Ext Groups Research Articles

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Let A denote the mod 2 Steenrod algebra. In this paper we make calculations to completely determine the Ext groups Ext A 4 , ∗ ( Z / 2 , Z / 2 ) and also to determine the structure of Z / 2 -submodule of decomposable elements in Ext A 5 , ∗ ( Z / 2 , Z / 2 ) .

Intertwining numbers; The n-rowed shapes

A fairly old problem in modular representation theory is to determine the vanishing behavior of the Hom groups and higher Ext groups of Weyl modules and to compute the dimension of the ℤ/(p)-vector space $$Hom_{\bar A_r } (\bar K_\lambda ,\bar K_\mu )$$ for any partitions λ, μ of r, which is the intertwining number. K. Akin, D. A. Buchsbaum, and D. Flores solved this problem in the cases of partitions of length two and three. In this paper, we describe the vanishing behavior of the groups $$Hom_{\bar A_r } (\bar K_\lambda ,\bar K_\mu )$$ and provide a new formula for the intertwining number for any n-rowed partition.

Sheaves on Artin stacks

We develop a theory of quasi–coherent and constructible sheaves on algebraic stacks correcting a mistake in the recent book of Laumon and Moret-Bailly. We study basic cohomological properties of such sheaves, and prove stack–theoretic versions of Grothendieck’s Fundamental Theorem for proper morphisms, Grothendieck’s Existence Theorem, Zariski’s Connectedness Theorem, as well as finiteness Theorems for proper pushforwards of coherent and constructible sheaves. We also explain how to define a derived pullback functor which enables one to carry through the construction of a cotangent complex for a morphism of algebraic stacks due to Laumon and Moret–Bailly. 1.1. In the book ([LM-B]) the lisse-etale topos of an algebraic stack was introduced, and a theory of quasi–coherent and constructible sheaves in this topology was developed. Unfortunately, it was since observed by Gabber and Behrend (independently) that the lisse-etale topos is not functorial as asserted in (loc. cit.), and hence the development of the theory of sheaves in this book is not satisfactory “as is”. In addition, since the publication of the book ([LM-B]), several new results have been obtained such as finiteness of coherent and etale cohomology ([Fa], [Ol]) and various other consequences of Chow’s Lemma ([Ol]). The purpose of this paper is to explain how one can modify the arguments of ([LM-B]) to obtain good theories of quasi–coherent and constructible sheaves on algebraic stacks, and in addition we provide an account of the theory of sheaves which also includes the more recent results mentioned above. 1.2. The paper is organized as follows. In section 2 we recall some aspects of the theory of cohomological descent ([SGA4], V) which will be used in what follows. In section 3 we review the basic definitions of the lisse-etale site, cartesian sheaves over a sheaf of algebras, and verify some basic properties of such sheaves. In section 4 we relate the derived category of cartesian sheaves over some sheaf of rings to various derived categories of sheaves on the simplicial space obtained from a covering of the algebraic stack by an algebraic space. Loosely speaking the main result states that the cohomology of a complex with cartesian cohomology sheaves can be computed by restricting to the simplicial space obtained from a covering and computing cohomology on this simplicial space using the etale topology. In section 5 we generalize these results to comparisons between Ext–groups computed in the lisse-etale topos and Ext–groups computed using the etale topology on a hypercovering. In section 6 we specialize the discussion of sections 3-5 to quasi–coherent sheaves. We show that if X is an algebraic stack and OXlis-et denotes the structure sheaf of the lisse-etale topos, then the triangulated category D qcoh(X ) of bounded below complexes of OXlis-et–modules with quasi–coherent cohomology sheaves satisfies all the basic properties that one would expect from the theory for schemes. For example we show in this section that if f : X → Y is a quasi–compact morphism of algebraic stacks and M is a quasi–coherent sheaf on X Date: November 2, 2005. 1

A homological interpretation of Jantzen's sum formula

For a split reductive algebraic group, this paper observes a homological interpretation for Weyl module multiplicities in Jantzen's sum formula. This interpretation involves an Euler characteristic χ built from Ext groups between integral Weyl modules. The new interpretation makes transparent for GLn (and conceivable for other classical groups) a certain invariance of Jantzen's sum formula under "Howe duality" in the sense of Adamovich and Rybnikov. For GLn a simple and explicit general formula is derived for χ between an arbitrary pair of integral Weyl modules. In light of Brenti's work on certain R-polynomials, this formula raises interesting questions about the possibility of relating Ext groups between Weyl modules to Kazhdan-Lusztig combinatorics.

Tilting objects in abelian categories and quasitilted rings

D. Happel, I. Reiten and S. Smarlo initiated an investigation of quasitilted artin K-algebras that are the endomorphism rings of tilting objects in hereditary abelian categories whose Hom and Ext groups are all finitely generated over a commutative artinian ring K. Here, employing a notion of *-objects, tilting objects in arbitrary abelian categories are defined and are shown to yield a version of the classical tilting theorem between the category and the category of modules over their endomorphism rings. This leads to a module theoretic notion of quasitilted rings and their characterization as endomorphism rings of tilting objects in hereditary cocomplete abelian categories.

On the Ext groups between Weyl modules for [formula omitted

This paper studies extension groups between certain Weyl modules for the algebraic group GL n over the integers. Main results include: (1) a complete determination of Ext groups between Weyl modules whose highest weights differ by a single root and (2) determination of Ext 1 between an exterior power of the defining representation and any Weyl module. The significance of these results for modular representation theory of GL n is discussed in several remarks. Notably the first result leads to a calculation of Ext groups between neighboring Weyl modules for GL n and also recovers the GL n case of a recent result of Andersen. Some generalities about Ext groups between Weyl modules and a brief overview of known results about these groups are also included.

Espaces symétriques de Drinfeld et correspondance de Langlands locale

We study the Galois action on the equivariant cohomology complex of Drinfeld's p-adic symmetric spaces and show how it encodes Langlands' correspondence for the so-called “principal elliptic” representations of GL d . This is the first stage of an expected generalization of Carayol's non-Abelian Lubin–Tate theory from supercuspidal to elliptic representations. In the process we obtain a new proof of Deligne's weight-monodromy conjecture for those varieties which admit p-adic uniformization by these spaces, we compute Ext groups and cup-products for elliptic representations, and we give a new computation of the compactly supported cohomology of p-adic symmetric spaces.

HOMOMORPHISMS AND HIGHER EXTENSIONS FOR SCHUR ALGEBRAS AND SYMMETRIC GROUPS

This paper surveys, and in some cases generalizes, many of the recent results on homomorphisms and the higher Ext groups for q-Schur algebras and for the Hecke algebra of type A. We review various results giving isomorphisms between Ext groups in the two categories, and discuss those cases where explicit results have been determined.

Symmetry theorems for Ext vanishing

It was proved by Avramov and Buchweitz that if A is a commutative local complete intersection ring with finitely generated modules M and N, then the Ext groups between M and N vanish from some step if and only if the Ext groups between N and M vanish from some step. This paper shows that the same is true under the weaker conditions that A is Gorenstein and that M and N have finite complete intersection dimension. The result is also proved if A is Gorenstein and has finite Cohen–Macaulay type. Similar results are given for non-commutative complete semi-local algebras.

Extensions of strict polynomial functors

We compute Ext-groups between Frobenius twists of strict polynomial functors. The main result concerns the groups Ext P ∗ ( D d ( i ) , F ( i ) ) where D d is the divided power functor, and F is an arbitrary functor. These groups are shown to be isomorphic to F ( A i ) for certain explicitly described graded space A i . We also calculate the groups Ext P ∗ ( W μ ( i ) , S λ ( i ) ) where W μ and S λ are respectively the Weyl and Schur functors associated to diagrams μ, λ of the same weight.

Gorenstein projective dimension for complexes

We define and study a notion of Gorenstein projective dimension for complexes of left modules over associative rings. For complexes of finite Gorenstein projective dimension we define and study a Tate cohomology theory. Tate cohomology groups have a natural transformation to classical Ext groups. In the case of module arguments, we show that these maps fit into a long exact sequence, where every third term is a relative cohomology group defined for left modules of finite Gorenstein projective dimension.

HOMOLOGICAL ALGEBRA OF TWISTED QUIVER BUNDLES

Several important cases of vector bundles with extra structure (such as Higgs bundles and triples) may be regarded as examples of twisted representations of a finite quiver in the category of sheaves of modules on a variety/manifold/ringed space. It is shown that the category of such representations is an abelian category with enough injectives by the construction of an explicit injective resolution. Using this explicit resolution, a long exact sequence is found that computes the Ext groups in this new category in terms of the Ext groups in the old category. The quiver formulation is directly reflected in the form of the long exact sequence. It is also shown that under suitable circumstances, the Ext groups are isomorphic to certain hypercohomology groups.

Open-string BRST cohomology for generalized complex branes

It has been shown recently that the geometry of D-branes in general topologically twisted (2,2) sigma-models can be described in the language of generalized complex (GC) structures. On general grounds, such D-branes (called GC branes) must form a category. We compute the BRST cohomology of open strings with both ends on the same GC brane. In mathematical terms, we determine spaces of endomorphisms in the category of GC branes. We find that the BRST cohomology can be expressed as the cohomology of a Lie algebroid canonically associated to any GC brane. In the special case of B-branes, this leads to an apparently new way to compute Ext groups of holomorphic line bundles supported on complex submanifolds: while the usual method leads to a spectral sequence converging to the Ext, our approach expresses the Ext group as the cohomology of a certain differential acting on the space of smooth sections of a graded vector bundle on the submanifold. In the case of coisotropic A-branes, our computation confirms a proposal of Orlov and one of the authors (A.K.).

Ext vanishing and infinite Auslander-Buchsbaum

A vanishing theorem is proved for Ext groups over non-commuta- tive graded algebras. Along the way, an “infinite” version is proved of the non-commutative Auslander-Buchsbaum theorem.

Deformation theory of algebraic stacks

We study the deformation theory of algebraic stacks. The 2-category of deformations using Ext groups of the cotangent complex is described. Then we show that an algebraic stack has a versal deformation under certain conditions.

The effect of the extrusion process on the digestibility of feed given to laying hens

The purpose of the study was to observe the effect of feeding extruded feed that contained rapeseed (that was extruded using the standard methods for large-scale production of extruded feed) on retention and coefficient of total tract apparent digestibility (CTTAD) in hens. The laying hens were divided into two groups NEX and EXT. The composition of the diets was the same in both groups (wheat 39%, rapeseed 13.5%, pea 13.5%, soya bean meal 6%, maize 14%, fish meal 3%, limestone 1%, premix 10%). But 0.45 kg of 1 kg of the diets EXT (wheat, pea, and rapeseed) was extruded by dry extrusion at temperature 137 °C and at moisture from 12 to 18% for 16 s. A balance assay was conducted with four laying hens from group NEX and eight from group EXT when they were 61 weeks old. The CTTAD of fat, calcium and amino-acids, and the retention of the nitrogen were estimated using the indicator method. The significance between the means was evaluated by a t-test. The average egg mass production during the assay was for group NEX 45 ± 3.9 g/hen/day and for group EXT 50 ± 5.0 g/hen/day. The extrusion process significantly increased ( P < 0.001) the CTTAD of fat from 0.807 (NEX) to 0.937 (EXT). The coefficient of nitrogen retention was higher in group EXT (0.629) than in group NEX (0.584). The extrusion significantly increased the CTTAD of arginine ( P < 0.05, NEX: 0.929, EXT: 0.944) and isoleucine ( P < 0.05, NEX: 0.870, EXT: 0.904). The extrusion had a tendency to increase the CTTAD of leucine ( P < 0.1, NEX: 0.885, EXT: 0.907), glutamic acid ( P < 0.1, NEX: 0.931, EXT: 0.945), glycine ( P < 0.06, NEX: 0.823, EXT: 0.859) and alanine ( P < 0.1, NEX: 0.806, EXT: 0.844). CTTAD of lysine was almost the same in both groups (NEX: 0.950 and EXT: 0.949). The extrusion process had a tendency to decrease the CTTAD of histidine ( P < 0.06, NEX: 0.906, EXT: 0.859) and tyrosine ( P < 0.07, NEX: 0.974, EXT: 0.921). The extrusion had a positive effect on the fat CTTAD, nitrogen retention and on the CTTAD of the essential amino-acids (except lysine and histidine). The decrease in the digestibility of these two amino-acids was not significant.

Spectra of D-branes with Higgs vevs

In this paper we continue previous work on counting open string states between D-branes by considering open strings between D-branes with nonzero Higgs vevs, and in particular, nilpotent Higgs vevs, as arise, for example, when studying D-branes in orbifolds. Ordinarily Higgs vevs can be interpreted as moving the D-brane, but nilpotent Higgs vevs have zero eigenvalues, and so their interpretation is more interesting -- for example, they often correspond to nonreduced schemes, which furnishes an important link in understanding old results relating classical D-brane moduli spaces in orbifolds to Hilbert schemes, resolutions of quotient spaces, and the McKay correspondence. We give a sheaf-theoretic description of D-branes with Higgs vevs, including nilpotent Higgs vevs, and check that description by noting that Ext groups between the sheaves modelling the D-branes, do in fact correctly count open string states. In particular, our analysis expands the types of sheaves which admit on-shell physical interpretations, which is an important step for making derived categories useful for physics.

D-branes, orbifolds, and Ext groups

In this note we extend previous work on massless Ramond spectra of open strings connecting D-branes wrapped on complex manifolds, to consider D-branes wrapped on smooth complex orbifolds. Using standard methods, we calculate the massless boundary Ramond sector spectra directly in BCFT, and find that the states in the spectrum are counted by Ext groups on quotient stacks (which provide a notion of homological algebra relevant for orbifolds). Subtleties that cropped up in our previous work also appear here. We also use the McKay correspondence to relate Ext groups on quotient stacks to Ext groups on (large radius) resolutions of the quotients. As stacks are not commonly used in the physics community, we include pedagogical discussions of some basic relevant properties of stacks.

Ext groups of Janet systems

In this paper we develop the basic ideas of Janet by using the theory of D -modules. For the so-called completely integrable systems (Janet Systems), we show that the higher Ext groups are null.

On the good filtration dimension of Weyl modules for a linear algebraic group

Let G be a linear algebraic group over an algebraically closed field of characteristic p whose corresponding root system is irreducible. In this paper we calculate the Weyl filtration di- mension of the induced G-modules,∇(�) and the simple G-modules L(�), fora regular weight. We use this to calculate some Ext groups of the form Ext � ∇(�),�(µ) � , ExtL(�),L(µ) � , and Ext � ∇(�),∇(µ) � , where �,µ are regular and �(µ) is the Weyl module of highest weight µ. We then deduce the projective dimensions and injective dimensions for L(�), ∇(�) and �(�) for � a regular weight in associated generalised Schur algebras. We also deduce the global dimension of the Schur algebras for GLn, S(n,r), when p > n and for S(mp,p) with m an integer.