**2**

votes

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**2**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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 ...