Homological algebra studies homology in a general algebraic setting. The purpose is extraction of information about structures involved in terms of tangible objects like rings groups and modules.

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Why $\bar{d_0}$ is injective in a minimal free resolution? [closed]

If $(R,\mathfrak{m},k)$ is a Noetherian local ring, $A$ a finite $R$-module. Let $L.:\cdots\rightarrow L_1\xrightarrow{d_1} L_0\xrightarrow{d_0} A\rightarrow 0$ be a minimal free resolution. $\bar{d_i}...
4
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1answer
84 views

Calculate $\operatorname{RHom}$ in a the derived category of graded $\mathbb{C}[x]$-Modules

I was trying to do the following exercise. Consider the category of graded $\mathbb{C}[x]$-Modules, it is clear that we can regard $\mathbb{C}[x]$ as a graded module setting $\operatorname{deg}(x)=1$. ...
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35 views

projective resolution for an $I$-torsion $R$-module

Let $R$ be a commutative Noetherian ring with non-zero identity, $I$ be an ideal of $R$ and $M$ be an $I$-torsion $R$-module. We know that there exists an injective resolution of $M$ in which each ...
2
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1answer
68 views

Is R/m a flat R-module?

Let $(R,\frak m)$ be a commutative Noetherian local ring. Is $R/\frak m$ a flat $R$-module? Thanks.
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1answer
70 views

Compute Ext with Macaulay2

I want to compute Ext with Macaulay2. I see in the website they write how to do but I can not do. Can anyone help me with an example? For example, let $S=k[x,y,z,t]$. How compute $\mathrm{Ext}^i_S (...
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3answers
200 views

Why are we interested in cohomology?

I've been studying algebraic topology for over half a year now and came across alot of different topics of it (fundamental groups, Van Kampen, singular homology, homology theory, Mayer Vietoris, ...
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44 views

Homomorphisms of Chain Complexes

Let $(K, d^{K})$ and $(L, d^{L})$ be chain complexes. For $n \in \mathbb{Z}$ define $$ \mathrm{Hom}(K, L)_{n} := \prod_{j \in \mathbb{Z}} \mathrm{Hom}(K_{j}, L_{j+n})$$ and $$ d_{n}^{K,L} \ \colon \...
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Is this functor representable?

Fix a group $G_0$ and $R$ a subset of $G_0$. Consider the functor $F$ from $\textbf{Grps}$ to $\textbf{Sets}$, sending every object $G$ in $\textbf{Grps}$ to $F(G)$, the subset of $\varphi \in \text{...
2
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1answer
40 views

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 \...
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1answer
39 views

Projective Dimension and Schanuel's Lemma

Let $R$ be a ring and $M$ a (say, left) $R$-module of projective dimension $n$. According to Noncommutative Noetherian Rings, any projective resolution of $M$ can be terminated at length $n$, and this ...
2
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48 views

Are product / coproduct projections / inclusions 'semistrict'?

Let $\mathbf{C}$ be a category with zero object, kernels, and cokernels. Then, a morphism $f\colon A\rightarrow B$ in $\mathbf{C}$ is semistrict iff the canonical map $\operatorname{Coker}(\ker (f))\...
2
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0answers
41 views

Proof of Birger Iversen “Cohomology of Sheaves” Theorem 6.8

I am having troubles completing the proof of theorem 6.8 (page 44) from Birger Iversen, Cohomology of Sheaves. (pdf here) Previously we had constructed a functor $\rho$ from $K^+(A)$ (the homotopy ...
3
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1answer
102 views

Does trivial cohomology imply trivial homology? Does $\operatorname{Hom}(A,\mathbb Z) = \operatorname{Ext}^1(A, \mathbb Z) = 0$ imply $A = 0$?

Is there a topological space $X$ such that $H^i(X; \mathbb{Z}) = 0$ for all $i > 0$, but $H_n(X; \mathbb{Z}) \neq 0$ for some $n > 0$? In his answer to the question Is homology determined by ...
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0answers
33 views

Any characterization for commutative rings over which “projective modules” equal “free modules”?

As far as I know, over any PID, an polynomial rings over a field, or an local ring, projective modules are always free. This kind of results make me curious about if there are any overall ...
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0answers
40 views

Augmented graded algebras - properties

Let $A$ be an augmented graded unital algebra over field $k$. Define $A_+=\bigoplus\limits_{i\ge 1}A^{(i)}$. I'm trying to show that $\sum\limits_{i+j>k}A_+^i\otimes A_+^j=\bigcap\limits_{l+m=k}...
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2answers
53 views

Some question in relative homology

When we consider the pair (X,A) in relative homology, do we assume A is a sub complex of X? And why don't we just consider X/A instead of (X,A)? Is there an better advantage to consider (X,A) ...
4
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1answer
112 views

Sheaf cohomology via resolutions vs. derived categories

So I know that when introducing sheaf cohomology, there are two main approaches via derived categories, and a perhaps more "down to earth" method of resolving by acyclic, fine, soft, sheaves. I'm ...
4
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0answers
33 views

Connection between cobar construction of DG-coalgebra and cobar construction from monad

Given a monad $M:C\to C$ we can construct a cobar resolution from it directly as a functor $\Delta\to [C,C]$ Given a DG-coalgebra $(C,d)$ we can construct a cobar resolution $\Omega C$ of it as ...
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1answer
28 views

Composition of stable-pseudomonomorphisms

Terminology Let $\mathbf{C}$ be a finitely-complete finitely-cocomplete category with zero object (not necessarily additive!). A morphism $f\colon A\rightarrow B$ is a pseudomonomorphism iff $(0\...
2
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1answer
106 views

Direct proof that infinite product of copies of $\mathbb{Z}$ is not projective

It is well-known that the abelian group $$A = \prod_{n=1}^\infty \mathbb{Z}$$ is not free (see, for example this MO question), and that over a PID being free is equivalent to being projective (see ...
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1answer
90 views

When is $\operatorname{Hom}(M, E)$ injective? [closed]

Let $R$ be a commutative Noetherian ring with non-zero identity, $M$ be an $R$-module, and $E$ be an injective $R$-module. When is $\operatorname{Hom}(M, E)$ an injective $R$-module?
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14 views

An algorithm for determining if a tensor is pure?

Suppose I have an category whose objects are free $R$-modules (R a polynomial ring) and whose morphism-spaces $\mathrm{Hom}(A,B)$ between objects $A$ and $B$ are spanned by a finite set of module-...
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1answer
54 views

Composition factors of injective indecomposable and projective indecomposable modules

Let $A$ be a finite-dimensional algebra over an arbitrary field $K$. Let $L_1$ and $L_2$ be simple modules such that $L_1 \not \cong L_2$. Let the $A$-module $Q_1$ be the injective hull of $L_1$, ...
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0answers
36 views

If $R$ is $I$ and $J$-adically complete, then it is $(I+J)$-adically complete.

Let $R$ be a Noetherian ring with ideals $I$ and $J$. I already proved the following: Lemma: If $I \subseteq J$ and $R$ is $J$-adically complete, then $R$ is $I$-adically complete. And now I'm ...
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1answer
30 views

Minimal graded free resolution and matrix representations

In a graded $R$-module, let $$0 \to C_p \xrightarrow{\phi_p} C_{p-1} \xrightarrow{\phi_{p-1}}C_{p-2} \to \dots \to C_1 \xrightarrow{\phi_1} C_0 \xrightarrow{\psi} M \to 0$$ be a minimal graded free ...
3
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1answer
43 views

finding high weight vector in Verma module

Let $\frak{g}$ be a (semi-)simple lie algebra. Let $\lambda$ be a dominant integral weight. Denote $L(\lambda)$ to be the irreducible representation of highest weight $\lambda$. From BGG resolution, ...
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2answers
55 views

Minimal graded free resolution of the ideal $I = (x^r, y^s) \subset k[x,y]$

What is the minimal free-graded resolution of the ideal $I = (x^r, y^s) \subset k[x,y]=R$ for $r,s \in \mathbb{N}$? I tried reducing this down to $r = s = 1$ and I think it is $$0 \to R(-2) \to R(-1)...
0
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1answer
21 views

Existence of homomorphism in diagram

Consider the following diagram where each complex is an $R-$module and the rows are exact and the maps $g,h$ are $R-$mod homomoprhisms and the right square commmutes $h(p(b)) = q(g(b)), \forall b \in ...
1
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1answer
32 views

Direct limit of a direct system looking like a cochain complex of objects.

I would like to ask you about a special kind of direct systems $ (A_i, f_{i}^{j} )_{ i,j \in ( I , \leq ) } $ looking like a cochaîn complex $ (A_i , f_{i}^{j} )_{ i,j \in ( \mathbb{N}^* , \leq ) } $ ...
4
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2answers
98 views

Kähler differential over a field

I have been working with Kähler differentials, and I have $\Omega^1_{B/k}$, where B is a commutative $k$-algebra, and $k$ is a field. I was wondering that for $d(b)=0$, does this imply that $b\in k$? ...
2
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1answer
85 views

Variation on localization of Tor

It is known that Let $R$ be a commutative ring with unit and $S \subset R$ a multiplicative sistem. If $M$ and $N$ are $R$-modules there is a isomorphism of $S^{-1}R$-modules: $$S^{-1}\mathrm{Tor}...
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0answers
45 views

Long exact sequence to short exact sequence [duplicate]

$$\dots\xrightarrow{p_*}\pi_{n+1}(B)\xrightarrow{\partial}\pi_n(F)\xrightarrow{\text{inc}_*}\pi_n(E)\xrightarrow{p_*}\pi_n(B)\xrightarrow{\partial}\pi_{n-1}(F)\xrightarrow{\text{inc}_*}\dots$$ From ...
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0answers
36 views

$\mathcal{T}^i(X/Y,\mathcal{F})$ forms a sheaf

In Hartshorne's Deformation Theory, given an $A$-algebra $B$ and a $B$-module $M$, he defines these functors $T^i$ for $i=0,1,2$ that outputs $B$-modules $T^i(B/A,M)$. In Exercise 3.5, he asks the ...
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1answer
44 views

I'm searching for $X$ and $V$ such that the evaluation map $f:H^n(X,A;V)\to \hom_\mathbb{Z}(H_n(X,A;\mathbb{Z}),V)$ is not injective

Let $V$ be a $\mathbb{Z}$-module $(X,A)$ a pair of topological spaces. We define the pairing-map in singular co- homology$$H^n(X,A;V)\times H_n(X,A;\mathbb{Z})\to V$$ $$([\xi ],[\alpha])\mapsto \xi (\...
2
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1answer
39 views

Contractibility of an exact chain complex

How can one prove that an exact (acyclic) chain complex of projective modules that is trivial in negative degrees is contractible? I would appreciate some nudges in the right direction more than ...
2
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1answer
78 views

A chain complex is split if and only if it splits as a direct sum.

This is the first part of Exercise 1.4.2 in An Introduction to Homological Algebra by Weibel. The first part is showing that a chain complex, $C$, with boundaries $B_n$ and cycles $Z_n$ in $C_n$ is ...
2
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2answers
62 views

Counterexample to, if $f$ is acyclic then $kerf$ and $cokerf$ acyclic.

This is the second half of exercise 1.3.5 in An Introduction to Homological Algebra by Weibel, it simply asks if this statement is true of false and I believe it is false but cannot construct a ...
3
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0answers
68 views

Why is Hom$_A(M,A)$ a right $\Gamma$ comodule?

I'm reading through appendix I (Hopf algebroids) of Ravenel's green book, and I came across a line I can't understand in a proof. The part of the lemma I'm interested in states: $\mathbf{Lemma A1....
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1answer
44 views

Baer Sum notation requires clearence.

I am working on Baer sum and I have my book by Rotman, Introduction to Homology, and also MacLanes book Homology and they use notation I am puzzled on. I have understood baer sum of extensions ...
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Compute directly that the mapping cone of a homotopy equivalence is contractible

Let's consider the category $Ch_R$ of cochain complexes of modules over a commutative ring $R$. I'm trying to prove that if the chain map $\phi:M\rightarrow N$ is a homotopy equivalence then its ...
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25 views

Automorphism of semisimple Lie algebra corresponding to a simple reflection

Let $\mathfrak{g}$ be a complex, finite-dimensional Lie algebra. Let $\mathfrak{h}\subset \mathfrak{g}$, $W$ and $\Pi$ be a Cartan subalgebra, its Weyl group and the set of all simple roots, ...
3
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1answer
62 views

Baer sum of $\mathbb{Z}_9$ and $\mathbb{Z}_9$

I am working on trying to figure out the third extension of $\mathbb{Z}_3$ by $\mathbb{Z}_3$, I know one is $\mathbb{Z}_9$ and the neutral element (with respect to baer sum) $\mathbb{Z}_3\oplus \...
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1answer
42 views

Getting a double complex that computes Ext

Suppose $C$ is an abelian category and I am trying to compute $Ext^i(M,N)$ for some objects $M,N$. Suppose there is an exact sequence $0 \rightarrow A_1 \rightarrow A_2 ... \rightarrow A_n \...
3
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0answers
54 views

$T^i$ functors in Hartshorne's Deformation Theory

In chapter 3 of Hartshorne's Deformation Theory, he defines functors $T^i$ for $i=0,1,2$ that take as input a ring homomorphism $A\rightarrow B$ and a $B$-module $M$ and outputs $T^i(B/A,M)$, a $B$-...
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0answers
58 views

An exact sequence of inverse systems of $R$-modules

Let $$0\longrightarrow \big\{A_n,f_{mn}\big\}_{m \leq n} \overset{\Phi}\longrightarrow \big\{B_n,g_{mn}\big\}_{m \leq n} \overset{\Psi}\longrightarrow \big\{C_n,h_{mn}\big\}_{m \leq n} \...
3
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1answer
42 views

An identity map which is not null-homotopic

I have some problems in understanding how the definition of a null-homotopic cochain map actually works. Maybe I lack concrete examples. Let $f^{.}:X^{.}\longrightarrow Y^{.}$ a cochain map of ...
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0answers
33 views

Analogue of the trivial extension for higher Ext.

I've been doing some homological algebra and some work on showing some extensions are equivalent, and a thought just came to me, which is that I didn't know how to write down what the analogue of the ...
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0answers
46 views

Hochschild homology of dgas with nontrivial differential

In this question, we see how to compute the Hochschild homology of a dga with zero differential: it's just the same as computing its Hochschild homology as a graded algebra. I want to know about ...
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1answer
49 views

Is $M_R\otimes _R {_R}N\cong M_{\mathbb Z}\otimes_{\mathbb Z} {_{\mathbb Z}}N$?

Suppose $M$ is a right $R-$module and $N$ is a left $R-$ module. Also $M$ and $N$ are naturally $Z-$ module, both in left and right side. So we will denote $M_R$, $M_{\mathbb Z}$, and $_RN, _{\mathbb ...
1
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1answer
75 views

Exact Sequences in algebraic geometry [closed]

A very basic question. I am going to take my first course in Algebraic Geometry next semester and I am now repeating some commutative algebra to be prepared. I just came up to the part of Homological ...