Capítulo 6: Plan Económico-Financiero
6.11 Estimación del flujo de caja o efectivo
descends to the contraderived categories of linear factorizations equipped with the con- traderived tensor product. In particular, given A/(w)-modules X, Y , there is a canonical isomorphism
σ wX ⊗LA −wY
∼
=−wX ⊗LA wY
in DctrLF(A, 0), hence also H∗
wX ⊗LA −wY
∼
= −wX ⊗LA wY canonically, since σ :
DctrLF(A, 0) → DctrLF(A, 0) doesn’t affect H∗
. In particular, the two candidates for the definition of stable Hochschild homology are canonically isomorphic.
Note, however, that in contrast to classical cohomologically Z-graded complexes, the sign change automorphism σ : DctrLF(A, 0) → DctrLF(A, 0) is in general not isomor-
phic to the identity functor: For example, taking A =k[ε]/(ε2) withk a field of charac-
teristic 6= 2, the matrix factorizations X := (ε : A A : ε) and σX = (−ε : A A : ε) are not contraderived weakly equivalent: if they where, they would be homotopy equiv- alent by Theorem I.4.1.5, but [X, σX] = 0 as a short calculation shows. ♦ Proposition I.4.5.5. Assume that one of the following holds:
(i) ∆ admits a bounded, finite rank bA-free resolution as a −w-curved mixed complex.b (ii) gl. dim( bA-Mod) < ∞.
Then there is a canonical isomorphism in DctrLF( bA,0):
−wb∆ ⊗LA bbwM ∼= R foldΠ
−wb∆ ⊗LA bbwM
Proof. This is a special case of Proposition I.4.4.8.
I.4.6. Ordinary versus stable Hochschild homology
In this section we describe the relation between ordinary and w-stable Hochschild ho- mology. We keep the notation from the beginning of the previous Section I.4.5.
Under the assumptions of Proposition I.4.5.5 the total w-stable Hochschild homology
wsHHA/k
t (X) of an bA/(w)-module X can be computed as the total cyclic homology (inb the sense of Remark I.2.1.8) of a bounded mixed complex P∆⊗A bbwX ∈ D
bMC( bA,0 d), the
underlying complex of which computes the ordinary Hochschild homology HHA/∗ k(X) of
X; here P∆→−wb∆ is a suitable resolution of the diagonal. The ordinary and total cyclic
homology associated with a bounded mixed complex are related through a converging spectral sequence which is a special case of the spectral sequence of a double complex. Instead of recalling the latter in full generality, we decided to describe the structure that arises in the application to mixed complexes directly. In particular, this section does not assume any knowledge about spectral sequences.
Chapter I.4. Some homotopy theory
Suppose X ∈ MC(A, 0d), i.e. (X, d, s) is a bounded 0-curved mixed complex over A
with s of q-degree d. Our goal is to define successive approximations to the total cyclic homology HC(X) – the homology ofL
nXn−nd2 with respect to the homogeneous dif-
ferential d +s of degree d2 – starting from the ordinary (Z-graded) homology H∗(X, d) of (X, d). For that, given an integer t ≥ 1 we call an element x ∈ X≤n:=L
k≤nXk−
kd 2
an approximative n-cycle of order t if (d +s)(x) ∈ X≤n+1−t; it is called an approximative
n-boundary of order t if there exists some y ∈ X≤n−1+(t−1) such that x ≡ (d +s)(y) modulo X≤n−1. The A-submodules of X≤n consisting of approximative n-cycles of or- der t resp. approximative n-boundaries of order t are denoted Zn
t and Bnt, respectively.
Their definition expresses the idea that, given an element x =P
kxk∈ X
≤n, instead of
requiring d xk−1+ sxk+1 to vanish for all k ∈ Z (which would mean that x is an honest
(d +s)-cycle) one can define weakenings by requiring the vanishing of d xk−1+ sxk+1 for
some konly, and indeed x ∈ Znt is equivalent to requiring it for k > n + 1 − t. Also note that, once t is large enough, the notions of approximative n-cycle of order t and actual (d +s)-cycle in X≤n coincide since X is bounded. Back to the definitions, the quotient (Zn
t +X≤n−1)/ Bnt ∼= Znt /(Bnt ∩ Znt) is called the n-th approximative cyclic homology of
order t of X and is denoted HCn
t(X). Finally, we denote HC≤n(X) ⊂ HC(X) the sub-
module of HC(X) consisting of those cohomology classes that can be represented by a cycle from X≤n.
Proposition I.4.6.1. The following properties hold: (i) There is a canonical isomorphism HCn
1(X) ∼= Hn(X, d)−nd2 .
(ii) The restriction of d +s to Znt induces a differential dnt : HCnt(X) → HCn+1−tt (X) of q-degree d2, satisfying ker(dnt)/ im(dtn−1+t) ∼= HCnt+1(X). In particular, dn1 van-
ishes, hence HCn
2(X) ∼= Hn(X, d)−nd2 canonically, too, and
dn2 : Hn(X, d)−nd 2
∼
= HCn2 → HCn−12 ∼= Hn−1(X, d)D−(n−1)d2 E is the differential induced by s.
(iii) For t 0 there is a canonical isomorphism HCnt(X) ∼= HC≤n(X)/ HC≤n−1(X). Moreover, the the formation of approximative cycles, boundaries and homologies, as well as the differentialsdnt and the isomorphisms in (ii) are functorial in(X, d, s) with respect to morphisms of mixed complexes.
Proof. This amounts to (hopefully) carefully writing out the definitions. In the fol- lowing, the groups Hn, Zn etc. will be taken with respect to X. (i) The embedding
Xnnd
2 X
≤n maps the d-cycles Zn into the approximative n-cycles Zn
1 of order 1
I.4.6. Ordinary versus stable Hochschild homology and the d-boundaries into the approximative n-boundaries Bn
1 of order 1, hence gives
rise to a well-defined map ψn : Hn → HCn
1 = (Zn1+X≤n−1)/ Bn1. We prove that this
map is injective and surjective. For injectivity, suppose that x ∈ Zn is an approxima-
tive n-boundary of order 1. Then, by definition there exists some y ∈ X≤n−1 such that (d +s)(y) ≡ x modulo X≤n−1. In particular, we conclude x = d yn−1, so x is a
d-boundary. For surjectivity, suppose that x ∈ Zn
1 ⊂ X≤n is an approximative n-cycle
of order 1. Then 0 = [(d +s)(x)]n+1 = d xn, so xn ∈ Zn defines a cohomology class
in [xn] ∈ Hn. Also x − xn ∈ X≤n−1 ⊂ B1n, so [x] = [xn] = ψn([xn]) in HCt1, prov-
ing surjectivity. (ii) If x ∈ Zt
n, then by definition we have (d +s)(x) ∈ X≤n+1−t. Also,
(d +s)(x) ∈ ker(d +s) since d +s is a differential, so (d +s)(x) ∈ Zn+1−tt represents an ap- proximative (n+1−t)-th cohomology class of order t. Further, if x ∈ Zn
t ∩ Bnt, there exists
y ∈ X≤n−1+(t−1)such that x = (d +s)(y)+x0for some x0 ∈ X≤n−1= X≤(n+1−t)−1+(t−1), so (d +s)(x) = (d +s)(x0) ∈ Bn+1−tt . This proves that d +s induces a well-defined differ- ential dn
t : HCnt → HCn+1−tt , and we prove next that there are canonical isomorphisms
ψtn: ker(dnt)/ im(dn−1+tt ) ∼= Hnt+1. For this, note that given x ∈ Znt, we have dnt([x]) = 0 if and only if (d +s)(x) ∈ Bn+1−tt , i.e. if and only if there exists some y ∈ X≤n−1such that (d +s)(x) ≡ (d +s)(y) modulo X≤n−t, which in turn is equivalent to x ∈ Zn
t+1+X≤n−1.
Since moreover Bnt ⊆ Bnt+1, it follows that the identity on representatives induces a well- defined map ˜ψtn : HCn
t ⊇ ker(dnt) → (Znt+1+X≤n−1)/ Bt+1n ∼= HCnt+1. Since Znt+1 ⊆ Znt,
it is clear that this map is surjective; also, if x ∈ Znt represents [x] ∈ ker(dnt), we have ˜ψtn([x]) = 0 in HCnt+1 if and only if x ∈ Bnt+1, i.e. if and only if there is some y ∈ X≤n−1+t such that x ≡ (d +s)(y) modulo X≤n−1. By definition, any such y belongs
to Zn−1+tt , so we conclude ker( ˜ψtn) = im(dn−1+tt ), and hence ˜ψtn induces an isomor- phism ψn
t : ker(dnt)/ im(dn−1+tt ) ∼= Hnt+1 as claimed. Finally for (iii) note that since X is
bounded we have Zn
t = ker(d +s)∩X≤nand Bnt = im(d +s)∩X≤n+X≤n−1for t 0, and
hence HCnt = Znt /Bnt ∩ Znt = ker(d +s) ∩ X≤n/(im(d +s) ∩ X≤n+ ker(d +s) ∩ X≤n−1) ∼= HC≤n/HC≤n−1 as claimed.
Remark I.4.6.2. The additional Z/2Z-grading on HCt(X) that we neglected in the
above construction of HC causes all odd differentials dn2t+1 on HC to vanish. ♦ We want to consider the above construction as a functor on the bounded derived category DbMC(A, 0
d) of mixed complexes. First, we define the appropriate target
category by abstracting from the properties of HC we established in Proposition I.4.6.1:
Definition I.4.6.3. A spectral complex (of q-degree d) over A is a tuple E = ((Ent,dnt, ψtn)n∈Z,t≥1,(E≤n∞, ψ∞n)n∈Z)
Chapter I.4. Some homotopy theory
(i) A t ≥ 1-indexed family of bounded A-complexes (En
t,dnt : Ent → En+1−2tt hd2i)n∈Z.
(ii) Isomorphisms ψn
t : Hn(E∗t) := ker(dtn)/ im(dn−1+2tt ) ∼= Ent+1 for n ∈ Z and t ≥ 1.
(iii) A Z-filtered A-module (E≤n∞)n∈Z= (· · · ⊆ E≤n∞ ⊆ E≤n+1∞ ⊆ · · · ).
(iv) Isomorphisms of A-modules ψn
∞: E≤n∞ /E≤n−1∞ ∼= lim−→tEnt for all n ∈ Z.
Here, the colimit lim−→
tE n
t is taken over the Z-diagram t 7→ Ent whose transition maps
En
t → Ent+1 are defined as 0 if dnt 6= 0 and as Ent = Znt Hn(E∗t,d∗t) ∼= Ent+1 if dnt = 0.
Note that this diagram is eventually constant since the Ent are uniformly bounded in n and the differentialdnt has cohomological degree1 − 2t.
The A-module underlying the filtered module (E≤n∞ )n∈Z is called the limit of E.
Given spectral complexes E and F, a morphism of spectral complexes f : E → F is a family of A-linear homomorphisms fn
t : Ent → Fnt together with a homomorphism of
Z-filtered A-modules (E≤n∞)n∈Z → (F≤n∞)n∈Z that are compatible with the differentials
d and the isomorphisms ψ. The resulting category of spectral complexes is denoted SCh(A). Assigning to a spectral complex E its underlying complex (E∗1,d∗1) defines a forgetful functor SCh(A) → Ch∗(A) from spectral complexes to (homologically graded)
chain complexes over A, which is faithful and reflects isomorphisms.
Proposition I.4.6.1 says that X 7→ HC(X) defines a functor from bounded mixed complexes to spectral complexes. Next we check that it factors over the derived category:
Fact I.4.6.4. Mapping a bounded mixed complex(X, d, s) to the spectral complex HC(X) descends naively to a functorHC : DbMC(A, 0
d) → SCh(A).
Proof. We have to check that for a quasi-isomorphism (with respect to d) of mixed complexes X → Y the induced morphism of spectral complexes HC(X) → HC(Y ) is an isomorphism, and this follows from HC∗1(−) ∼= H∗(−) and the fact that the forgetful functor SCh(A) → Ch∗(A) reflects isomorphisms.
Proposition I.4.6.5. Suppose the hypothesis of Proposition I.4.5.5 are met, and let M ∈ bA/(w) -Mod. Consider the following spectral complex over A:b
wSHHA/k(M ) := HC( −wb∆ ⊗
L b
A bwM) (I.4.3)
It has the following properties:
(i) The n-th component of the underlying complex of wSHHA/k(M ) is canonically
isomorphic to the shifted ordinary Hochschild homology group HHA/−nk(M )−nd 2 .
I.4.6. Ordinary versus stable Hochschild homology
(ii) The limit of wSHHA/k(M ) is canonically isomorphic to wsHHA/k
t (M ), the total
w-stable Hochschild homology of M over A.
Corollary I.4.6.6. Under the hypothesis of Proposition I.4.5.5, the w-stable Hochschild homology wsHHA/kt (M ) of an bA/(wb)-module M admits a natural, bounded Z-filtration.
Finally, we consider degeneration of the spectral complex HC(X):
Fact I.4.6.7. For a bounded mixed complex X ∈MC(A, 0d), the following are equivalent:
(i) HC(X) is degenerate, i.e. all differentials dnt for n ∈ Z and t ≥ 1 vanish.
(ii) Any d-cycle x ∈ Xn extends to a (d +s)-cycle ˜x:= x + x0 ∈ X≤n for x0∈ X≤n−1.
In this case,[x] 7→ [˜x] defines an isomorphism Hn(X, d)−nd 2
∼
= HC≤n(X)/ HC≤n−1(X). Proof. Consider an approximative n-cycle x ∈ Zn
t of order t. We have seen in the
proof of Proposition I.4.6.1 that dn
t([x]) = 0 is equivalent to x ∈ Znt+1+X≤n−1. Hence,
the vanishing of dnt is equivalent to every approximative n-cycle of order t having an approximative n-cycle of order t + 1 in its X≤n−1-coset. Since some x ∈ X≤n is an
approximative n-cycle of order 1 if and only if xn∈ Xn is a d-cycle, and since cycles of
order t agree with actual (d + s)-cycles for t 0, the equivalence of (i) and (ii) follows. The last claim follows by observing that in the chain of isomorphisms
Hn(X, d)h−nd 2 i ∼= HC n 1(X) = Hn(HC∗1(X), d∗1 ≡ 0) ∼ = HCn2(X) = Hn(HC∗1(X), d∗2 ≡ 0) .. . ∼ = HCn t(X) ∼= HC ≤n(X)/ HC≤n−1(X), t 0 :
each isomorphism is given on representatives by extending of an approximative cycle of order t to an approximative cycle of order t + 1, modulo X≤n−1.