### 2011-12 course

The year’s course has come to an end.  (I would like to give huge thanks to the people in the class — their detailed comments and suggestions have led to a vast number of improvements.)  A revised version is now posted at the usual place (the August 16, 2012 version).

I may later write a post with some thoughts on how the course went.  (In short: I think it showed that it is possible to cover the amount of material I wanted to, in a single year.  There were two topics I wanted to cover, but did not because I have not yet written up the exposition; but they were replaced with other useful facts.)  But right now, I want to post the latest version of the notes, with a substantially new exposition of the proof of Serre duality (the final chapter — the missing chapters are earlier) for projective varieties $X$.

I have reverted to an earlier proof I gave in versions years ago, in terms of finite (often flat) morphisms.  (I was partially prompted by an email discussion with Yuhao Huang at Berkeley — thank you Yuhao!)

There are a number of possible statements of Serre duality one might want, with the expected trade-offs:  better statements require more work.  I found productive (both personally and pedagogically) to discuss these trade-offs, and to make some decisions, and to see (in class) where approaches broke down, and where they worked.

I concentrated on several desiderata.  We want some some of duality involving a dualizing sheaf.  We will want the dualizing sheaf to be the determinant of the cotangent bundle.  This is surprisingly hard, and left to the end.

There will be potentially three versions of each desired type of duality.  There is the dimensional (lame) version, saying that the dimension of two cohomology groups are the same.  Better, there should be some duality between the cohomology groups, that should be functorial in the sheaf/bundle (“functorial Serre duality”).  And best of all, it should arise from a cup product of some sort (“trace version” of Serre duality), which requires defining the cup product.

The first kind of duality one might ask for (because we used it in discussing curves and surfaces) is Serre duality for vector bundles.  Better yet, one can have it for coherent sheaves.  (And we can get better still:  Serre duality for families; dualizing complexes; etc.  But the brutal demands of doing it all within a course, with proofs, means that we do not go there.)

In order to move toward proving these things (or even to move toward making some of the statements precise), we have to discuss Ext groups, which wasn’t hard given what we had done earlier.  As an aside (double-starred), I discussed the cup product for Ext, which of course is needed for the best statements of Serre duality.  (But it is not needed for what I prove, and what I need!)  The method of proof is to do Serre duality for projective space, and then to go to finite flat covers.  The key trick is to use a version of the “upper shriek” construction, an occasional adjoint to $\pi_*$ (even if isn’t quite “upper shriek”, so I denoted it $\pi^!_{sh}$ rather than $\pi^!$).  A complication is added by the fact that we are working with O-modules, but the construction I give only works for quasicoherent sheaves; and it works best for closed immersions.  I won’t say much more here, because it will only confuse you — it is best to read the notes.

With this approach, Serre duality (in the versions we use) ends up being surprisingly easy — it seems to come out of nowhere.  (This was independently stated to me by three people in the class, and I agree.)  But there is a surprising amount of subtlety hidden in the exposition here.  The subtlety isn’t in the arguments, or what is said; it is in the choice of what to say, and what paths to take.   There were many points where I tried to take a different route, and then something bad happened, forcing me to retract.  (The version from earlier this year proved Serre duality for projective schemes using closed immersions into projective space, but I got myself into difficulty, as observed by a number of people, including Yuhao Huang, Yuncheng Lin, Preston Wake, Charles Staat, and Yifei Zhu.  This approach ended up being much cleaner.)

Finally, this approach does not easily show that the dualizing sheaf of a smooth projective variety is the “sheaf of algebraic volume forms” (the determinant of the cotangent bundle).  To do this requires more work, and a different approach, which also allows us to prove the adjunction formula.  (However, it *is* possible to prove this directly using the “finite flat cover” approach.  Matt Baker and Janos Csirik worked it out in this note.  (I thank both Matt and Janos for permission to post this here.)

Aside:  I realized that alternative expositions can work as well, with different costs and benefits.  For example, the category of quasicoherent sheaves on a variety actually has enough injectives, so one can work directly in that category.

What comes next.  My to-do list has been finite and shrinking, but there are many things left to do.  There are a number of loose ends in the chapters already done.   I had hoped to finish the three final chapters this summer, and now I am just going hoping to finish a draft of one of them (on regular sequences).

A revised version is now posted at the usual place (the March 25, 2012 version). We have reached the end of the second quarter of our academic year, so I want to pause and look back on where we are, and fill in those who are just watching the notes evolve. (The course webpage is here.)

If we continue at the current breakneck pace, we will finish all the central material I have claimed can be covered in a single-year course. We may not succeed, but it will not be because the goal is impossible. (Instead: I have some material still to think through and prepare, and I may not manage it to my satisfaction.) I am well aware that I have 30 weeks to work with (longer than the academic year at most universities), and the people in the class are not typical, in many ways.

More precisely: in the notes, we’ve reached elliptic curves (we will begin the next quarter showing that they are group schemes). I consider everything up to 21.8 to be in very good shape. There are things that still need fixing, but I have an explicit finite list, which is large, but shrinking. I have no sections that (in my mind) need serious revision before 21.9.

Here are some rambling thoughts, both large and small, in the order in which they appear in the text. Before I begin, I should say that there are many many improvements, due to people in my class, but also a large number of sending emails from elsewhere on the globe, and also posting here. I want to repeatedly thank you for the huge number of comments you have sent in.

The section on valuative criteria (13.5) is now in potential “final form”. In other words, it is now self-contained, and open for criticism. I state the criteria (6 in total: valuative criteria for separatedness, universal closure, and properness, each in “DVR” and “general” versions), but do not prove them. I sketch the proof of the valuative criterion for separatedness in the DVR case (I basically give the proof). This is based on the discussion in the post on valuative criteria here. Please feel free to complain! (Any attempt to give a complete proof of the valuative criterion of properness ended up being longer than I wanted to include at this point.)

Fun fact (14.5.B): suppose you have a short exact sequence of quasicoherent sheaves. If the first and third are locally free, then so is the second. If the second and third are locally free and of finite rank, then so is the first. I had wondered about a counterexample if the “finite rank” hypotheses were removed. Daniel Litt has given me one, and posted it here. (Perhaps this or something like it is in the literature? Perhaps this should be added to the stacks project?)

I am mildly curious about the following (cf. 16.4). (Not curious enough that I’ve given it any thought, but curious enough that I’m hopeful someone has a very fast answer.) If $S_*$ is a graded ring, and $M_*$ is a graded $S_*$-module, if $M_*$ is finite type, is the corresponding quasicoherent sheaf finite type? And similarly for coherence? Presumably yes.  (Update June 29, 2012:  Fred Rohrer has explained this now, see below.)

The way in which I first discuss pullbacks has evolved (17.3); three different approaches all come into it (the affine-local picture; the universal property; and the “inverse image then tensor with structure sheaf” definition). (Feedback I’d earlier gotten: one expert prefers a more general approach, doing things for ringed spaces; two learners found the exercises surprisingly straightforward. So far I’m sticking with straightforward over general.)

The notion “generated by global sections” is slightly awkward, especially when relativized. I’m using the terms “globally generated” (16.3), “finitely globally generated” (16.3), and “relatively globally generated” (18.3.7). If this potentially bothers you, please complain. Ideally make a counteroffer, or at least an argument.

Relative Proj is now done differently (see 18.2). I am now quite happy with the approach, because I have (sadly) given up on dealing with any universal property, as without it, the construction is very easy (when done in the right way). If anyone reads it, please let me know what you think, and tell me what is still confusing. (Summary of feedback to date: people find this an uninspiring topic, but the exercises are gettable.)

In Exercise 18.3.B, we show that the composition of projective morphisms is projective if the final target is quasicompact. (That wacky hypothesis is part of the sign that the notion of projective notion is not great.) I am curious: does anyone know a counterexample without the quasicompactness hypothesis? This isn’t important (it will undoubtedly never come up for me in real life). [Update August 21, 2012: I’ve now asked it on mathoverflow.]

(Update March 27, 2012: there were many typos in the Chow’s Lemma section, so a revised version is now here.) In 20.8, I prove the following form of Chow’s Lemma: if $\pi: X \rightarrow \text{Spec} A$ is proper, and $A$ is Noetherian, then there exists $\rho: X' \rightarrow X$ surjective and projective, with $\pi \circ \rho$ also projective, and with $\rho$ an isomorphism on a dense open subset of $X$. I want to include all other versions that reasonable people (or even reasonably unreasonable people) might reasonably use — with references, but most likely without proofs. The versions I can think of are: (i) weaken “proper” to “finite type and separated”, and weaken the conclusion to “$\pi \circ \rho$ is quasiprojective” (rather than projective), and (ii) a generalization where $\text{Spec} A$ is replaced by a Noetherian scheme, and (iii) = (ii)+(i) (EGA II.5.6.1). If $X$ is reduced, or irreducible, or integral, then we can obviously take $X'$ to be as well. EGA II.5.6 has a variant where the target is quasicompact and separated, with a finite number of irreducible components. Are there any other variants I should care about?

A revised version is now posted at the usual place. The first quarter of our academic year is now over (meaning I’ve completed the first third of the course), so now is a good time to report how things went.

First, I’ll briefly explain changes since the last version (which was posted without much fanfare late in the quarter). (i) The long-promised starred section on geometrically connected (and irreducible and reduced and integral) is now added (10.5); if anyone tries any of it, please let me know how it goes. (ii) The section on (very) ample line bundles was pulled out of 16.3 (on globally generated and base-point-free line bundles) into a new section (17.6) because Giuliano Gagliardi pointed out that it used pullbacks Important Theorem 17.4.1 (describing maps to projective schemes in terms of line bundles — also known as the functorial description of projective space). Odds are unfortunately high that I’ve screwed up some dependencies, so if you find yourself in that part of the notes, please keep an eye out for unintended consequences. (iii) I’ve finally changed (open, closed, locally closed) “immersion” to (open, closed, locally closed) “embedding” throughout (the horror!).

Now back to a report on the class. I was pleased with how it went — we covered more than I’d hoped, reaching 12.2. This is far more than a third of the notes, even taking into account the chapters and sections that are not yet fully written. I should admit that the class was unusually strong this year, and thus not representative, but my hypothesis that one can cover a huge amount of the foundations of the subject in a single-year course looks like it might be vindicated. There were fewer comments in class than usual (has the exposition solidified too much?), but the problem set solutions were very strong. Partially by design, we ended with a “punchline” (a pleasant concluding topic that I think courses should ideally have — even if it is often not possible): I spent much of the final lecture discussing lines on surfaces in 3-space. Given what we know, we were able to prove interesting things, and also get a glimpse of geometrically interesting ideas in the future. I hope this helped give some relief from the intense barrage of formalism leading up to it.

I have two possible punchlines in mind for the second (winter) quarter: the theory of curves, or intersection theory. And while it is dangerous to predict two quarters in advance, one possible punchline for the final (spring) quarter is the topic of the 27 lines on “the” cubic surface.

Comments, suggestions, and corrections many of you have sent in (by commenting here, or by emailing me) have been very helpful. I am behind on responding to them, but I’m notably less behind than I was at the start of the quarter, and I expect this to continue. So please keep sending in comments (even highly opinionated ones)!

A revised version of the notes is now posted in the usual place.

I am continually learning more about how rich and complex the notion of associated points is. I had first understood them in terms of primary decomposition. My initial presentation was quite algebraic, and geometrically unmotivated. I later realized that associated points could be better understood without primary ideals, and the result would be streamlined and shorter. The result was in the version originally posted on this site (and that remained in the notes until the new version). I was pleased with myself: people could get through associated points without too much sweat, although without much motivation. (Feedback from readers and students confirmed this: people found this section more obscure than I would have liked.)

Just over a year ago, Matthew Emerton vigorously argued in this comment that there was a much better geometric point of view. My experience in the past is that when Matt makes a point like this, it causes a revolution in how I think about the topic, and indeed it happened again. (Charles Staats joined the discussion as well.)

As I’m about to discuss associated points in this year’s course, I’ve had a chance to carefully think over Matt’s point of view. I’m not surprised that he convinced me; but I am surprised at how big a difference his perspective made to me. (I also find striking that this is my third point of view on associated points, and I can even see a case for a fourth or a fifth depending on how you think. As one example, primary ideals are certainly absolutely central from many points of view.)

Section 6.5 now attempts to get across this perspective, and I offer it somewhat tentatively, because I may be missing some insights that will make it cleaner still.

Here is the current exposition. As always, the most important things for a reader/learner/expert to know are properties of a concept, not the definitions or proofs. I state the key properties first; they are quite different than the key properties I had in the previous incarnation. As I now see it, the key properties are this. Let $A$ be a Noetherian ring (for now), and $M$ a finitely generated $A$-module.

Following (my interpretation of) Matt’s inspiration: the most important property is:

(A) The associated points of $M$ are precisely the generic points of irreducible components of the supports of sections of $\tilde{M}$ (elements of $M$).

This leads to lots of useful properties by purely geometric thinking (as Matt points out). We could even take it as the definition, but in the rigorous development, I don’t. (Side point: a variant of this is the statement that the associated points of $M$ are precisely the generic points of supports that happen to be irreducible. I found this less useful, but perhaps I’m missing some insights. It’s possible Matt had this definition in mind.)

There are two other “first principles” to keep in mind.

(B) There are a finite number of associated points.

(C) A function is a zero-divisor if and only if it vanishes at an associated point.

I couldn’t get (B) and (C) to follow cheaply from (A). But lots of great properties follow from these (especially (A), sometimes with the help of (B)), including:

1. Generic points of irreducible components are associated.
2. The map from $M$ to the product of stalks/localizations at the associated points is injective.
3. The support of $m \in M$ is the closure of those associated points where it is supported.
4. The nonreduced locus of Spec $A$ is the closure of the nonreduced associated points.
5. The notion of associated points behaves well with respect to localization, so for example we can define associated points of (coherent sheaves on) locally Noetherian schemes.

In order to establish (A)(C), I need some algebra, which I do in a series of exercises. I’m not surprised that algebra is necessary, because Noetherianness (of the “sheaf”, not just the topology) needs to be used. But perhaps the exposition can be “geometrized” further, to make it more geometrically natural. (Any suggestions would be appreciated — not just for the notes, but for me!)

In particular, the definition of associated prime I take is a prime that is the annihilator of some element of $M$. I show (B) and (C) directly, as well as #2 and #5, which I then use to show (A).

Any comments on the exposition would be greatly appreciated. Am I missing some fundamental insights that would simplify things? Is the exposition readable to someone seeing it for the first time? Are the problems gettable by people seeing these ideas for the first time? I’ll get some feedback from my class, but they are not representative of a possible readership (as many of them know a ridiculous amount).

(I’m curious how much of this carries over out of the Noetherian or integral setting. If the various definitions then differ, which is the “right” one? But this is a secondary issue; I haven’t thought about it much, and it also is less relevant for most learners.)

A revised version is now posted at the usual place. There are no new sections. I’ll be posting revisions over the next academic year, in response to comments from the course, comments here (both new and old), and a large number of emails I’m gradually going through.

I won’t bother adding a new post in the future when there are only relatively minor revisions. But if you want to be informed, just let me know. And if people would prefer that I announce each revision, I’ll do so.

The course webpage is here (http://tinyurl.com/FOAG1112). The course notes are here.

The seventeenth version of the notes is the July 21 version in the usual place. This version has a complete exposition (i.e. everything I currently intend to say) of flatness (chapter 25), and a proof of Serre duality (chapter 28). Some content is added earlier (e.g. the Artin-Rees Lemma). The next post may appear in August, depending on baby constraints.

Status report.

There are only three more content chapters still to come, one on smooth/etale/unramified morphisms; one on formal functions and related issues (Zariski’s main theorem, Stein factorization, etc.); and one on regular sequences and related issues (local complete intersections, Cohen-Macaulayness, etc.). I’m 100% sure they will appear, but I’m not sure when (again, due to oncoming family constraints). Of course, a lot of work remains to be done to fill in holes and patch problems in the rest of the notes (and responding to old comments), so I may spend some time doing that.

I also want to take this opportunity to thank Sándor Kovács for advice throughout this project (and before), both technical and otherwise.

For learners.

Flatness is confusing the first time you see it. Also the second and the third. But with each iteration, you will digest and master more aspects of flatness. With most parts of algebraic geometry, when you learn a concept, you get used to one strange thing and then you’re good to go. That’s not true with flatness — when thinking over this chapter, I realized that there are many different types of results and arguments that come up. I’ve done my best to organize them, and to discuss no more than I find brutally necessary. (Many important flatness facts are left unproved or unstated, but hopefully by the end you will know enough to be able to read what you need elsewhere.)

The structure of the chapter is described in 25.1.1, so I won’t repeat it here.

I hope some of you read the proof of cohomology and base change — if you do, please let me know, and please let me know what is most confusing!

Here as always are some suggested problems. Of course, try every exercise marked “easy”. If it isn’t easy, let me know!

Here are twelve problems on flatness: 25.2.E (transitivity of flatnes); 25.2.G (relating flatness in algebra with flatness in scheme theory); 25.2.L (explicit examples), 25.2.M (cohomology commutes with flat base change — this looks hard but isn’t), 25.3.A (explicit and important Tor calculation), 25.3.F (practice with Tor), 25.4.D (flat = torsion-free for a PID), 25.4.F (“finite flat morphisms have locally constant degree”), 25.4.I (an explicit example that will come up later, involving two planes meet at a point), 25.5.D (going-down for flat morphisms), 25.5.F (fibers of flat morphisms have the “expected dimension”), 25.7.B (important! invariance of many important numbers in flat families), some 25.8.A-F (using cohomology and base change),

If you want to work through some pleasant explicit examples, I’d recommend 25.4.8 onwards, on flat limits. Another fun discussion that will help you see if you understand flatness well enough to do something is Hironaka’s example of a proper nonprojective nonsingular threefold, 25.8.6. If you get stuck, please let me know.

If you read 25.10 on flatness and completions, please let me know how it went.

Chapter 28 is a starred proof of Serre duality. I hope some of you try to read it — it is not double-starred, which means that I intend for this to be readable, and not just an indication that a proof exists. As with flatness, I try to prove no more than I really need to given this stage of the notes/course. If you read this, some good problems to try are 28.3.C (relations among Ext, sheaf Ext, and H^i), 28.3.H (Ext and vector bundles), and 28.3.J (the local-to-global spectral sequence for Ext).

For experts (and general discussion)

The Artin-Rees Lemma.

Greg Brumfiel explained the Artin-Rees Lemma to me in a way that made it very natural — enough so that I can no longer forget the proof. I’d never understood it well before. I hope I’ve gotten it across with some semblance of Greg’s clarity (13.6 and parts of 25.10).

Question: suppose $A$ is a Noetherian ring, and $I$ an ideal. Suppose $0 \rightarrow M \rightarrow N \rightarrow P \rightarrow 0$ is an exact sequence of $A$-modules. I find it entertaining that it remains exact when tensoring by $\hat{A}$, but that $0 \rightarrow \hat{M} \rightarrow \hat{N} \rightarrow \hat{P} \rightarrow 0$ need not be exact (without coherence hypotheses on the modules — perhaps just on $P$?). Does anyone have a( reference to a)n example where exactness does not hold? (And as a consequence, we’ll see an example where completion is not the same as tensoring with $\hat{A}$.) Update August 17, 2011: An answer is given in the section “Completion is not exact” in the “Examples” chapter of the Stacks Project. I’m not sure how to find out the tag. But I’ll add this example, in the version to be released around the end of August.

Flatness.

I found the flatness notes of Brian Lehmann (available on his webpage) very nice. Andrew Critch’s enlightening a postiori explanation of how to think about flatness is incorporated into
Remark 25.4.2, just before the equational criterion. (Side remark: we don’t use the equational criterion for anything.)

I hope someone looks closely at my exposition (and proof) of Cohomology and Base Change. That’s a topic where I think I learned the right perspective only by talking to people, and part of my goal is to translate some of the folklore into writing. If you have never bothered fully understanding the proof, and want to, please take a look and let me know where the exposition confuses you. (It is now divided up into some general facts about cohomology of complexes, and a very short argument for the theorem itself.)

Max Lieblich told me that he first figured out Cohomology and Base Change by translating to local rings, and working there, which has the advantange that you can make short exact sequences split. I could imagine that this would yield an even faster exposition. (One worry: the statement I want of cohomology and base change involves an honest Zariski open set, see part (i) of my statement. But I bet Max’s approach would give that too.) Partially because I’d already written this, I haven’t tried to piece together how Max’s argument should go. But if someone does, or someone thinks I should because it would make things more transparent, please let me know.

The flatness chapter is disappointingly long — and I even didn’t prove (for example) that the flat locus is open (under reasonable hypotheses). I didn’t prove Grothendieck’s generic freeness lemma because I didn’t use it (but I stated it). I didn’t prove the fibral flatness theorem, but stated it. Are there things that I really should include? Are there things I’ve included that you think could reasonably be tossed in a first course? (You’ll noticed that lots of the chapter is already starred or double-starred.) One fact that isn’t there but will be (in a later chapter) is what Brian Conrad calls “miracle flatness”, about a morphism $\pi: X \rightarrow Y$ to a nonsingular scheme, and relating the flatness of $\pi$, the equidimensionality of the fibers, and the Cohen-Macaulayness of $X$.

Serre duality

[Update August 2012: more discussion is in the 22nd post.]

I’m going to upset some people here, by not proving the “right” statement. My goal, given that this discussion comes at the end of a long set of notes, and at the end of a long course, is to prove just enough to justify the statements made earlier in the notes.

Here’s what gets used:
(i) we need a perfect pairing (28.1.1.1) $H^i(X, \mathcal{F}) \times H^{n-i}(X, \mathcal{F}^{\vee} \otimes \omega_x) \rightarrow k$ in good circumstances.
(ii) We need the dualizing sheaf to be the determinant of the cotangent bundle if $X$ is smooth.
(iii) Perfect pairing (i) often arises from something better (which I call Strong Serre duality) which is an isomorphism $\rm{Ext}^i_X(\mathcal{F}, \omega_X) \rightarrow H^{n-i}(X, \mathcal{F})^{\vee}$.
I show (iii), but don’t show that these maps are well-behaved in any way at all — for what we do, we don’t need the perfect pairing (28.1.1.1) to be “natural” in any way — we just need dimensions. The reason I can’t show any sort of naturality (in a pedagogically easy way) is that it isn’t worth the trouble of defining the natural maps $\rm{Ext}^i_X( \mathcal{F},\mathcal{G}) \times H^j(X, \mathcal{F}) \rightarrow H^{i+j}(X, \mathcal{G})$. In 28.3.4, I do mention where these maps come from, and outline the Yoneda cup product for Ext’s (following Grothendieck’s Theoreme de dualite pour les faiscaux algebriauqes coherents — apologies for lack of accents).

The advantage of my approach is that we can prove the statement we actually use relatively easily (although not so easily that I’d remove the star from the chapter). Keep in mind that we are the end of the course; I want to prove what we use as easily as possible.

(The disadvantage is that we clearly prove the wrong statement!)

Side remark: in an earlier version of the course, I proved Serre duality via duality for finite flat morphisms. This results in a proof which is much easier and shorter. (To apply it, we need the “miracle flatness theorem” I mentioned above; but that will be included.) The serious downside of this approach was that I was unable to prove (ii). So instead I decided to go with the current exposition, which requires more work.

Random questions for experts

1. I’ve proved uppersemicontinuity of fiber dimension on the target (for a projective morphism). But I haven’t proved uppersemicontinuity of fiber dimension on the source (for locally finite type morphisms to locally Noetherian schemes; or if you really care, for locally finitely presented morphisms in general, but that’s just an easy generalization once you’ve got the hard part). I don’t know an easy proof (i.e. short given what is already done in the notes). Does anyone know one (or have a reference)? It seems to be surprisingly hard work. (I also asked for a trick solution here.)

2. A reference questions about the category of O-modules on a scheme. I have heard that they don’t have enough projectives. (I asked a variant of this question here.) Does anyone have a reference (ideally with a proof)? I’ve heard that locally free sheaves on a scheme are not necessarily projective in the category of O-modules. Reference with proof? (Update August 2, 2011: see David Speyer’s comment below.)

3. (unimportant; maybe better suited to mathoverflow) It was in grad school that I first heard about the Lefschetz principle, allowing you to reduce all statements over an algebraically closed field of char 0 to $\mathbf{C}$. Even now I’m not sure precisely what this principle is supposed to be (except in a rather baby case, where it is basically elimination of quantifiers). Is there a reference somewhere? Here is an interesting article complaining about it. (Warning: you need access to jstor to access it. Bibliographic info: A. Seidenberg, Comments on Lefschetz’s Principle, The American Mathematical Monthly, Vol. 65, No. 9 (Nov., 1958), pp. 685-690.) Here is a possible reference (that I’ve not read): Frey, Gerhard and Rück, Hans-Georg, The strong Lefschetz principle in algebraic geometry, Manuscripta Math. 55 (1986), no. 3-4, 385–401.