# Tagged Questions

**3**

votes

**2**answers

149 views

### Definition of the differential of the Cone of a morphism of complexes [on hold]

Let $(F^\bullet,d_F)$ and $(G^\bullet,d_G)$ be two complexes in an abelian category $\mathbf{A}$.
The complex cone $Cone(\varphi)^\bullet$ of a morphism of complexes $\varphi:F^\bullet \to G^\bullet$ ...

**-4**

votes

**0**answers

43 views

### Global dimension of matrix algebra [on hold]

Let $k$ be a field, and $A=T_{n}(k)$.
$gldim(A) = 1$, and if
$B = A/rad(A)^{2}$, then $gldim(B) = n-1$.
Some indication!!
How can I prove that $gldim(A) = 1$, and $gldim(B) = n-1$ ?
Thank you!

**-1**

votes

**1**answer

156 views

### Distinguished triangle and short exact sequence [closed]

Forgive me for asking an elementary question.
Given coherent sheaves $A$, $B$, $C$ and morphisms $B\xrightarrow{f} C\xrightarrow{g} A$ which give rise to the distinguished triangle $A[-1] \rightarrow ...

**7**

votes

**0**answers

57 views

### Non-Standard Derived Equivalences of Non-Flat Algebras

I read that for algebras $R$ and $S$ (over a commutative ring), assuming that $R$ or $S$ is flat, the existence of a derived equivalence $\mathcal{D}(R) \to \mathcal{D}(S)$ implies the existence of an ...

**3**

votes

**2**answers

341 views

### Definition of E-infinity operad

What is the definition of $E_\infty$-operad in the category of chain complexes over $\mathbb{Z}/p\mathbb{Z}$? J. Smith in http://arxiv.org/abs/math/0004003 define it for complexes over $\mathbb{Z}$ ...

**3**

votes

**0**answers

75 views

### Differentially closed fields

Let F be a field. Recall that an additive map $d: F\rightarrow F$ is said to be a derivation if $d(ab)=ad(b)+d(a)b$.
Now let $F$ be a ring and let $d$ be a derivation of $F$. Examples I have in mind ...

**2**

votes

**0**answers

103 views

### Reference request for a “truncated version” of the de Rham algebra

Let's start on the $n$-torus for sake of simplicity.$\newcommand{\T}{\mathbb T}$
If I understand the relevant definitions correctly, the usual de Rham algebra of smooth differential forms on $\T^n$ is ...

**5**

votes

**1**answer

118 views

### Do $RHom(C,D)$ and $DG(C,D)$ have equivalent homotopy categories?

Toen in The homotopy theory of dg-categories and derived Morita theory Section 6 introduced the internal Hom's between dg-categories. Actually for two dg-categories $C$ and $D$, Toen defined
$$
...

**6**

votes

**0**answers

260 views

### Transgression map spectral sequence of Ext

Let $X$ be a scheme over $k$ and $p:\ X \to Spec(k)$ the structure morphism. If $M$ is an étale sheaf of abelian groups over $Spec(k)$ I have a Grothendieck spectral sequence ...

**7**

votes

**0**answers

232 views

+200

### Dimensions of dual vector spaces

Let $V_F$ be an infinite dimensional right $F$-vector space (over a field $F$, or even over a division ring). The dual space $V^{\ast}={\rm Hom}(V,F)$ is naturally a left $F$-vector space (coming ...

**3**

votes

**0**answers

98 views

### Are there necessary and sufficient conditions for a chain complex $0 \to C_0 \to C_1 \to C_2 \to 0$ to be Poincare?

I am looking for necessary and/or sufficient conditions for the chain complex $0 \to C_0 \to C_1 \to C_2 \to 0$ over a principal ideal domain to be Poincare in the sense that $H_0 \cong H^2$, $H_1 ...

**0**

votes

**1**answer

113 views

### Tensor products of simple modules over algebras [closed]

Let $A$ and $B$ be $\mathbb{C}$-algebras. Suppose that $M$ and $N$ are respectively simple $A$ and $B$ modules.
We can regard $M\otimes_{\mathbb C}N$ as $A\otimes_{\mathbb C} B$-modules in natural ...

**4**

votes

**0**answers

128 views

### References for bilinear forms on chain complexes?

I am looking for references that include general results and theorems for bilinear forms defined on chain complexes. That is, bilinear forms $\langle \cdot , \cdot \rangle_i : C_i \times C_i \to ...

**8**

votes

**2**answers

235 views

### derived categories as presentable DG-categories

Let $A$ be a ring. Is it true that the DG category of unbounded complexes of $A$-modules, localized by quasi-isomorphisms, is cocomplete and compactly generated? What would be a reference for that and ...

**1**

vote

**1**answer

117 views

### What can be said about $A$ and $B$ given the exact sequence $0 \to R^p \to A \to R^r \to R^q \to B \to 0$?

Let $A,B$ be two $R$-modules over a commutative ring $R$ (restrict to $R = \mathbb{Z}$ or $R= \mathbb{K}$ a field where appropriate). Suppose $A$ and $B$ fit into an exact sequence
$0 \to R^p \to ...

**4**

votes

**1**answer

233 views

### Homotopy classification of selfmaps of product of spheres?

Self-maps of n-torus $T^n=S^1\times ...\times S^1$ are classified by the induced homorphism of fundamental group $\pi_1 T^n=Z^n$.
Is a similar result true form self-maps of $S^k\times ...\times S^k$ ...

**15**

votes

**2**answers

492 views

### Grothendieck spectral sequence when one of the functors is contravariant

Let $f \colon X \rightarrow S$ be a morphism of schemes. I am interested in computing the cohomology groups of
$$
\mathbf{R}\mathscr{H}om(\mathbf{R}f_* \mathcal{O}_X, \mathcal{O}_S)
$$
in terms of ...

**3**

votes

**1**answer

98 views

### Homological characterisation of standardly stratified algebras using Ext

Let A be a finite dimensional algebra and $S_1,S_2,...,S_n$ the simple $A$-modules and $P_1,..,P_n$ the indecomposable projective $A$-modules. For $i=1,...,n$, define the standard module $\Delta_i$ as ...

**6**

votes

**2**answers

820 views

### Does module Hom commute with tensor product in the second variable?

Let $A$ be a commutative ring, and $L, M, N$ be $A$-modules. Then is it true that
$$\text{Hom}_A (L, M)\otimes_A N \cong \text{Hom}_A (L, M\otimes_A N)$$
as $A$-modules?
(Note that there is a ...

**1**

vote

**1**answer

149 views

### $E_{\infty}$ algebra in characteristic zero

I asked this question on MSE(http://math.stackexchange.com/q/1579026/239218).
Let $A^{\bullet}$ be a cosimplicial commutative algebra over a field $\Bbbk$. Denote with $N(A)^{\bullet}$ the ...

**6**

votes

**1**answer

327 views

### Is the sheaf of smooth functions flat?

Let $X$ be a smooth algebraic variety over $\mathbb{C}$. Is the sheaf of smooth functions on $X$ flat as an $\mathcal{O}_X$ module?

**3**

votes

**0**answers

70 views

### Classification of representation-finite algebras up to stable equivalence of Morita type

assume K is an algebraically closed field.
I wanted to ask if there is a classification of the representation-finite K-algebras up to stable equivalence of Morita type at least for some small numbers ...

**4**

votes

**0**answers

216 views

### Do differential objects form triangulated categories?

Let $\mathcal{A}$ be a (fixed) additive category. To a differential object $(A,a)$ for $\mathcal{A}$ (so, $a:A\to A$ and $a^2=0$) one may associate an $\mathcal{A}$-complex $\dots \to ...

**10**

votes

**1**answer

432 views

### teaching higher algebra

Has anyone ever (successfully or unsuccessfully) taught a course in higher algebra (in the $\infty$-categorical sense)?
I'm asking out of curiosity (and also hoping for more resources).
The ...

**2**

votes

**0**answers

122 views

### completion and convergence of spectral sequence

I would like to understand the connection between $p$-adic completion and the strong convergence of a spectral sequence. Precisely, suppose $E^2_{s,t}\implies G_{s+t}$ is a first quadrant strongly ...

**2**

votes

**0**answers

42 views

### Calculation of minimal right $\operatorname{add}(M)$-approximations

given a finite dimensional quiver algebra $A$ and a generator $M$ with $\operatorname{Ext}^1(M,M)=0$. By Wakamatsus lemma, for any $A$-module $N$ there exists a surjective $A$-linear map $f\colon M_1 ...

**4**

votes

**1**answer

174 views

### Information and intuition packed in the Chern character for coherent sheaves

even after quite some time learning it, I still get somehow puzzled by the Chern character. Let me recall some stuff to get notation and setting.
Let us consider a smooth projective algebraic variety ...

**3**

votes

**2**answers

185 views

### How do we get the quotient $Ext^1(N,M)/Hom(N,M)$?

If $0\longrightarrow M\longrightarrow E\longrightarrow N\longrightarrow 0$ is a short exact sequence of torsion free coherent sheaves on a surface. Here $M$ is a line bundle, $E$ a vector bundle of ...

**5**

votes

**0**answers

96 views

### Second homology of mapping class group via universal coefficient theorem

Let $S_g$ be the closed, orientable surface of genus $g\ge 2$ and $Mod(S_{g,1})$ the mapping class group of homeomorphisms fixing a base point $x_0\in S_g$ modulo isotopies fixing $x_0$.
It is known ...

**1**

vote

**1**answer

89 views

### dg-resolution of the polynomial algebra

I am intersted in constructing a cofibrant resolution of the commutative polynomial algebra in some number of variables in the category of dg-algebras(not necceserily commutative).
The resolutions ...

**20**

votes

**1**answer

679 views

### Lemma 2 from Beilinson's “Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra”, intuition?

This is a followup to here.
Consider Lemma 2 from Beilinson's paper "Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra", as follows.
Lemma 2. For any pair $i$, $j$ such that $0 ...

**0**

votes

**0**answers

61 views

### Homology of product of two groups [duplicate]

There is well known formula for the homology of product of two groups with coefficient in integers, that is
$0 \rightarrow \oplus_{p+q=n}H_p(G,\mathbb{Z}) \otimes H_q(H,\mathbb{Z}) \rightarrow H_n(G ...

**7**

votes

**0**answers

157 views

### Terminology for vanishing of Hochschild homology with symmetric coefficients?

In a title or abstract for a paper, if I say "Hochschild cohomology of this algebra $A$ vanishes in degrees two and above" then
it should hopefully be understood by most readers as saying ...

**0**

votes

**0**answers

48 views

### One-sided endomorphism rings of centred bimodules

Let R be an associative unital ring. An R-bimodule M is called centred bimodule if M = R*Z(M), where Z(M)={m:rm=mr,∀r∈R}, i.e., M is generated as an R-module by the set of R-centralizing elements. ...

**3**

votes

**1**answer

178 views

### Reference for constructing tensor products of finitely presented functors

I need references related to the construction of tensor product between functors
Let $k$ be a commutative ring, $C$ a small $k$-linear category and $A$ cocomplete abelian category. Let $A^C$ denote ...

**5**

votes

**1**answer

83 views

### Flat dimension of injectives over a Gorenstein ring

Let $A$ be a Gorenstein noetherian local ring, and let $M$ be an $A$-module of finite injective dimension.
If $M$ is a finite $A$-module, it is easy to show these assumptions imply that $M$ has ...

**3**

votes

**0**answers

115 views

### Seeking an unpublished manuscript by Tetsuro Okuyama

Several papers in representation theory attribute the notion of relatively projective modules to Tetsuro Okuyama's manuscript "A generalization of projective covers of modules over finite group ...

**4**

votes

**0**answers

108 views

### Cohomology of $Sym^m Q \otimes Sym^k Q \otimes L^p$

Let $V$ be a complex vector space. Let $L=\mathcal{O}(-1)$ and $Q=V/L$ be the quotient bundle over $\mathbb{P}V$.
I'm trying to compute the cohomologies with coefficients in $Sym^m Q \otimes Sym^k Q ...

**5**

votes

**1**answer

101 views

### Maps between products of symmetric powers

This question might be too elementary but it arises naturally as a part of a more complicated computation and I struggle to find the answer.
Let $V$ be an $n$-dimensional complex vector space. ...

**2**

votes

**1**answer

138 views

### Formal DG-algebra

Let $\mathcal{C}$ be a nice $k$-linear abelian category (the example I have in mind is the category of coherent sheaves on a smooth projective variety over $\mathbb{C}$). Let $B \in ...

**1**

vote

**0**answers

105 views

### Relation of primary decomposition of two ideals

I have a simple question: Let $R=\mathbb{C} \lbrace t,u \rbrace$ be the ring of formal series in two variables. Let $I,J \subset R$ ideals of heigth one, and $I \subset J$. What is the relation of the ...

**20**

votes

**2**answers

1k views

### Lemma 1 from Beilinson's “Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra”, intuition?

Consider Lemma 1 from Beilinson's paper "Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra", as follows.
Let $\mathcal{C}$ and $\mathcal{D}$ be triangulated categories, $F: ...

**8**

votes

**2**answers

341 views

### Confusion about Subcategories of Category $\mathcal{O}$

So, in learning about category $\mathcal{O}$ representations of a semisimple Lie algebra $\mathfrak{g}$, I've come across two natural kinds of subcategories, and I think I'm confused about their ...

**5**

votes

**1**answer

193 views

### To calculate $Tor_1^G(\mathbb{Z},N_{ab})$ and $Tor_1^Q(\mathbb{Z},N_{ab})$

Let $G$ be a finite $p$-group and $N$ be a normal subgroup of $G$. I wish to calculate $Tor_1^G(\mathbb{Z},N_{ab})$ and $Tor_1^Q(\mathbb{Z},N_{ab})$, where $Q=G/N$. I could not found any lecture notes ...

**3**

votes

**1**answer

131 views

### For what kind of sheaves can we always extend a sheaf map from a closed subset to the whole space?

Let $X$ be a topological space. We know that a sheaf on $X$ is call soft if for any closed subset $Z$ of $X$, a section on $Z$ can be always extend to a section on $X$.
Now we consider a similar ...

**1**

vote

**0**answers

104 views

### Is local-to-global spectral sequence functorial?

Consider a lower term of local-to-global spectral sequence
$0 \to H^1(X,\mathcal{Hom}(\mathcal{F},\mathcal{G})) \to Ext^1(\mathcal{F},\mathcal{G}) \to H^0(X,\mathcal{Ext}^1(\mathcal{F},\mathcal{G})) ...

**14**

votes

**1**answer

277 views

### Moduli space of boundary maps with prescribed chain and homology groups?

Let $R$ be a reasonable ring (maybe I mean a PID, or $\mathbb{Z}$, and when sufficiently desperate, a field). Now consider fixed sequences $C_n$ and $H_n$ of $R$-modules, which are tame in every ...

**3**

votes

**1**answer

167 views

### generalized universal coefficient sequence

Take the familiar Universal Coefficient Theorem for ordinary homology with $\mathbb{Z}$-coefficients and ordinary cohomology with coefficients in some abelian group $A$:$$0\rightarrow ...

**1**

vote

**0**answers

61 views

### Twisting cochains in non-unital settings

I'm reading chapter 2 of Loday and Vallette's book: Algebraic operads. My question is about how to translate some results in augmented settings into non unital settings.
Let $k$ be a field of ...

**2**

votes

**0**answers

77 views

### Flat resolutions of DG-schemes

Recall that a DG-scheme is a pair $(X,\mathcal{O}_X)$,
where $(X,\mathcal{O}^0_X)$ is a scheme, $\mathcal{O}_X$ is a sheaf of commutative DG-algebras over $(X,\mathcal{O}^0_X)$, and each ...