Let $V:=\oplus_{j\in\mathbb{Z}}V_j$ be a graded $\mathbb{F}$-vector space over the field $\mathbb{F}$. The graded tensor product of graded vector spaces is given by $V \otimes W: …

Let $\mathcal{C}$ be an abelian category, $\mathcal{Ab}$ the category of abelian groups and $Lex(\mathcal{A}, \mathcal{B})$ the category of left exact functors between abelian cate …

Let $\scr A$ be an abelian category with exact products and a cogenerator (e.g. $\scr A$ is a category of modules). Let ${\mathbf K}(\scr A)$ be the homotopy category of cochain c …

Let $A \to B$ be a finitely generated homomorphism between two commutative noetherian rings. As far as I understand, in various generalizations of this situation, such a map is ca …

Let $\mathcal{C}$ is a grothendiect category and consider all of what follows in $\mathcal{C}$. Let $${\varepsilon_i: 0\to A_i \to B_i \to C_i\to 0\ ,\ \phi_i^j}$$ be a direct sy …

What is $Ext_{\mathbb Z}^1 (\mathbb Z^{\mathbb N},\mathbb Z)$, where $\mathbb Z^{\mathbb N}$ stands for the infinite product $\prod_{n \in \mathbb N} \mathbb Z$?

Hi: When I read the book "An introduction to Homological algebra" by Weibel, the page 206, line 9 says that "Shapiro's Lemma tell us that $H_q(S_n(X)\otimes_{Z}A)$ is zero if $ …

Hy guys. I'm doing some independent analysis which makes use of the tensor product of modules (over commutative rings with unit 1, and ring homomorphisms map $1 \mapsto 1$). Let $\ …

Hi all, I am reading about minimal model of $A_\infty$ structure. So far, I find two different ways of the construction, given by Kadeishvili and Kontsevich, Soibelman. 1) The co …

This question is related to http://mathoverflow.net/questions/130008/lydon-hochschild-serre-spectral-sequence-and-cup-products. I have the followin result by J.S Milne in his book …

Lurie in Higher Algebra, section 1.3 associates a quasi-category to a dg-category A via the so called dg-nerve construction, extending the classical nerve. I have a feeling the hom …

What is the best way to show, that $Lex(\mathcal{A},\mathcal{Ab})$ is abelian, where $\mathcal{A}$ is an abelian category and $\mathcal{Ab}$ is the category of abelian groups from …

First here is my setup: Let $W$ be some group, and $C$ a normal subgroup of finite index, and let $W/C=G$. Now let $L$ be a a $G$-module on which $C$ acts trivially, so in particu …

Consider the following diagram which lives in the category of $R$-modules. $$ \begin{array}{ccccccccc} 0 & \xrightarrow{i} & A & \xrightarrow{f} & B & \xrighta …

Motivation. In my research I have a situation where a monoid $M$ is acting by nice cellular maps on a contractible cell complex and so the augmented chain complex is a resolution o …