THE FLAT MODEL STRUCTURE ON Ch (O)
JAMES GILLESPIE
Abstract.Let Ch(O) be the category of chain complexes of O-modules on
a topological space T (where O is a sheaf of rings on T). We put a Quill*
*en
model structure on this category in which the cofibrant objects are buil*
*t out
of flat modules. More precisely, these are the dg-flat complexes. Dually*
*, the
fibrant objects will be called dg-cotorsion complexes. We show that this*
* model
structure is monoidal, solving the previous problem of not having any mo*
*noidal
model structure on Ch(O). As a corollary, we have a general framework for
doing homological algebra in the category Sh(O)of O-modules. I.e., we ha*
*ve
a natural way to define the functors Extand Torin Sh(O).
1.Introduction
Let (O, T ) be a ringed space. That is, T is a topological space and O is a
sheaf of (commutative) rings (with 1) on T . In algebraic geometry we study the
category Sh(O) , of sheaves of O-modules. This category is a popular example of
a Grothendieck category. Consequently, it has a lot of properties similar to t*
*he
category Rmod , of R-modules, where R is a commutative ring with identity. One
main difference though is that Sh(O) doesn't have enough projective objects. Th*
*is
causes some difficulties when doing homotopy theory on the chain complex cate-
gory Ch (O). (I.e., doing homological algebra in Sh(O) .) Surely we can compute
ExtnO(-, -) in Sh(O) since Grothendieck categories have enough injectives. How-
ever, since we also have a tensor product on Sh(O), we would also like a TornO(*
*-, -)
functor analogous to the situation in Rmod . (In fact, Sh(O) is the same as Rm*
*od ,
when we take T to be the one point space and let O be the constant sheaf with
O(U) = R for all U T .) Since we do not have enough projectives we can not
define it in the same way we do for Rmod . The idea then is to use flat mod-
ules since they are tensor exact. But a flat resolution of a sheaf S need not *
*be
unique up to homotopy equivalence. Consequently, we can't uniquely define Tor
using such flat resolutions. Edgar Enochs and L. Oyonarte recently showed we can
define Torby showing the existence of flat resolutions for which all the kernel*
*s are
cotorsion [EO01 ]. Considering such resolutions allows us to show that any shea*
*f S
has a unique resolution up to homotopy equivalence.
We will deal with the problem in a more general way, and from a slightly diff*
*erent
perspective. We take the point of view that wherever there exists a notion of h*
*omo-
topy in some category D, then there should be a Quillen model structure [Qui67]
on D describing it. Furthermore, if D is a closed symmetric monoidal category (*
*i.e.,
if D has a tensor product), then ideally we would like a monoidal model structu*
*re
____________
Date: January 3, 2004.
The author thanks Mark Hovey of Wesleyan University and Edgar Enochs of the *
*University of
Kentucky.
1
2 JAMES GILLESPIE
on D. Loosely speaking, this just means the model structure is compatible with
the tensor product.
It was shown in [Gil03] that there is a new monoidal model structure on Ch (R)
which we call the "flat model structure". The point of this paper is to general*
*ize
this result to Ch (O). Again, this is a much more interesting and useful result*
* since
Sh(O) doesn't have enough projectives. In model category terms, this means there
is no projective model structure on Ch (O) analogous to the well known projecti*
*ve
model structure on Ch (R). There is an "injective model structureö n complexes
of sheaves, but it is not monoidal [Joy84].
One useful corollary to having this flat model structure is that it will impl*
*icitly
deal with the "flat resolution problem" mentioned above. The model structure wi*
*ll
provide the "right typeö f resolution, simply by taking a cofibrant replacemen*
*t of
the module S. In short, the flat model structure on Ch (O) gives us the proper
framework to define the derived functors Ext and Tor using flat resolutions. The
method boils down to taking resolutions by flat O-modules so that the kernels a*
*re
cotorsion. So the method of Enochs and Oyonarte mentioned above is encompassed
in Quillen's idea of a monoidal model category.
2.Preliminaries
This section is meant to serve as a reference for definitions, notation, and *
*the-
orems that we will refer to in later sections of the paper. In particular, we *
*will
discuss cotorsion pairs, which Hovey showed in [Hov02 ] are closely related to *
*model
structures on abelian categories. For the rest of the paper we assume that the *
*reader
has a basic understanding of model categories (see [DS95 ],[Hov99 ], and [Qui67*
*]),
Grothendieck categories (see [Sten75]), and sheaves (see [Har77]). For much more
information on cotorsion pairs see [EJ01].
Let D be an abelian category. A cotorsion pair (also called a cotorsion theor*
*y)
is a pair of classes of objects (A, B) of D such that A? = B and A = ?B. Here A?
is the class of objects X 2 D such that Ext1(A, X) = 0 for all A 2 A, and simil*
*arly
?B is the class of objects X 2 D such that Ext1(X, B) = 0 for all B 2 B. Two
simple examples of cotorsion pairs in Rmod are (P, A) and (A, I) where P is the
class of projectives, I is the class of injectives and A is the class of all R-*
*modules.
Another example of a cotorsion theory in Rmod is (F, C) where F is the class *
*of
flat modules and C are the so-called cotorsion modules. For a reference on coto*
*rsion
modules see [Xu96 ] and [EJ01].
The cotorsion pair is said to have enough projectives if for any X 2 D there *
*is a
short exact sequence 0 -!B -!A -!X -!0 where B 2 B and A 2 A. We say it has
enough injectives if it satisfies the dual statement. It is worth making a note*
* on this
terminology. The phrase "enough projectivesä nd "enough injectives" is standard
in reference to cotorsion pairs. Unfortunately we also use the phrase "enough
projectves/injectives" in reference to a category. This should never be confus*
*ing
since from the context we are always referring to either a category or a cotors*
*ion
pair. Note however that saying that the projective cotorsion pair, (P, A), has
enough projectives is equivalent to saying that the category has enough project*
*ives.
So in fact the terminology applied to a cotorsion theory is just a generalizati*
*on of
the usual terminology. In addition however, for any class of objects F in an ab*
*elian
category D, the author will use the terminology enough F-objects to mean for any
object X 2 D there exists an F 2 F and an epimorphism F -! X. Thus if (F, C) is
THE FLAT MODEL STRUCTURE ON Ch(O) 3
the "flat" cotorsion pair described above, saying we have enough F-objects means
we can find a surjection F -! M where F is flat. But we say (F, C) has enough
projectives to mean there exists a short exact sequence
0 -!C -!F -! M -!0
where C 2 C and F 2 F.
The central cotorsion pair we study in this paper is (F, C) where F is the se*
*t of
all flat O-modules and C = F? is the class of cotorsion O-modules. An O-module
F is called flat if the functor F O - is an exact functor. Recall that for two*
* O-
modules S1 and S2, we define S1 O S2 to be the sheafification of the presheaf w*
*hich
assigns each open set U T to S1(U) O(U)S2(U). We always assume our rings
are commutative with identity 1, so that S1(U) O(U) S2(U) is an O(U)-module
and therefore S1 O S2 is a sheaf of O-modules. Like most properties of sheaves,
flatness is a "stalkwise property". That is, F is flat iff Fp is a flat Op-modu*
*le for
each p 2 T . This follows from the well-known isomorphism
(S1 O S2)p ~=(S1)p Op (S2)p ,
which can be found in [Lit82]. In Sh(O), there are enough flat objects even tho*
*ugh
there are not enough projective objects. This fact is not hard and the proof ca*
*n be
found as Proposition 1.2 in [Har66].
We say a cotorsion theory is complete if it has enough projectives and enough
injectives. The three cotorsion pairs in Rmod described above are all examples*
* of
complete pairs. Proving that the "flat" pair (F, C) is complete is nontrivial *
*and
two different proofs were recently given by the three authors of [BBE01 ]. Simi*
*larly
we have the "flat" cotorsion pair on Sh(O), the category of sheaves of O-modules
where O is a ringed space on T . This cotorsion pair is also complete as follo*
*ws
from [EO01 ].
The next theorem is a special case of Theorem 2.2 from [Hov02 ] which relates
cotorsion pairs to model structures on abelian categories. It is this method wh*
*ich
we will use to find the flat model structure on Ch (O).
Theorem 2.1. (Hovey) Let D be an abelian category with all small limits and
colimits and Ch(D) be the category of chain complexes on D. Let E be the class
of all exact chain complexes and let Q and R be classes of chain complexes such
that (Q, R \ E) and (Q \ E, R) are complete cotorsion theories. Then there exis*
*ts a
model structure on Ch(D) , where the weak equivalences are H*-isomorphisms, the
cofibrations are the monomorphisms whose cokernels are in Q and the fibrations
are the epimorphisms whose kernels are in R.
Furthermore, in case D is a closed symmetric monoidal category, Hovey gives
conditions on the classes Q and R which guarantee that the model structure is
monoidal. So to find the flat model structure we will need to come up with the
classes Q and R. This leads us to the following definition which first appeared
in [Gil03].
Definition 2.2. Let (A, B) be a cotorsion pair on an abelian category D. Let X
be a chain complex.
(1) X is called an A complex if it is exact and ZnX 2 A for all n.
(2) X is called a B complex if it is exact and ZnX 2 B for all n.
(3) X is called a dg-A complex if Xn 2 A for each n, and every chain map
f :X -!B with B a B-complex, is homotopic to zero.
4 JAMES GILLESPIE
(4) X is called a dg-B complex if Xn 2 B for each n, and every chain map
f :A -!X with A an A-complex, is homotopic to zero.
We denote the class of A complexes by Aeand the class of dg-A complexes by
dgAe. Similarly, the B complexes are denoted by eBand the class of dg-B complex*
*es
are denoted by dgBe. These definitions are inspired from the dg-projective comp*
*lexes
and projective complexes in Ch (R), induced by the projective cotorsion pair.
Now let G be a Grothendieck category and (A, B) be a cotorsion pair. Corol-
lary 3.8 of [Gil03] tells us that if G has enough A-objects then we have the in*
*duced
cotorsion pairs (Ae, dgBe) and (dgAe, eB) of chain complexes.
A cotorsion pair (A, B) is called hereditary if one of the following hold:
(1) A is resolving. That is, A is closed under taking kernels of epis.
(2) B is coresolving. That is, B is closed under taking cokernels of monics.
(3) Exti(A, B) = 0 for any A 2 A and B 2 B and i 1.
See [GR99 ] for a proof that these are equivalent.
If a cotorsion pair (A, B) is hereditary and our category D has enough projec*
*tives
and injectives, then dgAe\ E = Aeand dgBe\ E = eB, where E be the class of all
exact chain complexes [Gil03]. In this case we say the induced cotorsion pairs
are compatible, and we have cotorsion pairs (dgAe, dgBe\ E) and (dgAe\ E, dgBe).
This makes the situation reminiscent of the hypothesis of Theorem 2.1 by making
Q = dgAe and R = dgBe. But for a general Grothendieck category G, we may
not have enough projectives. Therefore, to get the analogous result for heredit*
*ary
cotorsion pairs in G we will need a slight modification. This will be provided*
* in
Corollary 3.7.
The typical cotorsion pairs one comes across in practice are hereditary. For
example, all of the previous examples of cotorsion pairs are hereditary. For t*
*he
"flat" cotorsion pair (F, C) in Sh(O), a typical tensor product argument (like *
*the
one found in Proposition 3.4 of [Lan97]) will show that this is an hereditary c*
*otorsion
pair.
3. A simplification for Grothendieck categories
For the rest of the paper we let (F, C) denote the "flat" cotorsion pair in t*
*he
category Sh(O) . As we have already noted, it follows from [Gil03] that we have
two induced cotorsion pairs in Ch (O), which we denote (dgFe, eC) and (Fe, dgCe*
*). We
will call the complexes in the class eF, flat, and the complexes in the class d*
*gFe, dg-
flat. Similarly, we will use the terminology cotorsion and dg-cotorsion. To obt*
*ain
the "flat" model structure we will use Hovey's Theorem 2.2 from [Hov02 ], which
we have reprinted in section 2 in a more convenient form as Theorem 2.1. Thus
we wish to show first that (dgFe, eC) and (Fe, dgCe) are complete and second th*
*at
(dgFe, eC) and (Fe, dgCe) are compatible. By compatible we mean that dgFe\ E = *
*eF
and dgCe\ E = eCwhere E is the class of exact complexes.
In this section we solve the problem of showing the induced cotorsion pair
(dgFe, eC) is complete. We also show that (dgFe, eC) and (Fe, dgCe) are compat*
*ible.
Since the solutions are very general and could possibly be used for other situa*
*tions
we prove it for an arbitrary Grothendieck category with reasonable hypotheses on
the cotorsion pair.
THE FLAT MODEL STRUCTURE ON Ch(O) 5
In particular, we let (A, B) be an hereditary cotorsion pair and G be a Groth*
*endieck
category with enough A-objects. Again, this means that for any object X 2 G, we
have an epimorphism A -! X where A 2 A. We will also assume that (A, B) is
cogenerated by a set S. A cotorsion pair (A, B) is cogenerated by a set if ther*
*e is
a set S A (so not just a class) such that S? = C. This idea is fundamental to
the study of cotorsion pairs. It will come up again in section 4. These hypothe*
*ses
are all satisfied by the "flat" cotorsion pair in Sh(O). For example, to see th*
*at the
cotorsion pair is cogenerated by a set, see the proof of Theorem 3.1 in [EO01 ].
One thing that makes a Grothendieck category feel more öc ncrete" is the fact
that every object is ~-presented where ~ is some infinite cardinal. Since we wi*
*ll use
this idea heavily we start with some definitions.
Definition 3.1. Let ~ be a cardinal number. A nonempty category K is called
~-filtered if every small subcategory S K with |mor (S)| ~ is the base of a
cocone. (Here mor(S) is the set of morphisms of S.) We call a functor F :K -!D
a ~-filtered functor if K is a ~-filtered category. The colimit of a ~-filtered*
* functor
F :K -!D is called a ~-filtered colimit.
One specialization of the idea of a ~-filtered category is the notion of a ~-*
*filtered
ordinal. (For example, see [Hov99 ] pp. 28.) Of course this is an ordinal ff in*
* which
for any S ff with |S| ~, we have supS < ff (or to say the same thing, [S 2 *
*ff).
It is a fact that given any infinite cardinal ~, the smallest ~-filtered limit *
*ordinal is
~+ , the cardinal successor of ~. Also, any successor cardinal ~0> ~ is ~-filte*
*red.
Given a functor F :K -!D and an object X 2 D, we will denote the composition
of Hom D(X, -) with F by Hom D(X, F ). Recall that there is a natural map
colimHom D (X, F ) -!Hom D (X, colimF )
(assuming all of these colimits exist) induced by the universal property of a c*
*olimit.
Definition 3.2. Let ~ be a cardinal and let X be an object of a category D. We
say X is ~-presented (or ~-small) if for any ~-filtered functor F :K -! D (whose
colimit exists in D), the natural map colimHom D(X, F ) -! Hom D(X, colimF ) is
an isomorphism.
In particular, suppose ff is a ~-filtered ordinal and
X0 X1 . . .Xfl . . .
is an increasing sequence of objects indexed by ff. If X is ~-presented, then a*
*ny
map X -! [Xflfactors through some Xfl. Also, notice that if ~0 ~, then a
~-presented object X 2 D is also ~0-presented.
Now lets recall the following lemma of Grothendieck. See [Gro57] for a proof.
Lemma 3.3. (Grothendieck) If G is a Grothendieck category, then an object I 2 G
is injective iff for each subobject V U, where U is a generator, and each mor*
*phism
f :V -! I, f extends to a morphism U -!I.
In a Grothendieck category, G, every object is ~-presented for some ~ [Hov01 *
*].
Also the class of subobjects of an object is actually a set.(See [Sten75] pp 94*
*). Now
if we let U be a generator for G, then each subobject V U is ~V -presented for
some cardinal ~V . Let ~ = sup{~V : V U}. Then clearly all U and all of its
subobjects are ~-presented. Now if we let K be a ~-filtered category, and consi*
*der
a functor F :K -!G where F (d) is injective for all i 2 K, then for any subobje*
*ct
V U, and arrow i -!j in K, we get a commutative diagram
6 JAMES GILLESPIE
Hom (U, F (i))----!Hom (V, F (i))----!0
?? ?
y ?y
Hom (U, F (k))----!Hom (V, F (k))----!0
with exact rows. Thus
colimHom (U, F ) -!colimHom (V, F ) -!0
is exact. And since K is ~-filtered, the direct limits commute to give us
Hom (U, colimF ) -!Hom (V, colimF ) -!0
is exact. So Grothendieck's Lemma 3.3 implies the following lemma.
Lemma 3.4. Let G be a Grothendieck category. Then there exists a cardinal ~
such that for every ~-filtered functor F :K -!G with F (j) injective for all j *
*2 K,
we have colimF injective.
Lemma 3.5. Let G be a Grothendieck category and let (A, B) be a cotorsion pair
cogenerated by a set S. Then there exists a cardinal ~0such that for every ~0-f*
*iltered
functor F :K -!G with F (j) 2 B for all j 2 K, we have colimF 2 B.
Proof.Since (A, B) is cogenerated by a set S, there actually exists a single ob*
*ject
A which cogenerates (A, B). I.e. Ext1(A, B) = 0 iff B 2 B. Let ~ be as in the l*
*ast
lemma and let ~0 ~ be a cardinal for which A is ~0-presented.
Now let K be a ~0-filtered category and let F :K -! G be a functor such that
F (j) 2 B for all j 2 K. It follows from Corollary 6.6 of [Hov02 ], that in a
Grothendieck category we may take injective coresolutions functorially. There-
fore, we may take a functorial injective coresolution of the diagram F (K) as s*
*hown
below:
Ij ----! Ij0
x? x
? ??
F (j)----! F (j0)
Applying the functor Hom (A, -) to the diagram of injective complexes {Ij}j2K
gives us (in the obvious way), a diagram of complexes {Hom (A, Ij)}j2K By hy-
pothesis Ext1(A, F (j)) = H1 Hom (A, Ij) = 0 for all j 2 K. So it follows that
H1[colimj2KHom (A, Ij)] = 0. But since K is ~-filtered, the last lemma tells us
that colimF -! colimj2KIj is an injective coresolution. Also, since K is ~0-fil*
*tered,
Hom (A, colimj2KIj) ~=colimj2KHom (A, Ij).
Therefore,
Ext1(A, colimF ) = H1[Hom (A, colimj2KIj] = 0.
For a moment, let D be any cocomplete abelian category (not necessarily Groth*
*en-
dieck). Though it is an abuse of notation, we will also let D denote the class
of objects of D. Then for a complete cotorsion theory (A, B), clearly (A, B) =
(A \ D, B) = (A, B \ D). So Hovey's Theorem 2.2 from [Hov02 ] gives us a (trivi*
*al)
model structure on the category D with the cofibrations being the injective maps
i with coki 2 A and the fibrations being the surjective maps p with kerp 2 B. In
THE FLAT MODEL STRUCTURE ON Ch(O) 7
particular, any map f :X -! Y factors as f = pi where i is a cofibration and p *
*is
a fibration. We will use this to prove the next proposition. The author learned*
* the
idea of the proof from Edgar Enochs.
Proposition 3.6. Let (A, B) be a cotorsion pair in a Grothendieck category G wi*
*th
enough A-objects. Furthermore assume that (A, B) is cogenerated by a set. Then
the cotorsion pair of complexes (dgAe, eB) has enough injectives.
Proof.The assumption that G has enough A-objects is only used to guarantee that
(dgAe, eB) is indeed a cotorsion pair. (See Corollary 3.8 of [Gil03].)
Let X be any complex. We want to embed X as 0 -! X -! B -! A -! 0
where B 2 Beand A 2 dgAe. First let ~ be any ordinal and we will describe a
general embedding method. Later we will specify an ordinal which will prove the
proposition.
Let (nk)k2N be any sequence which maps N bijectively onto Z. Set X0 = X. Now
dn1
factor X0n1--! Zn1-1(X0) as pn1in1 where pn1: Yn01-!Zn1-1(X0) is a surjection
with kerpn12 B and in1: X0n1-!Yn01is an injection with cokin12 A. Then set
in1d 0 pn1 0
X1 = . .-.!X0n1+1---!Yn1 --! Xn1-1 -!. ...
Then X1 satisfies
(1) Zn1(X1) = kerpn1
(2) Hn1-1(X1) = 0
(3) X0 X1 and X1=X0 = . .-.!0 -!Yn01=X0n1-!0 -!. .i.s a dg-A complex.
Using this method, we continue by transfinite induction to build an increasing
sequence of chain complexes. Suppose ff is an ordinal and suppose that we have
already constructed
X = X0 X1 X2 . .X.fi . . .
for all fi < ff. If ff is a limit ordinal then we set ff = [fi 0. So ExtnO(F, Hom (S, Q)) = 0 for n > 0.
Proposition 5.6. A complex of O-modules F is dg-flat if and only if Fn is flat *
*for
each n, and F O E is exact for each exact complex E.
Proof.()) Let F be dg-flat and E any exact complex. We wish to show F E is
exact. By Lemma 5.5, E+ is cotorsion. So the global sections complex
Hom (F, E+ ) = Hom (F, E+ )(T )
is exact. But using adjoint associativity
Hom (F, E+ ) = Hom (F, Hom (E, S(Q)) ~=Hom (F E, S(Q)) = (F E)+ .
So the global sections complex
(F E)+ (T ) = . .-.!Sh((F E)n, Q) -!Sh ((F E)n+1, Q) -!. . .
must be an exact sequence (of O(T )-modules). Now one argues as in the proof of
Lemma 5.2 to conclude that F E is exact.
(() Suppose that we have a chain complex F in which every Fn is a flat O-
module and F E is exact for each exact complex E. By Corollary 3.7 we can find
a short exact sequence
0 -!F -! C -!F 0-!0
where C is cotorsion and F 0is dg-flat.
We claim that C is in fact flat. Using Lemma 5.4, it is enough to show that
C+ is exact and dg-injective. Certainly C+ is exact since C is exact. Also Cn is
flat since it is an extension of Fn and Fn0. This tells us that (C+ )n is injec*
*tive. To
finish showing C+ is dg-injective we need to show that every map E -!C+ , where
E is exact, is homotopic to zero. This follows from showing that the global sec*
*tions
complex Hom (E, C+ ) = Hom (E, C+ )(T ) is exact.
Now for any complex E we have a short exact sequence
0 -!F E -!C E -!F 0 E -!0
since 0 -! F -! C -! F 0-! 0 is "degreewise pure". Furthermore, if E is exact,
then by hypothesis F E is exact and since we have proved the ()) direction
of the Proposition, F 0 E is exact too. Therefore, by the fundamental lemma of
homological algebra, C also has the property that C E is exact whenever E is
exact.
THE FLAT MODEL STRUCTURE ON Ch(O) 19
Using this property, we go back to the problem of showing the global sections
complex Hom (E, C+ ) is exact whenever E is exact: For such an E, we have
Hom (E, C+ ) = Hom (E, Hom (C, S(Q)) ~=Hom (E C, S(Q)) = (E C)+
and the complex on the right is clearly exact. In fact, (E C)+ must be preshe*
*af
exact by the comments made after the proof of Lemma 5.3. Therefore, the global
sections complex Hom (E, C+ ) = Hom (E, C+ )(T ) is exact. This concludes the
proof that C is flat.
Now for any cotorsion complex C0, the sequence
0 -!Hom (F 0, C0) -!Hom (C, C0) -!Hom (F, C0) -!0
is exact. Indeed, in degree n we have the exact sequence
Y Y Y
Sh(O)(Ck, C0k+n) -! Sh (O)(Fk, C0k+n) -! Ext1O(Fk0, C0k+n) = 0.
k2Z k2Z k2Z
Since F 0and C are both dg-flat we see that Hom (F 0, C0) and Hom (C, C0) are e*
*xact.
Thus the fundamental lemma of homological algebra implies Hom (F, C0) is exact,
so that F is dg-flat.
Finally, we may now prove that the flat model structure is monoidal.
Theorem 5.7. Let (F, C) be the flat cotorsion pair in the category Sh(O). Then
the induced model structure on Ch (O) is monoidal with respect to the usual ten*
*sor
product of chain complexes.
Proof.We use Hovey's criteria listed in Proposition 5.1.
1) Cofibrations are monomorphisms with dg-flat cokernels. In particular, the
cokernel is flat in each degree. Therefore each cofibration is a pure injection*
* in each
degree. (See for example, the proof of Lemma XVI.3.1 in [Lan97].)
2) Let X, Y be dg-flat. We wish to show X Y is dg-flat. It is easy to s*
*ee
that (X Y )n is flat for each n. Now given an exact complex E, we know by
Proposition 5.6 that Y E is exact, and likewise X (Y E) is exact. But then
(X Y ) E is exact and using Proposition 5.6 again we get that X Y is dg-f*
*lat.
3) Let X be dg-flat and Y flat. We wish to show X Y is flat. Recall that
a complex is flat if and only if it is dg-flat and exact. So by 2) above, X Y*
* is
dg-flat. But since Y is exact, X Y is also exact. Therefore X Y is flat.
4) The unit of the monoidal structure is S(O), the complex consisting of O
in degree zero and 0 otherwise. Since O is a flat O-module, S(O) is dg-flat by
Lemma 3.4 of [Gil03].
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4000 University Drive, Penn State McKeesport, McKeesport, PA 15132-7698
E-mail address, James Gillespie: jrg21@psu.edu