The homological-algebra tag has no wiki summary.

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### Is there any relation between the simplicial $S^1$ and the Hochschild homology of a noncommutative algebras

Let $k$ be the base field and $A$ be a unital associative $k$-algebra. Let's review the Hochschild homology theory: we have the Hochschild chain comple $C_{\cdot}(A)$ where
$$
C_n(A):=A^{\otimes n+1}
...

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93 views

### Consistency of the u-invariant under field extension

A algebraic field extension L/k induces of homomorphism between the Wittrings. We get
$\phi: W(k) -> W(L)$. If every anisotropic isometry class of $W(k)$ stays anisotropic, the kernel of $\phi$ ...

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74 views

### Lifting Lie algebra cohomology class to Hochschild cochain

Let $g$ be a Lie algebra, $h\subset g$ an ideal. The associative algebra $N=U(h)\subset U(g)$ can be viewed as a $g$-module.
The Lie algebra cohomology ${\rm H}^*(g,U(g))$ is isomorphic to the ...

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199 views

### Is 'the' homotopy colimit of a sequence of exact triangles an exact triangle?

Let
$$
\begin{array}{rccccl}
A_0&\to& B_0&\to& C_0&\to\\
\downarrow & &\downarrow&&\downarrow\\
A_1&\to& B_1&\to& C_1&\to\\
\downarrow & ...

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97 views

### Sort of units for the Yoneda product (and/or in Hochschild cohomology)

In an abelian category $\mathcal A$ with enough projectives, we have the Yoneda pairing
$$\operatorname{Ext}^p_{\mathcal A}(Y,Z)\otimes \operatorname{Ext}_{\mathcal A}^q(X,Y)\longrightarrow ...

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**1**answer

195 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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231 views

### Sign problem for the shift functor on DG modules

Let $\mathcal{A}$ be a Differential graded $k$ category, where $k$ is a commutative ring. I want to extend the shift functor $[1]$ on chain complexes to dg modules over $\mathcal{A}$. Now a right dg ...

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**1**answer

83 views

### Kernel of the induced map of the wedge product

Let $A$ be a noetherian ring and let $M$ be a finitely generated $A$-module. Let $F$ be a free $A$-module and let $d: F \to M$ be a homomorphism which maps a basis of $F$ to a minimal set of ...

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votes

**1**answer

96 views

### Relative flasqueness?

It is known that a flasque sheaf on a topological space has trivial cohomology. Suppose that we are in a relative situation of a smooth fibration $\pi: X \to S$ and $F$ is a sheaf on $X$. Is there ...

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**1**answer

444 views

### A question about the universal coefficients theorems

This seemingly simple question stands unanswered on math.stackexchange.com for a couple of days (http://math.stackexchange.com/questions/680211/a-question-about-the-universal-coefficient-theorem), so ...

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365 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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votes

**2**answers

283 views

### Non-vanishing $\mathrm{lim}^1$-term for the cohomology of a CW-complex

Let $h$ be an additive cohomology theory. If we want to compute $h^*(X)$ for an infinite CW-complex $X$, a standard method is to use the Milnor sequence
$$ 0 \to \mathrm{lim}^1_k h^{n-1}(X^{(k)}) \to ...

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106 views

### semi-orthogonal decompositions and embeddings

This is most likely a stupid question, but I am curious. It is well known that $\mathbb{P}^n$ has a semi-orthogonal decomposition
$$D^b(X)=\langle \mathcal{O},\dots,\mathcal{O}(n)\rangle$$
Suppose ...

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**0**answers

96 views

### syzygy of a generalized cohen-macaulay module

Let $R$ be a local, noetherian ring of dimension $d$ and suppose it is generalized cohen-macaulay. Is it true that For any finitely generated $ R $-module $ M $, which is maximal generalized ...

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245 views

### generators for derived category

Let $G$ be a algabraic group $G$ over a field $k$. We denote by $D^b(\mathrm{Repr}(G))$ the derived category of finite dimensional representations. Under what kind of assmumptions one has a generating ...

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232 views

### 2 - Calabi Yau algebras and bimodule coherence

Let $\Pi:=\Pi(Q)$ be the preprojective algebra of a connected non-Dynkin quiver over an algebraically closed field of characteristic zero.
In H. Minamoto "Ampleness of two-sided tilting complexes", ...

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### origin of spectral sequences in algebraic topology

I have the following somewhat vague question. I am not sure if it is appropriate for this forum, please feel free to close (or migrate to stackexchange).
I have been "brought up" as an algebraic ...

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vote

**1**answer

247 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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**2**answers

274 views

### Why the term “monad” in homological algebra?

Which is the origin and the reason for the choice of the term "monad" in the sense of homological algebra?
Does this concept have any relation whatsoever to the "monads" from category theory?

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162 views

### Computing the Abelianization of an Automorphism Group

Setup: We are working in a Henselian local ring $(R, \mathfrak m, k)$ that way may assume is Cohen-Macaulay, admits a canonical module and is of finite type (so is an isolated singularity). Let ...

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**1**answer

103 views

### The Existence of Pure Resolutions, Given a Degree Sequence?

I have been trying to understand the proof of the following theorem for the last month, I read some basics of sheaves theory and their cohomology, but still can't get the idea of this important ...

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**1**answer

214 views

### Inverse limit of Gorenstein local rings is again Gorenstein?

If we have the system of surjective ring homomorphisms
$f_{i,i+1}: R_{i+1} \twoheadrightarrow R_i$
for an arbitrary $i \geq 0$ such that all $R_i$ are Gorenstein local ring. Let us put
...

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224 views

### Tensor product of d.g-algebras

I'd like to prove that the tensor product functor $- \otimes Y$, where $Y$ is a d.g-algebra over a field of characteristic 0, preserves finite products of d.g-algebras. This statement is in a paper by ...

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138 views

### Cone of morphism between semi-free DG modules

Let $A$ be a DG algebra. Recall that an $A$-module $M$ is free if it is isomorphic to a sum of shifts of $A$, or semi-free if it has an exhaustive filtration $0 = F_0 \subset F_1 \subset \cdots ...

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267 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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55 views

### Relationship of height zero hypercovers to co-cartesian condition on cosimplicial modules

Suppose given a cosimplicial ring $R^\bullet$ and a cosimplicial module $M^\bullet$ (i.e. a cosimplicial Abelian group such that $M^n$ is an $R^n$-(left/right/bi)module). I have seen it said that ...

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299 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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### An invariant number of modules over Auslander Gorenstein modules

Given an Auslander Gorenstein $R$ of injective dimension $\mu$, one can associate with each finitely generated module $M$ with a number $\varepsilon_{\mu}(M)$, which is the number of the ...

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214 views

### Semi-free resolutions

Let $\mathscr{C}$ be a DG category (not much will be lost if you assume that $\mathscr{C}$ has one object, i.e. is a DG algebra). One way to construct the unbounded derived category of ...

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### When for every module $M$, $|E(M)| = |M|$

Is there a non-semisimple ring $R$ such that for any left $R$-module $M$, $|E(M)| = |M|$ ? (where $E(M)$ is the injective hull of $M$ and $|M|$ is the cardinality of $M$)

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### 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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137 views

### arrows in the injective representations of quivers

Let $Q$ be a quiver (a directed graph) and $E$ be a representation of $Q$ by R-modules, that is, for each vertex $v$ in $Q$ there exists an R-module $E(v)$ and for each arrow $a:v\to w$ there exists a ...

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373 views

### Splitness of commutative diagrams

Consider the following commutative diagram in the category of $R$-modules where $R$ is an associative ring with identity and all modules are unital.
$
\newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ ...

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242 views

### Is there a practical criterion to determine whether the limit of a diagram of real chain complexes is also a homotopy limit?

Consider a diagram D: I→ChR of real connective chain complexes.
In the example I have in mind all chain complexes are concentrated in some fixed degree n.
There is a canonical map lim D → holim D ...

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183 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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**1**answer

184 views

### Algebra structure $Tor(A,A)$

This is a question i asked on math.stackexchange but i didn't get any answer.
Let $A$ be algebra over commutative ring $k$ and $P_{\bullet}=(P_i,d_i)\rightarrow A$, $k$ projective resolution. Then we ...

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**1**answer

116 views

### Localizations of hereditary rings

It is known that if a commutative Noetherian ring $R$ is hereditary then for any maximal ideal $M$ the localization $R_M$ is also hereditary. Is the Noetherian assumption necessary?

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298 views

### Bar construction vs. twisted tensor product

One may study the cohomology of a space $E$ expressed as a homotopy pullback of $X$ and $Y$ over $Z$ using either the Eilenberg-Moore spectral sequence or the Serre spectral sequence for the fibration ...

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votes

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260 views

### What is the correct definition of the (derived) tensor product over a dg-algebra?

Let $A_\bullet$ be a dg-algebra over a field $k$. Let $M_\bullet$ (resp. $N_\bullet$) be a right (resp. left) $A_\bullet$-module. There is then a notion of the derived tensor product:
...

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477 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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**1**answer

178 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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258 views

### Is the derived Fukaya category a derived category in the classical sense?

I am trying to read Seidel's book, and I am confused about the "derived" Fukaya category. If I understand properly, one starts from the $A_{\infty}$-category $\mathfrak{F}(M)$, enlarges it to the ...

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115 views

### Criteria for a finite-dimensional $k$-Algebra to be basic and elementary

I have the following question:
Suppose, I have a finite dimensional $k$-Algebra $A$ over an arbitrary field $k$ and a finite dimensional module $M$ that is a generator-cogenerator of mod-$A$.
I'm ...

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**1**answer

189 views

### Computing Ext: $\text{Ext}(i_* \mathcal{O}_X, i_* \mathcal{O}_X)$ for closed embedding $i:X \rightarrow Y$

Let $V$ be a vector bundle on $X$, and $Y = \text{Tot}(V)$ be the total space of this bundle; we have a closed embedding $i: X \rightarrow Y$. Why is the following result true?
$$ \text{Ext}^k(i_* ...

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224 views

### Ext groups of affine scheme

Let $A$ be a commutative ring. $\textbf{Spec}(A)$ is the the
spectrum of $A$. $M$ and $N$ are $A$-modules. $\tilde{M}$ and
$\tilde{N}$ are sheaves associated on $\textbf{Spec}(A)$
respectively.
I ...

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389 views

### zero homology of augmented Koszul complex implies the sequence is regular?

Let $A$ be a Noetherian ring, $M$ a finite $A$-module and $I=(y_1,\cdots,y_n)$ an ideal of $A$ such that $M \neq IM$. Denote by $H_i(y_1,\cdots,y_n;M)$ the homology at dimension $i$ of the augmented ...

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### Is this a pure monomorphism?

Let $M$ be a representation of a quiver $Q=(V, E)$ by $R$-modules. By $M^{+}$ we mean a representation of $Q^{op}$ with $M^{+}(v)=\mathrm{Hom}(M(v), \frac{Q}{Z})$. One can easily see that there is ...

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431 views

### History of Koszul complex

This is a question about history of commutative algebra. I'm curios why Koszul complex from commutative algebra is called Koszul complex? All Koszul's early papers are about Lie algebras and Lie ...

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### finitely presented representations

Let Q=(V,E) be a direct graph where V is the set of all its vertices and E denotes the set of all its arrows. $X$ is called a representation of Q by modules if it is a functor from Q to R-Mod. i.e. ...

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106 views

### moduli space of two-term complexes of vector bundles over a fixed variety

Let $X$ be a fixed algebraic manifold over $\mathbb{C}$ , $\{E_{a}\}$ be vector bundles over $X$. We can construct moduli space of $\{E_{a}\}$ by classical theory. My question is that if we consider ...