All Questions
Tagged with homological-algebra reference-request
271 questions
2
votes
0
answers
239
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Reference Request: A "Chevalley-Eilenberg"-style formulation of the $L_\infty$ algebra minimal model theorem?
The nicest definition of $L_\infty$-algebras ---which I will call a "Chevalley-Eilenberg" style definition after the obvious analogy with the Chevalley-Eilenberg differential of Lie algebras--- is the ...
3
votes
2
answers
214
views
History of an open problem on partial tilting modules
The following is an open problem:
Given a partial tilting module $T$ over a finite dimensional algebra $A$ (that is $Ext_A^i(T,T)=0$ for all $i \geq 1$ and $pd(T) < \infty$), then $T$ is a tilting ...
4
votes
1
answer
448
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 non-...
4
votes
1
answer
179
views
Reference for isomorphism $Ext_A^n(X,Y) \cong Ext_A^1(\Omega^{n-1}(X),Y)$
Let $A$ be a finite dimensional algebra and $X,Y$ indecomposable modules and $n \geq 2$.
We know that $Ext_A^n(X,Y) \cong Ext_A^1(\Omega^{n-1}(X),Y)$.
Now such an isomorphism should be given by ...
4
votes
0
answers
290
views
Generalized Postnikov square
Following Wikipedia (https://en.wikipedia.org/wiki/Postnikov_square), a Postnikov square is a certain cohomology operation from a first cohomology group $H^1$ to a third cohomology group $H^3$, ...
3
votes
0
answers
166
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Edge map in derived categories
Let $\mathscr{A},\mathscr{B}$ be abelian categories, the first with enough projectives, together with a right-exact functor $F\colon \mathscr{A}\to\mathscr{B}$ (in my example, it is a tensor product, ...
45
votes
8
answers
10k
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A down-to-earth introduction to the uses of derived categories
When I was learning about spectral sequences, one of the most helpful sources I found was Ravi Vakil's notes here. These notes are very down-to-earth and give a kind of minimum knowledge needed about ...
1
vote
0
answers
77
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n-Gorenstein algebras and tilting modules
Let $\Lambda$ be an artin algebra over a commutative artinian ring $R$. $\Lambda$ is said to be $n$-Gorenstein, for some natural number $n$, provided it have finite self-injective dimension at most $n$...
4
votes
1
answer
153
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The homological negligibility of certain subsets in compact manifolds
Let $n\ge 3$ and $X$ be a compact connected $n$-manifold (without boundary).
I need a reference to the following facts (which I believe are true at least in dimension $n=3$):
Fact 1. For every ...
7
votes
1
answer
220
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Is $C^{\infty}(E)$ a projective Frechet $C^{\infty}(M)$-module for a $C^{\infty}$-fiber bundle $E\to M$ with compact fiber?
The question is a special case of a previous question.
Let $M$ be a compact smooth manifold, then it is clear that $C^{\infty}(M)$ is a Frechet algebra with pointwise multiplication and a collection ...
5
votes
2
answers
285
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Is $C^{\infty}(M)$ a projective Frechet $C^{\infty}(N)$-module for a smooth map $M\to N$ between compact smooth manifolds?
Let $M$ be a compact smooth manifold, then it is clear that $C^{\infty}(M)$ is a Frechet algebra with pointwise multiplication and a collection of semi-norm defined by $p_{\alpha}(f):=\sup_{\beta\leq\...
8
votes
1
answer
1k
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Surjectivity of a map on inverse limits
(The following is crossposted from Math.SE, where the question did not receive any answers.)
I am looking for a proof of the following lemma from P. Gabriel's Des catégories abéliennes (Chap. IV, §3, ...
1
vote
0
answers
123
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Reference request for a simple homological fact
Let $\mathcal{A}$ be an abelian category,
and let $0 \to M \to N \to K \to 0$ be a short exact sequences of complexes with value in $\mathcal{A}$.
Then there is a long exact sequence in cohomology
$\...
11
votes
0
answers
310
views
Snake lemma for equivalence relation
A sequence $$E(\zeta) \stackrel{\theta}{\to} X^2 \rightrightarrows X \stackrel{\zeta}{\to} Y $$
where the unlabelled arrows are the two projection, is said to be exact iff
$\zeta$ is the ...
10
votes
1
answer
1k
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Equivalent descriptions of Coherent Groups
Attending a series of lectures, I have recently been exposed to the notion of Coherent groups, defined as following:
Def: A group $G$ is called Coherent if every finitely generated subgroup $H$ of $G$...
25
votes
1
answer
2k
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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 ...
2
votes
1
answer
255
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Does anyone have a copy of Salce's paper "Cotorsion theories for abelian groups"?
The paper "Cotorsion theories for abelian groups" by L. Salce, was published in 1979 in Symposia Math. 21, pages 1-21. According to Google Scholar, it's been cited 233 times, and I keep seeing ...
15
votes
2
answers
2k
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When is bar-cobar duality an equivalence?
Let $A$ be an augmented differential graded algebra over a field $k$. I will write $BA$ for its bar construction (whose homology is $Tor^A(k, k)$). This is a co-augmented differential graded ...
1
vote
1
answer
174
views
Reference for a result of Auslander about the global dimension
One of Auslanders famous theorems is that he proved that the global dimension of a semiprimary ring is equal to the maximum of the projective dimensions of the simple modules of the ring. This result ...
2
votes
0
answers
143
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Computing derived functor of a complex with non-acyclic terms
Let $A^\bullet =(\dots\to A^i\to A^{i+1}\to\dots)$ be a bounded below complex in an abelian category $\mathcal{A}$ with sufficiently many injectives. Let $F\colon \mathcal{A}\to \mathcal{B}$ be an ...
1
vote
0
answers
161
views
Projective dimension of a principal ideal
Let $R$ be a polynomial ring, $I$ a homogeneous ideal, and let $\operatorname{pd}(R/I)$ denote its projective dimension. Is there a characterization of homogeneous elements $a\in R\setminus I$ for ...
3
votes
2
answers
1k
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An alternative definition of pseudo-coherent complex
Let $(X,\mathcal{O}_X)$ be a scheme or a general ringed space. First recall that a complex of $\mathcal{O}_X)$-modules $\mathcal{E}^{\bullet}$ is called strictly perfect if $\mathcal{E}^{\bullet}$ is ...
1
vote
1
answer
186
views
Comparing self-equivalences of a triangulated category and automorphisms of its Grothendieck group
There is a homomorphism from the group of (isomorphism classes of) self-equivalences of a triangulated category to the automorphism group of its Grothendieck group. Is this homomorphism surjective? If ...
2
votes
0
answers
61
views
Question on outer Ext-products
For group algebras $A=KG$ over a field $K$ with finite group $G$ there exists an outer product on Ext:
$Ext_A^i(M,N) \otimes_K Ext_A^j(M',N') \rightarrow Ext_A^{i+j}(M \otimes_K M',N \otimes_K N')$.
...
4
votes
0
answers
432
views
Reference request: sheaf-theoretic operations in the classical topology?
Like many graduate students before trying to learn something about étale cohomology and Deligne's proof(s) of the Riemann hypothesis part of the Weil conjectures, I am hunting for references detailing ...
1
vote
0
answers
63
views
Reference request for formula on global dimension
Given a finite dimensional algebra $A$ over an algebraically closed field $K$.
Let $A^e=A^{op} \otimes_K A$ be the enveloping algebra of $A$.
Who noted first that the global dimension of $A$ is equal ...
15
votes
1
answer
961
views
Who conjectured the Cartan determinant conjecture
The Cartan determinant conjecture states that every finite dimensional algebra of finite global dimension has the property that the determinant of its Cartan matrix is equal to one. Who stated this ...
6
votes
1
answer
241
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For which exact couples do associated spectral sequences degenerate at $E_1$?
It is well known that a bigraded exact couple of objects of an abelian category yields a spectral sequence (cf. https://ncatlab.org/nlab/show/exact+couple#SpectralSequencesFromExactCouples). My ...
1
vote
0
answers
113
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Reference request. The adjunction $\hom_{CDGA}(\Omega^\bullet(A),B^\bullet)\cong\hom_{CA}(A,B^0)$
We have the adjunction
$$\hom_{CDGA}(\Omega^\bullet(A),B^\bullet)\cong\hom_{CA}(A,B^0)$$
where $CDGA$ is the category of commutative diffferential graded algebras and $CA$ is the category of ...
6
votes
1
answer
253
views
Is the hom in derived category of a dg-algebra compatible with base field extension?
Let $k$ be a field and $A$ be an ordinary $k$-algebra. Let $M$ and $N$ be two left $A$-modules then we could consider the Ext group $\mathrm{Ext}^i_A(M,N)$ which is actually a $k$-vector space. Let $l/...
5
votes
0
answers
321
views
Do differential objects form triangulated categories?
Let $\mathcal{A}$ be a (fixed) additive category. To a differential object $(A,a)$ for $\mathcal{A}$ (so, $a:A\to A$ and $a^2=0$) one may associate an $\mathcal{A}$-complex $\dots \to A\stackrel{a}{\...
7
votes
1
answer
911
views
The Mittag-Leffler condition as necessary and sufficient
Let $A_1\leftarrow A_2\leftarrow A_3\leftarrow\dotsb$ be a projective system of abelian groups with the projection maps $p_{ij}\colon A_j\to A_i$, $j\ge i$. The derived functor of projective limit $\...
7
votes
1
answer
837
views
Yoneda extensions in exact categories and their derived categories
If $\mathcal{A}$ is any abelian category, then for all objects $X,Y$ in $\mathcal{A}$ and for all integers $i \geq 0$, there is an natural isomorphism
$$\mathrm{Ext}_\mathcal{A}^i(X,Y) \simeq \mathrm{...
7
votes
0
answers
555
views
Background on Kontsevich's Work on Quantization
Where can I find background reading material necessary to be able to read about Maxim Kontsevich's work on quantization? I would like to able to follow the ongoing seminar of IHES, "Resurgence and ...
4
votes
2
answers
390
views
About the cone being unique up to non-unique isomorphism
In an answer to this MO question [link] Fernando Muro sais:
the mapping cone of a morphism in a triangulated category is unique up
to non-unique isomorphism. This fact has originated a lot of ...
7
votes
1
answer
472
views
Reference for dualizable chain complexes
Let $k$ be a commutative ring. Let $Ch(k)$ denote the monoidal category of chain complexes. I need a reference, including a proof, for the following "folklore fact" (see for example here after ...
5
votes
0
answers
303
views
Recovering an A-infinity structure on an Ext-algebra from a quiver presentation
Let $A=KQ/I$ be a basic finite dimensional algebra given by a quiver with relations. Let $S$ denote the direct sum of the corresponding simple modules.
According to [Keller: A-infinity algebras in ...
7
votes
1
answer
493
views
Pro-representability of deformation functor associated to a DG Lie algebra
Edit : There are several satisfying proofs in the case each $L^i$ is finite-dimensional. It is proven (for example, Hinich DG coalgebras as formal stacks) that for $A$ : local Artin ring then $\...
5
votes
0
answers
520
views
Spectral sequences coming from filtrations (Postnikov systems) in triangulated categories: references and convergence
Let $c^i: M^{\ge i+1}\to M^{\ge i}$ for $i\in \mathbb{Z}$ be an sequence of morphisms in a triangulated category $C$ and assume that $M^{\ge i}$ are equipped with compatible morphisms into an object $...
3
votes
0
answers
310
views
Functoriality of Leray homology spectral sequences of fibrations
Let $p\colon E\to B$ and $p'\colon E'\to B'$ be two fibrations. Assume for simplicity that $B,B'$ are simply connected. Let we have morphism of these fibrations, i.e. two continuous maps
$$f\colon E\...
4
votes
0
answers
218
views
map of Koszul cohomology
I am reading paper "Standard systems of parameters and their blowing-up rings", J. Reine Angew. Math. 344 (1983), 201--220 of Peter Schenzel. In proof of Theorem 3.9, page 209-the second diagram, he ...
3
votes
0
answers
205
views
Finitistic dimension via tilting modules
is the following true (all algebras and modules are assumed to be finite dimensional):
The finitistic dimension of an algebra is equal to the supremum of projective dimensions of tilting modules?
It ...
0
votes
1
answer
337
views
Homological dimensions of tensor products of algebras
Given two finite dimensional algebras $A$ and $B$ over a field. The Gorenstein dimension of an algebra A is defined as the injective dimension of the module A. The finitistic dimension of an algebra A ...
0
votes
1
answer
186
views
Simultaneous extension of modules
Let $R$ be a commutative ring. Suppose $R$-modules $X,A,B,C$ and $Y$ are given such that the outer two rows and the outer two columns in the following diagram are exact.
$\hskip1in$
Does it ...
6
votes
1
answer
623
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 $Mor_{C}(d,e)=\{0\}\ \forall d\in D$; ...
1
vote
1
answer
293
views
Homological dimension of pure coherent sheaves and specialization
Let $X$ be a projective variety, not necessarily smooth, $R$ a DVR with residue field $k$ (assume char$(k)=0$). I am looking for examples of a pure coherent sheaf, say $F$, on $X_R:=X \times_k \mathrm{...
2
votes
2
answers
2k
views
Torsion-free modules over a general ring
I want to know how to prove that a torsion-free module over a general ring is flat. In Lectures on Rings and Modules, T.Y. Lam proves this in the case where your ring is an integral domain. Can you ...
6
votes
0
answers
209
views
Classification of representation-finite algebras up to stable equivalence of Morita type
Assume $K$ is an algebraically closed field.
I wanted to ask if there is a classification of the representation-finite $K$-algebras up to stable equivalence of Morita type (at least for some small ...
0
votes
0
answers
391
views
Conditions for splitting of short exact sequence?
Assume $K$ is a number field and $E$ is an elliptic curve defined over $K$.
Are there conditions under which the short exact sequence
$$0\rightarrow E (K)/mE (K)\rightarrow H^1_{Sel}(K,E_m)\...
4
votes
0
answers
235
views
Universal enveloping algebra functor preserves quasi-isomorphism
Let $k$ be a field of characteristic 0. Let $\mathtt{DGA}_{k}$ denote the category of DG algebras and $\mathtt{DGLA}_{k}$ denote the category of DG Lie algebras. It is well known that there are model ...