**3**

votes

**0**answers

161 views

### Is there a Hochschild-Serre spectral sequence for unramified cohomology?

Similar to the Hochschild-Serre spectral sequence for etale cohomology ($H^p(G, H^q_{et}(X_L, \mathcal F|_{X_L})) \Rightarrow H^{p+q}_{et}(X, \mathcal F)$ for a Galois field extension $L/k$ with ...

**4**

votes

**0**answers

126 views

### Reference for Grothendieck's duality and Cousin, Dualizing and Residual complexes

I am a graduate student currently reading Hartshorne's Residues and Duality. In order to reach the construction of the right adjoint $f^!$ of $Rf_*$ for some special types of maps of locally ...

**2**

votes

**1**answer

66 views

### Morphisms $P \to M$ in the derived category of a dg-category, if $P$ is h-projective

Let $\mathbf A$ be a dg-category. Denote by $\mathsf{C}_{\mathrm{dg}}(\mathbf A)$ the dg-category of right $\mathbf A$-modules, and by $\mathsf{C}(\mathbf A) = Z^0(\mathsf{C}_{\mathrm{dg}}(\mathbf ...

**4**

votes

**1**answer

279 views

### On various “extension closures” and “orthogonals” in triangulated categories

A vague form of my question is the following one: for a class of objects $D$ of a triangulated category $C$ we consider the class $E$ of objects that satisfy $Mor_{C}(d,e)=\{0\}\ \forall d\in D$; ...

**4**

votes

**1**answer

176 views

### Different ways of having infinite global dimension

Is there any ring $R$ of infinite global dimension such that any $R$-module is a retract (i.e. direct summand) of some $\oplus_{i\in I}M_i$ where each $M_i$ has finite projective dimension?
I ask ...

**3**

votes

**1**answer

207 views

### Model structure on non-negative differential graded algebras with homological grading

I was wondering if there exists a model structure on the category of non-negative differential graded algebras with homological grading. To be more precise: Let $Ch_{k}$ the model category of ...

**13**

votes

**1**answer

363 views

### Homotopy transfer in the opposite direction

Let $X\rightleftarrows Y\circlearrowleft$ be a strong deformation retraction of chain complexes (a.k.a. contraction), i.e. $X\rightarrow Y\rightarrow X$ is the identity, $Y\rightarrow Y$ is a homotopy ...

**0**

votes

**1**answer

41 views

### Subgroups with some closed property

Assume $A$ and $B$ are infinite abelian groups, $B$ is a subgroup of $A$. Is it true that if every homomorphism from $B$ to $\mathbb Z$ can be extended to a homomorphism from $A$ to $\mathbb Z$, then ...

**5**

votes

**0**answers

123 views

### Matrix factorizations as a dg-category?

Matrix factorizations (in the graded case) give a triangulated category. I would imagine that there should be an underlying dg-category. Is there such a definition, and if so, where can I find it in ...

**0**

votes

**0**answers

35 views

### Exactness in the zeroth term and (relative) vector field

Recently I've asked the following question:
Contraction with a vector field and pullback bundle
When I already know how $i_X$ should be understood (please see the discussion; recall that we are ...

**1**

vote

**1**answer

172 views

### Dimension of Ext modules [closed]

Let $(R,m)$ be a noetherian local ring, and $M$ and $N$ be two finitely generated $R$-module. Then is it true that $\dim \text{Ext}^k(M,N)\leq \dim M-k$? If not does the reversed inequality hold?

**3**

votes

**1**answer

183 views

### When is the cohomology of a space Ext(k,k)?

If $X$ is aspherical, we know that $H^*(X,k) = \text{Ext}_R(k,k)$, with $R = k\pi_1$. For non-aspherical spaces, do we ever have $H^*(X,k) = \text{Ext}_R(k,k)$ for some ring $R$? Obviously we need ...

**1**

vote

**0**answers

49 views

### Antisymmetrization of the Hochschild cocycle

Let $A$ be a commutative (unital, complex) algebra and let $\varphi$ be a $n+1$-linear functional on $A$ (we will call it cochain). Define ...

**3**

votes

**1**answer

121 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

280 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

134 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

224 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

83 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

51 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

153 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 ...

**0**

votes

**0**answers

129 views

### Is it a correct description of the bounded above derived category of coherent sheaves?

Let $X$ be a (Noetherian) scheme. Let $D^{-}_{\text{coh}}(X)$ be the derived category of complexes of $\mathcal{O}_X$-modules with bounded above and coherent cohomologies.
Do we have the following ...

**6**

votes

**0**answers

47 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

180 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

194 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

175 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 ...

**0**

votes

**0**answers

67 views

### Is this quasi-coherent sheaf a subsheaf of $\ker f$?

Let $f: \mathcal{F}\to \mathcal{G}$ be a morphism of quasi-coherent sheaves over a scheme $X$. Let also $T_U$ be a submodule of $\ker f_U$ with $|T_U|\leq \kappa$ for each open subset $U$ of $X$ ...

**2**

votes

**1**answer

121 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

108 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 ...

**4**

votes

**1**answer

226 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 ...

**5**

votes

**0**answers

226 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

90 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

412 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

135 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

90 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

103 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

203 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

162 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 ...

**6**

votes

**1**answer

305 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

54 views

### Intersection and union of torsion classes

One of the main result in
Cassidy, C., M. Hébert, and G. M. Kelly. "Reflective subcategories, localizations and factorization systems." Journal of the Australian Mathematical Society (Series A) ...

**0**

votes

**0**answers

106 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

293 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 ...

**8**

votes

**0**answers

347 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 ...

**5**

votes

**2**answers

417 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

100 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

304 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

116 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.
...

**11**

votes

**0**answers

406 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

196 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

52 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 ...

**10**

votes

**1**answer

304 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 ...