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7
votes
2answers
282 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 ...
2
votes
0answers
104 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 ...
2
votes
0answers
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 ...
0
votes
0answers
242 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 ...
3
votes
0answers
227 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", ...
21
votes
4answers
1k views

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 ...
1
vote
1answer
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 ...
7
votes
2answers
272 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?
3
votes
0answers
161 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 ...
0
votes
1answer
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 ...
5
votes
1answer
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 ...
1
vote
0answers
222 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 ...
1
vote
0answers
136 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 ...
3
votes
1answer
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 ...
2
votes
0answers
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 ...
-2
votes
2answers
295 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
1
vote
0answers
62 views

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 ...
2
votes
1answer
200 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 ...
1
vote
1answer
96 views

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 $M$)
4
votes
0answers
81 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 ...
1
vote
1answer
135 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 ...
5
votes
1answer
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{\ \ ...
7
votes
1answer
241 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 ...
5
votes
0answers
182 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 ...
4
votes
1answer
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 ...
4
votes
1answer
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?
11
votes
0answers
296 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 ...
3
votes
2answers
255 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: ...
5
votes
1answer
474 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 ...
2
votes
1answer
177 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 ...
5
votes
1answer
256 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 ...
2
votes
0answers
114 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 ...
2
votes
1answer
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_* ...
4
votes
0answers
223 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 ...
6
votes
3answers
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 ...
1
vote
0answers
128 views

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 ...
14
votes
1answer
424 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 ...
1
vote
1answer
205 views

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. ...
1
vote
1answer
105 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 ...
1
vote
1answer
207 views

Pure monomorphism of functors-

Suppose that $F, G: Q\rightarrow R{\rm -Mod}$ be two covariant functors where $Q$ is an abelain category and $R$ is a commuatative ring. Also let $\eta: F\rightarrow G$ be a natural transformation ...
3
votes
0answers
155 views

Hochschild homology of a tensor algebra modulo a two-sided ideal

Let $V$ is a module over a field $k$, and $A=T(V)$ the tensor algebra generated by $V$. The Hochschild homology $HH_*(A)$ has been determined by Loday and Quillen in their paper "Cyclic homology and ...
0
votes
0answers
69 views

algebras of infinite injective dimension

Are there any connected graded Noetherian algebra of infinite injective dimension but has a balanced dualizing complex? Thanks a lot.
4
votes
1answer
204 views

Two-sided bar construction

On page 4 of this paper by H. Abbaspour, the author defines the two-sided bar construction $$B(A,A,A):=A\otimes T(s\bar{A})\otimes A$$ of a differential graded algebra $(A,d_A)$ (over a field). The ...
6
votes
1answer
118 views

Abelian groups injective over their endomorphism

Let $M$ be an abelian group and let $R = \mbox{End}_\Bbb{Z}(M)$. Under what conditions (on $M$), $_RM$ is injective!?
2
votes
1answer
250 views

Brutal truncation of indecomposable complexes

Let $P^{\bullet} = (P^i, d^i)_{i\leq 0}$ be an indecomposable object in category of complexes bounded above, where each $P_i$ is a finitely generated projective module over an finite dimensional ...
2
votes
0answers
198 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 ...
1
vote
0answers
49 views

injective dimension of dualizing complex

Let $R$ be a balanced dualizing complex of a Noetherian connected graded algebras $A$. Dose one always have $\text{id}_A R = \text{id}_{A^{op}} R$? Thanks a lot.
4
votes
0answers
160 views

What is known about this short exact sequence in Lie algebra cohomology?

In its most general form, I look at the following. Let $g$ be a dg Lie algebra and $Z$ be its center. The sequence $ Z \to g \to g/Z $ gives me a short exact $ 0\to C(g,Z) \to C(g,g) \to C(g,g/Z) \to ...
0
votes
0answers
108 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 ...
1
vote
0answers
44 views

Duality of morphisms induced by multiplication of regular normal element

I was confused by the sequence of modules in [Yekiutieli and Zhang, Rings with Auslander dualizing complex, p.33, l.-10]. The question is that: why is the second morphism right multiplication of $t$? ...