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(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.

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79 views

Are dualizable objects in the derived category of a ringed topos perfect?

Recall that an object $a$ in a symmetric monoidal category $(\mathcal{C}, \otimes, e)$ is dualizable if there exists an object $b$ and morphisms $\varepsilon\colon b \otimes a \to e$ and $\eta\colon e ...
4
votes
2answers
265 views

When is $\Omega^1$ an equivalence?

Let $C$ be an abelian category with enough projectives and $\underline{C}$ the stable category of $C$ that is obtained by factoring out projective modules. When is the functor $\Omega^1 : \underline{...
5
votes
0answers
117 views

Homotopy functor calculus vs functor calculus in additive categories

Consider the Goodwillie calculus of a homotopy functor $F : \mathrm{Sp} \to \mathrm{Sp}$, where $\mathrm{Sp}$ denotes an appropriate model for spectra (... orthogonal spectra for instance). Then ...
5
votes
0answers
40 views

Cluster-tilting object for a local non-selfinjective algebra

Let $A$ be a non-selfinjective (which is equivalent to non-Gorenstein) local finite dimensional algebra. Is there a known example of such an $A$ having a cluster-tilting object? Id be surprised to ...
3
votes
0answers
56 views

conditions for algebra cohomology finiteness

Let $A=\bigoplus\limits_{n\geq 0}A_n$ be a graded (unital associative) ring, say an algebra over a field with finite dimensional graded components and $A_0$ semisimple. Are there reasonable conditions ...
2
votes
0answers
31 views

Projective dimensions of the terms in a minimal injective resolution of the regular module

Let $A$ be a finite dimensional algebra with finite global dimension and with minimal injective coresolution $I_i$ of the regular module $A$. The study of the projective dimensions of the $I_i$ is an ...
4
votes
0answers
81 views

Number of hereditary modules of a hereditary algebra

Let $Q$ always denote a Dynkin quiver. Given a connected path algebra $A=kQ$ and a module $M$, is there a useful criterion on $M$ when $End_A(M)$ is again a connected quiver algebra? Call a module ...
3
votes
0answers
55 views

On ideals in Noetherian rings, isomorphic to the trace of some finitely generated module

Let $R$ be a Noetherian ring. For a finitely generated $R$-module $M$, let $tr_R(M):=Im(\tau_M)$, where $\tau_M:M\otimes Hom(M,R)\to R$ is the map defined as $\tau_M(m\otimes f)=f(m)$. Let $I$ be a ...
3
votes
1answer
99 views

Bockstein homomorphism and Square Operations: Their consistency formulas

Here are various ways to define "Bockstein homomorphism:" Let $\beta_p:H^*(-,\mathbb{Z}_p) \to H^{*+1}(-,\mathbb{Z}_p)$ be the Bockstein homomorphism associated to the extension $$\mathbb{Z}_p\to\...
2
votes
0answers
87 views

Pontryagin square on spin and non-spin manifold

The Pontryagin square, maps $x \in H^2({B}^2\mathbb{Z}_2,\mathbb{Z}_2)$ to $ \mathcal{P}(x) \in H^4({B}^2\mathbb{Z}_2,\mathbb{Z}_4)$. Precisely, $$ \mathcal{P}(x)= x \cup x+ x \cup_1 2 Sq^1 x. $$ ...
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vote
0answers
57 views

Reference request for Leibniz rule and spectral sequences

Suppose $A_*$,$B_*$, and $C_*$ are chain complexes equipped with filtrations and a map $m:A_* \otimes B_* \to C_*$ respecting these filtrations. I am looking for a reference for the fact that the map $...
5
votes
1answer
60 views

Derived equivalences of Artin algebras with finitistic dimension zero

Let $A$ be an Artin algebra of finitistic dimension zero and $B$ an algebra derived equivalent to $A$. Does $B$ also have finitistic dimension zero? In case this is true, this might generalise the ...
1
vote
0answers
24 views

Piecewise hereditary algebras of Dynkin type that are QF-3

Is there an easy classification of piecewise hereditary (which means derived equivalent to a hereditary algebra) algebras of Dynkin type that are Quasi-Frobenius-3 (meaning that the injective envelope ...
4
votes
0answers
59 views

Sum of all projective dimensions of simple modules

Let $X_{n,t}$ be the set of all finite dimensional algebras (we can assume they are given by a connected quiver and admissible relations) that have global dimension equal to $n$ and $t$ simple modules....
4
votes
0answers
85 views

Koszul duality between QLS algebras and cdg algebras

A Quadratic-Linear-Constant (QLC) algebra $U$ is an algebra which can be written as $T(V)/P$ where $T(V) = \bigoplus_{n \in \mathbb{N}} V^{\otimes n}$ is the tensor algebra and $P \subseteq k \oplus ...
4
votes
0answers
114 views

Question on syzygies

Given a finite dimensional algebra $A$ over a field $K$ with $\Omega^i(D(A)) \cong \Omega^{i+1}(D(A))$ for some $i \geq 1$, where $D(A)=\operatorname{Hom}_K(A,K)$. Do we then also have $\Omega^{-i}(A)...
3
votes
0answers
124 views

Edge map in derived categories

Let $\mathscr{A},\mathscr{B}$ be abelian categories, the first with enough projectives, together with a right-exact functor $F\colon \mathscr{A}\to\mathscr{B}$ (in my example, it is a tensor product, ...
9
votes
0answers
168 views

Continuous cohomology of a profinite group is not a delta functor

Let $G$ be a profinite group, then there is a general notion of continuous cohomology groups $H^n_{\text{cont}}(G, M)$ for any topological $G$-module $M$ (I require topological $G$-modules to be ...
0
votes
1answer
86 views

Valuation ring satisfying either a.c.c. or d.c.c. on prime ideals

If a commutative ring with unity has finite Krull dimension, then it satisfies a.c.c. and d.c.c. on prime ideals. The converse is not true in general, as can be seen from here An infinite dimensional ...
4
votes
1answer
154 views

Algebras: Homology vs. Resolution as a dg-algebra

My question is what is the relation (if any) between the following two notions. Starting from an augmented algebra $A$ over a field $k$, one way to compute the homology of $A$ is to find a projective ...
3
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0answers
58 views

Can we abelianize quasi-isomorphisms of dga?

Suppose that $A \leftarrow X_1 \to \dots \leftarrow X_m \to B$ is quiso of differential graded algebras, and $A, B$ happen to be (graded) commutative. Can we find commutative $Y_1, \dots, Y_n$ ...
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vote
2answers
104 views

Why is the flat cotorsion pair actually a cotorsion pair?

I asked this question some while ago on Stack Exchange but didn't get an answer (link), so I am trying it here as well. Fix a ringed space $(X,\mathcal{O})$ and denote by $\mathcal{F}$ the class of ...
4
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0answers
89 views

Is there a simple algebraic setup to accomodate fibres and cofibres at the same time?

If I understand it correctly, there are two mutually dual "leading principles" in homotopy theory: never perform quotients, add structure instead; never require subobjects, take fibres instead. ...
5
votes
0answers
112 views

Tannakian theory for Lie algebras

Let $G$ be a reductive (just in case) linear algebraic group over $\mathbb{C}$ and let $\mathfrak{g}$ be the Lie algebra of $G$. Consider the category $\operatorname{Rep}(G)$ of finite dimensional ...
0
votes
2answers
140 views

Noetherian module, over Noetherian ring, which is isomorphic to its double dual [duplicate]

Let $M$ be a finitely generated module over a Noetherian ring $R$ such that $M$ is isomorphic with its double dual $M^{**}=Hom(Hom(M,R),R)$. Then is the natural map $j:M \to M^{**}$ defined as $j(m)(...
3
votes
0answers
91 views

Can a acyclic quiver algebra be derived equivalent to a non-acyclic quiver algebra?

Can a quiver algebra with acyclic quiver be derived equivalent to a quiver algebra with non-acyclic quiver? (I moved this question from another thread Derived equivalences of Dyck paths , where the ...
1
vote
1answer
45 views

Decomposing semihereditary rings

Let $R$ be a commutative semihereditary ring (i.e. every finitely generated ideal of $R$ is projective https://en.wikipedia.org/wiki/Hereditary_ring). Then is $R$ a finite direct product of Prufer ...
2
votes
1answer
68 views

Decomposing Noetherian hereditary rings of Krull dimension $1$ into product of hereditary domains (i.e. Dedekind domains)

Let $R$ be a commutative Noetherian hereditary ring (https://en.wikipedia.org/wiki/Hereditary_ring) of Krull dimension $1$. Then is it true that $R$ is a finite direct product of Dedekind domains ?
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32 views

Connection between Coxeter-periodicity and periodicity of the trivial extension

Let $A$ be a finite dimensional quiver algebra with an acyclic quiver. There are two notions related to periodicity for an algebra. Call $A$ Coxeter-periodic in case the Coxeter matrix $\phi_A$ of $A$ ...
3
votes
0answers
80 views

Finite test for periodicity of a module

Let $A$ be a finite dimensional quiver algebra and $M$ a finite dimensional $A$-module. Assume we want to test whether $M$ is a periodic module, meaning that $\Omega^n(M) \cong M$ for some $n \geq 1$. ...
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vote
0answers
46 views

Rings with flat injective envelope and global dimension at most one

Is there a classification of rings $R$ with the following properties: -The injective envelope of $R$ is flat. -The global dimension of $R$ is at most one. In case $R$ is a finite dimensional ...
7
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0answers
137 views

Is $\text{DGA}^{-}$ a monoidal model category?

Let $\text{DGA}^{-}$ denote the category of non-positively graded differential graded algebras with differentials of degree $+1$. It is well-known that $\text{DGA}^{-}$ has a model structure with ...
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0answers
71 views

Derived equivalences and the Coxeter polynomial

Let $A$ be a quiver algebra with an acyclic quiver and primitive idempotents $e_i$. The Cartan matrix $C_A$ of $A$ is defined as the matrix with entries $dim(e_i A e_j)$ and the Coxeter matrix $\phi_A$...
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vote
0answers
25 views

A regular sequence in a quotient by a “half lattice” defined by a toric manifold

I am interested in some properties of polynomial algebras associated with smooth compact toric varieties. Recall that a toric manifold can be obtained as a quotient $$P^{-1}(p) / \mathbb{K}$$ by the ...
9
votes
1answer
268 views

Derived equivalences of Dyck paths

Call two Dyck paths $D_1$ and $D_2$ derived equivalent in case their corresponding Nakayama algebras are derived equivalent (The Dyck path of a Nakayama algebra with a linear quiver is just the top ...
8
votes
0answers
205 views

Dualizable objects in homotopy category of chain complexes

The proposition 1.9 from "Duality, Trace and Transfer" by Dold and Puppe states that: Given a commutative ring $R$, a chain complex of $R$-modules is strongly dualizable in $Ho(Ch(R))$, the homotopy ...
2
votes
0answers
137 views

Derived functors and derivatives

When looking at derived functors of a non-exact functor (e.g., ext, tor, sheaf cohomology groups) I am struck by their similarity to derivatives of a non-constant function in that they are both ...
2
votes
0answers
108 views

On finite dimensional commutative algebras and regular sequences

Let $\mathbb{C}[u]:= \mathbb{C}[u_1,...,u_n]$ the algebra of polynomial functions on $\mathbb{C}^n$. I consider two ideals $I$, and $J$, where $I$ is the ideal generated by the polynomials vanishing ...
5
votes
0answers
116 views

A Poincare dual spectral sequence for equivariant cohomology: extension to generalized cohomology?

This question is a follow-up to my previous question: "Rotated" version of the Atiyah-Hirzebruch spectral sequence In that question, I discussed two different spectral sequences for ...
2
votes
0answers
56 views

On cohomological algebras related to toric manifolds

I am interested in some cohomological algebras related to toric manifolds. We consider a toric manifold $M$ as a quotient $$M = P^{-1}(p) / \mathbb{K}, \quad P : \mathbb{C}^n \to \text{Lie}(\mathbb{K})...
3
votes
0answers
112 views

derived symmetric powers of an ideal

Let $R$ be the polynomial algebra in $n$ variables over a field $F$ of characteristic $0$. Let $m$ be the ideal of the origin: $m=(x_1,...,x_n)$. We have a canonical map $Lsym^k(m)\to m^k$ from the ...
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0answers
66 views

On the dimension of the cohomology of toric manifolds

Let $M$ be a toric manifold. I'm not sure what conditions on $M$ are required, but one can assume, if needed, that it is compact, smooth, etc. We consider $M$ as a quotient given by the momentum map $...
2
votes
0answers
70 views

On some finiteness properties of cohomological algebras of complex tori

Denote by $A := \mathbb{C}[u,u^{-1}]$, $u = (u_1,...,u_n)$, the algebra of polynomial functions on the complex torus $(\mathbb{C} \setminus \{ 0 \})^n$, which we consider as a $\mathbb{C}[u]$-module. ...
4
votes
1answer
175 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 "topological" ...
3
votes
0answers
174 views

Cell structure on $B\mathbb{G}$ and the bar resolution of $\mathbb{G}$

Consider $\mathbb{G}$, which can be viewed as a group, as well as a 2-group. (For example, given a short exact sequence $$ 1 \to BG_2 \to \mathbb{G} \to G_1 \to 1 $$ and the fiber sequence: $$ B^2G_2 ...
2
votes
0answers
161 views

Homological conjecture for finite dimensional algebras

In the theory of finite dimensional algebras there are many homological conjectures. When working over an algebraically closed field it is well known that any such algebra is Morita equivalent to a ...
10
votes
1answer
168 views

Is there a kind of Poincare duality for Borel equivariant cohomology?

Let $G$ be a finite (or discrete) group, $M$ a $d$-dimensional manifold with smooth $G$-action (I am interested in the case where the action is not free, so $M/G$ is not a manifold). For an Abelian ...
17
votes
2answers
387 views

Does Koszul duality between $Comm$ and $Lie$ imply the power series identity $\exp(\ln(1-z))-1 = -z$?

To a symmetric sequence $V_\bullet$ of vector spaces, associate the generating function $F_V(z) = \sum_n \frac{\dim(V_n)}{n!} z^n$. Then $$F_{Comm_\ast}(z) = \exp(z)-1 \qquad F_{Lie}(z) = \ln(1-z)$$ ...
4
votes
0answers
60 views

Question on $n$-regular modules

Let $A$ be finite dimensional connected algebra. A simple module $S$ is called $n$-regular in case $pd(S)=n$, $Ext_A^i(S,A)=0$ for $i=0,1,...,n-1$ and $Ext_A^n(S,A)$ being a simple $A$-left module. ...
2
votes
1answer
132 views

Extensions of lattices

Let $G$ be a finite group and $\phi:M\to N$ be a surjective homomorphism of $G$-lattices (i.e. finitely generated $\mathbb{Z}[G]$-modules, free as $\mathbb Z$-modules), with kernel $K$. For every $n\...