(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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6
votes
1answer
348 views

Is the sheaf of smooth functions flat?

Let $X$ be a smooth algebraic variety over $\mathbb{C}$. Is the sheaf of smooth functions on $X$ flat as an $\mathcal{O}_X$ module?
4
votes
0answers
82 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 numbers ...
5
votes
0answers
219 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 ...
10
votes
1answer
458 views

teaching higher algebra

Has anyone ever (successfully or unsuccessfully) taught a course in higher algebra (in the $\infty$-categorical sense)? I'm asking out of curiosity (and also hoping for more resources). The ...
2
votes
0answers
123 views

completion and convergence of spectral sequence

I would like to understand the connection between $p$-adic completion and the strong convergence of a spectral sequence. Precisely, suppose $E^2_{s,t}\implies G_{s+t}$ is a first quadrant strongly ...
2
votes
0answers
42 views

Calculation of minimal right $\operatorname{add}(M)$-approximations

given a finite dimensional quiver algebra $A$ and a generator $M$ with $\operatorname{Ext}^1(M,M)=0$. By Wakamatsus lemma, for any $A$-module $N$ there exists a surjective $A$-linear map $f\colon M_1 ...
4
votes
1answer
191 views

Information and intuition packed in the Chern character for coherent sheaves

even after quite some time learning it, I still get somehow puzzled by the Chern character. Let me recall some stuff to get notation and setting. Let us consider a smooth projective algebraic variety ...
3
votes
2answers
192 views

How do we get the quotient $Ext^1(N,M)/Hom(N,M)$?

If $0\longrightarrow M\longrightarrow E\longrightarrow N\longrightarrow 0$ is a short exact sequence of torsion free coherent sheaves on a surface. Here $M$ is a line bundle, $E$ a vector bundle of ...
5
votes
0answers
101 views

Second homology of mapping class group via universal coefficient theorem

Let $S_g$ be the closed, orientable surface of genus $g\ge 2$ and $Mod(S_{g,1})$ the mapping class group of homeomorphisms fixing a base point $x_0\in S_g$ modulo isotopies fixing $x_0$. It is known ...
1
vote
1answer
99 views

dg-resolution of the polynomial algebra

I am intersted in constructing a cofibrant resolution of the commutative polynomial algebra in some number of variables in the category of dg-algebras(not necceserily commutative). The resolutions ...
20
votes
1answer
699 views

Lemma 2 from Beilinson's “Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra”, intuition?

This is a followup to here. Consider Lemma 2 from Beilinson's paper "Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra", as follows. Lemma 2. For any pair $i$, $j$ such that $0 ...
0
votes
0answers
63 views

Homology of product of two groups [duplicate]

There is well known formula for the homology of product of two groups with coefficient in integers, that is $0 \rightarrow \oplus_{p+q=n}H_p(G,\mathbb{Z}) \otimes H_q(H,\mathbb{Z}) \rightarrow H_n(G ...
7
votes
0answers
159 views

Terminology for vanishing of Hochschild homology with symmetric coefficients?

In a title or abstract for a paper, if I say "Hochschild cohomology of this algebra $A$ vanishes in degrees two and above" then it should hopefully be understood by most readers as saying ...
0
votes
0answers
50 views

One-sided endomorphism rings of centred bimodules

Let R be an associative unital ring. An R-bimodule M is called centred bimodule if M = R*Z(M), where Z(M)={m:rm=mr,∀r∈R}, i.e., M is generated as an R-module by the set of R-centralizing elements. ...
3
votes
1answer
188 views

Reference for constructing tensor products of finitely presented functors

I need references related to the construction of tensor product between functors Let $k$ be a commutative ring, $C$ a small $k$-linear category and $A$ cocomplete abelian category. Let $A^C$ denote ...
5
votes
1answer
90 views

Flat dimension of injectives over a Gorenstein ring

Let $A$ be a Gorenstein noetherian local ring, and let $M$ be an $A$-module of finite injective dimension. If $M$ is a finite $A$-module, it is easy to show these assumptions imply that $M$ has ...
3
votes
0answers
122 views

Seeking an unpublished manuscript by Tetsuro Okuyama

Several papers in representation theory attribute the notion of relatively projective modules to Tetsuro Okuyama's manuscript "A generalization of projective covers of modules over finite group ...
4
votes
0answers
108 views

Cohomology of $Sym^m Q \otimes Sym^k Q \otimes L^p$

Let $V$ be a complex vector space. Let $L=\mathcal{O}(-1)$ and $Q=V/L$ be the quotient bundle over $\mathbb{P}V$. I'm trying to compute the cohomologies with coefficients in $Sym^m Q \otimes Sym^k Q ...
5
votes
1answer
103 views

Maps between products of symmetric powers

This question might be too elementary but it arises naturally as a part of a more complicated computation and I struggle to find the answer. Let $V$ be an $n$-dimensional complex vector space. ...
2
votes
1answer
154 views

Formal DG-algebra

Let $\mathcal{C}$ be a nice $k$-linear abelian category (the example I have in mind is the category of coherent sheaves on a smooth projective variety over $\mathbb{C}$). Let $B \in ...
1
vote
0answers
107 views

Relation of primary decomposition of two ideals

I have a simple question: Let $R=\mathbb{C} \lbrace t,u \rbrace$ be the ring of formal series in two variables. Let $I,J \subset R$ ideals of heigth one, and $I \subset J$. What is the relation of the ...
21
votes
2answers
1k views

Lemma 1 from Beilinson's “Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra”, intuition?

Consider Lemma 1 from Beilinson's paper "Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra", as follows. Let $\mathcal{C}$ and $\mathcal{D}$ be triangulated categories, $F: ...
8
votes
2answers
380 views

Confusion about Subcategories of Category $\mathcal{O}$

So, in learning about category $\mathcal{O}$ representations of a semisimple Lie algebra $\mathfrak{g}$, I've come across two natural kinds of subcategories, and I think I'm confused about their ...
5
votes
1answer
218 views

To calculate $Tor_1^G(\mathbb{Z},N_{ab})$ and $Tor_1^Q(\mathbb{Z},N_{ab})$

Let $G$ be a finite $p$-group and $N$ be a normal subgroup of $G$. I wish to calculate $Tor_1^G(\mathbb{Z},N_{ab})$ and $Tor_1^Q(\mathbb{Z},N_{ab})$, where $Q=G/N$. I could not found any lecture notes ...
3
votes
1answer
140 views

For what kind of sheaves can we always extend a sheaf map from a closed subset to the whole space?

Let $X$ be a topological space. We know that a sheaf on $X$ is call soft if for any closed subset $Z$ of $X$, a section on $Z$ can be always extend to a section on $X$. Now we consider a similar ...
1
vote
0answers
115 views

Is local-to-global spectral sequence functorial?

Consider a lower term of local-to-global spectral sequence $0 \to H^1(X,\mathcal{Hom}(\mathcal{F},\mathcal{G})) \to Ext^1(\mathcal{F},\mathcal{G}) \to H^0(X,\mathcal{Ext}^1(\mathcal{F},\mathcal{G})) ...
15
votes
1answer
290 views

Moduli space of boundary maps with prescribed chain and homology groups?

Let $R$ be a reasonable ring (maybe I mean a PID, or $\mathbb{Z}$, and when sufficiently desperate, a field). Now consider fixed sequences $C_n$ and $H_n$ of $R$-modules, which are tame in every ...
3
votes
1answer
184 views

generalized universal coefficient sequence

Take the familiar Universal Coefficient Theorem for ordinary homology with $\mathbb{Z}$-coefficients and ordinary cohomology with coefficients in some abelian group $A$:$$0\rightarrow ...
1
vote
0answers
65 views

Twisting cochains in non-unital settings

I'm reading chapter 2 of Loday and Vallette's book: Algebraic operads. My question is about how to translate some results in augmented settings into non unital settings. Let $k$ be a field of ...
2
votes
0answers
87 views

Flat resolutions of DG-schemes

Recall that a DG-scheme is a pair $(X,\mathcal{O}_X)$, where $(X,\mathcal{O}^0_X)$ is a scheme, $\mathcal{O}_X$ is a sheaf of commutative DG-algebras over $(X,\mathcal{O}^0_X)$, and each ...
1
vote
0answers
61 views

How to calculate Chern class for reflexive sheaf?

Let $(X,\omega)$ be a Kahler manifold of dimension $n$ and $\mathcal{F}$ a reflexive sheaf on $X$. Since there is no global resolution of sheaf by vector bundles in the non-projective manifolds. It ...
3
votes
0answers
140 views

Can one complete a morphism of commutative triangles to a “commutative cube” in a triangulated category?

This question is a continuation of Can one extend a morphism of commutative triangles to a morphism of octahedral diagrams?. I am deeply grateful for the contributions there; they roughly say that ...
4
votes
2answers
197 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
1answer
125 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
1answer
165 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
0answers
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
0answers
65 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
0answers
107 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
2answers
112 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
1answer
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
0answers
132 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
1answer
490 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
1answer
126 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
1answer
177 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
0answers
173 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
1answer
377 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
1answer
216 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
0answers
163 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
1answer
301 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
219 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 ...