Questions tagged [homological-algebra]

(Co)chain complexes, abelian Categories, (pre)sheaves, (co)homology in various (possibly highly generalized) settings, spectra, derived functors, resolutions, spectral sequences, homotopy categories. Chain complexes in an abelian category form the heart of homological algebra.

Filter by
Sorted by
Tagged with
5 votes
1 answer
911 views

An example where finitistic dimension does not equal right global dimension?

The (right) big finitistic dimension of a ring is Findim$(R) =$ sup{proj.dim(M) | $M$ a right $R$-module of finite projective dimension}. The (right) little finitistic dimension findim$(R)$ is the sup ...
David White's user avatar
  • 29.4k
5 votes
1 answer
930 views

Multiplicative Structures On Free Resolutions

Hello, this question is related to Differential graded structures on free resolution?. Given a regular local ring $S$ and $f\in{\mathfrak m}_S\setminus\{0\}$, I am interested in studying $R$-modules ...
Hanno's user avatar
  • 2,736
5 votes
1 answer
169 views

Restriction vs. multiplication by $n$ in Tate cohomology

$\DeclareMathOperator{\Res}{Res} \DeclareMathOperator{\Cor}{Cor}$ This question was asked in MSE. It got no answers or comments, and so I post it here. Let $H$ be a subgroup of a finite group $G$, and ...
Mikhail Borovoi's user avatar
5 votes
1 answer
282 views

Localization of a ring and the Hom functor

Let $R=\mathbb{Z}[x,x^{-1}]$ be the ring of Laurent polynomials in $x$, $\mathfrak{p}=(1-x)$ be an ideal in $R$ and $R_\mathfrak{p}$ be the localization. I want to know what $\text{Hom}_R(R_\mathfrak{...
user avatar
5 votes
1 answer
276 views

Is Koszul homology of a monomial ideal always generated by the "obvious" things?

Let $R = k[x_1 , \dots , x_n]$ be a polynomial ring over a field and $I$ a monomial ideal in $R$. Then, is it true that the Koszul homology of $R/I$ is always generated by elements of the form $$r e_{...
Rellek's user avatar
  • 401
5 votes
1 answer
293 views

Extension of $FP_{n}$ group

I am reading a paper (Finitely presented residually free groups by Bridson, Howie, Miller III, and Short, Theorem 5.2) where they write the following: Since $S_{0}$ is a group of type $FP_{n}(\mathbb{...
J.L.'s user avatar
  • 321
5 votes
1 answer
270 views

Manifold generators of O-bordism invariants

If I understand correctly, I can obtain the $O$-cobordism group of $$ \Omega^{O}_3(BO(3))=(\mathbb{Z}/2\mathbb{Z})^4, $$ The 3d cobordism invariants have 4 generators of mod 2 classes, are generated ...
annie marie cœur's user avatar
5 votes
1 answer
621 views

Hochschild homology of a category of modules over an algebra

Suppose $A$ is an algebra over some field, say the complex numbers if that helps. Then we can consider the category $\mathbf{C}_A$ of finite-dimensional modules over $A$. This category can be seen as ...
Lukas Woike's user avatar
  • 1,372
5 votes
1 answer
366 views

Tachikawa conjecture for commutative algebras proven?

The Tachikawa conjecture states that $Ext^i(M,M) \neq 0$ for some $i \geq 1$ for every non-projective module $M$ over a selfinjective finite dimensional algebra. In theorem 4.6. of http://maths.nju....
Mare's user avatar
  • 25.8k
5 votes
1 answer
244 views

Are finite-dimensional representations of groups of type $\text{FP}_{\infty}$?

Let $G$ be a group (possibly infinite) and $k$ be a field. A module $M$ over $k[G]$ is said to be of type $\text{FP}_{\infty}(k)$ if it has a projective resolution each of whose terms is finitely ...
Albert's user avatar
  • 53
5 votes
1 answer
182 views

Is this a description of the $\aleph_1$-localizing subcategory generated by a compact generator?

This should be obvious but I'm not seeing it: The $\mathfrak T$ be a triangulated category with coproducts and with a compact generator $A$ (that is, the functor $\mathfrak T(A,\_)$ preserves ...
Rasmus's user avatar
  • 3,144
5 votes
1 answer
1k views

How do I split a homotopy idempotent?

I want to check that the homotopy category of cochain complexes of an idempotent splitting, preadditive category is idempotent splitting. Let $a\xleftarrow{e}{}a$ be an idempotent chain map up to ...
Eitan Chatav's user avatar
5 votes
1 answer
792 views

Simplicial "universal extensions", the hammock localization, and Ext

Let $M,B$ be $R$-modules, and suppose we're given an n-extension $E_1\to\dots\to E_n$ of $B$ by $M$, that is, an exact sequence $$0\to M\to E_1\to\dots\to E_n \to B\to 0.$$ A morphism of $n$-...
Harry Gindi's user avatar
  • 19.4k
5 votes
1 answer
2k views

Question about hypercohomology / spectral sequence of a complex of "almost-acyclic" sheaves

I have a very particular situation involving a (non-exact) complex $K$ of coherent sheaves on a nonsingular projective variety $X$, and I need to compute the hypercohomology of the complex. The ...
user5395's user avatar
  • 545
5 votes
2 answers
298 views

In what degrees does Ext(S/(f),S) vanish?

Let $S=k[x_0,...,x_n]$ be the polynomial ring over a field $k$ and $f\in S$ non-zero and homogeneous. Is it true that $Ext^m(S/(f),S)$ is zero? This would help me to show that $Ext^m(S/fI,S)\cong Ext^...
Ida B.'s user avatar
  • 83
5 votes
1 answer
447 views

Do we have $\underline{\operatorname{Ext}}^i_\text{fppf}(\mathbb{G}_a,\mathbb{G}_m)=0$ for $i>0$?

Let $k$ be a characteristic zero field and consider the category $(\mathsf{Sch}/k)_\text{fppf}$ of schemes over $k$ with the fppf topology. I know that $\underline{\operatorname{Hom}}(\mathbb{G}_a,\...
Gabriel's user avatar
  • 943
5 votes
1 answer
311 views

Injective modules

Let $A$ be a finite dimensional $k$-algebra and let $I$ be an injective module。My question is whether $I$ is a direct sum of finite-dimensional injective modules。
Sun YongLiang's user avatar
5 votes
1 answer
143 views

Commutator of finite global dimension algebras

Let $A=KQ/I$ be a finite dimensional quiver algebra of finite global dimension. Is it true that the dimension of $A/[A,A]$ is equal to the number of simples of $A$? Here $[A,A]$ is the vector space ...
Mare's user avatar
  • 25.8k
5 votes
1 answer
308 views

Resolutions of $\mathbb{Z}_{(p)}$ as $\mathbb{Z}$-module

Are there any interesting canonical (maybe unbounded) projective resolutions of $\mathbb{Z}_{(p)}$ over $\mathbb{Z}$, for instance by tensoring together all the $\mathbb{Z}[x] \stackrel{qx-1}\to \...
Oren Ben-Bassat's user avatar
5 votes
1 answer
362 views

mod p (odd) cohomology of dihedral groups

I've been trying to find the cohomology for the trivial module for $\operatorname{PSL}_2(r^n)$ over $\mathbb{F}_p$ for $2 \neq p \neq r$ and have managed to reduce this to the cohomology of a maximal ...
Jacksbabypig's user avatar
5 votes
1 answer
292 views

Conceptual and practical reasons and consequences of inverting weak equivalences

Although dealing with this in one or other form for many years, to my shame this question only struck me now. One of the most radical differences between categories of "algebraic" and "...
მამუკა ჯიბლაძე's user avatar
5 votes
1 answer
493 views

A simple colimit in the derived category?

I have recently come across the following question : Let $X$ be a (bounded below)chain complex in an arbitrary abelian category, and denote ${\sigma_{\leq n}}$ the stupid truncations functors (i.e. ...
L.Guetta's user avatar
  • 175
5 votes
1 answer
707 views

Resolutions by free Differential Graded Algebras

I am struggling to find references for explicit computations of things like symmetric algebras and resolutions in the dg context. (any pointers in this direction would be highly appreciated!) I have ...
Yosemite Sam's user avatar
  • 1,869
5 votes
1 answer
546 views

Defining hom spaces in the derived category as limits of hom spaces in the homotopy category

Let $C$ be an abelian category and $K(C)$ the homotopy category of complexes in $C$. I've seen the following claimed in several sources (without proof): A. The following isomorphisms hold: $$\...
Saal Hardali's user avatar
  • 7,549
5 votes
1 answer
942 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 ...
user71111's user avatar
5 votes
1 answer
405 views

Inverse limit of Gorenstein local rings is again Gorenstein?

If we have the system of surjective ring homomorphisms $f_{i,i+1}: R_{i+1} \twoheadrightarrow R_i$ for an arbitrary $i \geq 0$ such that all $R_i$ are Gorenstein local ring. Let us put $R^{\...
Pierre's user avatar
  • 51
5 votes
1 answer
1k views

Two-sided bar construction

On page 4 of this paper by H. Abbaspour, the author defines the two-sided bar construction $$B(A,A,A):=A\otimes T(s\bar{A})\otimes A$$ of a differential graded algebra $(A,d_A)$ (over a field). The ...
Dave's user avatar
  • 281
5 votes
1 answer
487 views

Are any finitely generated reflexive module a 2nd syzygy?

Are any finitely generated reflexive module a second syzygy? (I´m thinking especially in normal noetherian domains) More general... Are any divisorial lattice a second syzygy? (I´m thinking ...
Hideyuki Kabayakawa's user avatar
5 votes
1 answer
175 views

Are module finite algebras over semiperfect rings again semiperfect?

Let $S$ be a Noetherian semiperfect ring (https://en.m.wikipedia.org/wiki/Perfect_ring). Let $R$ be a module finite associative $S$-algebra. Then, is $R$ also a semiperfect ring? (Clearly, $R$ is ...
uno's user avatar
  • 280
5 votes
1 answer
201 views

Equivalences of categories of complexes of modules

Let $k$ be an algebraically closed field of characteristic $0$. Let $R, S$ be two commutative $k$-algebras. Let $\operatorname{Ch}(R), \operatorname{Ch}(S)$ be the categories of complexes of $R$-...
Walterfield's user avatar
5 votes
1 answer
272 views

Axioms of derivators

I would like to enter the world of derivators. We can find a little history here and there about the limitations of triangulated categories and the motivation to enhance them, but also to compute ...
user234212323's user avatar
5 votes
1 answer
310 views

About a recent paper of Rickard on finitistic dimension

Apologizes if this is a basic question, but I am new to the area of finite dimensional algebras. I am reading the paper "Unbounded derived categories and the finitistic dimension conjecture" ...
Reading finitisitic's user avatar
5 votes
1 answer
188 views

Finite lattices that are Koszul

Let $L$ be a finite lattice and $A=KL$ the incidence algebra of $L$. It should be true that $L$ is modular if and only if the algebra $KL$ is quadratic (since being modular is equivalent to having no ...
Mare's user avatar
  • 25.8k
5 votes
1 answer
225 views

Concept of an exact ideal of a module category

Let $R$ be a ring and $\text{Mod}\,R$ the category of (left) $R$-modules. Consider an ideal $\mathcal{I}$ of $\text{Mod}\,R$. For $R$-modules $X$ and $Y$ let $\mathcal{I}(X,Y)$ be the collection of ...
kevkev1695's user avatar
  • 1,023
5 votes
1 answer
323 views

Unsplitting sequence of vector bundles

Let $V$ be a $n$-dimensional complex vector space. Using Grothendieck's notation, we define the Grassmannian $G(k,V)$ as the space of $k$-quotients of $V$ or, equivalently, as $$ G(k,V)=\{ \mathbb P W ...
Bobech's user avatar
  • 381
5 votes
1 answer
356 views

triviality of homology with local coefficients

Let $X$ be a manifold or a CW-complex. Let $\pi: \tilde X\longrightarrow X$ be a covering map. Let $\pi_1(X)$ be the fundamental group of $X$ and let $\rho: \pi_1(X)\longrightarrow O(n)$ be an ...
Shiquan Ren's user avatar
  • 1,970
5 votes
1 answer
697 views

What's the relationship between a $E_2$-Hochschild Cohomology module and a D-module?

Let's say for simplicity $A$ is a smooth algebra over a field $k$ ($A$ and $k$ are discrete commutative rings but from now on we are fully derived), and we will consider the $E_2$ algebra $HH^{\bullet}...
davik's user avatar
  • 2,035
5 votes
1 answer
209 views

Frobenius algebras from symmetric polynomials

Let $K$ be a field of characteristic 0 (maybe it works for more general fields) and $K[x_1,...,x_n]$ the polynomial ring in $n$ variables. Let $e_1,e_2,...,e_n$ denote the elementary symmetric ...
Mare's user avatar
  • 25.8k
5 votes
1 answer
1k views

Cohomology of derived tensor product of complexes and Künneth spectral sequence

Let $R$ be any commutative ring, let $V^\bullet$ and $W^\bullet$ be (co)chain complexes of $R$-modules, indexed cohomologically. We can also assume that they have both cohomology in nonpositive ...
Francesco Genovese's user avatar
5 votes
1 answer
375 views

$Ext^1$ for some modules over the polynomial ring in one variable

Let $M$ be a module over the polynomial ring $\mathbb C[x]$ such that $x+n$ is invertible in $M$ for every integer $n$. Let $N=\oplus _{n>0} \mathbb C_n$ where $\mathbb{C}_n := \mathbb C[x]/x+n$. ...
Alexander Braverman's user avatar
5 votes
1 answer
177 views

Decompose $MT(E(d)\times_{\mathbb Z_2} SU(2))$ as the wedge sum or smash product of spectra

Consider the extension $$1\to SU(2)\to X\to O\to1,$$ there are 4 possibilities for $X$: $X=O\times SU(2)$ or $E\times_{\mathbb{Z}_2}SU(2)$ or $Pin^+\times_{\mathbb{Z}_2}SU(2)$ or $Pin^-\times_{\...
annie marie cœur's user avatar
5 votes
1 answer
246 views

Derived Morita equivalence of associative algebras

An associative algebra $A$ is said to be Morita equivalent to another one $B$ if there is an equivalence $$\mathsf{Mod}_A\simeq \mathsf{Mod}_B$$ between its corresponding abelian categories of modules....
user avatar
5 votes
1 answer
351 views

Which triangulated categories are subcategories of compact objects "somewhere"?

Let $T$ be a small triangulated category. Under which conditions there exists a triangulated category $B$ closed with respect to (small) coproducts such that $T$ fully embedds into the subcategory of ...
Mikhail Bondarko's user avatar
5 votes
1 answer
362 views

Projective resolutions for commutative monoids

What is the right notion of a projective resolution of a commutative monoid? The category Mon of commutative monoids has plenty of projective (and even free) objects. Indeed, for every set $X$ one ...
Hannes Thiel's user avatar
  • 3,305
5 votes
1 answer
2k views

cohomology version of Cartan-Leray spectral sequence that deduces cup product

On page 338, A User's Guide to Spectral Sequences. 2nd Edition, by John McCleary, Theorem 8.9, there is a Cartan-Leray spectral sequence for homology: If $X$ is a connected pace on which the group $\...
Shiquan Ren's user avatar
  • 1,970
5 votes
1 answer
159 views

Relative flasqueness?

It is known that a flasque sheaf on a topological space has trivial cohomology. Suppose that we are in a relative situation of a smooth fibration $\pi: X \to S$ and $F$ is a sheaf on $X$. Is there ...
Vladimir Baranovsky's user avatar
5 votes
1 answer
192 views

Localizations of hereditary rings

It is known that if a commutative Noetherian ring $R$ is hereditary then for any maximal ideal $M$ the localization $R_M$ is also hereditary. Is the Noetherian assumption necessary?
M92's user avatar
  • 357
5 votes
3 answers
1k views

Künneth formula for Ext groups

Setup: Let $X,Y$ be quasi-compact quasi-separated schemes defined over a field $k$. If necessary, you can also assume that $X,Y$ are noetherian, but I don't want to assume that $X,Y$ have the ...
Martin Brandenburg's user avatar
5 votes
1 answer
381 views

Deforming ample line bundles vs cohomology group

Let X be a smooth projective variety over the complex numbers, of dimension at least two. $D$ is an ample divisor on X. Then we know for $m>>0$, $H^i(mD)=0$. Now suppose $E$ is another divisor ...
Ying Zhang's user avatar
  • 1,150
5 votes
1 answer
1k views

Explicit injective resolutions of (Laurent) polynomial rings

Hi, Despite being nothing but the dual notion of projective resolution, injective resolutions seem to be harder to grasp. For example, the general form of Poincaré-Lefschetz duality given in Iversen'...
Maxime Bourrigan's user avatar

1
17 18
19
20 21
53