Groupes et algébres de lie

ALGEBRAIC VARIETIES By ALEXANDER GROTHENDIEGK It is less than four years since eohomologieal methods (i.e. methods of Homologieal Algebra) were introduced into Algebraic Geometry in Serre's fundamental paper, and it seems already certain that they are to overflow this part of mathematics in the coming years, from the foundations up to the most advanced parts. All we can do here is to sketch briefly some of the ideas and results. None of these have been published in their final form, but most of them originated in or were suggested by Serre's paper. Let us first give an outline of the main topics of eohomologieal investigation in Algebraic Geometry, as they appear at present. The need of a theory of cohomology for 'abstract' algebraic varieties was first emphasized by Weil, in order to be able to give a precise meaning to his celebrated conjectures in Diophantine Geometry. Therefore the initial aim was to find the 'Weil cohomology' of an algebraic variety, which should have as coefficients something 'at least as good as a field of characteristic 0, and have such formal properties (e.g. duality, Runneth formula) as to yield the analogue of Lefschetz's 'fixed-point formula '. Serre's general idea has been that the usual ' Zariski topology ' of a variety (in which the closed sets are the algebraic subsets) is a suitable one for applying methods of Algebraic Topology. His first approach was hoped to yield at least the right Betti numbers of a variety, it being evident from the start that it could not be considered as the Weil cohomology itself, as the coefficient field for cohomology was the ground field of the variety, and therefore not in general of characteristic 0. In fact, even the hope of getting the 'true ' Betti numbers has failed, and so have other attempts of Serre's to get Weil's cohomology by taking the cohomology of the variety with values, not in the sheaf of local rings themselves, but in the sheaves of Witt-vectors constructed on the latter. He gets in this way modules over the ring W{h) of infinite Witt vectors on the ground field h, and W{h) is a ring of characteristic 0 even if h is of characteristic p =t= 0. Unfortunately, modules thus obtained over W{h) may be infinitely generated, even when the variety V is an abelian variety. Although interesting relations must certainly exist between these cohomology groups and the 'true ones', it seems certain 104 ALEXANDER GROTHENDIECK now that the Weil cohomology has to be defined by a completely different approach. Such an approach was recently suggested to me by the connections between sheaf-theoretic cohomology and cohomology of Galois groups on the one hand, and the classification of unramified coverings of a variety on the other (as explained quite unsystematically in Serre's tentative Mexico paper), and by Serre's idea that a 'reasonable' algebraic principal fiber space with structure group G, defined on a variety V, if it is not locally trivial, should become locally trivial on some covering of V unramified over a given point of V. This has been the startingpoint of a definition of the Weil cohomology (involving both 'spatial' and Galois cohomology), which seems to be the right one, and which gives clear suggestions how Weil's conjectures may be attacked by the machinery of Homological Algebra. As I have not begun these investigations seriously as yet, and as moreover this theory has a quite distinct flavor from the one of the theory of algebraic coherent sheaves which we shall now be concerned with, we shall not dwell any longer on Weil's cohomology. Let us merely remark that the definition alluded to has already been the starting-point of a theory of eohomologieal dimension of fields, developed recently by Tate. The second main topic for eohomologieal methods is the cohomology theory of algebraic coherent sheaves, as initiated by Serre. Although inadequate for Weil's purposes, it is at present yielding a wealth of new methods and new notions, and gives the key even for results which were not commonly thought to be concerned with sheaves, still less with cohomology, such as Zariski's theorem on 'holomorphic functions' and his 'main theorem'—which can be stated now in a more satisfactory way, as we shall see, and proved by the same uniform elementary methods. The main parts of the theory, at present, can be fisted as follows: (a) General finiteness and asymptotic behaviour theorems. (6) Duality theorems, including (respectively identical with) a eohomologieal theory of residues. (c) Riemann-Roch theorem, including the theory of Chern classes for algebraic coherent sheaves. (cu) Some special results, concerning mainly abelian varieties. The third main topic consists in the application of the eohomologieal methods to local algebra. Initiated by Koszul and Cartan-Eilenberg in connection with Hubert's 'theorem of syzygies', the systematic use of COHOMOLOGY THEORY 105 these methods is mainly due again to Serre. The results are the characterization of regular local rings as those whose global eohomologieal dimension is finite, the clarification of Cohen-Macaulay's equidimensionality theorem by means of the notion of eohomologieal codimension\ and especially the possibility of giving (for the first time as it seems) a theory of intersections, really satisfactory by its algebraic simplicity and its generality. Serre's result just quoted, that regular local rings are the only ones of finite global eohomologieal dimension, accounts for the fact that only for such local rings does a satisfactory theory of intersections exist. I cannot give any details here on these subjects, nor on various results I have obtained by means of a local duality theory, which seems to be the tool which is to replace differential forms in the case of unequal characteristics, and gives, in the general context of commutative algebra, a clarification of the notion of residue, which as yet was not at all well understood. The motivation of this latter work has been the attempt to get a global theory of duality in cohomology for algebraic varieties admitting arbitrary singularities, in order to be able to develop intersection formulae for cycles with arbitrary singularities, in a nonsingular algebraic variety, formulas which contain also a 'Lefschetz formula mod.p'. In fact, once a proper local formalism is obtained, the global statements become almost trivial. As a general fact, it appears that, to a great extent, the 'local' results already contain a global one; more precisely, global results on varieties of dimension n can frequently be deduced from corresponding local ones for rings of Krull dimension n + l. We will therefore turn now to giving some main ideas in the second topic, that is, the cohomology theory of algebraic coherent sheaves. First, I would like, however, to emphasize one point common to all of the topics considered (except perhaps for {d)), and in fact to all of the standard techniques in Algebraic Geometry. Namely, that the natural range of the notions dealt with, and the methods used, are not really algebraic varieties. Thus, we know that an affine algebraic variety with ground field h is determined by its co-ordinate ring, which is an arbitrary finitely generated ^-algebra without nilpotent elements; therefore, any statement concerning affine algebraic varieties can be viewed also as a statement concerning rings A of the previous type. Now it appears that most of such statements make sense, and are true, if we assume only J. to be a commutative ring with unit, provided we sometimes submit it to some mild restriction, as being noetherian, for instance. In 106 ALEXANDER GROTHENDIECK the same way, most of the results proved for the local rings of algebraic geometry, make sense and are true for arbitrary noetherian local rings. Besides, frequently when it seemed at first sight that the statement only made sense when a ground field k was involved, as in questions in which differential forms are considered, further consideration of the matter showed that this impression was erroneous, and that a better understanding is obtained by replacing k by a ring B such that A is a finitely generated J5-algebra. Geometrically, this means that instead of a single affine algebraic variety V (as defined by A) we are considering a 'regular map' or 'morphism' of V into another affine variety W, and properties of the variety V then are generalized to properties of a morphism V -> W (the 'absolute' notion for V being obtained from the more general 'relative' notion by taking W reduced to a point). On the other hand, one should not prevent the rings having nilpotent elements, and by no means exclude them without serious reasons. Now just as arbitrary commutative rings can be thought of as a proper generalization of affine algebraic varieties, one can find a corresponding suitable generalization of arbitrary algebraic varieties (defined over an arbitrary field). This was done by Nagata in a particular case, yet following the definition of schemata given by Chevalley he had to stick to the irreducible case, and with no nilpotent elements involved. The principle of the right definition is again to be found in Serre's fundamental paper, and is as follows. If A is any commutative ring, then the set Spec {A) of all prime ideals of A can be turned into a topological space in a classical way, the closed subsets consisting of those prime ideals which contain a given subset of A. On the other hand, there is a sheaf of rings defined in a natural way on Spec {A), the fiber of this sheaf at the point p being the local ring Ap. More generally, every module M over A defines a sheaf of modules on Spec {A), the fiber of which at the point p is the localized moduleMp over Ap. Now we call pre-schema a topological space X with a sheaf of rings Gx on X, called its structure sheaf, such that every point of X has an open neighborhood isomorphic to some Spec {A). If X and Y are two pre-schemas, a morphism f fro

Proceedings of the International Congress of Mathematicians

Author(s): Haglund, J; Haiman, M; Loehr, N; Remmel, J B; Ulyanov, A | Abstract: Let Rn be the ring of coinvariants for the diagonal action of the symmetric group Sn. It is known that the character of Rn as a doubly graded S-module can be expressed using the Frobenius characteristic map as nabla en, where en is the n-th elementary symmetric function and nabla is an operator from the theory of Macdonald polynomials. We conjecture a combinatorial formula for nabla en and prove that it has many desirable properties that support our conjecture. In particular, we prove that our formula is a symmetric function (which is not obvious) and that it is Schur positive. These results make use of the theory of ribbon tableau generating functions of Lascoux, Leclerc, and Thibon. We also show that a variety of earlier conjectures and theorems on nabla en are special cases of our conjecture. Finally, we extend our conjectures on nabla en and several on the results supporting them to higher powers nablam en.

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A combinatorial formula for the character of the diagonal coinvariants

In this paper we introduce and develop the theory of FI-modules. We apply this theory to obtain new theorems about: • the cohomology of the configuration space of n distinct ordered points on an arbitrary (connected, oriented) manifold; • the diagonal coinvariant algebra on r sets of n variables; • the cohomology and tautological ring of the moduli space of n -pointed curves; • the space of polynomials on rank varieties of n × n matrices; • the subalgebra of the cohomology of the genus n Torelli group generated by H 1 ; and more. The symmetric group S n acts on each of these vector spaces. In most cases almost nothing is known about the characters of these representations, or even their dimensions. We prove that in each fixed degree the character is given, for n large enough, by a polynomial in the cycle-counting functions that is independent of n . In particular, the dimension is eventually a polynomial in n . In this framework, representation stability (in the sense of Church–Farb) for a sequence of S n -representations is converted to a finite generation property for a single FI-module.

FI-modules and stability for representations of symmetric groups

We introduce generalized permutation patterns, where we allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We show that essentially all Mahonian permutation statistics in the literature can be written as linear combinations of such patterns. Almost all known Mahonian permutation statistics can be written as linear combinations of patterns of length at most 3. There are only fourteen possible such Mahonian statistics, which we list. Of these, eight are known and we give proofs for another three. The remaining three we conjecture to be Mahonian. We also give an explicit numerical description of the combinations of patterns a Mahonian statistic must have, depending on the maximal length of its patterns.

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Generalized permutation patterns and a classification of the Mahonian statistics

We prove a combinatorial formula for the Macdonald polynomial $\tilde{H}_{\mu }(x;q,t)$ which had been conjectured by Haglund. Corollaries to our main theorem include the expansion of $\tilde{H}_{\mu }(x;q,t)$ in terms of LLT polynomials, a new proof of the charge formula of Lascoux and Schutzenberger for Hall-Littlewood polynomials, a new proof of Knop and Sahi's combinatorial formula for Jack polynomials as well as a lifting of their formula to integral form Macdonald polynomials, and a new combinatorial rule for the Kostka-Macdonald coefficients $\tilde{K}_{\lambda \mu }(q,t)$ in the case that $\mu $ is a partition with parts $\leq 2$.

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A combinatorial formula for Macdonald polynomials

Introduction to $q$-analogues and symmetric functions Macdonald polynomials and the space of diagonal harmonics The $q, t$-Catalan numbers The $q, t$-Schroder polynomial Parking functions and the Hilbert series The shuffle conjecture The proof of the $q, t$-Schroder theorem The combinatorics and Macdonald polynomials The Loehr-Warrington conjecture Solutions to exercises Bibliography.

The q, t-Catalan numbers and the space of diagonal harmonics : with an appendix on the combinatorics of Macdonald polynomials

http://www.mathe2.uni-bayreuth.de/axel/papers/garsia:a_proof_of_the_q,t_catalan_positivity_conjecture.pdf

James Haglund

Papers

A combinatorial formula for the character of the diagonal coinvariants

A combinatorial formula for Macdonald polynomials

The q, t-Catalan numbers and the space of diagonal harmonics : with an appendix on the combinatorics of Macdonald polynomials

A proof of the q, t-Catalan positivity conjecture

Conjectured statistics for the q,t-Catalan numbers