Algebra, Algebraic Topology and their Interactions: Proceedings of a Conference held in Stockholm, Aug. 3 - 13, 1983, and later developments

407 Pages·1986·16.03 MB·English

by  Jan-Erik Roos

Lecture Notes ni
Mathematics
Edited by .A Dold and .B Eckmann
3811
Algebra, Algebraic ygolopoT
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Proceedings of a Conference
held ni Stockholm, Aug. 3-13, 1983,
and later developments
Edited Roos J.-E. by
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Editor
Jan-Erik Roos
Department of Mathematics, University of Stockholm
Box 6701, 311 85 Stockholm, Sweden
Mathematics Subject Classification (1980): 13-06, 13D03, 13E05, 13H99, 13J10,
14-06, 14F35, 16A24, 17B70, 18G15, 18G20, 20F05, 20F10, 55-06, 55P35,
55Q15, 55S30, 5?-xx
ISBN 3-540-16453-7 Springer-Verlag Berlin Heidelberg New York Tokyo
ISBN 0-387-16453-? Springer-Verlag New York Heidelberg Berlin Tokyo
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A MATHEMATICAL INTRODUCTION
These notes contain the outcome and later developments arising from a Nordic Summer
School and Research Symposium held in Stockholm, August 3-13 th, 1983 on "ALGEBRA,
ALGEBRAIC TOPOLOGY AND THEIR INTERACTIONS".
Let me first give a brief indication of the main ideas behind this symposium. During
the last decade several striking analogies between algebraic topology (at least
rational homotopy theory) and algebra (at least local algebra) had been observed. Let
me just give two examples. (More examples and details can be found in the paper
Z~irough the looking glass: A dictionary between rational homotopy theory and local
algebra by L. AVRAMOV and S. HALPERIN in these proceedings.)
First some preliminaries. Let X be a finite, simply-connected CW-complex, ~X the space
of loops on X and H.(~X,~) the rational homology algebra of ~X. (q~is algebra is even
a Hopf algebra.) At the same time, let (R,m) be a local commutative noetherian ring
R with maximal ideal m and residue field k = R/m, and let Ext~(k,k) be the graded
vector space @ Ext~(k,k) equipped with the algebra structure coming from the Yoneda
n>O
composition Ext~(k,k) @ Ext~(k,k) > Ext~+J(k,k). This Yoneda Ext-algebra
ExtR(k,k) is also a Hopf algebra and it is even the enveloping algebra of a certain
graded Lie algebra w*(R) over k. On the other hand, it is also known that H.(~X,~) is
the enveloping algebra of the rational homotopy Lie algebra ~.(~X)@z~. (Note that the
Samelson product on this Lie algebra corresponds under the isomorphism Wn_~(gLX)m~n(X)
to the Whitehead product on the ~ (X).) We are now ready for the examples:
n
Example I.- Let F > E > B be a Serre fibration and
"'" + Zn+1(B) ~ > Zn(F) ----> Zn(E) > Zn(B) ----> ... (I)
the corresponding homotopy exact sequence. In ]7[ Halperin proved (under some minor
extra conditions) that, if H*(F,~) is finite dimensional, then (I) breaks up into
exact sequences of 6 terms if we tensor it with ~. (More precisely, ~(Zodd(B)) is
torsion.) On the other hand, if A > B is a homomorphism of local commutative
noetherian rings such that B is A-flat and if B = B @A k is the "fibre" ring (assuming
for simplicity that the local rings have the same residue field k) then, using earlier
partial results of Oulliksen, Avramov proved in ]3[ that there is an exact sequence
n(~) > ~(H) > n(A ) 6 > n+~(~) )2(
...___> ____> ...
where 6 has properties similar to those of .~ (This time we do not have to tensor (2)
with anything.) It should be remarked that neither Avramov nor Halperin knew about the
other's work at the time. By now there are much more complete results and a common
IV
explanation in terms of differential graded algebras.
Example 2.- It was asked by Serre whether the series
dim$(Hn(~,~))-Z n , (3)
n>O
and by Kaplansky and Serre whether
[ dimk(Ext~(k,k)).zn (4)
n>0
behaved in a nice way, e.g. whether they were rational functions of Z. (X and R are as
in the preliminaries above.) I proved in []3] that~ for spaces X with dim X ~ 4 and
for local rings (R,m) with m_ 3 = O, the two questions were equivalent (even more precise
results were proved...). Thus, when Anick found a counterexample to the rationality
of (3), it was immediately obvious how to produce a eoLunterexample to the rationality
of (4). In [13] the algebra structures of H.(~X~) and ExtR(k,k) were also related to
each other. By now there are much more general results, at least about how the series
(3) and (4) are related. It has turned out that even for X arbitrary (finite, simply-
connected) and (R,m) arbitrary (local noetherian), the series (3) and (4) are all
"rationally related" )I to corresponding series (4) of local rings (S,[) with 3 = n 0
and to corresponding series (3) of finite Y:s with dim Y < 4 and thereby also
"rationally related" to series ~ dimk(Fn).z n , where ~ is a finitely presented graded
n>O
(l,2)-Hopf algebra, i.e. the quotient of a free associative algebra k< XI,...,X m > on
generators XI,...,X n of degree ,l by the two-sided ideal generated by some "quadratic
Hopf relations" :
j~ c]~j[Xi,X _ , where cij C k and where [Xi,X j] :~ XiXj + XjXi ' if i < j
J_<1 - LX~ ~ if i = j.
For more details, cf. [2].[In what follows, if @ V is a graded k-vector space, with
-- n>0 n
dimk(V n) < ~ for all n >_ O, we will call dir~(Vn)'Z n the Hilbert series of V = ~ V n.
n>0 n>0
With these and other examples in mind, it was clear that, if algebraists and
(algebraical) algebraical topologists could meet for a longer period of time, then a
fruitful interaction between their ideas might take place. Here are just a few examples
of results obtained at or after the Stockholm conference that are published here for
the first time:
In B~gvad-Ha!perinPs paper an algebraist and an algebraical topologist cooperate to
prove that, if H,(~X,$) ( recall that X is a finite, simply-connected CW-complex) is
noetherian (left or right noetherian does not matter, since we are dealing with a Hopf
I )For the meaning of "rationally related", cf. "LOOKING AHEAD" below.
V
algebra)~ then there are only a finite number of non-zero rational homotopy groups of
X. (The converse is evident.) On the other hand they also prove that, if (R,m) is a
local commutative noetherian ring with residue field k (no restrictions on k!), then
Ext~(k,k) is noetherian (if and) only if R is a local complete intersection. (In this
case only the two lowest 7i(R) can be different from 0.) The idea of the proof comes
from algebraic topology. The Lusternik-Schnirelmann (L.-S.) category (an old
topological concept from the 1930:s), which had been introduced quite recently in
rational homotopy theory (and thus in the theory of differential graded algebras) is
also used here for Avramov~s minimal models in a nice way. [~e paper by Lemaire in
these proceedings contains an up-to-date survey of L.-S. category, that completes and
goes beyond the beautiful earlier survey of I.M. James [9] from 1978.]
In order to present the next new result, I first have to recall an old result of Levin
[10], which combined with later results of Avramov and L~fwall (cf. these proceedings)
can be formulated as saying that, for any local commutative noetherian ring (R,m) , the
Lie algebra 7*(R) is closely related to the Lie algebra 7*(R/@n) of the artinian ring
R/m , n provided n is bi 6 enough (precisely how big n should be depends on the Artin-
Rees lemma, which Levin uses in [10] in a very clever way). More precisely, if n £
some n(R), then the natural Lie algebra map
*(R/m ) n 7" ~*(R)
_ -->
is onto, and the kernel of 7" is a free graded Lie algebra. (~ere are even more
precise results.) [In technical terms one says that R >~ R/m n is a Golod map.
A very ~eneral theor L of Golod maps is presented for the first time in the paper by
Avramov in these proceedings.] Here is one algebraical topological version of all this
(it is proved in the joint paper by Halperin and Levin in these proceedings): Let X be
a simply-connected CW-complex (not necessarily finite) with a finite number of cells
in each dimension and such that H*(X,~) is a finitely generated algebra (i.e. a
noetherian ring). Then there exists an no(X) such that, for all n ~ no(X), the rational
homotopy fibre of the inclusion of the n-skeleton n X > X is a wedge of spheres.
Note the analogy: the rational homology ring of the loop space of a wedge of spheres
is a free associative algebra. Results of this type had previously been known only for
X = BU(m) and, more generally~ for X = certain products of Eilenberg-MacLane spaces.
The earlier ideas of Levin are essential for the general proof.
There are many more examples of interaction between algebra and algebraic topology.
The analogy is often not perfect, and this inevitably leads to more work if one wants
to go from one side to another.
Let me say a few words about some other papers in these proceedings. LSfwall's paper
is a corrected version of about one half of his 1976 thesis, and this half was never
published, presumably because L6fwall first wanted to prove by his methods the
rationality of the series (4) in general. Now, as we have said above, we know better
as (4) is not always rational, but it was a genuine surprise when it turned out in
lV
1984 ]2[ that the special cases studied by LSfwall, and in particular finitely
presented graded (1,2)-Hopf algebras and their Hilbert series, were "rationally
related" to the general series (4) for general local rings (R,m). Thus with hindsight
one might say that in a sense L6fwallls thesis did treat the most general case.
L6fwall~s thesis has been used by many workers in the field and, in particular, by
LSfwall himself []]] in his construction of counterexamples to a conjecture by
v v
Kostrikin and Safarevic. The papers by Anick-L6fwall and FrSberg-Gulliksen-L6fwall in
these proceedings are recent studies of how finitely presented graded algebras and
their Hilbert series can behave. In particular the last paper can be used to prove that
there exists a finite simply-connected CW-complex X, whose H,(~X,~) has torsion of all
orders (buick and Avramov, to appear).
The reader may have noticed that, on the homotopy side, we often work over a field of
characteristic 0, whereas, in local algebra, we can have residue fields of all charac-
teristics. There are reasons for this ~__fc( .~ however with "LOOKING AHEAD" below, where
more optimistic comments are given). Indeed, in Torsten Ekedahlts paper in these
proceedings, we find the first theorems showing that the beautiful Deligne-Griffiths-
Morgan-Sullivan theory [6], that a K~hlerian compact manifold is "formal" over ~, i.e.
its real homotopy type is a formal consequence of its real cohomology ring, is false
in characteristic p. For the remaining papers in these proceedings (some in algebra,
some in algebraic topology and some being a mixture of both), we refer the reader to
the table of contents.
LOOKING AHEA])
Here are some further directions of research that seem to be fruitful:
)I Two formal power series P(Z) = ~ pn Z n (Po = ,I Pi integers) and Q(Z) = !0qnZn
n>0 n
(qo = 21 qi integers) are said to be "rationally related" if there exists a 2 x 2
matrix (Aik(Z)) whose entries are polynomials in Z with integral coefficients such
that det(Aik(Z)) @ 0 and such that
A11(Z)Q(Z) + A12(Z) All (Z) + A12(Z)
P(Z) = A21(Z)Q(Z) + A22(Z) ( thus A21(Z) + A22(Z) = I if Z = 0 ) .
These matrices modulo the diagonal ones {~ A(Z0 ) A(Z) 0 ) form a group under matrix
multiplication and it would be interesting to try to classify the orbits of this group
acting on, say, the set of power series that are rationally related Hilbert series of
finitely presented graded (1,2)-Hopf algebras. The old question of Kaplansky-Serre
mentioned above is equivalent to asking whether there is just one orbit. Now we know
that there are many orbits. Could we find nice representatives for them? Is there an
analog of the Serret theorem (cf. e.~. [12], p. 55)? Here we have only been talking
Vll
about rational relationship between Hilbert series of graded algebras. Is there an
underlying theory of "rational relationship" between the algebras themselves? If so,
it might be easier to get more precise results about the H,(~X,~) than in the papers
by Halperin et al. in these proceedings.
2) Torsten Ekedahl has recently developed the analog of rational homotopy theory for
spaces "over Z", using cosimplicial algebras. This theory seems very promising, but
nothing has yet been published about it.
3) Growth series and growth al~ebras of ~roups.
Let G be a finitely generated group, with a fixed set of generators S, where we
suppose that S is closed under the operation of taking inverses in G. Let k be a field
and introduce a filtration on the group ring k[G] by means of F-I(k[G]) = 0 ,
Fn(k[G]) = the sub vector space of k[G], spanned by products of ~ n (n ~ O) elements
from S. Then
¢ Fn(k[G])/ Fn-I(k[G]) dsf grs(k[G])
n>0
becomes a finitely generated graded algebra (the growth algebra of (G,S)) [h]. e~Tff
Hilbert series of this graded algebra is the growth series of (G,S). Under some
conditions .f~_c( e.g. [I] for the commutative noetherian case) there is a spectral
sequence of algebras
I E * = Ext* (k~k) => gr EXtk[G](k,k). (5)
grs(k[O])
Could (5) be useful in some cases to relate the growth series of G to the cohomology
of G? Another :![melborp It is known that, if G is fimitely presented, then grs(k[G])
is not necessarily so. Indeed, if grs(k[G]) is finitely presented, then its Hilbert
series is primitive recursive ]8[ and then ]5[ G must have a solvable word problem.
But there are finitely presented groups whose word problem is unsolvable. Thus we are
led to the following
PROBLEM: Is it true that the Hilbert series of finitely presented graded algebras are
always rationally related to growth series of finitely presented groups with a solvable
word problem (and conversely) ?
Stockholm, autumn 1985
JAN-ERIK ROOS
REFEEENCE :S
[I] R. ACHILLES - L. AVRAMOV, Relations between ~_~2perties of a ring and its
associated graded ring, Seminar Eisenbud, Singh, Vogel, vol. 23 Teubner-Texte
der Mathematik, vol. 48, 1982, 5-29, Teubner, Leipzig.
[2] D. ANICK - T. GULLIKSEN~ Rational dependence among Hilbert and Poincar@ series,
Journ. of Pure and Appl. Algebra, 38, 1985, 135-157.
IIIV
]3[ L. AVRAMOV, Homolq6y of local flat extensions and complete intersection defects,
Math. Ann., 228, 1977, 27-37.
]4[ N. BILLINGTON, Growth of ~roups and ~raded al~bras, Commun. in Algebra, 12,
1984, 2579-2588.(Correction later in the same journal.)
]5[ J.W. CANNON, The ~rowth of the closed surface groups and the compact hyperbolic
Coxeter groups (preprint, cf. Theorem 9.1).
[6] P. DELIGNE - .niP GRIFFITHS - J. MORGAN - D. SULLIVAN, Real homotopy theory of
K~hler manifolds, Invent. Math.~ 29, 1975, 245-274.
[7] .S HALPERIN, Rational fibrations~ minimal models and fibrin~s of homogeneous
spaces, Trans. Amer. Math. Sot., 244, ]978, 199-224.
]8[ C. JACOBSSON - V. STOLTENBERG-HANSEN, Poincar@-Betti series are primitive
recursive, Journ. London Math. Soc., ser. 2, 31, 1985, I-9.
]9[ I.M. JAMES, On__cate~ory, in the sense of Lusternik-Schnirelmann, Topology,
17, 1978, 331-348.
[10] G. LEVIN, Local rings and Golod homomorphisms, Journ. of Algebra, 37, 1975,
266-289.
[11] .C LOFWALL, Une al~bre nilpotente dont la s~rie de Poincar@-Betti est non
rationnelle, Comptes rendus Acad. Sc. Paris, 288, s$rie A, 1979, 327-330.
[12] O. PERRON, Die Lehre yon den Kettenbr~chen, Band I, Dritte Aufl., 1954, Teubner,
Stuttgart.
[13] J.-E. ROOS, Relations between the Poincar$-Betti series of loop spaces and of
local rings, Lecture Notes in Mathematics~ 740~ 1979, 285-322, Springer-Verlag,
Berlin, Heidelberg, New York.
Jan-Erik Roos
Department of Mathematics
University of Stockholm
Box 6701
S-113 58 MLOHKCOTS )NEDEWS(
ACKNOWLEDGEMENTS AND GENERAL INFORMATION
The Nordic Summer School and Research Symposium on "ALGEBRA, ALGEBRAIC TOPOLOGY AND
THEIR INTERACTIONS" received support from two sources:
)I The Swedish Natural Science Research Council (NFR)
and
2) The Nordic Governments, through "Nordiska Forskarkurser", which supports The Nordic
Summer School of Mathematics, an organization with one director from each of the
Nordic Countries and which works "with a minimum of bureaucracy" (these are the words
of the founder of the school) and selects subjects and sites for Summer Schools. The
founder and main animator is Professor Lars G~rding (Department of Mathematics,
University of Lund, LUND, Sweden). Since the start in ]966, 13 summer schools,
covering the following other subjects have been arranged: Harmonic Analysis (twice),
Several Complex Variables, Algebraic Topology, Pseudodifferential operators and
applications to index problems, Algebraic Geometry (twice), Discrete Groups and Quasi-
conformal maps, Operator Algebras and their Applications to Quantum Mechanics and
Group representations, Singularities, Value Distribution of Holomorphic maps into
Complex Projective Space (the Cartan-Ahlfors-Weyl theory) and Differential Geometry.
I wish to thank both NFR and "Nordiska Forskarkurser" for their generous support.
I also wish to thank Lars G~rding for his original (]966) initiative, which has turned
out to be so extremely useful and valuable.
The Summer School and Research Symposium took place at the University of Stockholm in
Frescati, August 3 - August 13th~ ]983. The morning sessions consisted mainly of
survey lectures, intended to bring the audience to the level of the research symposium
(in the afternoons), which successively grew more and more advanced. The following
survey lecture series were given:
David ANICK, Basic algebraic topology.
Luchezar AVRAMOV, Local algebra and algebraic topology.
David EISENBUD, Commutative algebra thr~gugh examplesinal~ebraicgeometry.
Tor H. GULLIKSEN, Local algebra and differential gra@ed algebra.
Stephen HALPERIN, Rational homotopy the or ~.
Melvin HOCHSTER, The homological con~cjtures for ipsal rings.
Christer LECH, Relations between a local rin~ and its completion.
Jean-Michel LEMAIRE, Lusternik-Schnirelmann cate~0r~ and related topics.
Rodney Y. SHARP, Basic commutative al~ebra.
Richard STANLEY, Commutative al~ebra and combinatorics.
In the afternoons there were both problem sessions (exercises) for some of the morning
lectures as well as lectures in the research symposium. The following research
symposium lectures did not lead to a publication in these proceedings:
R. FROBERG, Koszul algebras a~d Veronese embeddings.
.M FIORENTINI, Alg~bres gradu@es associ@es aux suites r@guliSres.
.T OGOMA, A note on unmixed domains, us jin6_Poincar@ series.
.D EISENBUD, Linear series on reducible curves and applications.
A.R. KUSTIN, Deformation and linkage of Gorenst_3in algebras.
.A HOLME, Chern numbers of smooth codimension 2 subvarieties of pN N( ~ 6)
.M HOCHSTER, Modules of finite homological dimension with negative intersection
mult ities. iplic
R.Y. SHARP, Generalized fractions and the monomial conjecture.
A.R. PRINCE, Local rings and finite projectij~e planes.
.M BRODMANN, Remarks on the connectedness of al~ebraic varieties.
.A SLETSJ~E, Toroidal embeddings and Poincar@ series.
.K BEHNKE, Infinitesimal deformations of cusp singularities.
.R STANLEY, Symmetric functions and representations of SL(n,~).
A. BJ~RNER, On the Stanley-Reisner.ring of a Tits building.
N. SUZUKI, d-sequences.
I wish to thank all the participants ni( total about 001 people) and in particular all
the lecturers for their interest in this meeting. I also wish to g~ve special thanks
to the following people who helped with practical details before, during and after the
conference: Maje ARONSSON, J6rgen BACKELIN, Rickard B~GVAD, Tnrsten EKEDAHL, Ir$ne
FLOD~N, Ralf FROBERG, Inez HJELM, Clas LO~WALL, June YAMAZAKI and Calle JACOBSSON
(main organizer of an excursion by boat ni the Stockholm archipelago). I also
thank Hubert SHUTRICK for linguistic help.
Finally I wish to thank Springer-Verlag for their cooperation. I hope their patience
will be rewarded.
Stockholm autumn 1985
JA~-ER~ ROOS