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Reviewed by V. Dlab

Reviewed by V. Dlab

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© Copyright American Mathematical Society 1996, 1998

The article begins by giving definitions and examples of algebraic groups and their representations, after which the most important results on weight space decompositions and root systems for reductive groups are proved. The classification of the finite-dimensional irreducible modules for connected reductive groups is given, relying on induction from a Borel subgroup. The derived functors of this induction and the representations of $\rm{SL}\sb 2$ are discussed as preparation for proving the Borel-Weil-Bott theorem and Serre duality.

To properly handle characteristic $p>0$, algebraic group functors are introduced. Representations in characteristic $p$ of Frobenius subgroups are studied, parallel to the development in characteristic zero (though most proofs are omitted here), and Kempf's vanishing theorem is proved.

The penultimate section introduces quantum groups and the quantized versions of many of the preceding results, emphasizing the similarity between the modular representation theory of algebraic groups and the representation theory of quantum groups at roots of unity. The conclusion is a section on tilting modules for both algebraic groups and quantum groups, including some results of the author that play a role in the construction of invariants of $3$-manifolds by N. Yu. Reshetikhin and V. G. Turaev [Invent. Math. 103 (1991), no. 3, 547--597; MR 92b:57024].

\{For the entire collection see MR 97d:00014.\}

Reviewed by Ben Ford

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"This paper has its origins in our efforts to understand the connection between the results of Dlab, Heath and Marko [op. cit.] and those of Soergel [op. cit.]. In its final form, it synthesizes the common features of all the above examples, and then goes beyond them. In approaching the paper, it may be helpful for the reader to think of our theory as organized in several `layers' of generality. At the simplest level, we have presented an abstract theory of `Specht modules', analogous to the theory of Specht modules for symmetric groups, and adequate (together with a small number of combinatorial arguments) to provide proofs of many known results. In addition, this theory gives new insight into tensor spaces and tilting modules. We have reorganized and formalized most of the symmetric group representation theory dealing with Specht modules and module filtrations with Specht module sections, cf. $§\S1.6$, 3.6, 3.9, 3.8, 5.2, as well as $§\S4.4$, 4.5, and 4.6. But the Specht/Weyl module correspondence which occurs in the symmetric group/general linear group case is too simple (at least when ${\rm char}\,k\not= 2$) to apply in the other two examples mentioned above. There, the role of the group algebra $k\germ S\sb r$ of the symmetric group is played by a commutative local ring, and all `Specht modules' are isomorphic. The correct definition is technical---the stratification hypothesis (3.1.1)---but gives a rich supply of non-isomorphic `Weyl modules', and produces new results, e.g., (5.1.5). In addition, the technicalities simplify in some important cases, cf. (3.6.5), (4.4.9), and (4.7.1). The technical version of the hypothesis also allows for a further generalization, motivated by the finite group theory examples presented in Chapter 6, where the endomorphism algebras involved are not quite quasi-hereditary. The required generalization involves the new notion of a stratified algebra in Chapter 2, which turns out to be a useful idea in its own right; an interesting example in the representation theory of algebraic groups for singular weights is given in $\S5.3$. Also, `recollement' (in the sense of the authors [J. Reine Angew. Math. 391 (1988), 85--99; MR 90d:18005] for module categories, which is modeled in earlier work [A. A. Beilinson, J. N. Bernstein and P. Deligne, in Analysis and topology on singular spaces, I (Luminy, 1981), 5--171, Asterisque, 100, Soc. Math. France, Paris, 1982; MR 86g:32015] on perverse sheaves) remains an important theme in stratification theory, and we make some effort to understand it in several contexts. Surprisingly, deformation theory---cast as the study of orders in semisimple algebras---becomes necessary for an adequate recollement theory for some of the algebras of interest in Lie theory. The very same deformation theory is necessary for Specht module theory of symmetric groups in characteristic 2. Thus, Chapter 4 develops this deformation theory in some detail. Finally, the simplicity of (4.4.9) and (4.7.1) suggests that the integral setting thus provided is the most natural for the stratification hypothesis. See also (6.4.7)."

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Reviewed by G. D. James

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Reviewed by H. H. Andersen

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As the author remarks, another account of Mathieu's work has been given by W. L. J. van der Kallen [Lectures on Frobenius splittings and $B$-modules, Tata Inst. Fund. Res., Bombay, 1993; MR 95i:20064]. More recently, a quite different approach using Lusztig's canonical basis has been found by J. Paradowski [in Algebraic groups and their generalizations: quantum and infinite-dimensional methods (University Park, PA, 1991), 93--108, Proc. Sympos. Pure Math., 56, Part 2, Amer. Math. Soc., Providence, RI, 1994; MR 95c:20060].

Reviewed by James E. Humphreys

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Reviewed by H. H. Andersen

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"Given a finite type Cartan datum one can associate at least four interesting categories to this. Namely, (1) a category of modules over the corresponding semisimple Lie algebra $\germ g$, (2) the category of rational modules of the corresponding semisimple, simply connected algebraic group defined over a field of positive characteristic, (3) the category of locally finite modules of the associated quantum algebra specialized at a root of unity, (4) the category of fixed level representations of the affine Kac-Moody algebra associated to $\germ g$. In each of the cases (1)--(3) we investigate a certain semisimple subcategory equipped with a `reduced' tensor product and we prove some `fusion rules' in each case; these are given in terms of the characters of the modules involved. Hence they are of course essentially old character formulas in a new guise. Our approach is in the framework of tilting modules."

Reviewed by Timo Neuvonen

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The second application is to the action by conjugation of a subgroup on the group itself. Let $G$ be a reductive group and let $H$ be a closed subgroup of $G$. The group $H$ acts on $G$ by $h· g=hgh\sp {-1}$. In the case where $H=G$, the invariants are called class functions and form an algebra $C(G)$ which is described (in the case where $G$ is semisimple and simply connected) using results of R. Steinberg on conjugacy classes. In particular, it is shown that $C(G)$ is the free algebra generated by $\chi\sb 1,\cdots, \chi\sb r$, where $\chi\sb i$ are the various trace functions on the irreducible representations of $G$ corresponding to the fundamental dominant weights. For an arbitrary closed subgroup $H$, some invariants in the algebra of invariants, $C(G,H)$, may be given in a similar fashion. Namely, consider the functions $\chi\sb \theta(g)={\rm trace}(R(g)\circ\theta)$ where $V$ is a finite-dimensional $G$-module, $R\colon G\to {\rm GL}(V)$ is the corresponding representation, and $\theta$ is an $H$-endomorphism of $V$. Such functions are called endofunctions. Of particular importance are those which arise for "tilting modules" $V$ (i.e., modules where both $V$ and its dual have good filtrations). Indeed, if $H$ is "saturated" (i.e., if $k[G/H]$ has a good filtration) then $C(G,H)$ is generated by such elements. This result has as a corollary the theorem on matrices given above.

\{For the entire collection see MR 95g:16001\}.

Reviewed by Frank D. Grosshans

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In case $n\geq r$ there exists an idempotent $e\in S$ such that $eSe$ is isomorphic to $k\Sigma\sb r$. Then the Schur functor ${\rm Hom}\sb S(Se,\text{--})$ relates $S$-mod and $k\Sigma\sb r$-mod, thus extending (and proving) the classical Frobenius-Schur duality between general linear and symmetric groups.

The present paper describes another approach to the problem of relating $S$ and $k\Sigma\sb r$ which avoids the restriction $n\geq r$. It is based on studying another finite-dimensional algebra, called the Ringel dual of $S$: By a result of Ringel, there exists a module $T$ (containing $(k\sp n)\sp {\otimes r}$ as a direct summand) over $S$ which has both a Weyl and a co-Weyl filtration and which is unique up to a choice of multiplicities. This module is called the characteristic tilting module over $S$. Its endomorphism algebra ${\rm End}\sb S(T)$ is called the Ringel dual of $S$. By a result of Donkin, for $n\geq r$, the algebras $S$ and ${\rm End}\sb S(T)$ are Morita equivalent. The desired relation between $S$ and $k\Sigma\sb r$ is now provided by Proposition 4.3, which states that ${\rm End}\sb S(T)$ (for a suitable choice of $T$) contains an idempotent, say $f$, such that $k\Sigma\sb r$ maps epimorphically onto $f{\rm End}\sb S(T)f$. Moreover, the Young and Specht modules over $k\Sigma\sb r$ can then be identified with direct summands of $T$ and with Weyl modules respectively, which explains some similarities in structure. If, in addition, the characteristic of $k$ is large relative to $n$, then (Theorems 4.4 and 4.6) one may choose $f$ to be 1.

As an application, a new proof of a result of G. James on decomposition numbers for two-part partitions is given. The paper uses the language of quasi-hereditary algebras and may serve as a good introduction to viewing Schur algebras as quasi-hereditary algebras.

\{For the entire collection see MR 95g:16001\}.

Reviewed by Steffen Konig

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In this paper the authors investigate an analogous situation in characteristic $p$. Let $G = G(k)$ be the simply connected Chevalley group associated to $I$, defined over an algebraically closed field $k$ of characteristic $p$. As discovered by Ringel, for any dominant weight $\lambda$ there is a unique indecomposable tilting module $P(\lambda)$ and any tilting module is a direct sum of such. This allows the authors to define multiplicities $V\sp {\nu}\sb {\lambda µ}$ by the requirement: $P(\lambda) \otimes P(µ) = \sum\sb { \nu \in P\sp {+}} V\sp {\nu}\sb {\lambda µ} · P(\nu)$. The main result of the paper (Theorem 4.10) gives an explicit formula for $V\sp {\nu}\sb {\lambda µ}$ in case $\nu$ belongs to the fundamental alcove $C\sb {p}$. If $\lambda + \rho$ and $µ + \rho$ also lie in the fundamental alcove and $p> n(I)$, a constant depending on the Dynkin diagram, we have $$V\sb {\lambda, µ}\sp {\nu} = \sum\sb {\{w \in W\sb {p}\vert w\sp {-1}(\nu + \rho) - \rho \in P\sp {+}\}} \epsilon(w) · K\sp {w\sp {-1}(\nu + \rho)-\rho}\sb {\lambda,µ} $$ where $W\sb {p}$ is the affine Weyl group and $\rho$ satisfies $2(\rho , \alpha\sb {i}) = (\alpha\sb {i} , \alpha\sb {i})$ for $i\in I$. Otherwise $V\sb {\lambda, µ}\sp {\nu} = 0$.

In the course of the proof, the authors consider a tensor category ${\scr P}\sb {\rm mod}$ in which the objects are the tilting modules, the morphisms are given by a functor $T$ and tensor product is the usual tensor product. The functor $T$ is defined so that there is a one-dimensional space of morphisms between isomorphic indecomposable modules of dimension coprime to $p$ and none between other pairs of indecomposables. The functor $T$ is used to prove a key lemma which states that if $M$ and $N$ are rational $G$-modules with $M$ indecomposable and of dimension divisible by $p$ then any direct summand of $M \otimes N$ also has dimension divisible by $p$. Note that the corresponding result for modular representations of finite groups also holds [D. J. Benson, Representations and cohomology. I, Cambridge Univ. Press, Cambridge, 1991; MR 92m:20005 (Proposition 5.8.1]. From this lemma, it follows that the multiplicities $V\sb {\lambda, µ}\sp {\nu}$ are the structure constants of an associative ring.

The main result implies that the tensor product multiplicities in the category ${\scr P}\sb {\rm mod}$ are the same as those of a category of certain integrable modules for a corresponding Kac-Moody Lie algebra endowed with a modified tensor product. The term "fusion ring" is borrowed from this setting. The authors expect that this coincidence may be explained through a lifting to quantum groups.

The authors also state a result (Theorem 4.8) in terms of Grothendieck rings. They show that the Grothendieck ring $K({\scr P}\sb {\rm mod})$ is a reduced quotient ring of the Grothendieck ring $K(G)$ of finite-dimensional rational $G$-modules. Interpreting ${\rm Spec}({\bf C}\otimes K(G))$ as the set of semisimple conjugacy classes in $G({\bf C})$, they show that ${\rm Spec}({\bf C}\otimes K({\scr P}\sb {\rm mod}))$ corresponds to the set of regular semisimple classes $[g]$ such that $g\sp {p}$ is central.

\{For the entire collection see MR 95e:00035\}.

Reviewed by Gregory Hill

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A central problem is to find filtrations of tensor products of induced modules, so that the quotients are induced modules. (In the context of quantum groups at a root of unity, this problem has implications for the study of tilting modules and fusion rules.) Most cases had already been settled by Jian-Pan Wang [J. Algebra 77 (1982), no. 1, 162--185; MR 84h:20032] and S. Donkin [Rational representations of algebraic groups, Lecture Notes in Math., 1140, Springer, Berlin, 1985; MR 87b:20054]. But Mathieu found a more unified approach based on the geometry of flag varieties, using the method of Frobenius splitting introduced by V. B. Mehta and A. Ramanathan [Ann. of Math. (2) 122 (1985), no. 1, 27--40; MR 86k:14038].

Frobenius reciprocity allows one to view the category of rational $G$-modules as a full subcategory of the category of rational $B$-modules, where $B$ is a fixed Borel subgroup. Accordingly, the author emphasizes more general questions about $B$-modules explored by P. Polo [Asterisque No. 173-174 (1989), 10--11, 281--311; MR 91b:20056; C. R. Acad. Sci. Paris Ser. I Math. 307 (1988), no. 15, 791--794; MR 91b:20055; C. R. Acad. Sci. Paris Ser. I Math. 308 (1989), no. 5, 123--126; MR 90f:20056] and himself [Math. Z. 201 (1989), no. 1, 19--31; MR 91a:20043]. The emphasis on $B$-modules arises in part from earlier work by A. Joseph and others on the Demazure character formula.

The Weyl group $W$ parametrizes the Schubert varieties $X\sb w$ in the flag variety $G/B$, with $G/B$ itself corresponding to the longest element of $W$. A $B$-module is called "excellent" if it has a filtration whose quotients are "dual Joseph modules" $H\sp 0(X\sb w,\scr{L})$, where $\scr{L}$ is an effective line bundle on $G/B$. Following Polo, the dual Joseph modules are injective objects in subcategories of $B$-modules with suitably bounded highest weights. A $G$-module is called "good" if it has a filtration whose quotients are dual Weyl modules $H\sp 0(G/B, \scr{L})$. Mathieu's arguments (completing case-by-case work of Wang and Donkin) show that the tensor product of two good modules is again good, as is the restriction of a good module to a parabolic subgroup. But an example found by the author shows that tensoring two excellent $B$-modules need not produce another excellent module. Instead, Mathieu's approach confirms a conjecture of Joseph (and its generalization proved in most cases by Polo): the tensor product of a good [resp. excellent] $B$-module with a one-dimensional $B$-module inducing an effective line bundle on $G/B$ is excellent.

The author brings his own perspective to these results, emphasizing Schubert filtrations, cohomological criteria for excellent filtrations, etc. He concludes by showing that the essential arguments can be done over an arbitrary base ring. This yields new information about $B$-modules, even in characteristic 0.

Reviewed by James E. Humphreys

\{For the entire collection see MR 95a:00012\}.

Reviewed by William M. McGovern

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The precise result is the following. It is known that $G$ has a unique simple module of degree $p\sp N$, where $N$ is the number of positive roots, called the basic Steinberg module. Any tensor product of distinct Galois twists of this module is called a partial Steinberg module. It follows from the Steinberg tensor product theorem that $V$ has a decomposition $V=R\otimes S$ where $S$ is a partial Steinberg module and $R$ has no factors that are twists of the basic Steinberg module. Write $S=\sigma\sb 1(St)\otimes\cdots\otimes\sigma\sb r(St)$ where the $\sigma\sb i$ are distinct automorphisms of $F$ and $\sigma\sb i(St)$ denotes the twist of $St$ by $\sigma$. Next, let $X$ be a root subgroup in $G$ and suppose $A$ is an elementary subgroup of $X$ of order $p\sp s$ generated by $x\sb \alpha(\lambda\sb 1), \cdots, x\sb \alpha(\lambda\sb s)$. The main result in this paper is the proof that $V$ restricted to $A$ is a free $A$-module if and only if $S$ restricted to $A$ is a free $A$-module if and only if the $r\times s$ matrix $(\sigma\sb i(\lambda\sb j))$ has rank $s$.

The argument is quite elegant and proceeds by proving the result first for $G={\rm SL}\sb 2$, and then using the ${\rm SL}\sb 2$ result to prove the general case. To prove the result for ${\rm SL}\sb 2$ the authors use their result mentioned above and a theorem of J. F. Carlson on shifted subgroups [J. Algebra 85 (1983), no. 1, 104--143; MR 85a:20004]. The general case is dealt with using Smith's theorem and recent results of S. Donkin on tilting modules [Math. Z. 212 (1993), no. 1, 39--60; MR 94b:20045].

The authors also discuss a variety whose $F$-rational points parametrize the conjugacy classes of elementary abelian subgroups of root subgroups, $A$, for which the restriction of $V$ to $A$ is not free.

Reviewed by J. Matthew Douglass

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Let $A$ be a finite-dimensional algebra over some field $k$ and $\{E(\lambda)\vert $ $\lambda\in\Lambda\}$ the set of simple $A$-modules. Assume that $\Lambda$ is ordered, or more generally an adapted partially ordered set. If $P(\lambda)$ is the projective cover of $E(\lambda)$ and $\Delta(\lambda)$ the maximal factor module of $P(\lambda)$ having only composition factors $E(µ)$ with $µ\leq\lambda$, then $\Delta(\lambda)$ is called standard and $\Delta=\{\Delta(\lambda)\vert \lambda\in\Lambda\}$ denotes the set of standard modules. The set $\nabla$ of costandard modules is defined dually. If $\theta=\{\theta\sb 1,\cdots,\theta\sb m\}$ is some finite set of $A$-modules, then $\scr F(\theta)$ denotes the full subcategory of $A$-mod having a $\theta$-filtration. The pair $(A,\Lambda)$, where $\Lambda$ is adapted, is called a quasi-hereditary algebra if all standard modules have trivial endomorphism ring and the following equivalent conditions hold (Theorem 1): (i) $\scr F(\Delta)$ contains ${}\sb A\!A$, (ii) $\scr F(\Delta)=\{X\vert {\rm Ext}\sp 1(X,\nabla)=0\}$, (iii) $\scr F(\Delta)=\{X\vert {\rm Ext}\sp i(X,\nabla)=0$ for all $i\geq 1\}$, (iv) ${\rm Ext}\sp 2(\Delta,\nabla)=0$.

If $\scr C$ is an abelian category and $\theta=\{\theta(\lambda)\vert \lambda\in \Lambda\}$ is a finite set of objects of $\scr C$ such that (1) all spaces ${\rm Hom}(\theta(\lambda),\theta(\omega)$) and ${\rm Ext}\sp 1(\theta(\lambda),\theta(µ))$ are finite-dimensional and (2) the associated quiver has no oriented cycles, then $\theta$ is called standardizable. The second condition defines a partial order on $\Lambda$. If $\scr F(\theta)$ denotes the full subcategory of objects of $\scr C$ with $\theta$-filtration, then Theorem 2 states the following: If $\theta$ is standardizable, then there exists a quasi-hereditary algebra $(A,\Lambda)$ such that $\scr F(\theta)$ and $\scr F(\Delta)$ are equivalent. As an application, the authors get for a quasi-hereditary algebra $A$ the existence of a (generalized) tilting-cotilting module $T$ with ${\rm add}\,T=\scr F(\Delta)\cap\scr F(\nabla)$. If, additionally, the projective dimension of any standard module and the injective dimension of any costandard module are each at most one, the pair $(\scr F(\nabla),\scr F(\Delta))$ can be described by the torsion pair $(\scr G(T),\scr H(T))$, induced by the tilting module $T$.

In the last section, the Auslander algebra $A\sb n$ of $k[T]/\langle T\sp n\rangle$, which is quasi-hereditary, is studied.

{For the entire collection see MR 93j:16002}.

Reviewed by Otto Kerner

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The author explores the indecomposable summands of certain tensor products. When two highest weight modules (of weights $\lambda$ and $µ$) are tensored, there is a unique indecomposable summand of highest weight $\lambda + µ$. In a number of special cases he is able to identify this summand with $M(\lambda + µ)$. His assumption is usually that $\lambda, µ, \lambda + µ$ are all bounded above by twice a Steinberg weight: $2(p\sp n-1) \rho$. (This is the setting in which projectives for the $n$th Frobenius kernel appear.)

Reviewed by James E. Humphreys

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Let $A$ be a finite-dimensional quasi-hereditary algebra with simple modules $L(\lambda)$, standard modules $\Delta(\lambda)$ and co-standard modules $\nabla(\lambda)$, and let $\scr T$ denote the class of modules which have both $\Delta$-filtrations and $\nabla$-filtrations. It was proved by C. M. Ringel [Math. Z. 208 (1991), no. 2, 209--223; MR 93c:16010] that there is a one-to-one correspondence $M(\lambda)\leftrightarrow L(\lambda)$ between the indecomposable modules in $\scr T$ and the simple modules. The direct sum $T\coloneq\bigoplus M(\lambda)$ is a (generalized) tilting and cotilting module. Moreover, if $A'\coloneq {\rm End}\sb A(T)$ then $A'$ is again quasi-hereditary (with standard modules $\Delta\sb {A'}(\lambda)={\rm Hom}\sb A(T,\nabla(\lambda))$).

In this paper, the author studies the implications of these results for reductive groups; these include the following. The class $\scr T$ is closed under tensor products (which was proved earlier). The modules $M(\lambda)$ behave well on truncation to Levi factors. Further, the author studies the relationship between the $M(\lambda)$ and injective modules for the infinitesimal subgroups $G\sb n$ of $G$. He conjectures that the injective indecomposables for $G\sb n$ are always restrictions of certain $M(\lambda)$. This is a refinement of an older conjecture (the injectives of $G\sb n$ have extensions to $G$-modules; this has been verified in many situations).

Let $G={\rm GL}\sb n(K)$; then the modules $M(\lambda)$ are precisely the indecomposable summands of tensor products of exterior powers of the natural module. It is known that one gets the Schur algebra $S(n,r)$ as an associated quasi-hereditary algebra. The author proves that its conjugate $S(n,r)'$ is a generalized Schur algebra, in the sense of earlier work [S. Donkin, J. Algebra 104 (1986), no. 2, 310--328; MR 89b:20084a; J. Algebra 111 (1987), no. 2, 354--364; MR 89b:20084b]; in particular, it is isomorphic to $S(n,r)$ if $r\leq n$. From this he obtains that the filtration multiplicity $[M(\lambda): \nabla(µ)]$ is equal to the decomposition number $d\sb {µ'\lambda'}=[\nabla(µ'):L(\lambda')]$ (where $\tau'$ is the transpose of the partition $\tau$).

Reviewed by Karin Erdmann

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Reviewed by Jie Du

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Reviewed by Michel Brion

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Several applications are given to quasi-hereditary algebras. Let $E(1),\cdots,E(n)$ be the simple $A$-modules. Let $P(i)$ be the projective cover of $E(i)$ and $U(i)$ the sum of all images of maps $P(j)\to P(i)$ with $j>i$ and $\Delta(i)=P(i)/U(i)$. Dually, the family of modules $\nabla=\{\nabla(1),\cdots,\nabla(n)\}$ is defined. The algebra $A$ is quasi-hereditary if ${}\sb AA$ belongs to $\scr F(\Delta)$ and ${\rm End}\sb A(\Delta(i))$ is a division ring for $1\leq i\leq n$. If $A$ is quasi-hereditary, the categories $\scr F(\Delta)$ and $\scr F(\nabla)$ have almost split sequences, the ${\rm Ext}$-projective objects in $\scr F(\Delta)$ are projective $A$-modules; moreover there is a tilting and cotilting module $T$ with ${\rm add}\,T=\scr F(\Delta)\cup\scr F(\nabla)$.

Reviewed by J. Antonio de la Pena

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© Copyright American Mathematical Society 1993, 1998

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Selected Matches for: Author=(Collingwood)

Reviewed by Chon Hu Cheng

© Copyright American Mathematical Society 1990, 1998

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Selected Matches for: Author=(Reiten,I*)

Let $\scr C$ be a subcategory of $\text{mod}\,\Lambda$. Then $\scr C$ is called contravariantly finite if each $\Lambda$-module $X$ admits a right approximation; this means that there is a module $C\in\scr C$ and a morphism $f\colon C\to X$ such that the induced morphism $\text{Hom}(M,f)$ is surjective for all $M\in\scr C$.

Of particular interest are contravariantly finite subcategories with additional properties. These are resolving categories which by definition are closed under extensions, kernels of surjections and contain the projective modules. For a contravariantly finite resolving subcategory the objects consist of the summands of modules which have a filtration with composition factors the minimal right approximations of the simple modules (3.8). This has the consequence that the finitistic dimension of the algebra is finite if the subcategory of modules of finite projective dimension is contravariantly finite (3.10). Note however that there are examples showing that this is not always the case [K. Igusa, Smalo and G. Todorov, Proc. Amer. Math. Soc. 109 (1990), no. 4, 937--941; MR 91b:16010].

It is shown in Section 5 that there is a one-to-one correspondence between isomorphism classes of basic cotilting modules and contravariantly finite resolving subcategories $\scr C$ such that each module admits a finite resolution by objects in $\scr C$ (5.5). In this part the methods developed in a paper by Auslander and R.-O. Buchweitz [Mem. Soc. Math. France No. 38 (1989), 5--37; MR 91h:13010] are used. As a consequence of this, one can describe all contravariantly finite resolving subcategories for hereditary algebras as those subcategories which are of the form $\text{Sub}\,T$ (the subcategory of modules cogenerated by $T$) for a cotilting module $T$.

It should be mentioned that this paper has stimulated several investigations.

Reviewed by Dieter Happel

© Copyright American Mathematical Society 1992, 1998