**3**

votes

**1**answer

158 views

### LES for relative cohomology via sheaves

I was unable to find a suitable answer for the following question:
Once one learns that singular cohomology is the same as cohomology with coefficients in locally constant sheaf, it is natural to try ...

**8**

votes

**0**answers

298 views

### End of the Ext functors

Let $R$ be a ring, and consider the hom functor $\hom\colon Mod(R)^\text{op}\times Mod(R)\to Mod(R)$; the end of $\hom$ is well-known to be the set of endomorphisms (endonatural transformations) of ...

**0**

votes

**1**answer

135 views

### $\inf\{i\in \mathbb N \cup \{0\}\cup\infty\mid Ext^i_R(R/I,R)\neq 0\}=0 ?$

Let $R := k[x_1, \cdots, x_n, \cdots]/(x_1^1, \cdots , x^n_n, \cdots),$ where $k$ is a field. Set $I:=(x_1, \cdots, x_n, \cdots)$. the questions are:
Is $\inf\{i\in \mathbb N \cup \{0\}\cup ...

**3**

votes

**1**answer

250 views

### A question on the cohomology of elliptic curves over local fields

Let $K$ be a number field,$\nu$ a nonarchimedian prime of $K$, $K_{\nu} $ the completion of $K $ at $\nu $ with maximal unramified extension $K_{\nu}^{unr} $. Let $E $ be an elliptic curve defined ...

**1**

vote

**0**answers

86 views

### Global dimension of graded Lie algebra

The rational global dimension of a graded algebra $A=(A_k)_{k\geq 0}$, with $A_0=\mathbb Q$, denoted here ${\rm gl}\dim A$ is defined to be the greatest integer $k$ (or $\infty$) such that ${\rm ...

**2**

votes

**0**answers

58 views

### Minimal free resolution of sum of ideals

Let $S$ be the polynomial ring in $n$ variables, and let $I_1$ and $I_2$ be ideals in $S$. What can be said about the $\mathbb{Z}$-graded minimal free resolution of $I_1+ I_2$ in terms of the ...

**2**

votes

**1**answer

176 views

### Decide two indices of Ext functor

This question is from the proof of Theorem 11.34 in the book: Twenty-four Hours of Local Cohomology.
Let $R$ and $S$ be CM local ring and $R\to S$ a local homomorphism such that $S$ is a finite ...

**6**

votes

**0**answers

55 views

### A “lower-central” filtration of Steenrod algebra?

$\renewcommand{\Atwo}{\mathcal{A}_2}$ So, a lot of good work has been accomplished by filtering the Steenrod algebras $\mathcal{A}_p$ in powers of the Augmentation ideal; For reasons partly ...

**2**

votes

**1**answer

210 views

### When is finding an explicit inverse of an isomorphism not possible

My question is about Shapiro's lemma. Consider the isomorphism $\phi: H^n(G, Hom_{ZH}(ZG, A))\cong H^n(H,A)$ of shapiro's lemma. I would like to describe this via cochains.
So the obvious map is ...

**3**

votes

**1**answer

255 views

### Is the derived category of perfect complexes idempotent complete?

Let $\mathcal{C}$ be a category. We call a morphism $\alpha: X\rightarrow X$ an idempotent if $\alpha^2=\alpha$ in $\mathcal{C}$. We call $\mathcal{C}$ is $\textit{idempotent complete}$ if any ...

**4**

votes

**0**answers

201 views

### When do we have $D_{\text{perf}}(\text{Qcoh}(X))\simeq D_{\text{perf}}(X)$?

Let $(X,\mathcal{O}_X)$ be a scheme (or more generally a ringed space). We know that in general the derived category of complexes of quasi-coherent modules $D(\text{Qcoh}(X))$ is not equivalent to the ...

**2**

votes

**1**answer

192 views

### An alternative definition of pseudo-coherent complex

Let $(X,\mathcal{O}_X)$ be a scheme or a general ringed space. First recall that a complex of $\mathcal{O}_X)$-modules $\mathcal{E}^{\bullet}$ is called strictly perfect if $\mathcal{E}^{\bullet}$ is ...

**2**

votes

**0**answers

128 views

### What kind of ringed space $X$ has the property that a locally free sheaf is projective in Qcoh$(X)$?

It is well known that for an affine scheme $X$, every finitely generated locally free sheaf $\mathcal{E}$ is projective in the category Qcoh$(X)$. i.e. the functor ...

**5**

votes

**1**answer

378 views

### Is a locally free sheaf projective in the category of $\mathcal{O}_X$-modules when $X$ is an affine scheme?

Let $X$ be an affine scheme and $\mathcal{E}$ a finitely generated locally free sheaf on $X$. It is obvious that $\mathcal{E}$ is a projective object in the category Qcoh$(X)$ since we can pass to ...

**6**

votes

**0**answers

475 views

### How can you see the minimal relations on a quiver from its bimodule resolution?

Suppose that you are given an algebra $KQ/I$, coming from a quiver Q, of finite global dimension. Suppose also that you know its minimal bimodule resolution over its enveloping algebra. Can you get a ...

**2**

votes

**0**answers

95 views

### project limit on $n$- simplical complex which is principal homogeneous with respect to an action

The setting:
Let G be compact locally $\Bbb{Q}_p$ analytic group. We fix a countable basis of open normal subgroups $G\supset G_1\supset ...G_r\supset...$
We suppose that we are given a system of ...

**2**

votes

**1**answer

435 views

### Recollement of multiple $t$-structures

Given a recollement
$$
\mathbf{D}^0 \underset{\underset{i_R}\leftarrow}{\overset{\overset{i_L}\leftarrow}\to} \mathbf{D} \underset{\underset{q_R}\leftarrow}{\overset{\overset{q_L}\leftarrow}\to} ...

**1**

vote

**0**answers

147 views

### Récollement of stable $t$-structures

Given a recollement
$$
\mathbf{D}^0 \underset{\underset{i_R}\leftarrow}{\overset{\overset{i_L}\leftarrow}\to} \mathbf{D} \underset{\underset{q_R}\leftarrow}{\overset{\overset{q_L}\leftarrow}\to} ...

**-1**

votes

**1**answer

99 views

### Is this apushout diagram [closed]

Let $A, B, C, E$ and $F$ be some objects in an abeleian category $\mathcal{C}$. Let we have a commutative diagram
\begin{array}{ccccccccc}
0 & \xrightarrow{} & A & \xrightarrow{f} & ...

**3**

votes

**1**answer

115 views

### Is the image of a idempotent morphism in $\mathcal{K}(\mathcal{A})$ defined in the naive way?

Let $\mathcal{A}$ be an abelian category and $\mathcal{K}(\mathcal{A})$ be the homotopy category of chain complexes in $\mathcal{A}$. It is well-known that $\mathcal{K}(\mathcal{A})$ is idempotent ...

**1**

vote

**0**answers

281 views

### How to prove that any perfect complex on an affine scheme is strictly perfect?

Let $(X,\mathcal{A})$ be a ringed space. A complex $\mathcal{S}^{\bullet}$ of $\mathcal{A}$-modules is $\textit{perfect}$ if for any point $x\in X$, there exists an open neighborhood $U$ of $x$ and a ...

**5**

votes

**1**answer

177 views

### Relations between functors in a recollement

Consider a recollement situation like the following
by the very definition of the various functors it follows that $i^* j_*=0$, and $j^! i_* = 0 = j^* i_!$. Also, $j^! i_! = 0 = j^* i_*$ by ...

**7**

votes

**1**answer

365 views

### Explict form of $E_\infty$-morphisms between differential graded commutative algebras

This is a partial duplicate to this MO question, I apologize for that. I'm asking since the answers there still do not allow me to work out an answer to my question, which is a bit more specific.
...

**0**

votes

**0**answers

112 views

### Relating overlapping simplicial complexes

Let $X$ be a simplicial complex and let $A,B\subset X$ be
subcomplexes such that $C=A\cap B$ is a non-empty simplicial
complex. Finally, let $C_{\cdot}(X)$, $C_{\cdot}(A)$, $C_{\cdot}(B)$,
...

**1**

vote

**1**answer

326 views

### Are chain complexes over a field always injective?

Question: Let $\mathbb{F}$ be an algebraically closed field of characteristic zero and let $\mathrm{Ch}_{\mathbb{F}}$ be the category whose objects are chain complexes (of $\mathbb{F}$-modules) and ...

**9**

votes

**0**answers

360 views

### Pairing of cohomology and homology Künneth formulas

Let $k$ be a field, and let $X$ and $Y$ be CW-complexes of finite type (although the question makes sense for $k$ a ring and for more general chain complexes of finitely generated free abelian ...

**6**

votes

**2**answers

538 views

### Applications of cohomology to probability and statistics

Are there interesting/useful applications of cohomology (and homological algebra in general) to probability and statistics, or information theory?
By "interesting/useful", I mean "not merely ...

**0**

votes

**0**answers

112 views

### $\text{Hom}(G,\mathbb{Z})$ [duplicate]

Fix a cardinal $\kappa$ and consider $\mathbb{Z}^\kappa$ with componentwise addition and the subgroup $$F_\kappa :=\{g:\kappa \to \mathbb{Z}: \{\alpha\in \kappa: g(\alpha)\neq 0\} \text{ is ...

**5**

votes

**1**answer

330 views

### Does homotopy invariance of homology follow from the structure of the simplex category $\Delta$?

Explicitly: Let $\Delta$ denote the simplex category, and $\mathscr{C}$ any small category, and fix a functor $F:\Delta \rightarrow \mathscr{C}$ such that $F\Delta^0$ is terminal. Also, assume ...

**0**

votes

**1**answer

118 views

### Do levelwise quasi-isomorphisms of bicomplexes induce a quasi-isomorphism between the total complexes?

Let $C^{p,q}$ be a bicomplex with differentials $d_h:C^{p,q} \to C^{p+1,q}$ and $d_v:C^{p,q} \to C^{p,q+1}$ where $d_h \circ d_v = d_v \circ d_h$. Let $D^{p,q}$ be another bicomplex defined similarly.
...

**13**

votes

**0**answers

449 views

### Lifting DG-categories to characteristic zero

The question of lifting (smooth projective) varieties from an algebraically closed field $k$ of characteristic $p$ to characteristic zero (i.e., to the Witt vectors $W(k)$) is a classical one. It's ...

**4**

votes

**0**answers

215 views

### Can we obtain a derived category from an additive category? Like a category of Banach modules?

Let $A$ be a Banach algebra, let $A$-mod be the category of left Banach modules (as defined in Helemskii's "Banach and locally convex algebras"), $A$-mod is an additive category, but not abelian ...

**1**

vote

**0**answers

57 views

### Turning left modules into right modules over a homotopy Gerstenhaber algebra

For simplicity's sake, let $A$ be a dg-algebra over $\mathbb{Z}/2\mathbb{Z}$.
In the case when $A$ is a commutative algebra, we can turn a left $A$ module into a right $A$ module trivially. Of course ...

**11**

votes

**1**answer

324 views

### An example of an object in $D^b_{\text{coh}}(\mathbb{P}^2)$ which is not formal

We know that for a curve $X$, any object $\mathcal{E}^{\bullet}$ in the derived category $D^b_{\text{coh}}(X)$ is formal, i.e. $\mathcal{E}^{\bullet}$ is quasi-isomporphic to the direct sum of its ...

**26**

votes

**2**answers

726 views

### A more natural proof of Dold-Kan?

The Dold-Kan correspondence gives an equivalence of categories between $SAb$, the category of simplicial abelian groups, and $Ch_{\geq 0}$, the category of non-negatively graded chain complexes of ...

**5**

votes

**0**answers

173 views

### homology theory for affine and projective algebraic sets?

Given $f_1,\ldots,f_r\in K[x_1,\ldots,x_m]$, resp. homogeneous $f_1,\ldots,f_r\in K[x_0,\ldots,x_m]$, is there a chain complex built from these polynomials, such that any polynomial map $\varphi\!: ...

**3**

votes

**2**answers

350 views

### What are cohomology of Lie algebra with coefficients geometrically?

I want to find analog of following two statements.
Let $G$ be a discrete group, $M$ is representation of $G$. Local systems on $BG$ are the same as $G$ representations (because $\pi_1 (BG) =G$). Let ...

**3**

votes

**1**answer

254 views

### Lyndon–Hochschild–Serre spectral sequence for not normal subgroup

Is there analog of Lyndon–Hochschild–Serre spectral sequence for not normal subgroup?
What can you say about it? Can you describe $E^{p, q}_1$ ? What is about $E^{p, q}_2$?
What is the best technique ...

**3**

votes

**1**answer

209 views

### Homogeneous polynomial vector fields tangent to the unit sphere

This question has something to do with that one.
Let $n\ge1$ and $d\ge1$ be two given integers. Consider the polynomial vector fields $v=(v_1,\ldots,v_n)$ whose components $v_j$ are homogeneous of ...

**0**

votes

**0**answers

98 views

### Computations of derivations, $d^2$, of a (Grothendieck) spectral sequence

The maps $d^1:E_1\rightarrow E_1$ have a nice description. Is there any text providing us with a description of the higher derivations $d^2:E_2\rightarrow E_2$ arising from a Grothendieck spectral ...

**4**

votes

**0**answers

103 views

### Multiplicity of $Ext^{d-t}(M,\omega_R)$, ($d=\dim R, t=\dim M$)

Let $R=\bigoplus_{i \geq 0} R_i$ be a Cohen-Macaulay graded ring ($R_0$ is a field and $R$ is generated by $R_1$) of dimension $d$ with canonical module $\omega_R$, and $M$ a graded Cohen-Macaulay ...

**0**

votes

**0**answers

293 views

### What is a Beilinson spectral sequence?

I'm writing to ask just a question. I would like to understand better what is the Beilinson's spectral sequence and how it can be used. Is there any useful and clear reference you advice to someone ...

**1**

vote

**0**answers

110 views

### A naturality question concerning the universal coefficient spectral sequence

I am reading Hillman's book "algebraic invariants of links" and on page 20 he mentions the following universal coefficient spectral sequence.
Let X be a connected finite CW complex.Let $H$ be a ...

**11**

votes

**0**answers

373 views

### Homological algebra is linearized homotopical algebra?

I have stumbled across statements like
Homological algebra is linearized homotopical algebra.
Chain complexes are linearizations of simplicial complexes.
The Dold-Kan correspondence was ...

**4**

votes

**2**answers

368 views

### Reference for homotopy (co)limits of (co)chain complexes via totalization of double complexes

It seems to be a well-known fact that homotopy (co)limits
of (co)simplicial diagrams of nonnegatively graded
(co)chain complexes in (Grothendieck) abelian categories
can be computed by using the ...

**0**

votes

**1**answer

315 views

### cohomology version of Cartan-Leray spectral sequence that deduces cup product

On page 338, A User's Guide to Spectral Sequences. 2nd Edition, by John McCleary, Theorem 8.9, there is a Cartan-Leray spectral sequence for homology:
If $X$ is a connected pace on which the group ...

**1**

vote

**0**answers

122 views

### Exact Functors from Perverse Sheaves

Pretty sure this is a simple question, but there's something I'm missing here.
For context, I'm reading 'An Elementary Construction of Perverse Sheaves' by MacPherson and Vilonen. A key aspect of the ...

**1**

vote

**0**answers

145 views

### Can you always find a regular sequence consisting of monomials?

Let $\mathbb{k}$ be a field, and let $S=\mathbb{k}[x_1,x_2,\ldots,x_n]$. Let $M$ be an $S$-module. A sequence $$f_1,f_2,\ldots,f_r$$ of polynomials in the maximal ideal $\langle x_1,\ldots,x_n\rangle$ ...

**1**

vote

**1**answer

129 views

### Hochschild cohomology of the skew group ring D(X)#G in the complex analytic case

Let $X$ be a n-dimensional complex compact manifold and let $G$ be a finite subgroup of $Aut(X)$ acting by biholomorphic maps on $X$. I would like to compute the Hochschild cohomology group ...

**3**

votes

**0**answers

77 views

### Projective dimension of ring over its center

If $A$ is a ring and $Z(A)$ is its center then what is a sufficient condition for the projective dimension of $A$ over $Z(A)$ (ie: $pd_{Z(A)}(A)$) to be finite?
(Assuming that $A\neq Z(A)$).