## Operads and Γ-Homology of Commutative Rings (1998)

Citations: | 2 - 0 self |

### BibTeX

@MISC{Robinson98operadsand,

author = {Alan Robinson and Sarah Whitehouse},

title = {Operads and Γ-Homology of Commutative Rings},

year = {1998}

}

### OpenURL

### Abstract

this paper, we construct and investigate the natural homology theory for coherently homotopy commutative dg-algebras, usually known as E1 -algebras. We call the theory \Gamma-homology for historical reasons (see, for instance, [3]). Since discrete commutative rings are E1 rings, we obtain by specialization a new homology theory for commutative rings. This special case is far from trivial. It has the following application in stable homotopy theory, which was our original motivation and which will be treated in a sequel to this paper. The obstructions to an E1 multiplicative structure on a spectrum lie (under mild hypotheses) in the \Gamma-cohomology of the corresponding dual Steenrod algebra, just as the obstructions to an A1 -structure lie in the Hochschild cohomology of that algebra [15]. The \Gamma-homology of a discrete commutative algebra B can be understood as a refinement of Harrison homology, which was originally defined as the homology of the quotient of the Hochschild complex by the subcomplex generated by nontrivial shuffle products. It is better defined as the homology of a related complex which one obtains by tensoring each term

### Citations

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Citation Context ...A\Omega B M ; H \Gamma 0 (B=A ; M) �� DerA (B; M) ; (2) H \Gamma 1 (B=A ; M) �� ExalcomA (B; M) . (Here ExalcomA (B; M ), the module of infinitesimal A-algebra extensions of B by M , is as def=-=ined in [11], 0 -=-IV x18.) Proof. (1) In the bicomplex (5.1), C \Gamma 0;0 =d 00 (C \Gamma 0;1 ) �� B\Omega M . The image of the horizontal differential d 0 : C \Gamma 1;0 ! C \Gamma 0;0 is spanned by the usual rel... |

234 |
Koszul Duality for Operads
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Citation Context ...ur bracket. Now we give the details of the proof of Theorem 6.1. The first ingredient is the following co-operad structure (which is closely related to a co-operad discussed by Ginzburg and Kapranov (=-=[10]-=-, x3.5)). 6.2 Lemma. The chain complexes f e Cs(T U 0 )[\Gamma2] ; U 0 2 S 1 g form a co-operad. Proof. We define ` V;W : e C \Gamma2 (T U 0 ) ! e C \Gamma2 (T V 01)\Omega e C \Gamma2 (T W 0 ), for U ... |

229 |
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(Show Context)
Citation Context ...\Gamma; \Gamma] : H \Gamma l (B=A ; B)\Omega H \Gamma m (B=A ; B) ! H \Gamma l+m (B=A ; B) : We begin by explaining the idea of the construction, which mimics the Lie bracket in Hochschild cohomology =-=[6]-=-. We recall that this is defined as a graded commutator of circle products, where the circle product f ffi g is an alternating sum over i of `substitution of g into f in the i-th place'. As in x5, rea... |

193 |
Homotopy invariant algebraic structures on topological spaces
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(Show Context)
Citation Context ...a n action, there is a corresponding operad in which the objects are the complexes of cyclically-labelled trees in the plane: it is the analogue of the topological operad of Stasheff polyhedra -- see =-=[5]-=-.) By combining the two constructions we obtain a cofibrant E1 cyclic operad, the tree operad T , as follows. We take T S to be the chain (bi)complex associated with the bisimplicial set in which a (k... |

136 |
Categories and cohomology theories, Topology 13
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Citation Context ...C V 0\Omega B\Omega V\Omega M , where V runs through the category S+ . (There is also a cyclic version.) Conceptually it resembles the realization of a simplicial object, or the analogue described in =-=[18]-=-. Because the realization sometimes has to be applied to species other than the standard B\Omega V\Omega M , it is worthwhile to formulate a definition of the kind of general object which can be reali... |

62 |
On the (co-)homology of commutative rings
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Citation Context ... and M a B-module. (Although very different in appearance and in construction, this will play in our theory the role analogous to that played in Andr'e-Quillen theory by the cotangent complex of [1], =-=[14]-=-). Our cotangent complex will be a filtered object obtained by glueing together the objects C V 0\Omega B\Omega V\Omega M , where V runs through the category S+ . (There is also a cyclic version.) Con... |

54 | André-Quillen cohomology of commutative S-algebras
- Basterra
- 1999
(Show Context)
Citation Context ...ry lecture by the first author to the Adams Memorial meeting in 1990. Some details subsequently appeared in the second author's thesis [19] and a preprint [16]. Meanwhile Kriz [13] and later Basterra =-=[4]-=- were developing, by methods very different from ours, an E1 cohomology theory for ring spectra, which is extremely likely to be equivalent to ours. The paper is organised as follows. Section 1 contai... |

35 |
A Hodge-type decomposition for commutative algebra cohomology
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Citation Context ... f ]. So the bracket is graded anti-commutative in cohomology. This completes the proof of Theorem 6.1.sRemarks. The bracket described above is compatible with the Lie product in Harrison cohomology (=-=[7]-=-, x5.7). For 0-cocycles it is simply the usual bracket of derivations. OPERADS AND \Gamma-HOMOLOGY OF COMMUTATIVE RINGS 25 If n is odd or the characteristic of B is 2, then the circle product g 7! g f... |

28 |
Obstruction theory and the strict associativity of Morava K-theories
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- 1988
(Show Context)
Citation Context ...spectrum lie (under mild hypotheses) in the \Gamma-cohomology of the corresponding dual Steenrod algebra, just as the obstructions to an A1 -structure lie in the Hochschild cohomology of that algebra =-=[15]-=-. The \Gamma-homology of a discrete commutative algebra B can be understood as a refinement of Harrison homology, which was originally defined as the homology of the quotient of the Hochschild complex... |

26 |
Methode Simpliciale en Algebre Homologique et Algebre Commutative
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(Show Context)
Citation Context ...ebra, and M a B-module. (Although very different in appearance and in construction, this will play in our theory the role analogous to that played in Andr'e-Quillen theory by the cotangent complex of =-=[1]-=-, [14]). Our cotangent complex will be a filtered object obtained by glueing together the objects C V 0\Omega B\Omega V\Omega M , where V runs through the category S+ . (There is also a cyclic version... |

24 |
Commutative algebras and cohomology
- Harrison
- 1962
(Show Context)
Citation Context ...\Gamma p+q\Gamma1 (B=A ; M) : 20 ALAN ROBINSON AND SARAH WHITEHOUSE 5.4 Theorem [19]. The edge q = 0 of the spectral sequence above is precisely the complex used in defining the Harrison (co)homology =-=[12]-=- Harrs(B=A ; M) of B (with a shift in degree). Therefore there are natural transformations H \Gamma p\Gamma1 (B=A ; M) ! Harr p (B=A ; M) ; H \Gamma p\Gamma1 (B=A ; M) / Harr p (B=A ; M) when B is fla... |

21 |
C–structures I: A free group functor for stable homotopy theory, Topology 13
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(Show Context)
Citation Context ...d investigate the natural homology theory for coherently homotopy commutative dg-algebras, usually known as E1 -algebras. We call the theory \Gamma-homology for historical reasons (see, for instance, =-=[3]-=-). Since discrete commutative rings are E1 rings, we obtain by specialization a new homology theory for commutative rings. This special case is far from trivial. It has the following application in st... |

19 |
homology, Hochschild homology and triples
- Barr, Harrison
- 1968
(Show Context)
Citation Context ...and reveals a non-trivial differential in the spectral sequence of 5.2. 5.7 Example. First take B = A = F 2 . Then 1\Omega 1\Omega 1\Omega 1 is a non-bounding Harrison 4-cycle, by the calculation in (=-=[2], x4). T-=-hus Harr 4 (F 2 =F 2 ; F 2 ) 6�� 0, and by 5.4 our element 1\Omega 1\Omega 1\Omega 1 exists in E 2 3;0 . Since 3.7 or the transitivity theorem 3.4 implies that H \Gamma 3 (F 2 =F 2 ; F 2 ) �� ... |

12 |
Γ-(Co)homology of commutative algebras and some related representations of the symmetric groups,” Doctoral dissertation
- Whitehouse
- 1994
(Show Context)
Citation Context ...g the 1980's, and outlined in various lectures, including a plenary lecture by the first author to the Adams Memorial meeting in 1990. Some details subsequently appeared in the second author's thesis =-=[19]-=- and a preprint [16]. Meanwhile Kriz [13] and later Basterra [4] were developing, by methods very different from ours, an E1 cohomology theory for ring spectra, which is extremely likely to be equival... |

7 | The tree representation of \Sigma n+1
- Robinson, Whitehouse
- 1996
(Show Context)
Citation Context ...not do: the faces of D S intersect in unacceptably large subcomplexes, so that @D S ! D S is not injective. On the other hand, we can form another cyclic operad by taking E S to be the space of trees =-=[17]-=- with ends labelled by the set S, and ffi s;t to be the operation of grafting the end labelled s to the end labelled t to produce a new edge of length 1 . This operad has every E S contractible, and i... |

2 |
Modular operads, Max-Planck-Institut preprint MPI-94-78
- Getzler, Kapranov
(Show Context)
Citation Context ...ly well, of course, have chosen simplicial modules.) Our principal definitions use Getzler and Kapranov's theory [8] of cyclic operads, but we require Markl's non-unital version which is described in =-=[9]-=-. 1.1 Operads. Let S denote the category of finite sets S and isomorphisms of sets, S+ the subcategory of non-empty sets, and S 1 the category of based finite sets and isomorphisms. (To avoid foundati... |

2 |
Towers of E1 ring spectra with an application to BP
- Kriz
- 1993
(Show Context)
Citation Context ...tures, including a plenary lecture by the first author to the Adams Memorial meeting in 1990. Some details subsequently appeared in the second author's thesis [19] and a preprint [16]. Meanwhile Kriz =-=[13]-=- and later Basterra [4] were developing, by methods very different from ours, an E1 cohomology theory for ring spectra, which is extremely likely to be equivalent to ours. The paper is organised as fo... |

2 |
Gamma-homology of commutative ring and of E1 ring spectra
- Robinson, Whitehouse
- 1996
(Show Context)
Citation Context ...tlined in various lectures, including a plenary lecture by the first author to the Adams Memorial meeting in 1990. Some details subsequently appeared in the second author's thesis [19] and a preprint =-=[16]-=-. Meanwhile Kriz [13] and later Basterra [4] were developing, by methods very different from ours, an E1 cohomology theory for ring spectra, which is extremely likely to be equivalent to ours. The pap... |

1 |
Cyclic operads and cyclic homology, MSRI preprint
- Getzler, Kapranov
(Show Context)
Citation Context ...egory of chain complexes (dg-modules) over a commutative ground ring K. (We might equally well, of course, have chosen simplicial modules.) Our principal definitions use Getzler and Kapranov's theory =-=[8]-=- of cyclic operads, but we require Markl's non-unital version which is described in [9]. 1.1 Operads. Let S denote the category of finite sets S and isomorphisms of sets, S+ the subcategory of non-emp... |