# Tagged Questions

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69 views

### Exposition of the Calabi complex

I am interested in a complex derived by Eugenio Calabi in his article "On compact Riemannian manifolds with constant curvature".
The complex is referenced as "Calabi complex" in various citing ...

**3**

votes

**1**answer

176 views

### Tor dimension in polynomial rings over Artin rings

I found this tricky problem in trying to understand some properties of local rings at non-smooth points of embedded curves. But this would be a very long story. So I make it short and I try to go ...

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334 views

### Singular chains generated by manifolds with corners — does it really work?

Usually we define singular homology via the complex of singular chains:
$$C_\ast(X)=\bigoplus_{n\geq 0}\mathbb Z\langle\sigma:\Delta^n\to X\rangle$$
where the right hand side denotes the free abelian ...

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vote

**1**answer

239 views

### Cohomology after completion

I've been scouring google and asking friend about something I was certain must be absolutely the easiest thing to people who do homological algebra, and none seem to know the answer to this, so if ...

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**1**answer

247 views

### What is the “higher version” of chain homotopy in singular homology?

In basic algebraic topology, we know the following well-known chain homotopy theorem:
Let $X$ be a topological space and $I=[0,1]$ be the unit interval. Let $S_*(X)$ and $S_*(X\times I)$ be the ...

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votes

**2**answers

281 views

### Are there some websites for self learining of advanced mathematics? [closed]

Are there some websites for self learining of advanced mathematics?
For example, some great lecture vedio of differential geometry, Lie group , Lie algebra, algebraic topology and so on. Thanks

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**0**answers

75 views

### bialgebra cohomology

It seems that the Gerstenhaber-Schack (bi)complex of an associative bialgebra carries a homotopy e_3-algebra structure and a degree 2 Lie algebra bracket, up to homotopy. Does anyone know about a ...

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**0**answers

165 views

### Lagrangian (classical) BRST cohomology groups

I'm trying to understand BRST complex in its Lagrangian incarnation i.e. in the form mostly closed to original Faddeev-Popov formulation. It looks like the most important part of that construction ...

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votes

**1**answer

358 views

### Resolution of a module as an $A_\infty$ module over resolution of an algebra

The following question looks like a basic question on resolutions over commutative rings, unfortunately I was not able to find a reference.
Let $A$ be a regular commutative noetherian ring (and ...

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votes

**1**answer

169 views

### Deformations of a complex trivial up to quasi-isomorphism

Let $(C, d_0)$ be some cochain complex over commutative ring $R$ and $R$ is an algebra over a field $k$, then I could consider complex $(C[t], d_0+td_1)$ over ring $R \otimes k[t]$ where $d_1$ is some ...

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195 views

### Global dimension of endomorphism algebra of a coherent sheaf

Let $X$ be a smooth projective variety over algebraically closed field of characteristic zero. In one of the versions of definition of tilting object $\mathcal{F}$ on $X$ there is a requirement that ...

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91 views

### Godement resolution

Let $k$ be a field, $X$ a reasonable space, $k_{X}$ the constant (1-dimensional) sheaf on $X$. Write $\mathcal{G}^{\bullet}(k_X)$ for the Godement resolution of $k_X$. For each $n>0$, is ...

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203 views

### Extension to a right exact functor

Setup. Let $k$ be a field, $\mathcal{C}$ be a $k$-linear abelian category, $\mathcal{D}$ an arbitrary $k$-linear category. Let $\mathcal{C}'$ be a full subcategory of $\mathcal{C}$ with the following ...

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319 views

### 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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**3**answers

464 views

### 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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**2**answers

217 views

### 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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vote

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127 views

### 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 ...

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**0**answers

121 views

### The use of $Ext^{1}_A(M, N)$.

In the paper Prime Representations from a Homological Perspective. The authors show that $Ext^{1}_{\hat{\mathcal{F}}}(V, V)$ is one-dimensional if and only if $V$ is prime for some modules $V$ of ...

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**4**answers

189 views

### Stratifications and Cohomology Computations

I am interested in references and suggestions concerning the use of stratifications in topology to inductively compute topological invariants. I would appreciate a fairly introductory reference on the ...

**4**

votes

**1**answer

242 views

### Additive functors and Derived Categories

I have been learning about derived categories from Hartshorne's "Residues and Duality". One of the main theorems, as it is presented there seems to be the canonical isomorphism $RF \circ RG \cong R(F ...

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**1**answer

303 views

### Sequences of maps between modules such that $\ker(d_n) \subseteq \text{im}(d_{n+1})$

Consider a sequence of maps between $R$ modules (where $R$ is a ring with unity) $$\cdots \rightarrow M_{n+1} \xrightarrow{d_{n+1}} M_{n} \xrightarrow{d_{n}} M_{n-1} \rightarrow \cdots$$ such that ...

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**0**answers

218 views

### Do exact functors commute with spectral sequences ?

Let $F: \mathcal{A} \to \mathcal{B}$ be an exact covariant functor of abelian categories and let
$$\mathscr{C}: A \to A \to B \to A$$ be an exact couple in $\mathcal{A}$ with corresponding spectral ...

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**2**answers

319 views

### Morphisms between $K_0$

I suppose this is a question with a well known answer. Suppose $A$ and $B$ are two algebras over some field and there is a map
$$
f: \operatorname{K_0}(A) \to \operatorname{K_0}(B)
$$
is it ...

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votes

**2**answers

338 views

### Recovering an abelian category out of its derived category

I'm trying to learn more about derived category stuff and my curiosity has made me to ask these questions. Sorry if I'm being sloppy, I'm a new learner.
In Wikipedia it has been stated that since ...

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**3**answers

287 views

### DG-projective vs. K-projective complexes

Hello!
I'm a student learning the basics of working with the unbounded derived category $D(\mathcal{A})$. I arrived at the natural question, "is every K-projective complex formed out of projective ...

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votes

**2**answers

358 views

### Examples of tilting objects that don't come from exceptional sequences

This is a question on geometric tilting theory. On smooth projective variety it is possible to define in general tilting object as perfect complex that satisfy some properties, but are there examples ...

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**0**answers

181 views

### Not isomorphic varieties with isomorphic tilting algebras

Let $X$ be a smooth projective variety over a field, than tilting object $T$ on $X$ is a perfect complex that is a compact generator of the derived category $\operatorname{D}(QCoh(X))$ and satisfies ...

**4**

votes

**1**answer

240 views

### Why is the transfer map Tate-dual to restriction ?

In one of their papers (before Theorem 7.2), Benson and Carlson state that the transfer map is Tate-dual to the restriction homomorphisms (also see Remark 1.3 of this recent paper).
More ...

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vote

**2**answers

287 views

### An explicit homotopy equivalence between the de Rham complex and the Cech-de Rham total complex

I'm currently in need an explicit formula in classical cohomology which I'm pretty sure is well known, but which I've been unable to find in the references I am aware of.
Let $X$ be a smooth ...

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**1**answer

122 views

### Interesting examples of a 4-torsion X in a triangulated category such that $2 End(X/2X)\neq 0$?

It is well-known that for the sphere spectrum $S$ in the ('topological') stable homotopy category the object $S/2S$ i.e. the cone of $S\stackrel{\times 2}{\to}S$, is not $2$-torsion.
So I wonder ...

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**0**answers

174 views

### Reference Request: The Categorification of $\mathbb{Z}$ as cochain complexes of vector spaces

Just as the fact that a categorification of $\mathbb{N}$ is the category of finite dimensional vector spaces, a categorification of $\mathbb{Z}$ (in my mind) is the category of bounded cochain ...

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votes

**1**answer

524 views

### Classification of long exact sequences

Let $\mathcal C$ be the category of long exact sequences of finitely generated abelian groups almost all of whose entries vanish.
The category $\mathcal C$ is naturally additive as a subcategory of ...

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**0**answers

182 views

### (Reduced) cyclic homology of a free product of unital algebras

Shameless upfloat of 1-year old question - the motivation is that in general the corresponding Banach version is false, so I am trying to see where the proof breaks down, and what (if anything) can be ...

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votes

**3**answers

1k views

### Where can I find a proof of the de Rham-Weil theorem?

Where can I find a proof of the de Rham-Weil theorem?
Does anyone know?

**12**

votes

**1**answer

406 views

### $A_\infty$ structure on Ext-algebras well defined?

Let $M$ be an object in an $k$-linear abelian category with enough projectives. Then one can construct an $A_\infty$-structure on the Ext algebra
$$Ext^\bullet(M,M)$$
as follows: One chooses ...

**3**

votes

**2**answers

260 views

### Earliest/most standard reference for derived categories of hereditary algebras

Let $A$ be a hereditary algebra and let $\mathcal{D}$ be the derived category of bounded complex of finitely generated $A$-modules. Then, for any complex $C_{\bullet}$ in $\mathcal{D}$, we have ...

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391 views

### Somewhat general question that includes: “Do quasi-isomorphic cdgas have quasi-isomorphic spaces of derivations?”

Question: Given two quasi-isomorphic dg commutative algebras (over a field of characteristic zero, if you like), to what extent do their various homological geometric data agree?
Example: Given a dg ...

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votes

**2**answers

641 views

### torsion free modules over general ring

i want to know how to prove a torsion free modules over general ring is flat. (in "lecture on ring and modules, T.Y.Lam prove in case R is interal domain). please help me prove it or give me some ...

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votes

**0**answers

448 views

### On the multiplicative structure in spectral sequences.

Let $f\colon X \rightarrow Y$ be a continuous map of sufficiently nice
topological spaces (say, smooth manifolds). Let ${\cal F}=(\dots\rightarrow
F_i \rightarrow F_{i+1}\to \dots)$ be a bounded ...

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votes

**2**answers

562 views

### Relation between Gerstenhaber bracket and Connes differential

Let $C$ be an arbitrary algebra (more generally, a linear 1-category). The following structures are well-known:
A degree-0 product on the Hochschild cohomology $HH^*(C)$
$$
HH^*(C) \otimes HH^*(C) ...

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votes

**1**answer

146 views

### Existence of a bounded finitely generated torsion-free resolution

I am looking for a reference for (or a proof of) the following fact:
Let $G$ be a profinite group.
Let $X^\bullet$ be a complex of discrete $G$-modules.
We assume that the cohomology $G$-modules of ...

**3**

votes

**1**answer

514 views

### Is there a way to define a prime ideal object via diagrams in the category of rings?

I like to think in terms of commutative diagrams rather than referring to elements. So to me a group is really a group object, i.e. an object with some maps satisfying certain commutative diagrams. ...

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votes

**1**answer

480 views

### When does the homological dimension of a tensor product equal the sum of dimensions?

The notion of dimension I prefer most is right global dimension, but the question can also be asked for other notions (e.g. weak dimension, injective dimension, Krull dimension). Letting $d$ be ...

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**0**answers

194 views

### Extensions of maps between graded modules

Let $R$ be a connected graded ring (like $R=\mathbb R[x_1,\dots,x_d]$ with the usual grading) and let $N \subset R^{\oplus n}$ be a graded submodule, i.e. $$N= \bigoplus_{i \in \mathbb N} (N \cap ...

**8**

votes

**1**answer

633 views

### Morphisms of Banach spaces

What is the standard name in English for bounded linear maps $f:E\to F$ between Banach spaces such that the kernel $\ker(f)$ has a complement, and $\text{im}(f)$ is closed, and has a complement?
...

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votes

**0**answers

265 views

### Lifting of product of a Banach algebra

Let $A$ be a non unital Banach algebra. The product induces a bounded linear map $T:A \otimes_{\gamma} A\to A$ where $\otimes_\gamma$ denotes the Banach projective tensor product.
A lifting of $T$ is ...

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votes

**4**answers

1k views

### References for sign conventions in homological algebra

There is no shortage of sign conventions in homological algebra. And once these conventions are set out, there is no shortage of diagrams where an obvious commutative diagram on the underlying ...

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votes

**2**answers

602 views

### Modern reference for integral homology of a finitely generated abelian group

I'm interested in the calculation of $H_n (G,\mathbb{Z})$ for G a finitely generated abelian group, particularly for $n=3$. It's carried out in the 1954/55 Séminaire Henri Cartan, titled ...

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**1**answer

1k views

### Freyd-Mitchell's embedding theorem

Freyd-Mitchell's embedding theorem states that: if A is a small abelian category, then there exists a ring R and a full, faithful and exact functor F: A → R-Mod.
I have been trying to find a proof ...

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votes

**1**answer

1k views

### Where can I easily look up / calculate (abelian) group cohomology?

For an example I'm trying to understand, I need to calculate some cohomology group of some $\mathbb Z$-module with coefficients in some other $\mathbb Z$-module (with no interesting actions). (In ...