André-Quillen homology of commutative algebras

Iyengar, Srikanth B.

doi:10.1090/conm/436/08410

Cited by 26 publications

(9 citation statements)

References 22 publications

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“…What we must show is that the kernel is generated by a regular sequence; once this is so, the length of the regular sequence must be dimD 1 W pkq pR, kq " b 1 . For this, see [18,Theorem 8.5] and its proof in particular.…”

Section: 41mentioning

confidence: 99%

Derived Galois deformation rings

Galatius¹,

Venkatesh²

2018

Advances in Mathematics

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We define a derived version of Mazur's Galois deformation ring. It is a pro-simplicial ring R classifying deformations of a fixed Galois representation to simplicial coefficient rings; its zeroth homotopy group π0R recovers Mazur's deformation ring.We give evidence that these rings R occur in the wild: For suitable Galois representations, the Langlands program predicts that π0R should act on the homology of an arithmetic group. We explain how the Taylor-Wiles method can be used to upgrade such an action to a graded action of π˚R on the homology.

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Section: 41mentioning

confidence: 99%

Derived Galois deformation rings

Galatius¹,

Venkatesh²

2018

Advances in Mathematics

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“…Thus, if we filter by the simplicial degree coming from X we get a spectral sequence with E 1 -term We now turn to the case of commutative algebras over a commutative ring R. The resulting homology theory is André-Quillen homology and it is the subject of a very extensive discussion elsewhere in these nots by Srikanth Iyengar [24], which the reader should turn to. We only include it here because (a) the construction of a suitable homology theory for commutative algerbas was one of the early successes of the theory of model categories and (b) the first author of this monograph is very fond of it.…”

Section: Then (Algmentioning

confidence: 99%

Model categories and simplicial methods

Goerss¹,

Schemmerhorn²

2007

Interactions Between Homotopy Theory and Algebra

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Abstract. There are many ways to present model categories, each with a different point of view. Here we'd like to treat model categories as a way to build and control resolutions. This an historical approach, as in his original and spectacular applications of model categories, Quillen used this technology as a way to construct resolutions in non-abelian settings; for example, in his work on the homology of commutative algebras [29], it was important to be very flexible with the notion of a free resolution of a commutative algebra. Similar issues arose in the paper on rational homotopy theory [31]. (This paper is the first place where the now-traditional axioms of a model category are enunciated.) We're going to emphasize the analog of projective resolutions, simply because these are the sort of resolutions most people see first. Of course, the theory is completely flexible and can work with injective resolutions as well.There are now any number of excellent sources for getting into the subject and since this monograph is not intended to be complete, perhaps the reader should have some of these nearby. For example, the paper of Dwyer and Spalinski [16] is a superb and short introduction, and the books of Hovey [22] and Hirschhorn [21] provide much more in-depth analysis. For a focus on simplicial model categories -model categories enriched over simplicial sets in an appropriate way -one can read [20]. Reaching back a bit further, there's no harm in reading the classics, and Quillen's original monograph [30] certainly falls into that category.

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“…Specialising modulo p n in §3, we construct a map from derived de Rham cohomology to crystalline cohomology in general, and show that it is an isomorphism in the case of an lci morphism (see Theorem 3.27). The main tool here is a derived Cartier theory (Proposition 3.5), together with some explicit simplicial resolutions borrowed from [Iye07].…”

Section: Introductionmentioning

confidence: 99%

p-adic derived de Rham cohomology

Bhatt

2012

Preprint

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This paper studies the derived de Rham cohomology of Fp and p-adic schemes, and is inspired by Beilinson's work [Bei]. Generalising work of Illusie, we construct a natural isomorphism between derived de Rham cohomology and crystalline cohomology for lci maps of such schemes, as well logarithmic variants. These comparisons give derived de Rham descriptions of the usual period rings and related maps in p-adic Hodge theory. Placing these ideas in the skeleton of [Bei] leads to a new proof of Fontaine's crystalline conjecture Ccrys and Fontaine-Jannsen's semistable conjecture Cst.2 If X extends to a proper smooth O K -scheme, then the monodromy operator on H * dR (X) 0 is trivial, and one expects the comparison isomorphism to be defined over a smaller Galois and Frobenius equivariant filtered subalgebra Bcrys ⊂ Bst; this is the crystalline conjecture Ccrys.3 In [Bhaf], we use the preceding comparison theorem and the conjugate filtration on derived de Rham cohomology to show that the crystalline cohomology groups of even very mildly singular projective varieties (such as stable singular curves) are infinitely generated. In fact, we fail to find a single example of a singular projective variety with finitely generated crystalline cohomology!

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André-Quillen homology of commutative algebras

Cited by 26 publications

References 22 publications

Derived Galois deformation rings

Derived Galois deformation rings

Model categories and simplicial methods

p-adic derived de Rham cohomology

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