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Lecture Vii: Extension to the Singular Case Via Derived Prismatic Cohomology

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Lecture Vii: Extension to the Singular Case Via Derived Prismatic Cohomology LECTURE VII: EXTENSION TO THE SINGULAR CASE VIA DERIVED PRISMATIC COHOMOLOGY Let (A; I) be a bounded prism. In this lecture, we explain how to use Quillen's formalism of non-abelian derived functors [1, 11, 10] to extend the notion of prismatic cohomology to arbitrary (i.e., not necessarily smooth) p-complete A=I-algebras R in a manner compatible with the Hodge- Tate comparison theorem. In the following lectures, we shall use this extension to formulate and prove the ´etalecomparison theorem for the prismatic cohomology of any p-complete A=I-algebra. 1. Non-abelian derived functors Let A be a commutative ring. Write CAlgA for the category of commutative A-algebras, and PolyA ⊂ CAlgA for the full subcategory spanned by polynomial A-algebras in finitely many vari- ables. Note that each such A-algebra P is a projective object in the category CAlgA, i.e., for any surjection B ! C in CAlgA and any map P ! C in CAlgA, there exists a lift P ! B. In analogy with homological algebra, it is therefore natural to \derive" functors defined on PolyA to functors defined on all of CAlgA by simply applying the original functor to a suitable resolution. We discuss this construction (in limited generality) next using the language of 1-categories; we refer the reader to Lurie's [8, x5.5.8] for an exhaustive modern treatement (see also [5, x9.2], [7, x2.1] for related expositions specialized to some examples also encountered below), Quillen's [11] for the original presentation via model categories. Construction 1.1 (Non-abelian derived functors). Consider a functor F : PolyA ! Ab on PolyA valued in the category Ab of abelian groups. Our goal is to explain why F admits a well-behaved ex- tension to all of CAlgA. To formulate the result economically, we use the language of 1-categories. Thus, let D(Ab) be the derived 1-category of abelian groups. Note that D(Ab) admits all limits and colimits (and, in particular, has functorial cones). The desired extension of F is then captured by the following result, applicable more generally to functors valued in D(Ab). Proposition 1.2. Let F : PolyA !D(Ab) be a functor. There exists a unique extension LF : CAlgA !D(Ab) of the functor LF characterized by the following properties: (1) LF commutes with filtered colimits. In particular, if A[S] is a polynomial algebra on a possi- bly infinite set S, then the canonical map colimΣ⊂S F(A[Σ]) ! LF(A[S]) is an equivalence, where the colimit ranges over all finite subsets Σ ⊂ S. (2) LF commutes with geometric realizations of simplicial resolutions, i.e., given B 2 CAlgA 1 with a simplicial resolution P• ! B by A-algebras, the geometric realization jLF(P•)j of P• is equivalent to LF(B) via the natural map. More categorically, the functor LF : CAlgA !D(Ab) is the left Kan extension of F : PolyA ! D(Ab) along the inclusion PolyA ⊂ CAlgA. We call LF the left derived functor of F. 1 op Given a simplicial object X• : ∆ !C in an 1-category C, its geometric realization jX•j is simply the colimit of X•. In our applications, we shall only apply this construction to a simplicial chain complex K• 2 Ch(Ab), viewed as a simplicial object in D(Ab). In this case, there is a much more explicit and classical description of the colimit: the geometric realization jK•j is simply the direct sum totalization of the bicomplex obtained from K• by making the simplicial direction the horizontal one. Note that this bicomplex is located in the 2nd and 3rd quadrants with standard cohomological conventions. In particular, there are potentially infinitely many terms along antidiagonals, so using the direct product totalization would give a different notion. 1 Remark 1.3 (Explicitly computing LF). In applications, the functor F is strict, i.e., it is obtained from a functor G : PolyA ! Ch(Ab) by passage to the derived category along the canonical functor Ch(Ab) !D(Ab). For such F, Proposition 1.2 gives a completely explicit recipe for computing LF(B) that can be phrased without 1-categories as follows. For any B 2 CAlgA, there is a functorial simplicial resolution P• ! B by (typically infinitely generated) polynomial A-algebras Pi determined by P0 = A[B], P1 = A[A[B]], etc. with natural transition maps. Using Proposition 1.2 (1) and (2), we obtain the following concrete formula for LF(B): it is computed by the geometric realization of the simplicial chain complex G(P•), where G(Pn) is defined as colimi G(Pn;i), where fPn;ig is the filtered collection of finitely generated polynomial subalgebras of Pn. For future use, note that if F takes values in D≤0(Ab), then the same holds true for LF: this follows from Proposition 1.2 (1) and (2) as D≤0(Ab) ⊂ D(Ab) is closed under all colimits. Remark 1.4 (Variants). Proposition 1.2 uses only basic category theory, and consequently has many variants. We discuss some relevant ones next. (1) (Replacing the target) The target 1-category D(Ab) can be replaced by any 1-category C admitting all colimits. In applications, we shall often work with C being the category of derived I-complete objects in the derived 1-category D(B) of B-modules, where B is a commutative ring and I ⊂ B is a finitely generated ideal. Note that in this case, the colimits appearing in Proposition 1.2 must be interpreted in the completed sense, i.e., they are computed by applying derived I-completion to the corresponding colimit in D(B). (2) (Enlarging the source) Proposition 1.2 also holds true if we replace CAlgA with the 1- category sCAlgA of simplicial commutative A-algebras. In fact, in this case, the construction F 7! LF establishes an equivalence between Fun(PolyA; C) and the full 1-subcategory of Fun(sCAlgA; C) spanned by functors commuting with sifted colimits. To avoid additional technical baggage, we shall largely avoid simplicial commutative rings in these lectures. However, the author finds it useful to work in the larger category sCAlgA when deriving functors as colimits (such as \reducing modulo p") behave better in this setting. (3) (Derived exterior powers) We may replace the inclusion PolyA ⊂ CAlgA with other inclu- sions. For example, there is an analog of Proposition 1.2 for the inclusion VectA ⊂ ModA of the category VectA of finite projective A-modules inside all A-modules. If the functor F is additive and right exact, then LF gives the usual derived functor from homological algebra. On the other hand, as there is no additivity assumption in Proposition 1.2, we are now allowed to derive non-additive functors. An example that will be relevant later is given i i by M 7! ^ M for any i ≥ 0; the corresponding derived functor L^ : ModA !D(A) give a notion of derived exterior power for A-modules that agrees with the usual notion for flat A-modules (but not in general). In fact, as in (2) above, we may even extend this to de- rived category, getting functors L^i : D≤0(A) !D(A) for i ≥ 0; in this derived setting, we simply write ^i instead of L^i if there is no potential for confusion. There are also variants for other tensorial constructions, such as symmetric powers Symi or divided powers Γi; see [10, xI.V.4] for more on these notions and the interrelationship between them. 2. The cotangent complex In this section, we discuss the first example where Construction 1.1 arose in algebraic geometry. Definition 2.1 (The cotangent complex). Let A be a commutative ring. The cotangent complex functor L−=A : CAlgA !D(A) is the left derived functor of the functor PolyA ! ModA ⊂ D(A) 1 given by B 7! ΩB=A. 2 Using the explicit description of left derived functor given following Proposition 1.2, it is not too ≤0 ≤0 difficult to see that LB=A is naturally an object of D (B) (and not just D (A)). We summarize the fundamental properties of this construction next, and refer to [10, 12] for proofs. 0 1 (1) (Recovering K¨ahlerdifferentials). We have H (LB=A) ' ΩB=A. 1 (2) (Acyclicity of smooth algebras). If A ! B is smooth, then LB=A ' ΩB=A[0]. Thus, smooth algebras are \acyclic" for L−=A. In particular, if A ! B is ´etale,then LB=A ' 0. (3) (Transitivity triangle). Given A ! B ! C, we have an exact triangle L LB=A ⊗B C ! LC=A ! LC=B in D(C), extending the standard low degree exact sequences for K¨ahlerdifferentials. −1 2 (4) (Behaviour on quotients). If A ! B is surjective, then H (LB=A) ' I=I , where I ⊂ A is i the kernel. If I is generated by a regular sequence, then H (LB=A) ' 0 for all i 6= −1; in particular, LB=A[−1] is a finite projective B-module. (5) (Base change). Given maps A ! B and A ! C that are Tor independent, we have a natural base change isomorphism L LC=A ⊗A B ' LC⊗AB=B: In particular, this applies if one of the map A ! B or A ! C is flat. (More generally, a similar assertion holds true without any Tor independence conditions provided one uses the L simplicial commutative ring C ⊗A B in lieu of C ⊗A B.) Although this plays no role in our lectures, we remark for the sake of completeness that the cotangent complex has a natural interpretation in the world of derived algebraic geometry: it classifies derivations in the derived sense, see [9, Chapter 17].
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