring (not necessary commutative) and let $P_{\bullet}$ be a perfect $R$-bimodule (chain complex). I will denote the category of perfect right $R$-chain complexes by $\textbf{Perf}(R)$. ...
- 43
5
votes
1
answer
2k
views
Cohomology of derived tensor product of complexes and Künneth spectral sequence
Let $R$ be any commutative ring, let $V^\bullet$ and $W^\bullet$ be (co)chain complexes of $R$-modules, indexed cohomologically. We can also assume that they have both cohomology in nonpositive ...
- 2,079
9
votes
2
answers
991
views
Reference for homotopy colimit = total complex
I'm looking for a reference for the following fact:
take a simplicial chain complex $ X:\Delta^{op}\to Ch_{\geq 0}(\mathcal A)$ for $\mathcal A$ a nice abelian category (say, cocomplete with enough ...
- 15.8k
1
vote
1
answer
379
views
Splitting of short exact sequence in the category of finitely generated modules over a commutative Noetherian ring
In the category of finitely generated modules over a commutative Noetherian ring, the splitting of a short exact sequence can be checked locally at the maximal ideals of the ring. One reference for ...
- 642
5
votes
2
answers
683
views
Model categories and chain complexes
I'm fairly new to thinking about homological algebra and chain complexes in their own right, i.e outside of isolated examples such as for constructing simplicial homology, or for computing $Ext$ ...
- 168
5
votes
0
answers
168
views
Higher analogue of the Auslander-Bridger transpose
Let $A$ be an Artin algebra and $M$ a module with $Ext^i(M,A)=0$ for $i=1,...,n-2$.
Then in case $P_{n-1} \rightarrow ... \rightarrow P_0 \rightarrow M \rightarrow 0$ is the beginning of a minimal ...
- 26.5k