Articles of homological algebra

Left exact functors and long exact sequences

I wonder whether in any Abelian category $\mathcal{C}$ when we have a long exact sequence $0\to M_1\to M_2\cdots\to M_n\to 0$ and a (covariant) left exact functor $F$ we have $0\to FM_1\to FM_2\to \cdots FM_n$. I know this is certainly true if $n=3$ (i.e. we have a short exact sequence), but what about in general, may […]

Cohomological dimension of direct product

Let $cd$ denote the cohomological dimension of a group, i.e. the minimal length of a projective resolution of $\mathbb{Z}$ over the group ring. Let $G_1$ and $G_2$ be groups. It is easy to see that $$cd(G_1 \times G_2) \leq cd(G_1)+cd(G_2) \ , $$ (using the tensor product of resolutions), but is there a clear criterion […]

Exact sequence and torsion

I’ve come across another exact sequence, where (I guess) I need to deduce the result using some properties of torsion. I am calculating the homology of the Klein bottle using attaching maps. I start by defining $\Phi:I \times I \to K$ as the natural map and denote $\partial(I \times I)$ as the boundary, then let […]

$\mathrm{Hom}(M,F)$ can't be determined by the underlying sets of $M,F$? where $F$ is a free module, $M$ is not.

$\mathrm{Hom}(M,F)$ can’t be determined by the underlying sets of $M,F$? where $F$ is a free module, $M$ is not a free module. The question arises from the claim: let $G : \mathbf{Mod}_R\to\mathbf{ Set}$ be the forgetful functor which assigns to each $R$- module its underlying set. Then the functor $G$ does not have a right […]

Question about the $\mathrm{Tor}$ functor

Assume we want to define $\mathrm{Tor}_n (M,N)$ where $M,N$ are $R$-modules and $R$ is a commutative unital ring. We take a projective resolution of $M$: $$ \dots \to P_1 \to P_0 \to M \to 0$$ Now does it matter whether we apply $-\otimes N$ or $N \otimes -$ to this? It shouldn’t because we have […]

Finite injective dimension

Let $A$ be a commutative noetherian ring. Is it true that if $A$ is regular then any module over it has a finite injective dimension? What if $A$ is Gorenstein? Any reference who discuss this?

Inductive vs projective limit of sequence of split surjections II

This question is a follow-up of this earlier question I asked. Let $$ A_1\twoheadrightarrow A_2\twoheadrightarrow A_3\twoheadrightarrow A_4\twoheadrightarrow \cdots $$ be an inductive sequence of abelian groups, the connecting homomorphisms of which are surjective and split, that is, we have embeddings $A_{n+1}\rightarrowtail A_n$ such that the diagram \begin{array}{ccccccccc} A_n & \twoheadrightarrow & A_{n+1}\\ \uparrow & & […]

A short exact sequence that cannot be made into an exact triangle. (Weibel 10.1.2)

The following exercise is in Weibel Chapter 10. Regard the groups $\mathbb{Z}/2\mathbb{Z}$ and $\mathbb{Z}/4\mathbb{Z}$ as cochain complexes in degree 0. Show that the short exact sequence $$ 0 \rightarrow \mathbb{Z}/2\mathbb{Z} \stackrel 2 \rightarrow \mathbb{Z}/4\mathbb{Z} \stackrel 1 \rightarrow \mathbb{Z}/2\mathbb{Z} \rightarrow 0 $$ cannot be made into an exact triangle in the homotopy category of chain complexes. […]

Homology and topological propeties

i have this theorem with it’s proof but i don’t understand the last part They use this proposition: My question is Why $\varphi^c\cap U_i$ is closed and pairwise disjoint ? where $\varphi^c=\lbrace x, \varphi(x)\leq c\rbrace$ Please thank you.

Right exactness on a dense subcategory

Let $F : C \to D$ be a $k$-linear functor between cocomplete $k$-linear categories, which preserves directed colimits (in particular arbitrary direct sums). Let $C’ \subseteq C$ be a dense full subcategory such that the restriction $F|_{C’}$ is right exact (i.e. preserves coequalizers). Is it possible to conclude that $F$ is right exact? If not, […]