"Summer letters" of the year 2002 (English translation, April 2023)
Russian original is here -- https://posic.livejournal.com/190336.html
The beginning (four Summer 2000 letters) is here -- https://posic.livejournal.com/2773832.html
From: Leonid Positselski < posic@mpim-bonn.mpg.de >
To: roma, hippie
Date: Fri, 11 Oct 2002 18:56:58 +0200 (MEST)
Subject: definition of semi-infinite homology (corrected)
Hi Roma and Serezha,
The following questions related to the definition of the functor
of semi-infinite tensor product, i.e., Tor^{\infty/2} in Serezha's
notation, are treated below.
I: which abelian categories of modules are considered;
II: what functor on these categories should be derived;
III: the equivalence relation on the complexes of modules,
i.e., which complexes are to be viewed as trivial;
IV: semiflat complexes of S-modules;
V: constructions of resolutions;
VI: the definition of the derived functor.
You will see that the most convoluted explanations and complicated
formulas appear when one attempts to rewrite all of this in
the language of the algebras A and A^#, i.e., without explicit use
of comodules and semialgebras. At the same time, expressed in terms
of the coalgebra C, the semialgebra C, and (co)modules over them,
all the operations look rather simple and clear (though the somewhat
intricate calculations in part V may be an exception).
I-1. Suppose given a coalgebra C and a semialgebra S over it, i.e.,
S is an algebra object in the tensor category of bicomodules over C
with respect to the cotensor product. It is assumed that S is
injective over C both on the left and on the right. The arguments
of the semi-infinite homology functor will be the complexes of left
and right S-modules (i.e., resp., left and right C-comodules endowed
with an action of S).
I-2. If C is finite-dimensional, then mentioning coalgebras etc. can
be avoided. Instead, one considers an algebra A and
a finite-dimensional subalgebra N in it. It is assumed that
a. A is a projective left N-module; and
b. the tensor product S = N^*\ot_N A is
an injective right N-module.
Put A^# := Hom_{N-right}(N^*, S). By construction, A^# is
an N-bimodule endowed with a bimodule map N \to A^#. In fact, there
is an algebra structure on A^# (as explained in the letter number 2
of the previous series, i.e., from Summer 2000) and N is
a subalgebra in it.
The arguments of the semi-infinite homology functor are complexes
of left A^#-modules and complexes of right A-modules.
I-3. The correspondence between the sets of data 1. and 2. consists
in setting C = N^*, S = N^*\ot_N A = A^#\ot_N N^*. Conversely,
we have A = Hom_{N-left}(N^*,S), A^# = Hom_{N-right}(N^*,S). Right
S-modules are the same things as right A-modules, while the left
S-modules are the left A^#-modules. In particular, S itself is
a bimodule, over A^# on the left and over A on the right.
Remark: I do not think that one can rigorously correctly avoid
the use of coalgebras and comodules in the situation when C
(equivalently, N) is infinite-dimensional.
II-1. The semi-infinite homology is the derived functor of
the functor of tensor product of S-modules. This functor is
defined as follows. If M and L are a right and a left S-module,
then the vector space M\ot_S L is the cokernel of the map
M\oc_C S\oc_C L \to M\oc_C L,
where \oc_C denotes the cotensor product over C. So the tensor
product over a semialgebra is a quotient space of a certain
subspace of the tensor product of vector spaces. This functor is
neither left nor right exact.
II-2. In terms of the algebras A and A^# with the subalgebra N
the desired functor can be expressed in several ways. Firstly,
in the cokernel formula above one can rewrite M\oc_C S\oc_C L
as Hom_N(N, M\ot_N A\ot_k L) = Hom_N(N, M\ot_k A^#\ot_N L)
and M\oc_C L as Hom_N(N, M\ot_k L). In this form, one can try
to compare this construction to Sevastyanov's definition of
the semitensor product as ostensibly the image of the cotensor
product in the tensor product (though in fact such a comparison
does not work very well, and Sevastyanov's definition seems
to be not quite right to me).
In my opinion, Serezha's formulas from his paper "Semi-infinite
cohomology of quantum groups II" (see Subsection 4.7 titled "Choice
of resolutions") are much more interesting. Let M be a right
A-module and L be a left A^#-module. Then if M is injective
over N, then M\ot_S L = Hom_{A-right}(S,M) \ot_{A^#} L; and if
L is injective over N, then M\ot_S L = M \ot_A Hom_{A^#-left}(S,L).
The (not quite self-obvious) proof of these assertions is left to
the readers.
In the exposition below we do not use the language of A and A^#.
I will speak exclusively about complexes of left or right modules
over a semialgebra S (over a coalgebra C).
III. In my view, the correct definition of the equivalence
relation on complexes of modules to which the functor of
semi-infinite homology is applied is a certain mixture of two
kinds of equivalence relations which I always told you about as
"the derived categories D and D-prime". Let me recall that
if E is a DG-algebra or a DG-coalgebra, then there are two main
ways to define the derived category of DG-(co)modules over E.
Specifically, in the derived category "D" a DG-module is trivial
if its cohomology (with respect to its differential) are trivial.
In the category "D-prime", the trivial DG-modules are the ones
which can be obtained from the total modules of exact triples of
DG-modules using the operations of cone and infinite direct sum.
The derived category that is needed for semi-infinite homology
is sort of a category "D-prime" along the coalgebra C, but
the category "D" along the complementary directions in S (so to
speak, "D-prime" along the N-half, but "D" along the B-half).
There is one aspect here that the semialgebra S (as well as
the algebras A and A^#) carry no nontrivial differential, but
the theory of D and D-prime can be not quite trivial even for
such DG-algebras that are in fact usual algebras (even
concentrated in the homological degree zero).
The formal definition is uncomplicated: suppose given
a (generally speaking, doubly unbounded and in all respects
infinite-dimensional) complex of modules X over a semialgebra S.
Let us apply the forgetful functor and consider X as a complex of
comodules over the coalgebra C. A complex X is called trivial if
it is trivial as a complex of comodules over C, in the sense of
the following definition.
A complex of comodules X over C is called trivial if it satisfies
the following two equivalent conditions:
(a) for any complex of injective (for example, cofree) comodules I
over C we have Hom(X,I)=0 in the homotopy category of C-comodules;
or
(b) X as a complex of comodules can be obtained from the total
complexes of exact triples of complexes of comodules using
the operations of cone and infinite direct sum.
For example, if C is trivial, i.e., C=k=N, then we have S=A=A^#=B
and the above-defined trivial complexes of S-modules are the most
usual (doubly unbounded) acyclic complexes of modules over
the algebra A (or, which is the same, B).
Definition: by the derived category of left or right modules over
a semialgebra S one means the quotient category of the homotopy
category of complexes of S-modules by the subcategory of trivial
complexes defined above. Analogously one defines the derived
category of (left or right) C-comodules.
IV. I recall that on S-modules there is the operation of tensor
product over S, which was discussed in part II. As usual, this
operation is extended to complexes of S-modules in the obvious way.
Definition. A complex of left S-modules F is called semiflat if
for any trivial complex of right S-modules T the tensor product
complex T\ot_S F is acyclic as a c-s of vector spaces. Analogously
one defines semiflat complexes of right S-modules.
A complex of left C-comodules E is said to be coflat if for any
trivial complex of right C-comodules U the cotensor product complex
U\ot_C E is acyclic as a complex of vector spaces; analogously for
right C-comodules.
A claim: any semiflat complex of S-modules is coflat as a complex
of C-comodules. Proof: let F be a semiflat complex of left
S-modules; we are interested in a cotensor product U\oc_C F. One
needs to use the induction functor assigning to every C-comodule P
the induced S-module P\ot_C S. It is not difficult to see that
the induction functor takes trivial complexes of C-comodules to
trivial ones (for example, the definition (b) of trivial complexes
is convenient to use). Now we have U\oc_C F = (U\oc_C S) \ot_S F
and the semiflatness implies coflatness.
V. Theorem: the quotient category of the homotopy category of
semiflat complexes of S-modules by the thick subcategory of semiflat
trivial complexes is equivalent, via the natural projection, to
the quotient category of all complexes of S-modules by the trivial
ones (in other words, to the derived category of S-modules).
I will not prove here this theorem about triangulated categories
in its full strength, but will restrict myself only to proving
the surjectivity of the functor in question on the objects. In
other words, I will show that any complex of S-modules is connected
with a certain semiflat complex by a chain (in fact, a chain of
length two, i.e., just a "roof") of maps with trivial cones.
The latter fact can be proved in various ways -- for example, one
can use Koszul duality in the spirit of the way Serezha did it
in his early papers on the topic -- but I will give here a direct,
relatively elementary proof. It is based on the following lemma.
Lemma. There exists a functor assigning to every S-module its
embedding into a C-injective S-module.
A comment: clearly, this is the very same question that in
the stardard expositions of this subject (by both of you, for
example, as well as by Sevastyanov) was solved by "induction
from the subalgebra B". I do not have any such "subalgebra B"
here, so I need a direct way.
V-1. Proof of Lemma. Let M be an arbitrary S-module. Consider
M as a C-comodule and embed into an injective C-comodule P=P(M).
Clearly, one can do it in a functorial way, for example, just
choosing P(M)=C\ot_k M. Denote by Q(M) the quotient module of
the S-module S\oc_C P(M) by its submodule which is the image of
the kernel of the map S\oc_C M \to M under the inclusion
S\oc_C M \to S\oc_C P(M). It is not difficult to see that
the composition M \to P(M) \to S\oc_C P(M) \to Q(M) is a map
of S-modules (while the maps being composed are only C-comodule
maps). Besides, the map from M to Q(M) is injective. However,
Q(M) need not be an injective C-comodule.
One overcomes the latter problem by iterating the construction.
Consider the chain of embeddings M \to Q(M) \to Q(Q(M)) \to ...
I claim that the inductive limit of this chain is injective as
a C-comodule. Indeed, this limit, viewed as a C-comodule, is
also the inductive limit of the chain of injective comodules
S\oc_C P(M) \to S\oc_C P(Q(M)), where the maps in the chain
are the compositions through Q(M), Q(Q(M)), etc. However, it is
not difficult to see that the inductive limits of chains preserve
injectivity of comodules over an arbitrary coalgebra, since any
comodule is a union of finite-dimensional ones. Indeed,
it suffices to extend a map to the direct limit from
a finite-dimensional subcomodule to a finite-dimensional comodule;
now the map from a finite-dimensional comodule factorizes through
one of Q^n(M), and consequetly, through S\oc_C P(Q^n(M)). Thus
the lemma is proved.
V-2. Now let us show that for any complex of S-modules X there is
a map f: X \to Y such that all the terms of the complex Y are
cofree C-comodules, while the cone of f is a trivial complex in
the sense of the definition from part III. This is easy to do.
Using the lemma, embed the complex X into some complex of
C-injective S-modules J, take the quotient complex J/X, embed it
similarly one again, etc. Proceeding in this way, construct
a complex of complexes X \to J \to J_1 \to J_2 \to ... Consider
the total complex Y=Tot(J\to J_1\to...) formed by taking infinite
direct sums along the diagonals.
Obviously, Y is a complex of C-injective S-modules, while f: X\to Y
is a morphism of complexes of S-modules. It remains to show that
after taking the forgetful funtor to complexes of C-comodules
the cone of the morphism f becomes a trivial complex. This is done
in the standard way, just as one proves the similar result about
the category D-prime for comodules over an arbitrary (DG-)coalgebra.
One uses the construction of the homotopy direct limit
("the telescope") and the definition of triviality in the form (b)
from part III.
V-3. It remains to show that for any complex of C-injective
S-modules Y there is a semiflat complex Z together with a morphism
g: Z\to Y such that the cone of g is trivial. Actually, we will
prove more. Specifically:
1. All complexes of S-modules induced (using the operation
S\oc_C *) from complexes of injective C-comodules are semiflat;
2. Semiflat complexes form a triangulated subcategory closed
under infinite direct sums in the homotopy category of complexes
of S-modules;
3. For any complex of C-injective S-modules Y there is
a morphism of complexes of S-modules g: Z\to Y such that
i. the cone of g, viewed as a complex of C-comodules,
is contractible;
ii. the complex Z is constructed from the complexes of type 1.
using the operations of cone and infinite direct sum.
Assertion 1. is obvious, as the cotensor product of a complex of
injective C-comodules with a trivial complex of C-comodules is
an acyclic complex of vector spaces, as one can easily see.
Assertion 2. is clear. Finally, assertion 3. is provable by
taking the total complex of the relative bar-construction over
S relative to C. Specifically, one needs to write
... \to S\oc_C S \oc_C Y \to S\oc_C Y \to Y
and take Z to be the total complex of this whole thing except
the final Y. Item i. is checked by presenting an explicit
canonical homotopy which is intrinsic to bar-constructions of
this kind; while item ii. is provable in the standard way,
using the "telescope" construction again.
The main theorem of part V can be considered to have been proved
(with the caveats made in the beginning of the part).
VI. Now that all the difficult work is over, the definition of
the semi-infinite homology functor can be given by waving one's
hands. We want to define a certain functor of two arguments
which range over the categories of complexes of right and left
S-modules. In fact, this functor will be defined on
the quotient categories of complexes of S-modules by the thick
subcategories of trivial complexes (they are thick, because
any triangulated subcategory closed under countable direct sums
is also closed under direct summands). However, according to
the theorem from part V, the quotient categories we are
interested in are equivalent to the quotient categories of
semiflat complexes of S-modules by the trivial semiflat complexes.
Now on the semiflat complexes the functor of semi-infinite
homology is defined straightforwardly as the tensor product of
modules over the semialgebra S (see part II). Obviously, this
functor on the homotopy categories of semiflat complexes descends
to the quotient categories by trivial (semiflat) complexes
(see part IV). End of the argument.
So I have given the definition of the functor of semi-infinite
homology on the categories of modules over a semialgebra.
P.S. Remark (added in April 2006). In fact, the subcategory in
the homotopy category of complexes of S-modules generated using
the cones and direct sums by the complexes of S-modules induced
from complexes of injective C-comodules (in the next letter,
it is called the resolution subcategory) is equivalent, via
the natural projection, to the derived category of S-modules.
It suffices to show that the intersection of the resolution
subcategory with the subcategory of trivial complexes is zero.
Let X be a complex belonging to the intersection of the two
subcategories. Then X as a complex of C-comodules is
simultaneously a complex of injective C-comodules and a trivial
complex of C-comodules, hence X is contractible over C. Now
X is simultaneously contractible over C and belonging to
the subcategory generated using cones and infinite direct sums
by the complexes induced from C, hence it follows that X is
contractible over S. Moreover, if follows from the arguments
above that the resolution subcategory coincides with the full
subcategory in the homotopy category of S-modules consisting of
all the complexes of injective C-comodules that are K-projective
(in the sense of Spaltenstein) over S relative to C (i.e.,
belong to the left othogonal subcategory to the subcategory of
all complexes contractible over C).
From: Leonid Positselski < posic@mpim-bonn.mpg.de >
To: roma, hippie
Date: Fri, 11 Oct 2002 18:59:59 +0200 (MEST)
Subject: semi-infinite COhomology and contramodules (corrected and supplemented)
Hi Roma and Serezha,
in my previous letter from August 3 of this year what I consider
to be the correct definition of semi-infinite homology of
associative algebraic structures was given. The aim of
the present letter is to define the semi-infinite cohomology,
i.e., Ext_{\infty/2} in Serezha' notation. In my exposition
I will approximately follow the plan sketched in the previous
letter (with the exception of the new part VII of the present
letter, which has no analogue for semi-infinite homology).
I: which abelian categories of modules are considered;
II: what functor on these categories should be derived;
III: the equivalence relation on the complexes of modules,
i.e., which complexes are to be viewed as trivial;
IV: semiprojective complexes of S-modules, semiinjective
complexes of S-contramodules, and coinjective C-contramodules;
V: constructions of resolutions;
VI: the definition of the derived functor;
VII: the equivalence of derived categories; the semi-infinite
cohomology as Hom in a triangulated category.
I will rely in the most essential way on letter number four
(the last one) from the series of "Summer letters" of
the year 2000, in which the definition of the category of
contramodules, and in particular, the notion of a contramodule
over a semialgebra was spelled out.
I-1. Suppose given a coalgebra C and a semialgebra S over it;
it is assumed that S is injective over C both on the left and
on the right. The first arguments of the functor of
semi-infinite Ext will be complexes of left S-modules (see
the previous letter). The second arguments of the functor of
semi-infinite Ext will be complexes of left S-contramodules
(see letter number four from the Summer series of 2000).
I-2. If C is finite-dimensional, then instead of a semialgebra S
one can speak about a special kind of pair of associative algebras
A and A^# endowed with a common subalgebra N (dual to C).
In this case, the right S-modules are the right A-modules,
the left S-modules are the left A^#-modules, while the left
S-contramodules are the left A-modules, and finally, the right
S-contramodules are the right A^#-modules. The first arguments
of the functor of semi-infinite Ext turn out to be complexes of
left A^#-modules, the second ones are complexes of left A-modules.
I-3. As one can see from the above, it is impossible to avoid
using contramodules in the construction of the theory of
semi-infinite COhomology in terms of a semialgebra S even in
the case when both C and S, as well as all the modules under
consideration, are finite-dimensional. Although, if one is
speaking about contramodules over a coalgebra only, then,
of course, left C-comodules and left C-contramodules are
the same things if C is finite-dimensional.
II-1. The semi-infinite cohomology are the derived functor of
the functor of cohomomorphisms Cohom over a semialgebra S.
The letter functor assigns to every left S-module L and every
left S-contramodule P the vector space Cohom_S(L,P) of
"cohomomorphisms over S from L to P". This functor is defined
as "the kernel of a certain map from the cokernel of a certain
map to the cokernel of a certain map"; for this reason, it is
obviously neither left nor right exact with respect to either
the first or the second argument.
II-2. There is the following connection between my definition of
the functor Cohom_S and Serezha's formulas for the semi-infinite
Ext from Subsection 4.7 ("Choice of resolutions") of the paper
"Semi-infinite cohomology of quantum groups II". Let L be
a left A^#-module and P be a left A-module. Then Cohom_S(L,P) =
Hom_{A^#}(L, S\ot_A P) if P is projective as a left N-module,
and Cohom_S(L,P) = Hom_A(Hom_{A^#}(S,L), P) if L is injective
as a left N-module.
Moreover, for any left S-module L and left S-contramodule P,
over any semialgebra S, there are the identities
Cohom_S(L,P) = Hom_S(L, C\ocn_C P) if P is a projective
C-contramodule, and Cohom_S(L,P) = Hom_{S-contra}(Hom_S(S,L), P)
if L is an injective C-comodule. Here P \mapsto C\ocn_C P
and L \mapsto Hom_C(C,L) = Hom_S(S,L) are the functors between
the categories of left S-modules and left S-contramodules
introduced in the end of the fourth letter of the 2000 series.
The proof or refutation is left to the readers. This is
the functor whose derived functor will be the semi-infinite
cohomology.
III. Now I have to define the correct equivalence relation on
the complexes that are the arguments of the purported derived
functor. The equivalence relation on complexes of S-modules was
introduced in the previous letter; it remains to spell out the case
of S-contramodules. For the latter ones, the correct equivalence
relation is a mixture of the "relation D-second" along
the C-contramodule structure and "the relation D in
the perpendicular direction". Let me recall that in the categories
"D-second" the trivial objects are the complexes that can be
obtained from the totalizations of exact triples of complexes using
the operations of taking the cone and infinite direct product (as
opposed to the infinite direct sum for the categories D-prime).
The formal definition: a complex of S-contramodules (generally
speaking, unbounded in all directions) is called trivial if it is
trivial as a complex of C-contramodules (i.e., after forgetting
the rest of the structure). A complex of C-contramodules X
is called trivial if it satisfies the following two equivalent
conditions:
(a) for any complex of projective (for example, free) contramodules
P over C we have Hom(P,X)=0 in the homotopy category of
C-contramodules; or
b) X as a complex of contramodules can be obtained from the total
complexes of exact triples of complexes of contramodules using
the operations of cone and infinite product.
Definition: by the derived category of left contramodules over
a semialgebra S one means the quotient category of the homotopy
category of complexes of left S-contramodules by the subcategory
of trivial complexes defined above. Analogously one defines
the derived category of left C-contramodules.
IV-1. One extends the operation Cohom_S from (contra)modules
to complexes of (contra)modules by the following obvious rule:
Cohom_S(L,P) is the complex obtained by taking infinite products
along the diagonals of the bicomplex Cohom_S(L^i, P^j). Let me
point out that for operations of "homomorphism type", unlike
for operations of "tensor product type", one has to take
the infinite product (rather than the direct sum) along
the diagonals in this context.
Two definitions: a complex of left S-modules F is called
semiprojective if for any trivial complex of left S-contramodules
T the complex Cohom_S(F,T) is acyclic as a complex of vector
spaces. A complex of left S-contramodules P is called
semiinjective if for any trivial complex of left S-modules T
the complex Cohom_S(T,P) is acyclic.
Analogously, a complex of left C-comodules is called coprojective
if, for any trivial complex of left C-contramodules, the complex
Cohom_C between them is acyclic. A complex of left C-contramodules
is called coinjective if, for any trivial complex of left
C-comodules, the corresponding complex of vector spaces Cohom_C
is acyclic.
IV-2. It is clear that for complexes of C-comodules the three
properties -- injectivity (in the sense of the resp. derived
category), coflatness (in the sense of the previous letter) and
coprojectivity (in the sense of the last definition) -- are all
rather close to each other. It is not difficult to see that
any complex of injective comodules is coprojective and any
coprojective complex is coflat. Analogously, any complex of
projective C-contramodules is coinjective.
The most interesting question: how are the properties of
semiflatness and semiprojectivity related to each other for
complexes of S-modules? A partial answer: all semiprojective
complexes are semiflat (proof: use the fact that the dual
vector space to a right S-module is a left S-contramodule,
and the formula Cohom_S(L, M^*) = (M\ot_S L)^*).
Two assertions: any semiprojective complex of left S-modules
is coprojective as a complex of left C-comodules. Any
semiinjective complex of left S-contramodules is coinjective
as a complex of left C-contramodules. Proofs: in the former
case use the following coinduction functor for contramodules:
to every C-contramodule P the coinduced S-contramodule
Cohom_C(S,P) is assigned. In the latter case use the induction
functor from the previous letter.
IV-3. In fact, for C-co/contramodules all of this is somewhat
simpler. As we know, the main property of the derived categories
D-prime and D-second is that the adjustedness properties of
complexes in them can be checked termwise. For example, a complex
of C-comodules is injective (right orthogonal to trivial complexes
with respect to the usual Hom functor in the homotopy category
of comodules) if and only if it is homotopy equivalent to a complex
of injective objects. The same holds for contramodules and
projective complexes. So it makes sense to give several further
definitions.
A left C-comodule L is called coflat if the functor - \oc_C L is
exact on the abelian category of right C-comodules. A left
C-comodule L is called coprojective if the functor Cohom_C(L,-) is
exact on the abelian category of left C-contramodules. In fact,
it is easy to see that all the three properties of injectivity,
coflatness, and coprojectivity of C-comodules are equivalent
(a hint: one should restrict oneself to finite-dimensional second
arguments of the functors Hom, \oc, and Cohom, respectively, and
compare in this situation). Now one could, of course, notice that
any complex of coflat comodules is coflat and any complex of
coprojective ones is coprojective. These are undoubtedly true
but, in view of the above, trivial assertions.
A left C-contramodule P is called coinjective if the functor
Cohom_C(-,P) is exact on the abelian category of left C-comodules.
A left C-contramodule P is called contraflat if the functor
of contratensor product - \ocn_C P is exact on the category of
right C-comodules. It is not difficult to see that any
projective C-contramodule is coinjective and any coinjective
C-contramodule is contraflat (the converse assertion is discussed
in part VII). Any complex of coinjective C-contramodules is
a coinjective complex.
V. Teorema 1: the analogue of the main theorem from part V of
the previous letter holds for semiprojective complexes of
S-modules in lieu of the semiflat ones. The proof is exactly
the same as in the previous letter.
Teorema 2: the quotient category of the homotopy category of
semiinjective complexes of S-contramodules by the thick subcategory
of semiinjective trivial complexes is equivalent, via the natural
projection, to the quotient category of the homotopy category of
all S-contramodules by the trivial complexes.
Here one has to rewrite the argument from the previous letter
replacing comodules by contramodules. I will do it, cutting down
on some details.
Lemma. There exists a functor assigning to every S-contramodule
a surjective map onto it from a C-coinjective S-contramodule.
Of course, the closest analogue of Lemma from part V of
the previous letter would be existence of such a surjection from
a C-projective S-contramodule. The only problem is that I am
unable to prove that the inverse limit preserves projectivity of
contramodules -- while for coinjectivity I know how to check it
in the context we need. For the purposes of the present section,
C-coinjectivity is sufficient, while in part VII, where
C-projective resolutions will be needed, I will have to use
a certain conjecture.
V-1. Proof of Lemma. Let P be a left S-contramodule. Consider
it as a C-contramodule and present it as an epimorphic image of
a projective C-contramodule F. This can be easily done in
a functorial way, for example, taking F(P) = Hom_k(C,P).
Denote by Q(P) the kernel of the composition of maps
Cohom_C(S, F(P)) \to Cohom_C(S,P) \to Cohom_C(S,P)/im P,
where P \to Cohom_C(S,P) is the structure map of
the S-contramodule P. It is not difficult to see that
the composition Q(P) \to Cohom_C(S, F(P)) \to F(P) \to P
is a map of S-contramodules, while the maps being composed are
only maps of C-contramodules. Besides, the map from Q(P) to P
is surjective, while the C-contramodule Cohom_C(S, F(P)) is
projective. Now the assertion of the lemma follows from
the next sublemma.
Sublemma. Let ... \to Q_2 \to T_2 \to Q_1 \to T_1 \to Q_0
be a projective system of C-contramodules, where the contramodules
T_i are coinjective, while in the projective subsystem consisting
of Q_i only, all the maps are surjective. Then the inverse limit
lim Q_i = lim T_i is a coinjective contramodule. Proof of
Sublemma: one needs to check that for any exact triple of left
C-comodules L_1 \to L_2 \to L_3 one has an exact triple of vector
spaces Cohom(L_3, lim T_i|Q_i) \to Cohom(L_2, lim)
\to Cohom(L_1, lim). Let us first prove that for any
C-comodule L we have Cohom(L, lim) = lim Cohom(L, T_i|Q_i).
For any contramodule X, write the bar-construction
... \to Hom_k(C\ot_k C\ot_k L, X) \to Hom_k(C\ot_k L, X)
\to Hom_k(L,X); then Cohom_C(L,X) is, by definition,
the degree-zero homology of this complex. Now suppose that we
are given a projective system of (homological) complexes where
all the complexes with odd numbers have homology in degree zero
only, while the complexes with even numbers form a projective
subsystem with surjective transition maps. Then I claim that
the inverse limit complex has homology in degree zero only,
and it is equal to the inverse limit of the degree-zero homology
of the complexes in the system. Indeed, the projective systems
in which the subsystems with even numbers have surjective
transition maps are acyclic objects for the derived functor of
inverse limit lim^1. The rest of the proof of the sublemma is
left to the readers.
V-2. On the next step we need to show that for any complex of
S-contramodules X there is a map f: Y\to X such that all the terms
of the complex Y are conjective C-contramodules, while the cone
of f is a trivial complex. This is done exactly in the same way
as in item V-2 of the previous letter, with the difference that
the bicomplex is being totalized using direct products (and not
direct sums) along the diagonals. Then one uses the homotopy
inverse limit, and generally everything happens as it usually
does in D''.
V-3. On the last step it remains to prove that for any complex of
C-coinjective S-contramodules Y there is a semiinjective complex
of S-contramodules Z together with a morphism g: Y\to Z such that
the cone of g is trivial (in fact, even C-contractible). This is
done exactly in the same way as in section V-3 of the previous
letter, except that instead of the induced S-modules one uses
the coinduced S-contramodules Cohom_C(S,-) and the total complex
(unlike in the context of the previous letter) is, once again,
constructed by taking infinite products. Generally, if
the "constructions from sections V-2" are typical for
the derived categories D' and D'', then quite similarly
the "constructions from sections V-3" are typical for the more
classical (Spaltenstein's) derived categories D.
VI. Now the functor of semi-infinite cohomology can be defined
in the way completely analogous to the definition of
the semi-infinite homology in the previous letter.
VII. The aim of this part is to prove two assertions:
(1) the above-defined derived categories of left S-modules and
left S-contramodules are naturally equivalent; (2) under this
equivalence, the semi-infinite cohomology functor corresponds
to the Hom functor.
VII-1. I will start with a certain comment on Theorems 1-2 from
part V. In fact, the proof of these theorems proves more than
claimed in their formulation. Specifically, consider the full
triangulated subcategory in the homotopy category of complexes of
S-modules consisting of all the complexes that can be obtained
by cones, shifts, and infinite direct sums from the complexes
induced from complexes of injective C-comodules. Let us call it
the resolution subcategory in the homotopy category of complexes
of S-modules. Analogously, the resolution subcategory in
the homotopy category of complexes of S-contramodules consists of
all the complexes that can be obtained by cones, shifts, and
infinite products from the complexes coinduced from complexes of
coinjective C-contramodules. The proof of Theorems 1-2 actually
proves the following: any full triangulated subcategory in
the homotopy category of complexes of S-modules or S-contramodules
which contains the resolution subcategory, after taking its
quotient category by the intersection with the full subcategory
of trivial complexes, becomes equivalent, via the natural
projection, to the respective derived category.
Correction (added in November 2006). The generalization of
Theorems 1-2 stated here is a bit too general, and I do not know
if whether it can be proved. A correct assertion would be, for
example, this: any full subcategory in the homotopy category
of complexes of S-modules contained in the full subcategory of
all complexes of C-injective S-modules and containing
the resolution subcategory, and analogously any full subcategory
of complexes of S-contramodules contained in the subcategory of
of all complexes of C-projective S-contramodules, after taking
its quotient by the interesection with the full subcategory of
trivial complexes, becomes equivalent, via the natural projection,
to the respective derived category.
In particular, this is true for the full subcategory consisting of
all complexes of C-injective S-modules. Its intersection with
the full subcategory of trivial complexes consists of all
complexes of C-injective S-modules that are contractible over C.
Thus the derived category of S-modules is equivalent to
the quotient category of the homotopy category of complexes of
C-injective S-modules by the full subcategory of C-contractible
C-injective complexes.
VII-2. In order to claim the same for S-contramodules, I will need
to use the following conjecture.
Conjecture 1. For any coalgebra C, the classes of contraflat,
coinjective, and projective C-contramodules coincide. In other
words, every contraflat C-contramodule is projective.
Example. Suppose that the coalgebra C is finite-dimensional.
Then C-comodules and C-contramodules are simply modules over
the finite-dimensional algebra N dual to C. The contraflat
contramodules are the flat modules, and it is well-known that
any flat module over a finite-dimensional algebra is projective
[see e.g. Bass' paper in Trans. AMS v.95 for 1960].
I have the feeling that Conjecture 1 may be deducible from
the following much more fundamental assertion about contramodules.
Conjecture 2. For any contramodule P over a coalgebra C,
the intersection of the images of the vector spaces Hom_k(C/V, P)
under the structure map Hom_k(C,P) \to P, taken over all
the finite-dimensional subspaces (or, if one wishes, subcoalgebras)
V in C, is equal to zero in P.
Example. In the fourth letter of the year 2000 series, a question
was asked, how to prove that any finite-dimensional C-contramodule
is a contramodule over a finite-dimensional subcoalgebra in C. One
can see that this assertion is a particular case of Conjecture 2.
VII-3. Using Conjecture 1, one can claim that the full subcategory
in the homotopy category of S-contramodules consisting of all
complexes of C-projective contramodules contains the resolution
subcategory. Its intersection with the full subcategory of trivial
complexes consists of all the complexes of C-projective
S-contramodules that are contractible over C. Therefore,
the derived category of S-contramodules is equivalent to
the quotient category of the homotopy category of complexes of
C-projective S-contramodules by the full subcategory of
C-contractible C-projective complexes.
In the end of the 4th letter from the "Summer series" of the year
2000 it was shown that the additive categories of injective
C-comodules and projective C-contramodules are equivalent, and
S-module structures on C-comodules correspond bijectively to
S-contramodule structures on C-contramodules under this equivalence
-- so the categories of C-injective S-modules and C-projective
S-contramodules are also equivalent. Now we immediately obtain
from this an equivalence of the derived categories of S-modules
and S-contramodules.
VII-4. It is claimed that this equivalence transforms the functor
of semi-infinite cohomology into the Hom functor. This follows
from the formulas written down in section II-2 of the present
letter. More precisely, one can argue as follows. Let an object
of the derived category of S-modules be represented by a complex
of C-injective S-modules L, while an object of the derived category
of S-contramodules be represented by a complex of C-projective
S-contramodules P. Then the derived functor of the functor Cohom
on the objects L and P can be computed by choosing maps with
trivial cones L_1 \to L and P \to P_1, where L_1 and P_1 belong
to the resolution subcategories, and computing Cohom(L_1,P_1),
or Cohom(L_1,P), or Cohom(L,P_1) (which is the same). According
to the formula from section II-2, we know that Cohom(L_1,P_1) =
Hom(\psi(L_1), P_1), where \psi denotes the functor from
C-injective S-modules to C-projective S-contramodules, while
the Hom is taken in the homotopy category. Now it remains to show
that \psi(L_1) belongs to the left orthogonal subcategory and
P_1 to the right orthogonal subcategory to the subcategory of
C-contractible complexes in the homotopy category of complexes of
C-projective S-contramodules (or at least one of these two
assertions -- in fact, both are true). Indeed, let X be such
a C-contractible complex; then Hom(\psi(L_1), X) = Cohom(L_1, X)
= 0, since the complex L_1 is semiprojective. Analogously
Hom(X, P_1) = Cohom(\psi^{-1}(X),P_1) = 0, since the complex
psi^{-1}(X) is also C-contractible, while P_1 is semiinjective.
Looking into this argument, one can notice that even without
the assumption of Conjecture 1 it proves existence of a functor
\Psi from the derived category of S-modules to the derived
category of S-contramodules such that the derived functor of
Cohom is equal to the composition of \Psi with the Hom functor
in derived category of contramodules. Conjecture 1 is needed
in order to show that \Psi is a category equivalence. When
the coalgebra C is finite-dimensional, we know this.
P.S. Addition (May 2006). Here is the simplest counterexample to
Conjecture 2 (another counterexample was given in an Addition in
the fourth letter for the year 2000). Consider the coalgebra C
dual to the pro-finite-dimensional algebra of formal power series
in one variable C^* = k[[x]]. Then the C-contramodules are
the k-vector spaces P endowed with an operatoin of summation of
sequences of vectors with the formal coefficients x^n, i.e.,
for any p_0, p_1, ... in P the sum \sum x^i p_i is defined as
an element of P. Consider the free contramodule generated by
a sequence e_0, e_1, ... ; its elements are the formal sums
\sum a_i(x)e_i, where the formal power series a_i(x) have
the property that ord_x a_i(x) tends to infinity as i grows.
Now consider a homomorphism from the free contramodule F generated
by f_1, f_2, ... to the free contramodule E generated by
e_0, e_1, e_2, ... -- generally, such homomorphisms correspond
bijectively to arbitrary sequences of elements of E -- the images
of f_i -- and we are intereseted in the homomorphism taking f_i
to x^i e_i - e_0. An arbitrary element \sum a_i(x)f_i of
the contramodule F is taken by this map to \sum x^i a_i(x) e_i -
(\sum a_i(x))e_0. It is immediately clear from this that
the element e_0 does not belong to the image of this homomorphism.
Consider the quotient contramodule P=E/im F; it is claimed that
the coset of the element e_0 in P belongs to im Hom(C/V,P) for
all finite-dimensional subcoalgebras V in C. Indeed, for any V
there is i such that x^i as an element of C^* annihilates V;
so the expression x^i e_i represents an element of Hom(C/V,P)
whose image in P is equal to e_0.
P.P.S. Addition (June 2006). The following weaker version of
Conjecture 2 is true: for any nonzero C-contramodule P there is
a finite-dimensional subcoalgebra V in C (which can be chosen
to be simple, i.e., containing no nontrivial subcoalgebras)
such that the image of Hom(C/V,P) in P is not equal to P.
The proof consists of two lemmas.
Lemma 1 (Nakayama's lemma for contramodules). Let C^ss denote
the maximal semisimple subcoalgebra of C. Then for any nonzero
C-contramodule P the image of Hom(C/C^ss,P) in P is not equal to P.
Proof: notice that the coalgebra without counit D=C/C^ss is
conilpotent, i.e., any element of D is annihilated by the iterated
comultiplication map D\to D^{\ot i} for i large enough. Let us
show that for any contramodule P over a conilpotent coalgebra
without counit D surjectivity of the map Hom(D,P)\to P implies P=0.
Indeed, assume that P = im Hom(D,P). Let p be some element of P;
it comes from some map f_1: D\to P. Since the map Hom(D,P)\to P
is surjective, the map f_1 can be lifted to a certain map
D \to Hom(D,P), which leads to a map f_2: D\ot D \to P, etc. So
one constructs a sequence of maps f_i: D^{\ot i} \to P such that
f_{i-1}=m'_1(f_i), where m' is the contraaction map Hom(C,P) \to P
and m'_1 denotes the application of m' along the first tensor
factor in D^{\ot i}. Put g_i=m_{2..i}(f_i), i=2,3,..., where m is
the comultiplication map D \to D\ot D and m_{2..i} denotes
the substitution of the iterated comultiplication D \to D^{\ot i-1}
into the components numbered 2 to i in the tensor product D^{\ot i}.
Then g_i are maps D\ot D \to P. We have
m'_1(g_i)=m_{1..i-1}(f_{i-1}) i m_{1..2}(g_i)=m_{1..i}(f_i).
Notice that, as it follows from conilpotency of the coalgebra D,
the series \sum_{i=2}^\infty g_i converges in the sense of pointwise
limit of functions D\ot D \to P, and even in the sense of pointwise
limit of functions D \to Hom(D,P). (Here, as above,
the identification Hom(X, Hom(Y,Z)) = Hom(Y\ot X, Z) is presumed.)
Thus m'_1(\sum g_i) = \sum m_{1..i-1}(f_{i-1}) i m_{1..2}(\sum g_i)
= \sum m_{1..i}(f_i), hence we obtain m'_1(\sum g_i) -
m_{1..2}(\sum g_i) = f_1, so m'(f_1)=0. Lemma 1 is proved.
Lemma 2. Suppose that a coalgebra C is a direct sum of a family
of coalgebras C_a. Then any C-contramodule is a direct product of
contramodules over C_a. Proof: obvivously, the assertion of Lemma
holds for any free C-contramodule. Now let a C-contramodule P be
a direct product of C_a-contramodules P_a. Let us show that any
subcontramodule R in P is the direct product of its images R_a under
the projections P\to P_a. Suppose given a collection of elements
r_a in R. Consider the linear map f: C\to R whose restriction to
C_a is equal to the composition C_a\to k\to R, where the first map
being composed is the counit of the coalgebra C_a, while the second
map takes 1 to r_a. Denote by r the image of the functional f under
the contraaction map Hom(C,R)\to R. Then it is clear that the image
of r under the projection P\to P_a is equal to the image of r_a
under this projection. Thus R is the direct product of R_a. Now
it remains to notice that any C-contramodule is the quotient
contramodule of a free contramodule by some subcontramodule.
P.P.P.S. Addition (June 2006). Here is a proof of Conjecture 1
based on the results from the previous Addition. For any
C-contramodule X and any subcoalgebra V in C denote by
X_V = X/im Hom_k(C/V,X) = Cohom_C(V,X) the maximal subcontramodule
of X which is a contramodule over V. Let C^ss denote the maximal
semisimple subcoalgebra of C.
Lemma. For any C^ss-contramodule T there is a projective
C-contramodule P such that P_{C^ss} is isomorphic to T.
Proof: According to Lemma 2 above, T is a direct product of
contramodules over the simple components C_a of the semisimple
coalgebra C^ss. Any C_a-contramodule, in turn, is a direct sum of
some number of copies of the unique irreducible C_a-contramodule.
Hence one can easily conclude that it suffices to consider the case
when T is an irreducible C_a-contramodule. Let e_a be an idempotent
of the algebra C_a^* such that T is isomorphic to C_a^*e_a.
Consider the idempotent linear functional e_ss on the coalgebra C^ss
equal to e_a on C_a and zero on C_b for b not equal to a. It is
well-known that for any surjective ring homomorphism A\to B whose
kernel is a nil ideal in A, any idempotent element in B can be
lifted to an idempotent in A. Using this fact for
finite-dimensional algebras and Zorn's lemma, it is not difficult
to show that any idempotent linear functional on C^ss can be
extended to an idempotent linear functional on C. Let e be such
an idempotent element of C^* extending e_ss; set P = C^*e. It is
easy to see that the C^ss-contramodule P_{C^ss} is isomorphic to T.
The lemma is proved.
Now let Q be a contraflat C-contramodule; let us show that it is
projective. Consider a projective C-contramodule P for which
P_{C^ss} is isomorphic to Q_{C^ss}. Since P is projective,
the surjective map P\to Q_{C^ss} can be lifted to a homomorphism of
contramodules f: P\to Q. Since (coker f)_{C^ss} = coker(f_{C^ss})
= 0, it follows from Lemma 1 above that the homomorphism f is
surjective. It remains to show that f is injective. Notice that
for any right comodule M over a subcoalgebra V of the coalgebra C
there is an isomorphism M\ocn_C Q = M\ocn_V Q_V implying that
the V-contramodule Q_V is contraflat. Now assume that V is
finite-dimensional; then Q_V is a flat V^*-module. Consider
the map f_V: P_V\to Q_V and denote its kernel by K. For any
right V^*-module M we have a short exact sequence
0 \to M\ot_{V^*}K \to M\ot_{V^*} P_V \to M\ot_{V^*} Q_V \to 0.
In particular, since for any simple subcoalgebra V_a in V the map
V_a^*\ot_{V^*}f_V = f_{V_a} is an isomorphism, we can conclude that
the module V_a^*\ot_{V^*}K = K_{V_a} is zero. It follows that K=0
and f_V is an isomorphism. Finally, let R be the kernel of the map
f: P\to Q. Since f_V is an isomorphism, the subcontramodule R is
contained in the image of Hom(C/V,P) in P for any finite-dimensional
subcoalgebra V in C; but the intersection of all such images is
zero, since the C-contramodule P is projective.
The beginning (four Summer 2000 letters) is here -- https://posic.livejournal.com/2773832.html
From: Leonid Positselski < posic@mpim-bonn.mpg.de >
To: roma, hippie
Date: Fri, 11 Oct 2002 18:56:58 +0200 (MEST)
Subject: definition of semi-infinite homology (corrected)
Hi Roma and Serezha,
The following questions related to the definition of the functor
of semi-infinite tensor product, i.e., Tor^{\infty/2} in Serezha's
notation, are treated below.
I: which abelian categories of modules are considered;
II: what functor on these categories should be derived;
III: the equivalence relation on the complexes of modules,
i.e., which complexes are to be viewed as trivial;
IV: semiflat complexes of S-modules;
V: constructions of resolutions;
VI: the definition of the derived functor.
You will see that the most convoluted explanations and complicated
formulas appear when one attempts to rewrite all of this in
the language of the algebras A and A^#, i.e., without explicit use
of comodules and semialgebras. At the same time, expressed in terms
of the coalgebra C, the semialgebra C, and (co)modules over them,
all the operations look rather simple and clear (though the somewhat
intricate calculations in part V may be an exception).
I-1. Suppose given a coalgebra C and a semialgebra S over it, i.e.,
S is an algebra object in the tensor category of bicomodules over C
with respect to the cotensor product. It is assumed that S is
injective over C both on the left and on the right. The arguments
of the semi-infinite homology functor will be the complexes of left
and right S-modules (i.e., resp., left and right C-comodules endowed
with an action of S).
I-2. If C is finite-dimensional, then mentioning coalgebras etc. can
be avoided. Instead, one considers an algebra A and
a finite-dimensional subalgebra N in it. It is assumed that
a. A is a projective left N-module; and
b. the tensor product S = N^*\ot_N A is
an injective right N-module.
Put A^# := Hom_{N-right}(N^*, S). By construction, A^# is
an N-bimodule endowed with a bimodule map N \to A^#. In fact, there
is an algebra structure on A^# (as explained in the letter number 2
of the previous series, i.e., from Summer 2000) and N is
a subalgebra in it.
The arguments of the semi-infinite homology functor are complexes
of left A^#-modules and complexes of right A-modules.
I-3. The correspondence between the sets of data 1. and 2. consists
in setting C = N^*, S = N^*\ot_N A = A^#\ot_N N^*. Conversely,
we have A = Hom_{N-left}(N^*,S), A^# = Hom_{N-right}(N^*,S). Right
S-modules are the same things as right A-modules, while the left
S-modules are the left A^#-modules. In particular, S itself is
a bimodule, over A^# on the left and over A on the right.
Remark: I do not think that one can rigorously correctly avoid
the use of coalgebras and comodules in the situation when C
(equivalently, N) is infinite-dimensional.
II-1. The semi-infinite homology is the derived functor of
the functor of tensor product of S-modules. This functor is
defined as follows. If M and L are a right and a left S-module,
then the vector space M\ot_S L is the cokernel of the map
M\oc_C S\oc_C L \to M\oc_C L,
where \oc_C denotes the cotensor product over C. So the tensor
product over a semialgebra is a quotient space of a certain
subspace of the tensor product of vector spaces. This functor is
neither left nor right exact.
II-2. In terms of the algebras A and A^# with the subalgebra N
the desired functor can be expressed in several ways. Firstly,
in the cokernel formula above one can rewrite M\oc_C S\oc_C L
as Hom_N(N, M\ot_N A\ot_k L) = Hom_N(N, M\ot_k A^#\ot_N L)
and M\oc_C L as Hom_N(N, M\ot_k L). In this form, one can try
to compare this construction to Sevastyanov's definition of
the semitensor product as ostensibly the image of the cotensor
product in the tensor product (though in fact such a comparison
does not work very well, and Sevastyanov's definition seems
to be not quite right to me).
In my opinion, Serezha's formulas from his paper "Semi-infinite
cohomology of quantum groups II" (see Subsection 4.7 titled "Choice
of resolutions") are much more interesting. Let M be a right
A-module and L be a left A^#-module. Then if M is injective
over N, then M\ot_S L = Hom_{A-right}(S,M) \ot_{A^#} L; and if
L is injective over N, then M\ot_S L = M \ot_A Hom_{A^#-left}(S,L).
The (not quite self-obvious) proof of these assertions is left to
the readers.
In the exposition below we do not use the language of A and A^#.
I will speak exclusively about complexes of left or right modules
over a semialgebra S (over a coalgebra C).
III. In my view, the correct definition of the equivalence
relation on complexes of modules to which the functor of
semi-infinite homology is applied is a certain mixture of two
kinds of equivalence relations which I always told you about as
"the derived categories D and D-prime". Let me recall that
if E is a DG-algebra or a DG-coalgebra, then there are two main
ways to define the derived category of DG-(co)modules over E.
Specifically, in the derived category "D" a DG-module is trivial
if its cohomology (with respect to its differential) are trivial.
In the category "D-prime", the trivial DG-modules are the ones
which can be obtained from the total modules of exact triples of
DG-modules using the operations of cone and infinite direct sum.
The derived category that is needed for semi-infinite homology
is sort of a category "D-prime" along the coalgebra C, but
the category "D" along the complementary directions in S (so to
speak, "D-prime" along the N-half, but "D" along the B-half).
There is one aspect here that the semialgebra S (as well as
the algebras A and A^#) carry no nontrivial differential, but
the theory of D and D-prime can be not quite trivial even for
such DG-algebras that are in fact usual algebras (even
concentrated in the homological degree zero).
The formal definition is uncomplicated: suppose given
a (generally speaking, doubly unbounded and in all respects
infinite-dimensional) complex of modules X over a semialgebra S.
Let us apply the forgetful functor and consider X as a complex of
comodules over the coalgebra C. A complex X is called trivial if
it is trivial as a complex of comodules over C, in the sense of
the following definition.
A complex of comodules X over C is called trivial if it satisfies
the following two equivalent conditions:
(a) for any complex of injective (for example, cofree) comodules I
over C we have Hom(X,I)=0 in the homotopy category of C-comodules;
or
(b) X as a complex of comodules can be obtained from the total
complexes of exact triples of complexes of comodules using
the operations of cone and infinite direct sum.
For example, if C is trivial, i.e., C=k=N, then we have S=A=A^#=B
and the above-defined trivial complexes of S-modules are the most
usual (doubly unbounded) acyclic complexes of modules over
the algebra A (or, which is the same, B).
Definition: by the derived category of left or right modules over
a semialgebra S one means the quotient category of the homotopy
category of complexes of S-modules by the subcategory of trivial
complexes defined above. Analogously one defines the derived
category of (left or right) C-comodules.
IV. I recall that on S-modules there is the operation of tensor
product over S, which was discussed in part II. As usual, this
operation is extended to complexes of S-modules in the obvious way.
Definition. A complex of left S-modules F is called semiflat if
for any trivial complex of right S-modules T the tensor product
complex T\ot_S F is acyclic as a c-s of vector spaces. Analogously
one defines semiflat complexes of right S-modules.
A complex of left C-comodules E is said to be coflat if for any
trivial complex of right C-comodules U the cotensor product complex
U\ot_C E is acyclic as a complex of vector spaces; analogously for
right C-comodules.
A claim: any semiflat complex of S-modules is coflat as a complex
of C-comodules. Proof: let F be a semiflat complex of left
S-modules; we are interested in a cotensor product U\oc_C F. One
needs to use the induction functor assigning to every C-comodule P
the induced S-module P\ot_C S. It is not difficult to see that
the induction functor takes trivial complexes of C-comodules to
trivial ones (for example, the definition (b) of trivial complexes
is convenient to use). Now we have U\oc_C F = (U\oc_C S) \ot_S F
and the semiflatness implies coflatness.
V. Theorem: the quotient category of the homotopy category of
semiflat complexes of S-modules by the thick subcategory of semiflat
trivial complexes is equivalent, via the natural projection, to
the quotient category of all complexes of S-modules by the trivial
ones (in other words, to the derived category of S-modules).
I will not prove here this theorem about triangulated categories
in its full strength, but will restrict myself only to proving
the surjectivity of the functor in question on the objects. In
other words, I will show that any complex of S-modules is connected
with a certain semiflat complex by a chain (in fact, a chain of
length two, i.e., just a "roof") of maps with trivial cones.
The latter fact can be proved in various ways -- for example, one
can use Koszul duality in the spirit of the way Serezha did it
in his early papers on the topic -- but I will give here a direct,
relatively elementary proof. It is based on the following lemma.
Lemma. There exists a functor assigning to every S-module its
embedding into a C-injective S-module.
A comment: clearly, this is the very same question that in
the stardard expositions of this subject (by both of you, for
example, as well as by Sevastyanov) was solved by "induction
from the subalgebra B". I do not have any such "subalgebra B"
here, so I need a direct way.
V-1. Proof of Lemma. Let M be an arbitrary S-module. Consider
M as a C-comodule and embed into an injective C-comodule P=P(M).
Clearly, one can do it in a functorial way, for example, just
choosing P(M)=C\ot_k M. Denote by Q(M) the quotient module of
the S-module S\oc_C P(M) by its submodule which is the image of
the kernel of the map S\oc_C M \to M under the inclusion
S\oc_C M \to S\oc_C P(M). It is not difficult to see that
the composition M \to P(M) \to S\oc_C P(M) \to Q(M) is a map
of S-modules (while the maps being composed are only C-comodule
maps). Besides, the map from M to Q(M) is injective. However,
Q(M) need not be an injective C-comodule.
One overcomes the latter problem by iterating the construction.
Consider the chain of embeddings M \to Q(M) \to Q(Q(M)) \to ...
I claim that the inductive limit of this chain is injective as
a C-comodule. Indeed, this limit, viewed as a C-comodule, is
also the inductive limit of the chain of injective comodules
S\oc_C P(M) \to S\oc_C P(Q(M)), where the maps in the chain
are the compositions through Q(M), Q(Q(M)), etc. However, it is
not difficult to see that the inductive limits of chains preserve
injectivity of comodules over an arbitrary coalgebra, since any
comodule is a union of finite-dimensional ones. Indeed,
it suffices to extend a map to the direct limit from
a finite-dimensional subcomodule to a finite-dimensional comodule;
now the map from a finite-dimensional comodule factorizes through
one of Q^n(M), and consequetly, through S\oc_C P(Q^n(M)). Thus
the lemma is proved.
V-2. Now let us show that for any complex of S-modules X there is
a map f: X \to Y such that all the terms of the complex Y are
cofree C-comodules, while the cone of f is a trivial complex in
the sense of the definition from part III. This is easy to do.
Using the lemma, embed the complex X into some complex of
C-injective S-modules J, take the quotient complex J/X, embed it
similarly one again, etc. Proceeding in this way, construct
a complex of complexes X \to J \to J_1 \to J_2 \to ... Consider
the total complex Y=Tot(J\to J_1\to...) formed by taking infinite
direct sums along the diagonals.
Obviously, Y is a complex of C-injective S-modules, while f: X\to Y
is a morphism of complexes of S-modules. It remains to show that
after taking the forgetful funtor to complexes of C-comodules
the cone of the morphism f becomes a trivial complex. This is done
in the standard way, just as one proves the similar result about
the category D-prime for comodules over an arbitrary (DG-)coalgebra.
One uses the construction of the homotopy direct limit
("the telescope") and the definition of triviality in the form (b)
from part III.
V-3. It remains to show that for any complex of C-injective
S-modules Y there is a semiflat complex Z together with a morphism
g: Z\to Y such that the cone of g is trivial. Actually, we will
prove more. Specifically:
1. All complexes of S-modules induced (using the operation
S\oc_C *) from complexes of injective C-comodules are semiflat;
2. Semiflat complexes form a triangulated subcategory closed
under infinite direct sums in the homotopy category of complexes
of S-modules;
3. For any complex of C-injective S-modules Y there is
a morphism of complexes of S-modules g: Z\to Y such that
i. the cone of g, viewed as a complex of C-comodules,
is contractible;
ii. the complex Z is constructed from the complexes of type 1.
using the operations of cone and infinite direct sum.
Assertion 1. is obvious, as the cotensor product of a complex of
injective C-comodules with a trivial complex of C-comodules is
an acyclic complex of vector spaces, as one can easily see.
Assertion 2. is clear. Finally, assertion 3. is provable by
taking the total complex of the relative bar-construction over
S relative to C. Specifically, one needs to write
... \to S\oc_C S \oc_C Y \to S\oc_C Y \to Y
and take Z to be the total complex of this whole thing except
the final Y. Item i. is checked by presenting an explicit
canonical homotopy which is intrinsic to bar-constructions of
this kind; while item ii. is provable in the standard way,
using the "telescope" construction again.
The main theorem of part V can be considered to have been proved
(with the caveats made in the beginning of the part).
VI. Now that all the difficult work is over, the definition of
the semi-infinite homology functor can be given by waving one's
hands. We want to define a certain functor of two arguments
which range over the categories of complexes of right and left
S-modules. In fact, this functor will be defined on
the quotient categories of complexes of S-modules by the thick
subcategories of trivial complexes (they are thick, because
any triangulated subcategory closed under countable direct sums
is also closed under direct summands). However, according to
the theorem from part V, the quotient categories we are
interested in are equivalent to the quotient categories of
semiflat complexes of S-modules by the trivial semiflat complexes.
Now on the semiflat complexes the functor of semi-infinite
homology is defined straightforwardly as the tensor product of
modules over the semialgebra S (see part II). Obviously, this
functor on the homotopy categories of semiflat complexes descends
to the quotient categories by trivial (semiflat) complexes
(see part IV). End of the argument.
So I have given the definition of the functor of semi-infinite
homology on the categories of modules over a semialgebra.
P.S. Remark (added in April 2006). In fact, the subcategory in
the homotopy category of complexes of S-modules generated using
the cones and direct sums by the complexes of S-modules induced
from complexes of injective C-comodules (in the next letter,
it is called the resolution subcategory) is equivalent, via
the natural projection, to the derived category of S-modules.
It suffices to show that the intersection of the resolution
subcategory with the subcategory of trivial complexes is zero.
Let X be a complex belonging to the intersection of the two
subcategories. Then X as a complex of C-comodules is
simultaneously a complex of injective C-comodules and a trivial
complex of C-comodules, hence X is contractible over C. Now
X is simultaneously contractible over C and belonging to
the subcategory generated using cones and infinite direct sums
by the complexes induced from C, hence it follows that X is
contractible over S. Moreover, if follows from the arguments
above that the resolution subcategory coincides with the full
subcategory in the homotopy category of S-modules consisting of
all the complexes of injective C-comodules that are K-projective
(in the sense of Spaltenstein) over S relative to C (i.e.,
belong to the left othogonal subcategory to the subcategory of
all complexes contractible over C).
From: Leonid Positselski < posic@mpim-bonn.mpg.de >
To: roma, hippie
Date: Fri, 11 Oct 2002 18:59:59 +0200 (MEST)
Subject: semi-infinite COhomology and contramodules (corrected and supplemented)
Hi Roma and Serezha,
in my previous letter from August 3 of this year what I consider
to be the correct definition of semi-infinite homology of
associative algebraic structures was given. The aim of
the present letter is to define the semi-infinite cohomology,
i.e., Ext_{\infty/2} in Serezha' notation. In my exposition
I will approximately follow the plan sketched in the previous
letter (with the exception of the new part VII of the present
letter, which has no analogue for semi-infinite homology).
I: which abelian categories of modules are considered;
II: what functor on these categories should be derived;
III: the equivalence relation on the complexes of modules,
i.e., which complexes are to be viewed as trivial;
IV: semiprojective complexes of S-modules, semiinjective
complexes of S-contramodules, and coinjective C-contramodules;
V: constructions of resolutions;
VI: the definition of the derived functor;
VII: the equivalence of derived categories; the semi-infinite
cohomology as Hom in a triangulated category.
I will rely in the most essential way on letter number four
(the last one) from the series of "Summer letters" of
the year 2000, in which the definition of the category of
contramodules, and in particular, the notion of a contramodule
over a semialgebra was spelled out.
I-1. Suppose given a coalgebra C and a semialgebra S over it;
it is assumed that S is injective over C both on the left and
on the right. The first arguments of the functor of
semi-infinite Ext will be complexes of left S-modules (see
the previous letter). The second arguments of the functor of
semi-infinite Ext will be complexes of left S-contramodules
(see letter number four from the Summer series of 2000).
I-2. If C is finite-dimensional, then instead of a semialgebra S
one can speak about a special kind of pair of associative algebras
A and A^# endowed with a common subalgebra N (dual to C).
In this case, the right S-modules are the right A-modules,
the left S-modules are the left A^#-modules, while the left
S-contramodules are the left A-modules, and finally, the right
S-contramodules are the right A^#-modules. The first arguments
of the functor of semi-infinite Ext turn out to be complexes of
left A^#-modules, the second ones are complexes of left A-modules.
I-3. As one can see from the above, it is impossible to avoid
using contramodules in the construction of the theory of
semi-infinite COhomology in terms of a semialgebra S even in
the case when both C and S, as well as all the modules under
consideration, are finite-dimensional. Although, if one is
speaking about contramodules over a coalgebra only, then,
of course, left C-comodules and left C-contramodules are
the same things if C is finite-dimensional.
II-1. The semi-infinite cohomology are the derived functor of
the functor of cohomomorphisms Cohom over a semialgebra S.
The letter functor assigns to every left S-module L and every
left S-contramodule P the vector space Cohom_S(L,P) of
"cohomomorphisms over S from L to P". This functor is defined
as "the kernel of a certain map from the cokernel of a certain
map to the cokernel of a certain map"; for this reason, it is
obviously neither left nor right exact with respect to either
the first or the second argument.
II-2. There is the following connection between my definition of
the functor Cohom_S and Serezha's formulas for the semi-infinite
Ext from Subsection 4.7 ("Choice of resolutions") of the paper
"Semi-infinite cohomology of quantum groups II". Let L be
a left A^#-module and P be a left A-module. Then Cohom_S(L,P) =
Hom_{A^#}(L, S\ot_A P) if P is projective as a left N-module,
and Cohom_S(L,P) = Hom_A(Hom_{A^#}(S,L), P) if L is injective
as a left N-module.
Moreover, for any left S-module L and left S-contramodule P,
over any semialgebra S, there are the identities
Cohom_S(L,P) = Hom_S(L, C\ocn_C P) if P is a projective
C-contramodule, and Cohom_S(L,P) = Hom_{S-contra}(Hom_S(S,L), P)
if L is an injective C-comodule. Here P \mapsto C\ocn_C P
and L \mapsto Hom_C(C,L) = Hom_S(S,L) are the functors between
the categories of left S-modules and left S-contramodules
introduced in the end of the fourth letter of the 2000 series.
The proof or refutation is left to the readers. This is
the functor whose derived functor will be the semi-infinite
cohomology.
III. Now I have to define the correct equivalence relation on
the complexes that are the arguments of the purported derived
functor. The equivalence relation on complexes of S-modules was
introduced in the previous letter; it remains to spell out the case
of S-contramodules. For the latter ones, the correct equivalence
relation is a mixture of the "relation D-second" along
the C-contramodule structure and "the relation D in
the perpendicular direction". Let me recall that in the categories
"D-second" the trivial objects are the complexes that can be
obtained from the totalizations of exact triples of complexes using
the operations of taking the cone and infinite direct product (as
opposed to the infinite direct sum for the categories D-prime).
The formal definition: a complex of S-contramodules (generally
speaking, unbounded in all directions) is called trivial if it is
trivial as a complex of C-contramodules (i.e., after forgetting
the rest of the structure). A complex of C-contramodules X
is called trivial if it satisfies the following two equivalent
conditions:
(a) for any complex of projective (for example, free) contramodules
P over C we have Hom(P,X)=0 in the homotopy category of
C-contramodules; or
b) X as a complex of contramodules can be obtained from the total
complexes of exact triples of complexes of contramodules using
the operations of cone and infinite product.
Definition: by the derived category of left contramodules over
a semialgebra S one means the quotient category of the homotopy
category of complexes of left S-contramodules by the subcategory
of trivial complexes defined above. Analogously one defines
the derived category of left C-contramodules.
IV-1. One extends the operation Cohom_S from (contra)modules
to complexes of (contra)modules by the following obvious rule:
Cohom_S(L,P) is the complex obtained by taking infinite products
along the diagonals of the bicomplex Cohom_S(L^i, P^j). Let me
point out that for operations of "homomorphism type", unlike
for operations of "tensor product type", one has to take
the infinite product (rather than the direct sum) along
the diagonals in this context.
Two definitions: a complex of left S-modules F is called
semiprojective if for any trivial complex of left S-contramodules
T the complex Cohom_S(F,T) is acyclic as a complex of vector
spaces. A complex of left S-contramodules P is called
semiinjective if for any trivial complex of left S-modules T
the complex Cohom_S(T,P) is acyclic.
Analogously, a complex of left C-comodules is called coprojective
if, for any trivial complex of left C-contramodules, the complex
Cohom_C between them is acyclic. A complex of left C-contramodules
is called coinjective if, for any trivial complex of left
C-comodules, the corresponding complex of vector spaces Cohom_C
is acyclic.
IV-2. It is clear that for complexes of C-comodules the three
properties -- injectivity (in the sense of the resp. derived
category), coflatness (in the sense of the previous letter) and
coprojectivity (in the sense of the last definition) -- are all
rather close to each other. It is not difficult to see that
any complex of injective comodules is coprojective and any
coprojective complex is coflat. Analogously, any complex of
projective C-contramodules is coinjective.
The most interesting question: how are the properties of
semiflatness and semiprojectivity related to each other for
complexes of S-modules? A partial answer: all semiprojective
complexes are semiflat (proof: use the fact that the dual
vector space to a right S-module is a left S-contramodule,
and the formula Cohom_S(L, M^*) = (M\ot_S L)^*).
Two assertions: any semiprojective complex of left S-modules
is coprojective as a complex of left C-comodules. Any
semiinjective complex of left S-contramodules is coinjective
as a complex of left C-contramodules. Proofs: in the former
case use the following coinduction functor for contramodules:
to every C-contramodule P the coinduced S-contramodule
Cohom_C(S,P) is assigned. In the latter case use the induction
functor from the previous letter.
IV-3. In fact, for C-co/contramodules all of this is somewhat
simpler. As we know, the main property of the derived categories
D-prime and D-second is that the adjustedness properties of
complexes in them can be checked termwise. For example, a complex
of C-comodules is injective (right orthogonal to trivial complexes
with respect to the usual Hom functor in the homotopy category
of comodules) if and only if it is homotopy equivalent to a complex
of injective objects. The same holds for contramodules and
projective complexes. So it makes sense to give several further
definitions.
A left C-comodule L is called coflat if the functor - \oc_C L is
exact on the abelian category of right C-comodules. A left
C-comodule L is called coprojective if the functor Cohom_C(L,-) is
exact on the abelian category of left C-contramodules. In fact,
it is easy to see that all the three properties of injectivity,
coflatness, and coprojectivity of C-comodules are equivalent
(a hint: one should restrict oneself to finite-dimensional second
arguments of the functors Hom, \oc, and Cohom, respectively, and
compare in this situation). Now one could, of course, notice that
any complex of coflat comodules is coflat and any complex of
coprojective ones is coprojective. These are undoubtedly true
but, in view of the above, trivial assertions.
A left C-contramodule P is called coinjective if the functor
Cohom_C(-,P) is exact on the abelian category of left C-comodules.
A left C-contramodule P is called contraflat if the functor
of contratensor product - \ocn_C P is exact on the category of
right C-comodules. It is not difficult to see that any
projective C-contramodule is coinjective and any coinjective
C-contramodule is contraflat (the converse assertion is discussed
in part VII). Any complex of coinjective C-contramodules is
a coinjective complex.
V. Teorema 1: the analogue of the main theorem from part V of
the previous letter holds for semiprojective complexes of
S-modules in lieu of the semiflat ones. The proof is exactly
the same as in the previous letter.
Teorema 2: the quotient category of the homotopy category of
semiinjective complexes of S-contramodules by the thick subcategory
of semiinjective trivial complexes is equivalent, via the natural
projection, to the quotient category of the homotopy category of
all S-contramodules by the trivial complexes.
Here one has to rewrite the argument from the previous letter
replacing comodules by contramodules. I will do it, cutting down
on some details.
Lemma. There exists a functor assigning to every S-contramodule
a surjective map onto it from a C-coinjective S-contramodule.
Of course, the closest analogue of Lemma from part V of
the previous letter would be existence of such a surjection from
a C-projective S-contramodule. The only problem is that I am
unable to prove that the inverse limit preserves projectivity of
contramodules -- while for coinjectivity I know how to check it
in the context we need. For the purposes of the present section,
C-coinjectivity is sufficient, while in part VII, where
C-projective resolutions will be needed, I will have to use
a certain conjecture.
V-1. Proof of Lemma. Let P be a left S-contramodule. Consider
it as a C-contramodule and present it as an epimorphic image of
a projective C-contramodule F. This can be easily done in
a functorial way, for example, taking F(P) = Hom_k(C,P).
Denote by Q(P) the kernel of the composition of maps
Cohom_C(S, F(P)) \to Cohom_C(S,P) \to Cohom_C(S,P)/im P,
where P \to Cohom_C(S,P) is the structure map of
the S-contramodule P. It is not difficult to see that
the composition Q(P) \to Cohom_C(S, F(P)) \to F(P) \to P
is a map of S-contramodules, while the maps being composed are
only maps of C-contramodules. Besides, the map from Q(P) to P
is surjective, while the C-contramodule Cohom_C(S, F(P)) is
projective. Now the assertion of the lemma follows from
the next sublemma.
Sublemma. Let ... \to Q_2 \to T_2 \to Q_1 \to T_1 \to Q_0
be a projective system of C-contramodules, where the contramodules
T_i are coinjective, while in the projective subsystem consisting
of Q_i only, all the maps are surjective. Then the inverse limit
lim Q_i = lim T_i is a coinjective contramodule. Proof of
Sublemma: one needs to check that for any exact triple of left
C-comodules L_1 \to L_2 \to L_3 one has an exact triple of vector
spaces Cohom(L_3, lim T_i|Q_i) \to Cohom(L_2, lim)
\to Cohom(L_1, lim). Let us first prove that for any
C-comodule L we have Cohom(L, lim) = lim Cohom(L, T_i|Q_i).
For any contramodule X, write the bar-construction
... \to Hom_k(C\ot_k C\ot_k L, X) \to Hom_k(C\ot_k L, X)
\to Hom_k(L,X); then Cohom_C(L,X) is, by definition,
the degree-zero homology of this complex. Now suppose that we
are given a projective system of (homological) complexes where
all the complexes with odd numbers have homology in degree zero
only, while the complexes with even numbers form a projective
subsystem with surjective transition maps. Then I claim that
the inverse limit complex has homology in degree zero only,
and it is equal to the inverse limit of the degree-zero homology
of the complexes in the system. Indeed, the projective systems
in which the subsystems with even numbers have surjective
transition maps are acyclic objects for the derived functor of
inverse limit lim^1. The rest of the proof of the sublemma is
left to the readers.
V-2. On the next step we need to show that for any complex of
S-contramodules X there is a map f: Y\to X such that all the terms
of the complex Y are conjective C-contramodules, while the cone
of f is a trivial complex. This is done exactly in the same way
as in item V-2 of the previous letter, with the difference that
the bicomplex is being totalized using direct products (and not
direct sums) along the diagonals. Then one uses the homotopy
inverse limit, and generally everything happens as it usually
does in D''.
V-3. On the last step it remains to prove that for any complex of
C-coinjective S-contramodules Y there is a semiinjective complex
of S-contramodules Z together with a morphism g: Y\to Z such that
the cone of g is trivial (in fact, even C-contractible). This is
done exactly in the same way as in section V-3 of the previous
letter, except that instead of the induced S-modules one uses
the coinduced S-contramodules Cohom_C(S,-) and the total complex
(unlike in the context of the previous letter) is, once again,
constructed by taking infinite products. Generally, if
the "constructions from sections V-2" are typical for
the derived categories D' and D'', then quite similarly
the "constructions from sections V-3" are typical for the more
classical (Spaltenstein's) derived categories D.
VI. Now the functor of semi-infinite cohomology can be defined
in the way completely analogous to the definition of
the semi-infinite homology in the previous letter.
VII. The aim of this part is to prove two assertions:
(1) the above-defined derived categories of left S-modules and
left S-contramodules are naturally equivalent; (2) under this
equivalence, the semi-infinite cohomology functor corresponds
to the Hom functor.
VII-1. I will start with a certain comment on Theorems 1-2 from
part V. In fact, the proof of these theorems proves more than
claimed in their formulation. Specifically, consider the full
triangulated subcategory in the homotopy category of complexes of
S-modules consisting of all the complexes that can be obtained
by cones, shifts, and infinite direct sums from the complexes
induced from complexes of injective C-comodules. Let us call it
the resolution subcategory in the homotopy category of complexes
of S-modules. Analogously, the resolution subcategory in
the homotopy category of complexes of S-contramodules consists of
all the complexes that can be obtained by cones, shifts, and
infinite products from the complexes coinduced from complexes of
coinjective C-contramodules. The proof of Theorems 1-2 actually
proves the following: any full triangulated subcategory in
the homotopy category of complexes of S-modules or S-contramodules
which contains the resolution subcategory, after taking its
quotient category by the intersection with the full subcategory
of trivial complexes, becomes equivalent, via the natural
projection, to the respective derived category.
Correction (added in November 2006). The generalization of
Theorems 1-2 stated here is a bit too general, and I do not know
if whether it can be proved. A correct assertion would be, for
example, this: any full subcategory in the homotopy category
of complexes of S-modules contained in the full subcategory of
all complexes of C-injective S-modules and containing
the resolution subcategory, and analogously any full subcategory
of complexes of S-contramodules contained in the subcategory of
of all complexes of C-projective S-contramodules, after taking
its quotient by the interesection with the full subcategory of
trivial complexes, becomes equivalent, via the natural projection,
to the respective derived category.
In particular, this is true for the full subcategory consisting of
all complexes of C-injective S-modules. Its intersection with
the full subcategory of trivial complexes consists of all
complexes of C-injective S-modules that are contractible over C.
Thus the derived category of S-modules is equivalent to
the quotient category of the homotopy category of complexes of
C-injective S-modules by the full subcategory of C-contractible
C-injective complexes.
VII-2. In order to claim the same for S-contramodules, I will need
to use the following conjecture.
Conjecture 1. For any coalgebra C, the classes of contraflat,
coinjective, and projective C-contramodules coincide. In other
words, every contraflat C-contramodule is projective.
Example. Suppose that the coalgebra C is finite-dimensional.
Then C-comodules and C-contramodules are simply modules over
the finite-dimensional algebra N dual to C. The contraflat
contramodules are the flat modules, and it is well-known that
any flat module over a finite-dimensional algebra is projective
[see e.g. Bass' paper in Trans. AMS v.95 for 1960].
I have the feeling that Conjecture 1 may be deducible from
the following much more fundamental assertion about contramodules.
Conjecture 2. For any contramodule P over a coalgebra C,
the intersection of the images of the vector spaces Hom_k(C/V, P)
under the structure map Hom_k(C,P) \to P, taken over all
the finite-dimensional subspaces (or, if one wishes, subcoalgebras)
V in C, is equal to zero in P.
Example. In the fourth letter of the year 2000 series, a question
was asked, how to prove that any finite-dimensional C-contramodule
is a contramodule over a finite-dimensional subcoalgebra in C. One
can see that this assertion is a particular case of Conjecture 2.
VII-3. Using Conjecture 1, one can claim that the full subcategory
in the homotopy category of S-contramodules consisting of all
complexes of C-projective contramodules contains the resolution
subcategory. Its intersection with the full subcategory of trivial
complexes consists of all the complexes of C-projective
S-contramodules that are contractible over C. Therefore,
the derived category of S-contramodules is equivalent to
the quotient category of the homotopy category of complexes of
C-projective S-contramodules by the full subcategory of
C-contractible C-projective complexes.
In the end of the 4th letter from the "Summer series" of the year
2000 it was shown that the additive categories of injective
C-comodules and projective C-contramodules are equivalent, and
S-module structures on C-comodules correspond bijectively to
S-contramodule structures on C-contramodules under this equivalence
-- so the categories of C-injective S-modules and C-projective
S-contramodules are also equivalent. Now we immediately obtain
from this an equivalence of the derived categories of S-modules
and S-contramodules.
VII-4. It is claimed that this equivalence transforms the functor
of semi-infinite cohomology into the Hom functor. This follows
from the formulas written down in section II-2 of the present
letter. More precisely, one can argue as follows. Let an object
of the derived category of S-modules be represented by a complex
of C-injective S-modules L, while an object of the derived category
of S-contramodules be represented by a complex of C-projective
S-contramodules P. Then the derived functor of the functor Cohom
on the objects L and P can be computed by choosing maps with
trivial cones L_1 \to L and P \to P_1, where L_1 and P_1 belong
to the resolution subcategories, and computing Cohom(L_1,P_1),
or Cohom(L_1,P), or Cohom(L,P_1) (which is the same). According
to the formula from section II-2, we know that Cohom(L_1,P_1) =
Hom(\psi(L_1), P_1), where \psi denotes the functor from
C-injective S-modules to C-projective S-contramodules, while
the Hom is taken in the homotopy category. Now it remains to show
that \psi(L_1) belongs to the left orthogonal subcategory and
P_1 to the right orthogonal subcategory to the subcategory of
C-contractible complexes in the homotopy category of complexes of
C-projective S-contramodules (or at least one of these two
assertions -- in fact, both are true). Indeed, let X be such
a C-contractible complex; then Hom(\psi(L_1), X) = Cohom(L_1, X)
= 0, since the complex L_1 is semiprojective. Analogously
Hom(X, P_1) = Cohom(\psi^{-1}(X),P_1) = 0, since the complex
psi^{-1}(X) is also C-contractible, while P_1 is semiinjective.
Looking into this argument, one can notice that even without
the assumption of Conjecture 1 it proves existence of a functor
\Psi from the derived category of S-modules to the derived
category of S-contramodules such that the derived functor of
Cohom is equal to the composition of \Psi with the Hom functor
in derived category of contramodules. Conjecture 1 is needed
in order to show that \Psi is a category equivalence. When
the coalgebra C is finite-dimensional, we know this.
P.S. Addition (May 2006). Here is the simplest counterexample to
Conjecture 2 (another counterexample was given in an Addition in
the fourth letter for the year 2000). Consider the coalgebra C
dual to the pro-finite-dimensional algebra of formal power series
in one variable C^* = k[[x]]. Then the C-contramodules are
the k-vector spaces P endowed with an operatoin of summation of
sequences of vectors with the formal coefficients x^n, i.e.,
for any p_0, p_1, ... in P the sum \sum x^i p_i is defined as
an element of P. Consider the free contramodule generated by
a sequence e_0, e_1, ... ; its elements are the formal sums
\sum a_i(x)e_i, where the formal power series a_i(x) have
the property that ord_x a_i(x) tends to infinity as i grows.
Now consider a homomorphism from the free contramodule F generated
by f_1, f_2, ... to the free contramodule E generated by
e_0, e_1, e_2, ... -- generally, such homomorphisms correspond
bijectively to arbitrary sequences of elements of E -- the images
of f_i -- and we are intereseted in the homomorphism taking f_i
to x^i e_i - e_0. An arbitrary element \sum a_i(x)f_i of
the contramodule F is taken by this map to \sum x^i a_i(x) e_i -
(\sum a_i(x))e_0. It is immediately clear from this that
the element e_0 does not belong to the image of this homomorphism.
Consider the quotient contramodule P=E/im F; it is claimed that
the coset of the element e_0 in P belongs to im Hom(C/V,P) for
all finite-dimensional subcoalgebras V in C. Indeed, for any V
there is i such that x^i as an element of C^* annihilates V;
so the expression x^i e_i represents an element of Hom(C/V,P)
whose image in P is equal to e_0.
P.P.S. Addition (June 2006). The following weaker version of
Conjecture 2 is true: for any nonzero C-contramodule P there is
a finite-dimensional subcoalgebra V in C (which can be chosen
to be simple, i.e., containing no nontrivial subcoalgebras)
such that the image of Hom(C/V,P) in P is not equal to P.
The proof consists of two lemmas.
Lemma 1 (Nakayama's lemma for contramodules). Let C^ss denote
the maximal semisimple subcoalgebra of C. Then for any nonzero
C-contramodule P the image of Hom(C/C^ss,P) in P is not equal to P.
Proof: notice that the coalgebra without counit D=C/C^ss is
conilpotent, i.e., any element of D is annihilated by the iterated
comultiplication map D\to D^{\ot i} for i large enough. Let us
show that for any contramodule P over a conilpotent coalgebra
without counit D surjectivity of the map Hom(D,P)\to P implies P=0.
Indeed, assume that P = im Hom(D,P). Let p be some element of P;
it comes from some map f_1: D\to P. Since the map Hom(D,P)\to P
is surjective, the map f_1 can be lifted to a certain map
D \to Hom(D,P), which leads to a map f_2: D\ot D \to P, etc. So
one constructs a sequence of maps f_i: D^{\ot i} \to P such that
f_{i-1}=m'_1(f_i), where m' is the contraaction map Hom(C,P) \to P
and m'_1 denotes the application of m' along the first tensor
factor in D^{\ot i}. Put g_i=m_{2..i}(f_i), i=2,3,..., where m is
the comultiplication map D \to D\ot D and m_{2..i} denotes
the substitution of the iterated comultiplication D \to D^{\ot i-1}
into the components numbered 2 to i in the tensor product D^{\ot i}.
Then g_i are maps D\ot D \to P. We have
m'_1(g_i)=m_{1..i-1}(f_{i-1}) i m_{1..2}(g_i)=m_{1..i}(f_i).
Notice that, as it follows from conilpotency of the coalgebra D,
the series \sum_{i=2}^\infty g_i converges in the sense of pointwise
limit of functions D\ot D \to P, and even in the sense of pointwise
limit of functions D \to Hom(D,P). (Here, as above,
the identification Hom(X, Hom(Y,Z)) = Hom(Y\ot X, Z) is presumed.)
Thus m'_1(\sum g_i) = \sum m_{1..i-1}(f_{i-1}) i m_{1..2}(\sum g_i)
= \sum m_{1..i}(f_i), hence we obtain m'_1(\sum g_i) -
m_{1..2}(\sum g_i) = f_1, so m'(f_1)=0. Lemma 1 is proved.
Lemma 2. Suppose that a coalgebra C is a direct sum of a family
of coalgebras C_a. Then any C-contramodule is a direct product of
contramodules over C_a. Proof: obvivously, the assertion of Lemma
holds for any free C-contramodule. Now let a C-contramodule P be
a direct product of C_a-contramodules P_a. Let us show that any
subcontramodule R in P is the direct product of its images R_a under
the projections P\to P_a. Suppose given a collection of elements
r_a in R. Consider the linear map f: C\to R whose restriction to
C_a is equal to the composition C_a\to k\to R, where the first map
being composed is the counit of the coalgebra C_a, while the second
map takes 1 to r_a. Denote by r the image of the functional f under
the contraaction map Hom(C,R)\to R. Then it is clear that the image
of r under the projection P\to P_a is equal to the image of r_a
under this projection. Thus R is the direct product of R_a. Now
it remains to notice that any C-contramodule is the quotient
contramodule of a free contramodule by some subcontramodule.
P.P.P.S. Addition (June 2006). Here is a proof of Conjecture 1
based on the results from the previous Addition. For any
C-contramodule X and any subcoalgebra V in C denote by
X_V = X/im Hom_k(C/V,X) = Cohom_C(V,X) the maximal subcontramodule
of X which is a contramodule over V. Let C^ss denote the maximal
semisimple subcoalgebra of C.
Lemma. For any C^ss-contramodule T there is a projective
C-contramodule P such that P_{C^ss} is isomorphic to T.
Proof: According to Lemma 2 above, T is a direct product of
contramodules over the simple components C_a of the semisimple
coalgebra C^ss. Any C_a-contramodule, in turn, is a direct sum of
some number of copies of the unique irreducible C_a-contramodule.
Hence one can easily conclude that it suffices to consider the case
when T is an irreducible C_a-contramodule. Let e_a be an idempotent
of the algebra C_a^* such that T is isomorphic to C_a^*e_a.
Consider the idempotent linear functional e_ss on the coalgebra C^ss
equal to e_a on C_a and zero on C_b for b not equal to a. It is
well-known that for any surjective ring homomorphism A\to B whose
kernel is a nil ideal in A, any idempotent element in B can be
lifted to an idempotent in A. Using this fact for
finite-dimensional algebras and Zorn's lemma, it is not difficult
to show that any idempotent linear functional on C^ss can be
extended to an idempotent linear functional on C. Let e be such
an idempotent element of C^* extending e_ss; set P = C^*e. It is
easy to see that the C^ss-contramodule P_{C^ss} is isomorphic to T.
The lemma is proved.
Now let Q be a contraflat C-contramodule; let us show that it is
projective. Consider a projective C-contramodule P for which
P_{C^ss} is isomorphic to Q_{C^ss}. Since P is projective,
the surjective map P\to Q_{C^ss} can be lifted to a homomorphism of
contramodules f: P\to Q. Since (coker f)_{C^ss} = coker(f_{C^ss})
= 0, it follows from Lemma 1 above that the homomorphism f is
surjective. It remains to show that f is injective. Notice that
for any right comodule M over a subcoalgebra V of the coalgebra C
there is an isomorphism M\ocn_C Q = M\ocn_V Q_V implying that
the V-contramodule Q_V is contraflat. Now assume that V is
finite-dimensional; then Q_V is a flat V^*-module. Consider
the map f_V: P_V\to Q_V and denote its kernel by K. For any
right V^*-module M we have a short exact sequence
0 \to M\ot_{V^*}K \to M\ot_{V^*} P_V \to M\ot_{V^*} Q_V \to 0.
In particular, since for any simple subcoalgebra V_a in V the map
V_a^*\ot_{V^*}f_V = f_{V_a} is an isomorphism, we can conclude that
the module V_a^*\ot_{V^*}K = K_{V_a} is zero. It follows that K=0
and f_V is an isomorphism. Finally, let R be the kernel of the map
f: P\to Q. Since f_V is an isomorphism, the subcontramodule R is
contained in the image of Hom(C/V,P) in P for any finite-dimensional
subcoalgebra V in C; but the intersection of all such images is
zero, since the C-contramodule P is projective.