## Simplicial Degrees Of Functors

Citations: | 2 - 1 self |

### BibTeX

@MISC{Pirashvili_simplicialdegrees,

author = {Teimuraz Pirashvili},

title = {Simplicial Degrees Of Functors},

year = {}

}

### OpenURL

### Abstract

this paper is to show that if G is a simplicial group of finite length, then H n G also has finite length. Here the length of a simplicial group means the length of the corresponding Moore normalization and H n G is a simplicial abelian group given by [k] 7! H n G k . A similar fact is true if we replace G by a simplicial ring and we take the algebraic K-functors instead of group homology. The origin of such results goes back to the classical paper of Dold and Puppe (see Hilfsatz 4.23 of [DP]), where the following was proved: let

### Citations

378 |
Homotopical algebra
- Quillen
- 1967
(Show Context)
Citation Context ...ion NGsis a chain complex of (nonabelian) groups defined by N n Gs= " 0i!n Ker @ i ; whose boundary operator is the restriction of @ n . According to Lemma 5 and Proposition 2 of section 3 of ChI=-=I of [QHA]-=-, the functor N n : S ! Groups is exact, for any ns0. 2.1. Definition. We say that length of Gsis n if N i Gs= 0, for i ? n and N n Gs6= 0. Let S (k) be the category of such simplicial groups whose le... |

303 |
P.: Simplicial Objects in Algebraic Topology
- May
- 1967
(Show Context)
Citation Context ...LICIAL GROUPS OF FINITE LENGTH Here we give the definition and some elementary facts about simplicial groups of finite lengths. We use the standard notation and terminology on simplicial objects (see =-=[M]). Le-=-t S be the category of simplicial groups. We recall that for a simplicial group Gs, the Moore normalization NGsis a chain complex of (nonabelian) groups defined by N n Gs= " 0i!n Ker @ i ; whose ... |

69 |
On the (co)-homology of commutative rings
- Quillen
- 1970
(Show Context)
Citation Context ...eover, if F 2 P n (C), then deg(F ) = Sdeg(F ): Proof. Let G 2 C. Consider S 1\Omega G, where S 1 is a pointed simplicial circle, with n+ 1 points in degree n and tensor product is pointed version of =-=[QCR]-=-. Clearly (S 1\Omega G) ab = K(G ab ; 1): So l(S 1\Omega G ab ) = 1 and hence l(F (S 1\Omega G))sSdegF: By definition N i F (S 1\Omega G) = F (G 1 j :::: j G i ) with G 1 = ::: = G i = G. Hence, l(F (... |

41 |
Relative algebraic K-theory and cyclic homology
- Goodwillie
- 1986
(Show Context)
Citation Context ... normalization, so sdeg(GL) = 1. Rationally Ks(R) is the primitive part of HsGL(R). Hence Theorem 5.1 and Proposition 4.3 yield sdeg(K n\Omega Q)sn. Based on Goodwillie's theorem on relative K-theory =-=[Go]-=- one easily shows that sdeg(K n\Omega Q) = n; ns1: Now we return to the integral case. Since sdeg(GL) = 1 we have sdeg(E) = 1 and sdeg(K 1 ) = 1 and the fundamental theorem of algebraic K-theory shows... |

24 |
On the groups H
- Eilenberg, MacLane
- 1954
(Show Context)
Citation Context ...23 of [DP]), where the following was proved: let T : A ! B be a functor between abelian categories of degree d, meaning that the (d + 1)-st cross-effect functor in the sense of Eilenberg and Mac Lane =-=[EM]-=- vanishes. Then l(T Xs)sdl(Xs) for any simplicial object Xsin A. Here l(Xs) denotes the length of Xs. Actually, this property characterizes the degree of functors in the abelian case (see Lemma 3.6). ... |

19 |
Quadratic endofunctors of the category of groups
- Baues, Pirashvili
- 1999
(Show Context)
Citation Context ...tes the length of Xs. Actually, this property characterizes the degree of functors in the abelian case (see Lemma 3.6). One can modify the notion of the cross-effects in the nonabelian framework (see =-=[BaP]-=-) and define the notion of degree of functors. One can also use the above inequality to define the simplicial degree in the nonabelian set-up. However in this way we get generally different invariants... |

13 |
Homology of 2-types
- Ellis
- 1992
(Show Context)
Citation Context ...We do not know whether the functor H k : Groups ! Ab; k ? 2 determines or not all H n : Groups ! Ab; n ? k. ii) Ellis considered homology of crossed modules, i.e. simplicial groups of length one (see =-=[E]-=-). Our Theorem 5.1 shows that for the spectral sequence of Corollary 3 of loc. cit., one has E 2 pq = 0 for p ? q. Using this fact T. Datuashvili in [D] extended some results of Ellis. 6. MORE EXAMPLE... |

7 |
Higher K-theory of rings
- Gersten
- 1973
(Show Context)
Citation Context ...y conclude that sdeg(K 3 )s6. However we do not know what is the exact value of sdeg(K 3 ). In order to get the information for higher K-theory we use a trick of Dror Farjoun (see Proposition 2.20 of =-=[Ge]-=-). Let X 3 be the space as defined in loc. cit. Recall that it fits in a fibration X 3 ,! B StR \Gamma! K(K 3 R; 3) and H 4 X 3 = K 4 (R). Using Serre spectral sequence we get an exact sequence Z=2Z\O... |

4 |
Homologie nicht-additiver Functoren
- Dold, Puppe
- 1961
(Show Context)
Citation Context ...f we replace Gsby a simplicial ring and we take the algebraic K-functors instead of group homology. The origin of such results goes back to the classical paper of Dold and Puppe (see Hilfsatz 4.23 of =-=[DP]-=-), where the following was proved: let T : A ! B be a functor between abelian categories of degree d, meaning that the (d + 1)-st cross-effect functor in the sense of Eilenberg and Mac Lane [EM] vanis... |

4 |
Koszul duality for operads
- Ginsburg, Kapranov
- 1994
(Show Context)
Citation Context ...y (= Andr'e-Quillen homology in characteristic 0). More generally, the simplicial degree of the n-dimensional homology of P-algebras is n, where P is Koszul operad over characteristic zero field (see =-=[GK]-=-). Less trivial situations arise, when one considers algebraic K-functors, the homology of groups or Andr'e-Quillen homology in positive charactreristic, because in the last cases, the chains have inf... |

4 |
Lane homology and topological Hochschild homology
- Mac
- 1992
(Show Context)
Citation Context ... Mult p+1 tor q (A; \Delta \Delta \Delta ; A) =) Shukla p+q (A): Hence, sdeg Shukla n = max 0pn sdegMult p+1 tor n\Gammap = n + 1 and the claim is proved. According to the spectral sequence 4.2(4) of =-=[PW]-=-, the same relation is true for Mac Lane homology. 6.4. Andr'e-Quillen homology. In what follows k denotes a commutative ring with unit. For a commutative k-algebra A, one can define Andr'e-Quillen ho... |

2 |
Twisted (co)homological stability for monoids of endomorphisms
- Betley, Pirashvili
- 1993
(Show Context)
Citation Context ...Gamma! F 2 T (c) \Gamma! 0: According to Proposition 4.3 ii) our Lemma is a consequence of the long exact sequence for homotopy groups. Proof of Proposition 6.5. We use the same method as was used in =-=[BP]-=-, by induction on d. If d = 1, then F is additive functor. For an object c 2 C, one puts Xs= T (c). One can use the universal coefficient theorem for the functor F and for the chain complex associated... |

2 |
Spectral sequence for epimorphism
- Pirashvili
- 1982
(Show Context)
Citation Context ...uivalence. Therefore one has a spectral sequence E 1 pq = H q (C p (f); M) =) H p+q (G 00 ; M); where M is a G 00 -module. For more about this spectral sequence see Corollary 5.6, Proposition 5.7 and =-=[P]-=-. iv) Let G be a group. Then for each ns0 there exists unique (up to isomorphism) simplicial group T (G; n), whose normalization is trivial in all dimensions except n and n + 1, N n T (G; n) = G; N n+... |

1 |
Modules crois'es g'en'eralis'ee de longuer 2
- Conduch'e
(Show Context)
Citation Context ...plicial degree. Our Conjecture 4.7 claims that this should be always the case. In the second section we deal with simplicial groups of finite length, where we essentially use the results of Conduch'e =-=[C]-=-. Based on [BaP], in the next section, we define the cross-effects and the notion of degree of functors in the nonabelian framework. In the section 4, we introduce simplicial degrees and give some ine... |

1 |
On homology of 2-types and crosed modules
- Datuashvili
(Show Context)
Citation Context ...ules, i.e. simplicial groups of length one (see [E]). Our Theorem 5.1 shows that for the spectral sequence of Corollary 3 of loc. cit., one has E 2 pq = 0 for p ? q. Using this fact T. Datuashvili in =-=[D]-=- extended some results of Ellis. 6. MORE EXAMPLES 6.1. Algebraic K-functors. Main observation of this section is the fact that algebraic K-functors, when considered as functors from the category of ri... |

1 |
On cental group extensions and homology
- Eckmann, Hilton, et al.
(Show Context)
Citation Context ...e 2.2 iii) one has a slightly longer exact sequence. One can check (see loc. cit.) that in this way we obtain an exact sequence, which for central extensions, gives the well-known exact sequence from =-=[EHS]-=-. Proposition 5.7 was also established in [P], but we reproduce it here because of the beautiful consequence: the functor H 2 : Groups ! Ab determines all higher homology. In order to give a more prec... |

1 |
Double relative K-theory and relative K 3
- Keune
(Show Context)
Citation Context ...plicial ring of length one. Thus length of StCs(f) is two and �� i StCs(f) = 0 for i ? 2. This fact using direct calculation with Steinberg relations was established before by Keune (see Lemma 5.3=-= of [K]-=-). 6.3. Shukla homology and Mac Lane homology. We recall that homology is a kind of derived Hochschild homology (see [QCR]). More precisely. Let k be a commutative ring with unit and A be an associati... |

1 |
Spaces with finitily many non-trivial homotopy groups
- Loday
(Show Context)
Citation Context ...old-Puppe equivalence, such object is unique up to isomorphism. By definition l(K(G; n)) = n. ii) The category of simplicial groups of length one is isomorphic to the category of crossed modules (see =-=[L]-=-). Similarly the category of simplicial groups of length two is isomorphic to the category of 2-crossed modules in the sense of Conduch'e (see [C]). iii) Let f : G ! G 00 be an epimorphism of groups. ... |