(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.

learn more… | top users | synonyms

1
vote
0answers
23 views

Is this map representable? what is the fiber?

Consider the following map of stacks: Let $S$ be the stack whose $A$ points are diagrams of the form \begin{array}{ccccc} {} & {} & U & {} \\ {} & {} & \downarrow & \searrow \\...
9
votes
1answer
294 views

Understanding the purely formal part of the sheaf theoretic (cohomological) framework for representation theory

By now I have the impression that many statements in representation theory can be phrased a lot more elegantly using cohomological language. Therefore I'm trying to understand a bit better the sheaf ...
1
vote
0answers
68 views

Question on Hochschild cohomology

Let given ring $A$ without zero divisors and subring $\mathbf{k}\subset Z(A)$. Let also given $M~-$ $A~-$ bimodule, such that $xm = mx, \forall x\in \mathbf{k}, m\in M$. Is it true that if for $\...
1
vote
0answers
72 views

Monomial algebras and depth

Let $R:=k[x_1,\ldots, x_n]$ be the standard polynomial ring. Let $I\subseteq R$ be a monomial ideal of height $\ge 2,$ and $\{\ell_1, \ldots, \ell_{t}\}\subseteq R_1$ an $R$-regular sequence. Assume $...
7
votes
2answers
524 views

When is a functor a right derived functor?

Suppose we have Grothendieck abelian categories $\mathcal{A}, \mathcal{B}$. Suppose also we have given an exact functor of triangulated categories $$ F \colon D(\mathcal{A}) \to D(\mathcal{B}) $$ ...
13
votes
1answer
1k views

Where am I suppose to actually learn how to compute hypercohomology?

I'm reading about algebraic de Rham cohomology over characteristic zero which is constructed using hypercohomology. Already, constructing injective resolutions is difficult, and coupling this with ...
4
votes
0answers
226 views

The derived version of the Grothendieck spectral sequence

Consider the (very well known) Grothendieck spectral sequence for composition of functors $\mathcal F: \mathcal A \to \mathcal B$ and $\mathcal G: \mathcal B \to \mathcal C$ between abelian categories ...
1
vote
0answers
83 views

What is the definition of pure exact sequences in the category of chain complexes?

Let ‎$‎‎\mathcal{C}‎$ ‎be a ‎closed ‎symmetric ‎monoidal‎ ‎Grothendieck ‎category. ‎Then ‎there ‎are ‎two ‎general ‎notions ‎of ‎purity ‎in ‎‎$‎‎\mathcal{C}‎$‎, ‎the ‎‎$‎‎\lambda‎‎$‎-purity and the ‎$‎...
2
votes
1answer
85 views

Normalization of Hochschild cocycles

Let $A$ be a unital algebra over $\mathbb{C}$. Let $C^n(A)$ be the space of all $n+1$-linear maps $f:A^{n+1} \to \mathbb{C}$ (to be called $n$-cochains). Define $b:C^n(A) \to C^{n+1}(A)$ by the ...
3
votes
1answer
77 views

What is known about the morphism $H^*_{Lie}(L,L)\to H^*_{Lie}(L,UL)$ induced by $L\hookrightarrow UL$

Let $L$ be a (differential) graded Lie algebra over a field $k$ of characteristic 0, and let $UL$ be the universal enveloping algebra of $L$. The inclusion $L\hookrightarrow UL$ induces a morphism of ...
2
votes
0answers
92 views

Reversing the arrows-dual theorems

When one studies homological algebra one learns some basic stuff-diagram chasing, long exact sequences associated to short exact sequence of complexes and so on. Usually one works out the details with ...
2
votes
0answers
121 views

Inner automorphisms acts as identity on Hochschild homology

Let $A$ be a unital algebra and $u \in A$ be an invertible element. Let us consider $u_n(a_0 \otimes a_1 \otimes ... \otimes a_n):=ua_0u^{-1} \otimes ua_1u^{-1} \otimes ... \otimes ua_nu^{-1}$. Then $(...
0
votes
0answers
65 views

Hochschild cohomology of smooth commutative algebra

The beautiful Hochschild-Kostant-Rosenberg theorem tells us $HH^*(A,A)=\wedge^* \text{Der}(A)$ for a smooth affine algebra $A$. I wonder what happens when we consider $HH^*(A,A\otimes A)$, where $A\...
1
vote
0answers
61 views

Is the pair (C-flat modules, C-cotorsion modules) a cotorsion theory?

As we know, the pair (flat modules, cotorsion modules) is a cotorsion theory. Is the pair (C-flat modules, C-cotorsion modules) a cotorsion theory? Here C is a semidualizing module. Thanks!
7
votes
1answer
190 views

Isomorphism in cyclic cohomology vs isomorphism in Hochschild cohomology

Let $A$ be a unital algebra over a field $K$, $C^n(A)$ a space of all $n+1$ linear maps into scalar field $k$ (I'm interested in case $k=\mathbb{C}$) and $$(bf)(a_0,...,a_{n+1})=\sum_{i=0}^n(-1)^if(...
0
votes
0answers
95 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 ...
0
votes
1answer
227 views

Spectral sequences to compute Hom's in derived category

Does anybody have a good reference that lists spectral sequences that may be used to compute Hom sets in derived categories (of coherent sheaves, say)?
7
votes
0answers
93 views

Homology of inverse limits over inverse systems more complicated than towers

Most textbooks discussions of homology of inverse limits of chain complexes consider only “towers,” i.e. inverse systems indexed by the natural numbers. I’d like to find a reference that explains what ...
5
votes
1answer
183 views

Bijection modeling isomorphism of infinite-dimensional vector spaces

Let $T : V \to W$ be an isomorphism of vector spaces with bases $B_V$ and $B_W$, which may be of any cardinality. Does there exist a bijection $f : B_V \to B_W$ such that, for each $b_V \in B_V$,...
6
votes
0answers
268 views

Description of connecting maps of Derived functors

Let $C$ be an abelian category with enough injectives and $F$ be a left exact additive functor. Consider the short exact sequence $0 \to A' \to A \to A ''\to 0$. Therefore, we have connecting maps $\...
3
votes
0answers
76 views

Non-homotopic cdga maps with same effect on cohomology

Let $f, g : \mathcal{A} \to \mathcal{B}$ be maps of commutative differential graded algebras. An easy way to tell if they are homotopic is by looking at their effect on cohomology. However, if $f$ ...
6
votes
1answer
207 views

Can the Hochschild cochain complex be given the structure of a “homotopy BV algebra”?

In a 1993 letter, Deligne posed the following (paraphrased from a paper of Gerstenhaber and Voronov's): Conjecture (Deligne). The Hochschild cochain complex $CC^*(A)$ of an associative ...
2
votes
1answer
139 views

Milnor patching for general modules

The Milnor patching theorem for projective modules is the following statement. Given a pullback diagram of rings $$ \begin{array}{} R & \xrightarrow{f_2} & R_2 \\ \downarrow{f_1} & &...
1
vote
0answers
97 views

depth of ideal in polynomial ring

Let $R$ be a Noetherian local ring. Then the "depth" of an ideal $I$ measures the maximal length of regular sequence inside $I$. And $depth (R/I)$ measures the maximal length of regular sequence in ...
0
votes
0answers
87 views

forgetful on ext functor : reference request

learning about ext functors I encountered the following statement (source : https://en.m.wikipedia.org/wiki/Ext_functor) : "For $\mathbb{F}_p$ the finite field on $p$ elements, we also have ...
0
votes
1answer
86 views

Projective dimension and tensorial product of two modules

Let $M$,$N$ be two modules over commutative ring $R$. Can we say that $pd (M \otimes N )= $ $pd( M) + pd( N)$? Thanks!
4
votes
1answer
98 views

When is the category of Gorenstein projective $R$-modules Frobenius?

Let $R$ be a ring (associative with unit, but not necessarily commutative, and definitely not necessarily Noetherian.) Then the category $\operatorname{GP}(R)$ consists of those $R$-modules having a ...
2
votes
1answer
127 views

Why $k[x,y]$ is not a formally smooth algebra?

We could talk about the formal smoothness of an algebra. See for example Ginzburg's lecture notes For an associative algebra $A$ over a field $k$ we define $$ D(A)=T(A+\bar{A})/(\bar{ab}=a\bar{b}+\bar{...
1
vote
0answers
85 views

The $\mathrm{Hom}(E_2,E_1)$-action on $\mathrm{Ext}^1(E_2,E_1)$ [closed]

Let $E_1$ and $E_2$ be objects in an abelian category $\mathcal{C}$. How can I see the natural action of $\mathrm{Hom}(E_2,E_1)$ on $\mathrm{Ext}^1(E_2,E_1)$?
2
votes
0answers
60 views

Acyclic extension of free DGA-modules

I want to find some method to do the following: given an DGA-module $M$ over some commutative ring $k$, positively graded ($M_i=0$ if $i<0$), where each component $M_i$ has a free action by the ...
8
votes
1answer
273 views

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 ...
5
votes
0answers
104 views

Is the bar resolution of complexes dg-functorial?

Let $k$ be a commutative ring, and let $V$ be a complex of $k$-modules (more in general, we can take an $\mathcal A$-dg-module, where $\mathcal A$ is a dg-category. We can construct the bar resolution ...
2
votes
0answers
67 views

Regular sequence, and projective dimension

Let $I\subset J$ be two ideals of $k[x_{1},..,x_{n}]$, where $I$ is generated by a regular sequence. What can we say about the projective dimension of $J/I$ Thank you!
6
votes
0answers
97 views

mod $p$ homology module of unordered configuration spaces of the projective plane

Let $M$ be a manifold and $k$ be a positive integer. Let $F(M,k)$ be the $k$-th ordered configuration space over $M$, consisting of all ordered $k$-tuples of distinct points in $M$. Let $\Sigma_k$ be ...
0
votes
0answers
94 views

Projective dimension of modules

Let $M$, $N$ be two modules over commutative ring $R$. Suppose that they have finite projective dimension. Can we say something about the projective dimension of the $R-$module $Hom_{R}(M,N)$? Thanks!
2
votes
0answers
53 views

Surjections to free commutative dgas

Consider the category $C$ of commutative dgas (unbounded in both degrees) over a ring $R$ (in practice the integers or the integers mod $p$). Let $F$ be the free functor from chain complexes to $C$ ...
0
votes
1answer
105 views

$Ext$ functor over a product of groups

Let $G_1$ and $G_2$ be two groups (of some kind, e.g. finite groups). Let $M_1, N_1$ be $G_1$-modules, and $M_2, N_2$ be $G_2$ modules, always with coefficients in $\mathbb{C}$. Write $G = G_1 \...
4
votes
0answers
104 views

Partial formality of A-infinity structure implies formality

Let $A$ be a (finite dimensional, unital, associative) $k$-algebra, where $k$ is a (algebraically closed) field. Let $M$ be a (finite dimensional) $A$-module. Then, it is known that $\operatorname{Ext}...
12
votes
2answers
526 views

1-st cohomology of multiplicative group in a vector space

Let $\mathbb k$ be a field of characteristic $p$ and let $\mathbb k_n$ be a 1-dimensional representation of $\mathbb k^\times$, where the action is given by $t\circ v= t^n v$. Is it known what are the ...
3
votes
0answers
202 views

the origin of the differential of Spectral Sequence? [closed]

I'm wondering why the differential in homological Spectral Sequences $(E^*_{p,q},d^r)$ is defined as; $d^r_{p,q}: E^{r}_{p,q} \rightarrow E^{r}_{p-r,q+r-1}$ . From Weibel's homological algebra(p122), ...
1
vote
1answer
238 views

Free Symmetric Operads and $\mathbb{S}$-modules

In the definition of operads, if we restrict our attention to $\mathbb{S}$-modules where the action by the symmetric groups is free, then the free operads have still an underlying free $\mathbb{S}$-...
1
vote
0answers
54 views

Tor functor in the case of algebra of smooth functions

Let $A=C^{\infty}(\mathbb{S}^{1})$, let $B$ be the sub-algebra $C^{\infty}(0,1)$. Here we identify $\mathbb{S}^{1}$ by $\mathbb{R}/2\pi \mathbb{Z}$. I want to ask if there is a way I can decompose $A$ ...
10
votes
0answers
460 views

Local proof of Grothendieck-Riemann-Roch theorem

There is a theorem by Feigin and Tsygan(Theorem 1.3.3 here) which they call "Riemann-Roch" theorem. Given a smooth morphism $f:S\to N$ of relative dimension $n$ and a vector bundle $E/S$ of rank $k$ ...
10
votes
0answers
215 views

Finite dimensionality of Ext(M,N)

Let $K$ be a field of characteristic $0$ and let $R$ be a (noncommutative) Noetherian $K$-algebra. Let $M$ and $N$ be simple left $R$-modules and assume further the following conditions on $R$: (...
1
vote
0answers
20 views

Reference needed for exact sequence of ACM curves with homogeneous coordinate ring.

Let $C\subset\Bbb{P}^n$ be an ACM (arithmetically Cohen-Macaulay) curve with homogeneous coordinate ring $R$. Then there is an exact sequence $$0\to \text{Tor}_i(R,\Bbb{C})_k\to H^1(C,\wedge^{i+1}...
5
votes
1answer
189 views

Auslander-Reiten theory for Gorenstein algebras

In the paper "Cohen-Macauley and Gorenstein artin algebras", Auslander and Reiten have a short section about Auslander-Reiten theory in Gorenstein algebras (I always assume we have an artin algebra ...
4
votes
0answers
100 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 ...
3
votes
1answer
134 views

Isomorphisms between extension group and $\mathfrak{u}$-cohomology

Let $\mathcal{O}^{\mathfrak{p}}$ be a parabolic subcategory (see, Humphrey's BGG's Ch.9) of the BGG category $\mathcal{O}$ (w.r.t the Cartan $\mathfrak{h}$ and Borel $\mathfrak{b}$) over a finite-...
0
votes
0answers
74 views

Right split for homomorphism onto $S_\infty$

Let $G$ be a closed subgroup of $S_\infty$ and let $f:G\rightarrow S_\infty$ be a continuous surjective homomorphism. Under which conditions $f$ has a right split, i.e. there exists some $g:S_\infty\...
5
votes
1answer
323 views

Known results in the Cohomology of finite groups

I am learning to compute cohomology of finite groups and came across this survey article http://www.ams.org/notices/199707/adem.pdf "Recent Developments in the cohomology of finite groups" by ...