(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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2
votes
0answers
33 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
170 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 $b_V \in B_V$,...
5
votes
0answers
212 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
62 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
152 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
116 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
86 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
86 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
80 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
88 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
115 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
84 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
57 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
244 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
101 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
60 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
92 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
90 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
49 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
103 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
96 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
514 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
201 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
236 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
52 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
443 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
189 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
19 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
178 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 ...
3
votes
0answers
89 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
117 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
313 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 ...
6
votes
2answers
414 views

Socle of tilting modules in the BGG category $\mathcal{O}$ over a semisimple Lie algebra

Suppose that $\mathfrak{g}$ is a finite dimensional, complex, semisimple Lie algebra. Let $\mathcal{O}$ be the BGG category over $\mathfrak{g}$. Tilting module theory play an important role in the ...
8
votes
1answer
436 views

Cohomology theory “from” Grothendieck's six operations?

How, precisely, (as suggested by grothendieck) cohomology theory naturally follows from the Grothendieck's six operations associated to the category derived from the topos? I would like some ...
3
votes
0answers
44 views

Fracturing $t$-structures

$\def\tee{\mathfrak{t}}$ Let $\tee_1,\tee_2$ be two $t$-structures on a triangulated category $\cal T$; call them fracturing if the two fiber sequences $\tau^\le_1X\to X\to \tau^\ge_1X$ and $\tau^\...
3
votes
1answer
91 views

t-structure induced on the Verdier quotient ${\cal T}/\cal S$

Let $\mathfrak t$ be a $t$-structure on a triangulated category $\cal T$. Let $\cal S$ be a thick (or even non-thick) triangulated subcategory, and ${\cal T}/\cal S$ the Verdier quotient. Is there a ...
4
votes
1answer
78 views

Are product / coproduct projections / inclusions 'semistrict'?

Let $\mathbf{C}$ be a category with zero object, kernels, and cokernels. Then, a morphism $f\colon A\rightarrow B$ in $\mathbf{C}$ is semistrict iff the canonical map $\operatorname{Coker}(\ker (f))\...
8
votes
1answer
163 views

If a faithfully flat extension of dg/A_$\infty$-algebra is formal, is the original algebra formal (over positive characteristic)?

Proposition 6.2 of Formality of DG algebras (after Kaledin) by Lunts reads (with a few additions to clarify notation): Let $k$ be a field of characteristic 0. Let $A$ be an $A_\infty$ algebra ...
24
votes
2answers
550 views

Complete the following sequence: point, triangle, octahedron, . . . in a dg-category

Let $\mathcal C$ be a pre-triangulated dg-category (or a stable $\infty$-category, if you wish). An object $X$ in $\mathcal C$ gives a "point": $$X$$ A morphism $X\xrightarrow f Y$ in $\mathcal C$ ...
1
vote
0answers
112 views

Example of non-locally finite stability condition

I am trying to work out example 5.6 in Bridgeland's paper "Stability conditions on triangulated categories". http://annals.math.princeton.edu/wp-content/uploads/annals-v166-n2-p01.pdf A standard ...
1
vote
0answers
53 views

Complex integration over 1-singular chain

Let $f$ be a continuous function on $\mathbb{C}$ and assume that $\lim_{z\to \infty} zf(z) = \lambda.$ Let us note for all natural $n$ $$C_n = \{z \in \mathbb{C} : |z|=n\}.$$ Then, a usual fact of ...
6
votes
1answer
330 views

Why are quasi-isomorphisms of homotopy algebras only defined for arity 1?

When reading about homotopy algebras (e.g. $L_\infty$-algebras, $A_\infty$-algebras), an $\infty$-morphism $f$ is called an $\infty$-quasi-isomorphism if $f_1$ is a quasi-isomorphism. Recall/Example (...
0
votes
1answer
109 views

Projective resolutions of torsion modules [closed]

Let $l$ be a prime number, $n\in \mathbb{Z}$. Is it true that any finitely generated $\mathbb{Z}/l^n\mathbb{Z}$-module has a finite (left) resolution by free finitely generated $\mathbb{Z}/l^n\mathbb{...
2
votes
1answer
168 views

Third (co-) homology of Cyclic groups

Is there a general simple theorem for the third cohomology of cyclic groups $H_3(\mathbb{Z}_n, U(1))= ?$. In particular, I am interested in finding $H_3(\mathbb{Z}_8, U(1))$. I know the answer can be ...
1
vote
0answers
29 views

Is an weakly finite R-module Serre subcategory of the category of R-modules?

A definition for weakly finite $R$-modules is as follow: Definition: Let ($R$,$m$) a local ring. Let $S$ be the largest class of $R$-modules satisfying the following four properties: (1) If $M \in S$...
0
votes
1answer
121 views

When Hom(M,E) is injective? [closed]

Let $R$ be a commutative Noetherian ring with non-zero identity, $M$ be an $R$-module and $E$ be an injective $R$-module. When $Hom(M,E)$ is injective? Thanks.
1
vote
0answers
58 views

The projective resolution of a direct summand

For an $R$-module $M$ fix a projective resolution $P^\bullet\to M$. If $N$ is a direct summand of $M$, that is there is $L$ such that $M=N\oplus L$, then is there a projective resolution of $N$ which ...
21
votes
0answers
300 views

Does the Tate construction (defined with direct sums) have a derived interpretation?

Any abelian group M with an action of a finite group $G$ has a Tate cohomology object $\hat H(G;M)$ in the derived category of chain complexes. There are several ways to define this. One is as the ...
12
votes
1answer
320 views

Cohomology of central extensions of groups

Let G be a central extension of a finite group H by $Z/2$. I need an explicit description of the differentials $d_2$ and $d_3$ in the Lyndon-Hochschild-Serre spectral sequence which converges to the ...