## Deformations via simplicial deformation complexes. arXiv math.AG/0311168 (2005)

Citations: | 12 - 1 self |

### BibTeX

@MISC{Pridham05deformationsvia,

author = {J. P. Pridham},

title = {Deformations via simplicial deformation complexes. arXiv math.AG/0311168},

year = {2005}

}

### OpenURL

### Abstract

### Citations

478 |
Categories for the working mathematician
- MacLane
- 1971
(Show Context)
Citation Context ...vant deformation problem. In particular, all operads give rise to monadic adjunctions. The frequent existence of such adjunctions, and their rôle in cohomology and homology, are well documented. (e.g.=-=[Mac71]-=- Ch.VII §6, and Notes at the end of Ch VI) It is thus unsurprising that monadic adjunctions should be useful in deformation theory, and they immediately enable the construction of SDCs for many deform... |

269 |
An introduction to homological algebra
- Weibel
- 1994
(Show Context)
Citation Context ...d FB⊗A = (B ⊗ A) ⊗A Symm A = B ⊗Λ Symm A. Let ⊤ = FU and ⊥ = UF. We have the unit and co-unit: and ηM : M = S 1 (M) ֒→ Symm AM → B ⊗Λ Symm AM = ⊤M, εR : ⊥R → b[r1 ⊗ ... ⊗ rn] ↦→ R b · r1 ...rn. As in =-=[Wei94]-=-, we may form the canonical augmented cosimplicial complex M �⊤M �� ��⊤2M � ⊤3 M · · ⊤4M ...... with face operators ⊤ i η⊤ n−i : ⊤ n M → ⊤ n+1 M and degeneracy operators ⊤ i UεF ⊤ n−i : ⊤ n+1 M → ⊤ n ... |

223 |
Rational homotopy theory
- Quillen
- 1969
(Show Context)
Citation Context ...functors. The reason for these problems is that differential graded objects are usually natural things to consider only in characteristic zero. This statement is implied by omission in papers such as =-=[Qui69]-=-, showing that DG objects suffice when considering homotopy theory over the rationals. In non-zero characteristic, it is necessary to use simplicial objects in their stead. This equivalence of DG and ... |

170 |
Complexe cotangent et déformations I
- Illusie
(Show Context)
Citation Context ...In particular, there is an isomorphism of cosimplicial complexes C • (E) ∼ = DerB0 (⊥•R0,R0) = HomR0 (LR0/B0 • ,R0), where L R0/B0 • is one presentation of the cotangent complex. Illusie ([Ill71] and =-=[Ill72]-=-) and André ([And67]) define the cotangent complex using the resolution arising from the adjunction between algebras and sets, whereas we have used the adjunction between flat algebras and flat module... |

155 |
Topologie algébrique et théorie des faisceaux
- Godement
- 1958
(Show Context)
Citation Context ...now in precisely the same situation as the previous section, and the SDC is u ∗ E n = HomΛ(N ,(u ∗ u∗) n (N ))α n, where αn = u∗ (α(u∗u∗) n−1M0 ◦ ... ◦ αM0 ). To see this more clearly, observe, as in =-=[God58]-=-, Remark II.4.3.2, that an element n ∈ Γ(U,u−1u∗N ) can be represented by a function of the form n(x0,x1) ∈ Nx1 , 12� � defined on a set of the form x0 ∈ U; x1 ∈ U(x0), where U(x0) is some open étale... |

155 |
Functors of Artin Rings
- Schlessinger
- 1968
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Citation Context ...CΛ to be the category of Artinian local Λ-algebras with residue field k We require that all functors on CΛ satisfy (H0) F(k) = •, the one-point set. We take the following definitions and results from =-=[Sch68]-=-: Definition A.1. For p : B → A in CΛ surjective, p is a small extension if ker p = (t), a principal ideal, such that mB(t) = (0). Note that any surjection can be be factorised as a composition of sma... |

112 | Deformations of algebras over operads and the Deligne conjecture
- Kontsevich, Soibelman
(Show Context)
Citation Context ... are constructed for geometric problems, the methods used are predominantly analytic. This proves inadequate for the more algebraic problems, such as deformations of an algebraic variety. However, in =-=[KS00]-=-, Kontsevich and Soibelman go some way towards constructing DGLAs for algebraic problems, by providing DGLAs to govern deformations over an operad. In non-zero characteristic, DGLAs do not, in general... |

60 |
On the (co-)homology of commutative rings
- Quillen
(Show Context)
Citation Context ...omotopic free complexes, it follows that Der(Q∗,k) → Der( ¯ N ∗ W ¯ N(Q)∗,k) gives an isomorphism on cohomotopy groups, i.e. π i (e ∗ ) ∼ = π i (e(L(E)) ∗ ), the proof being the same as the result in =-=[Qui70]-=- that the quasi-isomorphism class of the tangent complex is independent of the choice of cofibrant resolution. Hence E ∗ → E(L(E)) ∗ is a quasi-isomorphism of SDCs. Corollary 5.22. For any DGLA L • , ... |

52 |
Hopf Algebras, Mathematics Lecture Note Series
- Sweedler
- 1969
(Show Context)
Citation Context ...joint Functor Theorem (since A−FCoAlg has all colimits, and V preserves these). Note that in this case the free functor is hard to write down explicitly, but this is unnecessary for our purposes. See =-=[Swe69]-=- for such a description. By the Theorem 0.35, this adjunction is comonadic, so that deformations of C0 are given by the SDC E n = HomΛ(M, ⊥ n M) U(α⊤ n−1 C0 ◦...◦αC 0 ). 2.2 The general approach In ge... |

31 |
Deformation theory via differential graded Lie algebras, In: “Seminari di Geometria Algebrica 1998–1999”, Scuola Normale Superiore
- Manetti
- 1999
(Show Context)
Citation Context ...nctor if: 1. η is surjective whenever B ։ A. 2. η is an isomorphism whenever A = k. Note that a deformation functor satisfies conditions (H1) and (H2). The following results are proved by Manetti (in =-=[Man99]-=-): Theorem A.11. (Standard Smoothness Criterion) Given φ : F → G a morphism of deformation functors, with (V,ve) φ′ −→ (W,we) a compatible morphism of obstruction theories, if (V,ve) is complete, V φ′... |

25 |
Méthode simpliciale en algèbre homologique et algèbre commutative. Lecture
- André
(Show Context)
Citation Context ... is an isomorphism of cosimplicial complexes C • (E) ∼ = DerB0 (⊥•R0,R0) = HomR0 (LR0/B0 • ,R0), where L R0/B0 • is one presentation of the cotangent complex. Illusie ([Ill71] and [Ill72]) and André (=-=[And67]-=-) define the cotangent complex using the resolution arising from the adjunction between algebras and sets, whereas we have used the adjunction between flat algebras and flat modules. However, Quillen ... |

8 |
Techniques de descente cohomologique. In Théorie des topos et cohomologie étale des schémas
- Saint-Donat
- 1972
(Show Context)
Citation Context ...pected. In the Zariski case we may also use the Godement resolution, taking X ′ = ∐ x∈X x. Of course, if we work on topoi where the Godement or Čech resolutions are inappropriate, then the methods of =-=[SD72]-=- usually allow us to choose a different simplicial resolution of X which will work. 2 An Algorithm for Computing SDCs In this section we make formal the approach which has been used so far to compute ... |

7 |
Benson,“Representations and Cohomology I
- J
- 1991
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Citation Context ...eometric point ¯x → X, and a ring Λ as in Section 0.1, with finite residue field k. Throughout, A will denote a ring in CΛ. Denote π1(X, ¯x) by Γ. 394.1 Representations to GLn It is well known (e.g. =-=[Mil80]-=- Theorem 5.3), that there is an equivalence of categories between finite π1(X, ¯x)-sets, and locally constant étale sheaves on X, with finite stalks. There is thus an equivalence of categories between... |

6 | Topics in algebra — deformation theory. Lecture Notes, available at http://www.math.brown.edu/∼abrmovic/kontsdef.ps
- Kontsevich
- 1994
(Show Context)
Citation Context ...poid ❀ Deformation Functor. However, there are several respects in which DGLAs are not wholly satisfactory. Finding a DGLA to govern a given problem is not easy — Kontsevich describes it as an “art” (=-=[Kon94]-=-), so the arrow Deformation Problem ❀ DGLA above is merely aspirational. Kontsevich also remarks that, where DGLAs are constructed for geometric problems, the methods used are predominantly analytic. ... |

1 | Deformations via DGLAs in mixed characteristic - Pridham - 2003 |

1 |
The split pro-algebraic fundamental group of a compact Kähler manifold
- Pridham
- 2005
(Show Context)
Citation Context ... just the expected ring Λ[[T1,... ,Tm]]. Proof. This is very similar to the proof of Theorem 0.8. A full proof in the analogous case of nilpotent Lie algebras (rather than Artinian rings) is given in =-=[Pri05]-=- Theorem 1.24. Thus an SDC E∗ is equivalent to a system of smooth local complete pro-Artinian rings Q∗, with the dual structures to those described in Section 1.1. In particular, since we are only con... |