(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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43 views

Tor functor in the case of algebra of smooth functions

Let $A=C^{\infty}(\mathbb{S}^{1})$, let $B$ be the sub-algebra $C^{\infty}(0,1)$. Here we identify $\mathbb{S}^{1}$ by $\mathbb{R}/2\pi \mathbb{Z}$. I want to ask if there is a way I can decompose $A$ ...
8
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338 views
+100

Local proof of Grothendieck-Riemann-Roch theorem

There is a theorem by Feigin and Tsygan(Theorem 1.3.3 here) which they call "Riemann-Roch" theorem. Given a smooth morphism $f:S\to N$ of relative dimension $n$ and a vector bundle $E/S$ of rank $k$ ...
9
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0answers
136 views

Finite dimensionality of Ext(M,N)

Let $K$ be a field of characteristic $0$ and let $R$ be a (noncommutative) Noetherian $K$-algebra. Let $M$ and $N$ be simple left $R$-modules and assume further the following conditions on $R$: ...
1
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0answers
17 views

Reference needed for exact sequence of ACM curves with homogeneous coordinate ring.

Let $C\subset\Bbb{P}^n$ be an ACM (arithmetically Cohen-Macaulay) curve with homogeneous coordinate ring $R$. Then there is an exact sequence $$0\to \text{Tor}_i(R,\Bbb{C})_k\to ...
4
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80 views

Auslander-Reiten theory for Gorenstein algebras

In the paper "Cohen-Macauley and Gorenstein artin algebras", Auslander and Reiten have a short section about Auslander-Reiten theory in Gorenstein algebras (I always assume we have an artin algebra ...
3
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0answers
85 views

Universal enveloping algebra functor preserves quasi-isomorphism

Let $k$ be a field of characteristic 0. Let $\mathtt{DGA}_{k}$ denote the category of DG algebras and $\mathtt{DGLA}_{k}$ denote the category of DG Lie algebras. It is well known that there are model ...
3
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65 views

Isomorphisms between extension group and $\mathfrak{u}$-cohomology

Let $\mathcal{O}^{\mathfrak{p}}$ be a parabolic subcategory (see, Humphrey's BGG's Ch.9) of the BGG category $\mathcal{O}$ (w.r.t the Cartan $\mathfrak{h}$ and Borel $\mathfrak{b}$) over a ...
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74 views

Right split for homomorphism onto $S_\infty$

Let $G$ be a closed subgroup of $S_\infty$ and let $f:G\rightarrow S_\infty$ be a continuous surjective homomorphism. Under which conditions $f$ has a right split, i.e. there exists some ...
5
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1answer
289 views

Known results in the Cohomology of finite groups

I am learning to compute cohomology of finite groups and came across this survey article http://www.ams.org/notices/199707/adem.pdf "Recent Developments in the cohomology of finite groups" by ...
6
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2answers
319 views

Socle of tilting modules in the BGG category $\mathcal{O}$ over a semisimple Lie algebra

Suppose that $\mathfrak{g}$ is a finite dimensional, complex, semisimple Lie algebra. Let $\mathcal{O}$ be the BGG category over $\mathfrak{g}$. Tilting module theory play an important role in the ...
8
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1answer
414 views

Cohomology theory “from” Grothendieck's six operations?

How, precisely, (as suggested by grothendieck) cohomology theory naturally follows from the Grothendieck's six operations associated to the category derived from the topos? I would like some ...
3
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42 views

Fracturing $t$-structures

$\def\tee{\mathfrak{t}}$ Let $\tee_1,\tee_2$ be two $t$-structures on a triangulated category $\cal T$; call them fracturing if the two fiber sequences $\tau^\le_1X\to X\to \tau^\ge_1X$ and ...
3
votes
1answer
81 views

t-structure induced on the Verdier quotient ${\cal T}/\cal S$

Let $\mathfrak t$ be a $t$-structure on a triangulated category $\cal T$. Let $\cal S$ be a thick (or even non-thick) triangulated subcategory, and ${\cal T}/\cal S$ the Verdier quotient. Is there a ...
4
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1answer
77 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 ...
7
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1answer
157 views

If a faithfully flat extension of dg/A_$\infty$-algebra is formal, is the original algebra formal (over positive characteristic)?

Proposition 6.2 of Formality of DG algebras (after Kaledin) by Lunts reads (with a few additions to clarify notation): Let $k$ be a field of characteristic 0. Let $A$ be an $A_\infty$ algebra ...
24
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2answers
521 views

Complete the following sequence: point, triangle, octahedron, . . . in a dg-category

Let $\mathcal C$ be a pre-triangulated dg-category (or a stable $\infty$-category, if you wish). An object $X$ in $\mathcal C$ gives a "point": $$X$$ A morphism $X\xrightarrow f Y$ in $\mathcal C$ ...
1
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0answers
85 views

Example of non-locally finite stability condition

I am trying to work out example 5.6 in Bridgeland's paper "Stability conditions on triangulated categories". http://annals.math.princeton.edu/wp-content/uploads/annals-v166-n2-p01.pdf A standard ...
1
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0answers
50 views

Complex integration over 1-singular chain

Let $f$ be a continuous function on $\mathbb{C}$ and assume that $\lim_{z\to \infty} zf(z) = \lambda.$ Let us note for all natural $n$ $$C_n = \{z \in \mathbb{C} : |z|=n\}.$$ Then, a usual fact of ...
6
votes
1answer
317 views

Why are quasi-isomorphisms of homotopy algebras only defined for arity 1?

When reading about homotopy algebras (e.g. $L_\infty$-algebras, $A_\infty$-algebras), an $\infty$-morphism $f$ is called an $\infty$-quasi-isomorphism if $f_1$ is a quasi-isomorphism. Recall/Example ...
0
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1answer
103 views

Projective resolutions of torsion modules [closed]

Let $l$ be a prime number, $n\in \mathbb{Z}$. Is it true that any finitely generated $\mathbb{Z}/l^n\mathbb{Z}$-module has a finite (left) resolution by free finitely generated ...
2
votes
1answer
167 views

Third (co-) homology of Cyclic groups

Is there a general simple theorem for the third cohomology of cyclic groups $H_3(\mathbb{Z}_n, U(1))= ?$. In particular, I am interested in finding $H_3(\mathbb{Z}_8, U(1))$. I know the answer can be ...
1
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0answers
27 views

Is an weakly finite R-module Serre subcategory of the category of R-modules?

A definition for weakly finite $R$-modules is as follow: Definition: Let ($R$,$m$) a local ring. Let $S$ be the largest class of $R$-modules satisfying the following four properties: (1) If $M \in ...
0
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1answer
114 views

When Hom(M,E) is 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 $Hom(M,E)$ is injective? Thanks.
1
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56 views

The projective resolution of a direct summand

For an $R$-module $M$ fix a projective resolution $P^\bullet\to M$. If $N$ is a direct summand of $M$, that is there is $L$ such that $M=N\oplus L$, then is there a projective resolution of $N$ which ...
20
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290 views

Does the Tate construction (defined with direct sums) have a derived interpretation?

Any abelian group M with an action of a finite group $G$ has a Tate cohomology object $\hat H(G;M)$ in the derived category of chain complexes. There are several ways to define this. One is as the ...
9
votes
0answers
187 views

Cohomology of central extensions of groups

Let G be a central extension of a finite group H by $Z/2$. I need an explicit description of the differentials $d_2$ and $d_3$ in the Lyndon-Hochschild-Serre spectral sequence which converges to the ...
3
votes
0answers
193 views

N-periodic derived categories

I have some seemingly basic questions about $N$-periodic derived categories to which I have not found answers in any of the usual places. Let $R$ be a ring, and let $D(R)_{\mathbb Z/N\mathbb Z}$ ...
2
votes
0answers
79 views

Lie algebra (co)homology of the Lie algebra of differential operators

Let $X$ be an algebraic variety (or a complex manifold) over $\mathbb C$. Let $D(X)$ be its algebra of differential operators. Mostly I am interested in algebraic differential operators, but the case ...
2
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147 views

Could Partial Tiltings be studied as Almost Complete Tiltings?

The first part of what follows is a brief recap of the definitions, setting and motivations for my questions. Experts can find the questions at the end. Here $k$ denotes an algebraically closed ...
3
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1answer
130 views

On the ordered set of real numbers, does sheaf+cosheaf imply constant?

I have a technical question about unbounded chain complexes. I couldn't think of a descriptive title for it. Let $P$ be a chain complex of contravariant functors on $\mathbf{R}$ (the real numbers ...
16
votes
1answer
380 views

A spectral sequence for computing cohomology of a space from that of its strata

Let $X$ be a smooth complex variety (not necessarily compact) and let $D$ be a normal crossings divisors with components $D_1$, $D_2$, ..., $D_N$. For a set of indices $I$, let $D_I = \bigcap_{i \in ...
0
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0answers
40 views

Equivalence between recollements

Suppose you have a pair of recollements $$ \mathcal{A}' \stackrel{\overset{i^*}{\longleftarrow}}{\stackrel{\overset{i_*}{\longrightarrow}}{\underset{i^!}{\longleftarrow}}} ...
4
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1answer
82 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 $X$ ...
4
votes
1answer
80 views

Filtered Colimit of associative $k$-algebras that are domains

Let $C$ be a filtered subcategory of the category of commutative algebras over a fixed field $k$ whose objects are all integral domains. Then the colimit of the obvious diagram is an integral domain. ...
5
votes
1answer
198 views

a question about Bockstein spectral sequence

I find the following theorem for Bockstein spectral sequence at http://pages.vassar.edu/mccleary/files/2011/04/MC10.fin_.pdf, page 459: Question. for a fixed $k$, if $\beta$ does not hit ...
9
votes
1answer
221 views

A general version of the 5 lemma

Suppose you have an abelian category $\bf A$, and $A\to B\to C$, $A'\to B'\to C'$ two exact sequences, in a diagram $$ \begin{array}{cccccccc} 0 &\to & A &\to& B &\to& C ...
5
votes
1answer
316 views

A Hom-Tensor identity - $\text{Hom}_{R}(P,B)\otimes _SC \cong \text{Hom}_{R}(P,B \otimes_S C) $

let $R,S$ be associative algebras over $\mathbb{C}$. Let $\mathcal{C} \subseteq$ $R$-Mod be a full abelain subcategory of $R$-Mod which is the category of $R$-modules. Let $B$ and $C$ be, a ...
4
votes
2answers
188 views

Tensor product of monomorphisms is a monomorphism?

Given a commutative ring $k$ and for $i = 1,2$ a homomorphism of $k$-modules $X_i \overset {f_i} \longrightarrow Y_i$ with $X_i$ flat over $k$. Is the following conclusion true for general $k$? If ...
6
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180 views

Is there an obstruction which classifies “quasi-isomorphism but not chain equivalence”?

Fix a ring $R$ and let $C_\bullet$, $D_\bullet$ be (possibly unbounded) chain complexes of $R$-modules. Assume that $f_\bullet:C_\bullet \to D_\bullet$ is a quasi-isomorphism: that is to say, $f$ is a ...
5
votes
0answers
143 views

Switching left and right adjoints in recollement situations

In Fascieaux pervers, Beilinson, Bernstein and Deligne define a recollement situation as a triple of triangulated categories $\mathcal{D}_U,\mathcal{D}_F$ and $\mathcal{D}$, together with functors $$ ...
4
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0answers
171 views

Compactly supported distributions as a projective G-module

For a Lie group $G$ and a locally convex space $V$ let $\mathcal{E}(G,V)$ be the locally convex space of smooth functions from $G$ to $V$, and accordingly $\mathcal{E}_c^\prime(G,V)$ the space of ...
7
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1answer
240 views

Cap product on Leray-Serre spectral sequences

Let $\ p:X\to B\ $ be a fibration of spaces with fiber $F$. Then there are homological and cohomological Leray-Serre spectral sequences $E^{r}_{pq}$ and $E_r^{pq}$ that converge to $H_*(X)$ and ...
7
votes
1answer
253 views

Weight filtration on certain Galois representations

Let $G$ be the absolute Galois group of a number field $K$. Let $\ell$ be a prime number. There are representations $\mathbb{Z}_\ell(n)$ of $G$ on the group of $\ell$-adic integers given by the ...
3
votes
2answers
185 views

Definition of the differential of the Cone of a morphism of complexes [closed]

Let $(F^\bullet,d_F)$ and $(G^\bullet,d_G)$ be two complexes in an abelian category $\mathbf{A}$. The complex cone $Cone(\varphi)^\bullet$ of a morphism of complexes $\varphi:F^\bullet \to G^\bullet$ ...
7
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0answers
73 views

Non-Standard Derived Equivalences of Non-Flat Algebras

I read that for algebras $R$ and $S$ (over a commutative ring), assuming that $R$ or $S$ is flat, the existence of a derived equivalence $\mathcal{D}(R) \to \mathcal{D}(S)$ implies the existence of an ...
3
votes
2answers
439 views

Definition of E-infinity operad

What is the definition of $E_\infty$-operad in the category of chain complexes over $\mathbb{Z}/p\mathbb{Z}$? J. Smith in http://arxiv.org/abs/math/0004003 define it for complexes over $\mathbb{Z}$ ...
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86 views

Differentially closed fields

Let F be a field. Recall that an additive map $d: F\rightarrow F$ is said to be a derivation if $d(ab)=ad(b)+d(a)b$. Now let $F$ be a ring and let $d$ be a derivation of $F$. Examples I have in mind ...
2
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117 views

Reference request for a “truncated version” of the de Rham algebra

Let's start on the $n$-torus for sake of simplicity.$\newcommand{\T}{\mathbb T}$ If I understand the relevant definitions correctly, the usual de Rham algebra of smooth differential forms on $\T^n$ is ...
5
votes
2answers
169 views

Do $RHom(C,D)$ and $DG(C,D)$ have equivalent homotopy categories?

Toen in The homotopy theory of dg-categories and derived Morita theory Section 6 introduced the internal Hom's between dg-categories. Actually for two dg-categories $C$ and $D$, Toen defined $$ ...
6
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274 views

Transgression map spectral sequence of Ext

Let $X$ be a scheme over $k$ and $p:\ X \to Spec(k)$ the structure morphism. If $M$ is an étale sheaf of abelian groups over $Spec(k)$ I have a Grothendieck spectral sequence ...