Winfried Bruns Udo Vetter Determinantal Rings - Uni-osnabrueck.de

1. PreliminariesThis section serves two purposes. Its Subsections A and B list the ubiquitous basicnotations. In C and D we introduce the principal objects of our investigation and relatethem to their geometric counterparts.A. Notations and ConventionsGenerally we will use the terminology of [Mt] which seems to be rather standardnow. In some inessential details our notations differ from those of [Mt]; for example wetry to save parentheses whenever they seem dispensable. A main difference is the use ofthe attributes “local” and “normal”: for us they always include the property of beingnoetherian. In the following we explain some notations and list the few conventions thereader is asked to keep in mind throughout.All rings and algebras are commutative and have an element 1. Nevertheless wewill sometimes list “commutative” among the hypotheses of a proposition or theorem inorder to signalize that the ring under consideration is only supposed to be an arbitrarycommutative ring. A reduced ring has no nilpotent elements. The spectrum of a ring A,Spec A for short, is the set of its prime ideals endowed with the Zariski topology. Theradical of an ideal I is denoted Rad I. The dimension of A is denoted dim A, and theheight of I is abbreviated ht I.All the modules M considered will be unitary, i.e. 1x x for all x M . Ann M isthe annihilator of M , and the support of M is given bySupp M {P Spec A : MP 6 0}.We use the notion of associated prime ideals only for finitely generated modules overnoetherian rings:Ass M {P Spec A : depth MP 0}.The depth of a module M over a local ring is the length of a maximal M -sequence in themaximal ideal. The projective dimension of a module is denoted pd M . We remind thereader of the equation of Auslander and Buchsbaum for finitely generated modules overlocal rings A:if pd M pd M depth M depth A(cf. [Mt], p. 114, Exercise 4). If a module can be considered a module over different rings(in a natural way), an index will indicate the ring with respect to which an invariantis formed: For example, AnnA M is the annihilator of M as an A-module. Instead ofMatsumura’s depthI (M ) we use grade(I, M ) and call it, needless to say, the grade of Iwith respect to M ; cf. 16.B for a discussion of grade. The rank rk F of a free module F isthe number of elements of one of its bases. We discuss a more general concept of rank in16.A: M has rank r if M Q is a free Q-module of rank r, Q denoting the total ring of

21. Preliminariesfractions of A. The rank of a linear map is the rank of its image. The length of a moduleM is indicated by λ(M ).The notations of homological algebra concerning Hom, , and their derived functorsseem to be completely standardized; for them we refer to [Rt]. Let A be a ring, M andN A-modules, and f : M N a homomorphism. We putM HomA (M, A)andf HomA (f, A) : N M .M and f are called the duals of M and f .For the symmetric and exterior powers of M (cf. [Bo.1] for multilinear algebra) weuse the symbolsi MandSj (M )iiVVresp. Sometimes we shall have to refer to bases of F , F and F , given a basise1 , . . . , en of the free module F . The basis of F dual to e1 , . . . , en is denoted by e 1 , . . . , en .For I (i1 , . . . , ik ) the notation eI is used as an abbreviation of ei1 · · · eik , wherease I expands into e i1 · · · ei k . (The notation eI will be naturally extended to arbitraryfamilies of elements of a module.)We need some combinatorial notations. A subset I Z also represents the sequenceof its elements in ascending order. For subsets I1 , . . . , In Z we letσ(I1 , . . . , In )denote the signum of the permutation I1 . . . In (given by iuxtaposition) of I1 . . . In relative to its natural order, provided the Ii are pairwise disjoint; otherwise σ(I1 , . . . , In ) 0.A useful formula:σ(I1 , . . . , In ) σ(I1 , . . . , In 1 )σ(I1 . . . In 1 , In ).For elements i1 , . . . , in Z we defineσ(i1 , . . . , in ) σ({i1 }, . . . , {in }).The cardinality of a set I is denoted I . For a set I we letS(m, I) {J : J I, J m}.Last, not least, by1, . . . , bi, . . . , nwe indicate that i is to be omitted from the sequence 1, . . . , n.

3B. Minors and Determinantal IdealsB. Minors and Determinantal IdealsLet U (uij ) be an m n matrix over a ring A. For indices a1 , . . . , at , b1 , . . . , btsuch that 1 ai m, 1 bi n, i 1, . . . , t, we put ua1 b1 .[a1 , . . . , at b1 , . . . , bt ] det .uat b1······ u a1 b t. . uat btWe do not require that a1 , . . . , at and b1 , . . . , bt are given in ascending order. Thesymbol [a1 , . . . , at b1 , . . . , bt ] has a twofold meaning: [a1 , . . . , at b1 , . . . , bt ] A as justdefined, and[a1 , . . . , at b1 , . . . , bt ] Nt Ntas an ordered pair of t-tuples of non-negative integers. Clearly [a1 , . . . , at b1 , . . . , bt ] 0if t min(m, n). For systematic reasons it is convenient to let[ ] 1.If a1 · · · at and b1 · · · bt we say that [a1 , . . . , at b1 , . . . , bt ] is a t-minor of U . Ofcourse, as an element of A every [a1 , . . . , at b1 , . . . , bt ] is a t-minor up to sign. We call tthe size of [a1 , . . . , at b1 , . . . , bt ].Very often we shall have to deal with the case t min(m, n). Our standard assumption will be m n then, and we use the simplified notation[a1 , . . . , am ] [1, . . . , m a1 , . . . , am ].The m-minors are called the maximal minors, those of size m 1 the submaximalminors. (In section 9 the notion “maximal minor” will be used in a slightly more generalsense.)The ideal generated by the t-minors of U is denotedIt (U ).The reader may check that It (U ) is invariant under invertible linear transformations:It (U ) It (V U W )for invertible matrices V, W of formats m m and n n resp.Sometimes we will need the matrix of cofactors of an m m matrix:Cof U cij ,cij ( 1)i j [1, . . . , bj, . . . , m 1, . . . , bi, . . . , m].

41. PreliminariesC. Determinantal Rings and VarietiesLet B be a commutative ring, and consider an m n matrix X11 · · · X1n . X . Xm1 · · · Xmnwhose entries are independent indeterminates over B. The principal objects of our studyare the residue class ringsRt (X) B[X]/It (X),B[X] of course denoting the polynomial ring B[Xij : i 1, . . . , m, j 1, . . . , n]. The idealIt (X) is generated by the t-minors of X, cf. B. Whenever we shall discuss properties ofRt (X) which are usually defined for noetherian rings only (for example the dimension orthe Cohen-Macaulay property), it will be assumed that B is noetherian.Over an algebraically closed field B K of coefficients one can immediately associatea geometric object with the ring Rt (X). Having chosen bases in an m-dimensional vectorspace V and an n-dimensional vector space W one identifies HomK (V, W ) with the mndimensional affine space of m n matrices, of which K[X] is the coordinate ring. Underthis identification the subvariety defined by It (X) corresponds toLt 1 (V, W ) { f HomK (V, W ) : rk f t 1 }.We want to associate the letter r with “rank”, and so we replace t by r 1. Furthermorewe put L(V, W ) HomK (V, W ).It is not surprising that the geometry of Lr (V, W ) reflects certain properties of thelinear maps f Lr (V, W ). Let us consider the following two elementary statementswhich will lead us quickly to some nontrivial information on Lr (V, W ): (a) The map fcan be factored through K r . (b) Let U V be a vector subspace of dimension r ande a supplement of V , i.e. V U Ue ; if f U is injective, then there exist unique linearUe U , h : U W such that f (u uemaps g : Ue) h(u) h(g(eu)) for all u U , ue U(in fact, h f U ).Statement (a) shows that the morphismL(V, K r ) L(K r , W ) Lr (V, W ),given by the composition of maps, is surjective. Being an epimorphic image of an irreducible variety, Lr (V, W ) is irreducible itself. An application of (b): It is easy to see thatthe subsetM { f Lr (V, W ) : f U injective }is a nonempty open subvariety of Lr (V, W ): One chooses a basis of V containing abasis of U ; then M is the union of subsets of Lr (V, W ) each of which is defined by thenon-vanishing of a determinantal function. By property (b) we have an isomorphism e , U ) L(U, W ) \ Lr 1 (U, W ) M.L(U

5C. Determinantal Rings and Varietiese , U ) L(U, W ), we conclude atSince the variety on the left is an open subvariety of L(Uonce that e , U ) L(U, W ) (m r)r rndim Lr (V, W ) dim M dim L(U mr nr r2 .Furthermore M is non-singular. Varying U one observes that all the points f Lr (V, W )\Lr 1 (V, W ) are non-singular:(1.1) Proposition. (a) Lr (V, W ) is an irreducible subvariety of L(V, W ).(b) It has dimension mr nr r2 .(c) It is non-singular outside Lr 1 (V, W ).The only completely satisfactory information on Rr 1 (X) we can draw from (1.1),is its dimension:dim Rr 1 (X) mr nr r2Part (a) only shows that the radical of Ir 1 (X) is prime, and unfortunately there seemsto be no easy way to prove that Ir 1 (X) is a radical ideal itself (over every reducedring B of coefficients). Once this is known one can of course directly reverse (c): Thegenerators of the ideal of Lr (V, W ) have all their partial derivatives in Ir (X), and theJacobi criterion (or the definition of non-singularity, depending on ones point of view)implies in conjunction with (c) that Lr 1 (V, W ) is the singular locus of Lr (V, W ).Proposition (1.1) and its proof have been included not only in order to enrich theseintroductory considerations by some substantial results. We shall encounter algebraicversions of the ideas underlying its proof several times again.It would be very difficult (for us, at least) to investigate the rings Rt (X) withoutviewing them as the most prominent members of a larger class of rings of type B[X]/Iwhich we call determinantal rings. Their defining ideals I can be described as follows:Given integers1 u1 · · · up m,0 r1 · · · rp m,1 v1 · · · vq n,0 s1 · · · sq n,andthe ideal I is generated by the(ri 1)-minors of the first ui rowsand the(sj 1)-minors of the first vj columns,i 1, . . . , p, j 1, . . . , q. Later on we shall introduce a systematic notion for determinantal rings which is hard to motivate at this stage.In order to relate the general class of determinantal rings just introduced to the geometric description of Rr 1 (X) given above, one chooses bases d1 , . . . , dm and e1 , . . . , enof V and W resp., K being an algebraically closed field, V and W vector spaces ofdimensions m and n. LetVk kXKdiandWk i 1(e1 , . . . , e nis the basis dual to e1 , . . . , en , cf. A above).kXi 1Ke i

61. PreliminariesThen the ideal I above defines the determinantal variety{ f HomK (V, W ) :rk f Vui ri , rk f Wv j sj ,i 1, . . . , p, j 1, . . . , q }.The reader may try to find and to prove the analogue of (1.1) for the variety just defined.It will of course be included in the main results of the Sections 5 and 6.D. Schubert Varieties and Schubert CyclesIn the sections 4–9 we shall treat a second class of rings simultaneously with thedeterminantal rings: the homogeneous coordinate rings of the Schubert varieties (generalized to an arbitrary ring of coefficients) which we call Schubert cycles for short. Thereare two reasons for our treatment of Schubert cycles: (i) They are important objects ofalgebraic geometry. (ii) Their combinatorial structure is simpler than that of determinantal rings, and most often it is easier to prove a result first for Schubert cycles and todescend to determinantal rings afterwards. Algebraically one can consider every determinantal ring as a dehomogenization of a Schubert cycle (cf. 16.D and (5.5)). In geometricterms one passes from a (projective) Schubert variety to an (affine) determinantal varietyby removing a hyperplane “at infinity”.The first step in the construction of the Schubert varieties is the description of theGrassmann varieties in which they are embedded as subvarieties. While a projectivespace gives a geometric structure to the set of one-dimensional subspaces of a vectorspace, a Grassmann variety does this for the set of m-dimensional subspaces, m fixed.Let K be an algebraically closed field, V an n-dimensional vector space over K, ande1 , . . . , en a basis of V . In a first attempt to assign “coordinates” to a vector subspaceW , dim W m, one chooses a basis w1 , . . . , wm of W and represents w1 , . . . , wm aslinear combinations of e1 , . . . , en :wi nXxij ej ,i 1, . . . , m.j 1Unfortunately the assignment W (xij ) is not well-defined, since (xij ) depends on thebasis w1 , . . . , wm of W . Exactly the matricesT · (xij ),T GL(m, K),represent W . However, the Plücker coordinatesp ([a1 , . . . , am ] : 1 a1 · · · am n)formed by the m-minors of (xij ) remains almost invariant if (xij ) is replaced by T · (xij );it is just replaced by a scalar multiple: The point of projective space with homogeneouscoordinates p depends only on W ! Thus one has found a well-defined map nP : { W V : dim W m } PN (K),N 1.mIt is called the Plücker map.

7D. Schubert Varieties and Schubert CyclesThis construction can of course be given in more abstract terms. With each subspaceW , dim W m, one associates the embeddingiW : W V.Then the m-th exterior powerm iW :m W m VmmVVW onto a one-dimensional subspace of V which in turn corresponds to a pointmVin P( V ) PN (K).f ). ForIt is easy to see that the Plücker map is injective. Let p P(W ) P(Wreasons of symmetry we may assume that the first coordinate of p is nonzero. Then wef resp. such thatcan find bases w1 , . . . , wm and we1 , . . . , wem of W and Wmapswi ei nXj m 1xij ej ,we i ei nXj m 1eij ej ,xi 1, . . . , m.Looking at the m-minors [1, . . . , bi, . . . , m, k] of the m n matrices of coefficients appearingei for i 1, . . . , m, hencein the preceding equations one sees immediately that wi wf.W WIt takes considerably more effort to describe the image of P. The map P is inducede of affine spaces; Pe assigns to each m n matrix the tuple of its m-minors.by a morphism PLet X be an m n matrix of indeterminates, and let Y[a1 ,.,am ] , 1 a1 · · · am n,denote the coordinate functions of AN 1 (K). Then the homomorphism of coordinatee is given asrings associated with Pϕ : K[Y[a1 ,.,am ] : 1 a1 · · · am n] K[X],Y[a1 ,.,am ] [a1 , . . . , am ],[a1 , . . . , am ] specifying an m-minor of X now. We denote the image of ϕ byG(X);it is the K-subalgebra of K[X] generated by the m-minors of X. By construction it iseclear that the affine variety defined by the ideal Ker ϕ is the Zariski closure of Im P,whereas the corresponding projective variety is the closure of Im P. Much more is true:(1.2) Theorem. (a) P maps the set of m-dimensional subspaces of V bijectivelyonto the projective variety with homogeneous coordinate ring G(X).e maps the mn-dimensional affine space of m n matrices over K surjectively onto(b) Pthe affine variety with coordinate ring G(X).Part (a) obviously follows from (b). In order to prove (b) one first has to describethe variety belonging to G(X) as a subvariety of AN 1 (K). This problem will be solved

81. Preliminariese a question which will naturallyin (4.7). Secondly one has to show the surjectivity of P,come across us in Section 7, cf. (7.14).The projective variety appearing in (1.2),(a) is usually denoted by Gm (V ) and calledthe Grassmann variety of m-dimensional subspaces of V . (A different choice of a basis forV only yields a different embedding into PN (K); all these embeddings are projectivelyequivalent.)The argument which showed the injectivity of P helps us to determine the dimensionof Gm (V ): the open affine subvariety of Gm (V ) complementary to the hyperplane givenby the vanishing of Y[1,.,m] , is isomorphic to the affine space of dimension m(dim V m),hencedim Gm (V ) m(dim V m).(Note that we are using (1.2) here !) Varying the hyperplane one furthermore sees thatGm (V ) is non-singular. The non-singularity of Gm (V ) can also be deduced from anotherbasic fact. The group GL(V ) of automorphisms of V acts transitively on Gm (V ), sincetwo m-dimensional subspaces of V differ by an automorphism of V only. On the othermmVVhand this action is induced by the natural action of GL(V ) on P( V ) (via V ); soGL(V ) operates transitively as a group of automorphisms on the Grassmann varietyGm (V ).(1.3) Theorem. Gm (V ) is a non-singular variety of dimension m(dim V m).To define the Schubert subvarieties one considers the flag of subspaces associatedwith the given basis e1 , . . . , en of V taken in reverse order:Vj nXi n j 1Kei ,0 V0 . . . Vn V.Let 1 a1 · · · am n be a sequence of integers. Then the Schubert subvarietyΩ(a1 , . . . , am ) of Gm (V ) is defined byΩ(a1 , . . . , am ) { W Gm (V ) : dim W Vai i fori 1, . . . , m }.The varieties thus defined of course depend on the flag of subspaces chosen. But theautomorphism group of V acts transitively on the set of flags, and its action induced onGm (V ) makes corresponding Schubert subvarieties differ by an automorphism of Gm (V )only. Hence Ω(a1 , . . . , am ) is essentially determined by (a1 , . . . , am ). It is indeed justifiedto call Ω(a1 , . . . , am ) a variety:(1.4) Theorem. Ω(a1 , . . . , am ) is the closed subvariety of Gm (V ) defined by thevanishing of all the coordinate functionsY[b1 ,.,bm ] ,bi n am i 1 1for somei, 1 i m.Proof: The proof is simpler if we dualize our notations first. Let ci n aiPjand Wj k 1 Kek . Then V Vn j Wj and there is a projection πj : V Wj ,Ker πj Vn j . By definitionΩ(a1 , . . . , am ) { W Gm (V ) : dim πci (W ) m ifor i 1, . . . , m }.

9E. Comments and ReferencesAfter the choicePn of a basis w1 , . . . , wm , the subspace W is represented by the matrix(xuv ), wu v 1 xuv ev . One obviously hasdim πci (W ) m i Im i 1 (xuv : 1 v ci ) 0,and in case this condition holds, every m-minor which has at least m i 1 of its columnsamong the first ci columns of (xuv ), vanishes. Thus all the coordinate functions named inthe theorem vanish on Ω(a1 , . . . , am ). Conversely, if Im i 1 (xuv : 1 v ci ) 6 0, thenthere is an m-minor of (xuv ) different from zero and having at least m i 1 of itscolumns among the first ci ones of (xuv ). —For arbitrary rings B of coefficients the Schubert cycle associated with Ω(a1 , . . . , am )is the residue class ring of G(X) with respect to the ideal generated by all the minors[b1 , . . . , bm ] such that bi n am i 1 1 for some i.E. Comments and ReferencesThe references given below have been included to manifest the geometric significanceof determinantal and Schubert varieties. We have restricted ourselves to books (with oneexception) since any selection of research articles would inevitably turn out superficialand random. (After all, the AMS classification scheme contains the keys “Determinantalvarieties” and “Schubert varieties”.)The classical source for “the geometry of determinantal loci” is Room’s book [Rm]. Itgives plenty of information on the early history of our subject. The decisive treatment ofSchubert varieties has been given by Hodge and Pedoe in their monograph [HP]. Amongthe recent books on algebraic geometry those of Arabello, Cornalba, Griffiths, and Harris[ACGH], Fulton [Fu], and Griffiths and Harris [GH] contain sections on determinantaland/or Schubert varieties. Kleiman and Laksov’s article [KmL] may serve as a pleasantintroduction.

2. Ide

This book is now out of print. The authors are grateful to Springer-Verlag for the permission to make this postscript file accessible. ISBN 3-540-19468-1 Springer-Verlag Berlin Heidelberg New York ISBN -387-19468-1 Springer-Verlag New York Berlin Heidelberg "c Springer-Verlag Berlin Heidelberg 1988