Communications in Algebra Resolutions of monomial ...

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Communications in Algebra

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Resolutions of monomial almost complete intersections and generalizations Hara Charalambous a;Alyson Reeves b a Dept. of Math, SUNY at Albany, Albany, NY b Supercomputing Research Center, Bowie, MD

To cite this Article Charalambous, Hara andReeves, Alyson(1995) 'Resolutions of monomial almost complete intersections

and generalizations', Communications in Algebra, 23: 11, 4087 — 4099 To link to this Article: DOI: 10.1080/00927879508825451 URL: http://dx.doi.org/10.1080/00927879508825451

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COMMUNICATIONS IN ALGEBRA, 23(1 I), 4087-4099 (1995)

RESOLUTIONS O F MONOMIAL ALMOST COMPLETE INTERSECTIONS AND GENERALIZATIONS Hara Charalambous * Dept. of Math., SUNY at Albany, Albany, NY 12222 [email protected] Alyson Reeves Supercomputing Research Center, Bowie, MD 20715 [email protected] Abstract Let S = k [ x 1 , . . .,x,] be a polynomial ring over a field k and let I be a monomial ideal of S. The main result of this paper is an explicit minimal resolution of k over R = S/I when I is a monomial almost complete intersection ideal of S. We also compute an upper bound on the multigraded resolution of k over a generalization of monomial almost complete intersection rings.

' T h e a u t h o r s a r e grateful t o t h e N S F for s u p p o r t d u r i n g t h e preparation of this paper. Both a u t h o r s would like t o t h a n k t h c M a t l ~ e m a ~ i cDse p a r t m e n t at Rrandeis University for their kind hospitality d u r i n g t h e preparation of this nlauuscript. T h e second a u t h o r was supported by a n N S F Postdoctoral I:cllo\\,sl~ip.

4087 Copyright O 1995 by Marcel Dekker, Inc.

CHARALAMBOUS AND REEVES

1

Introduction

Throughout, S = k [ x l , . . . ,x,,] is the polyno~nialring in n variables over a field k. The resolution of the residue class fieid k over algebras of the form R = 311 and its c o r r ~ s ~ o n d i nPoincarC g series 11;tvcIxen the subject of many investigat,ions ovcr the years. As a complctc list. of references would take the for a short survey and a list better part of a page, we refer the reader to of related problems, and to [ I for structures of resolutions. ~ t crnultigraded resolution When I is a monomial ideal, o n t car, c o ~ ~ i p rtlic: of k over .!?/I and the mr~ltigracledPoincar6 series of R. It is clear that the latter offers much more information than the general PoincarC series of R. When I is such a rnonornial ideal. Backelill sl~o\vedthat the Poincark series m

[a

[a

P,x(z) =

dirnk('r'orr(k,k ) ) z t


I =0

is rational. His proof actually shows that the multigraded Poincare series (see definition below) has tlle form n

+

n(l z ~ i ) l Q ~ i=l

for some polynomial QR E Z [ Y ~.,.., yn][z], and that the degree of QR is bounded by the sum C nz,, where xy' divides some monomial generator of I, but x;"' does not. While it is often possible to deduce certain characteristics of the minimal resolution of X: over R from its Poincark series, generally the PoincarG series does not pro\.ide enough information to determine the resolution explicitly. In section 3, we provide the minimal resolution of k when S I I is an almost complete monomial intersection. This is one of the cases where the explicit formula for the general PoirlcarC series is known. It is contained in results of (Proposition 4.4 when nz' # 1, and Theorem 6.10 when mi = 1 since (2:' , . . . ,x:', xy' . . . xpr) = (x;', . . . , x:.) : (x;'-"', . . . ,x : ~ - ~ ' ) , see Theorem 5 for notation). T h e essential new point is that one can choose the generators of the homology so that the Massey operations are actually zero, (Theorem 5). As a result we can describe the resolution in terms of multigraded generators with explicit fornlulas for the derivations. In section 4, we show that the nlultigraded betti numbers of the ~ninimal resolution of k are a subset of t.he multigraded betti numbers of some other resolutions. In Theorem 1 1 we show that in some c.ases we can compute explicitly the set of allowable m111t.idcgrcesand point. out that computational evidence suggests that this is a gc~ieralpI~c.ilo~~ic~iion. I n section 3,we recall some pertinent defini t.io11sand rc~sult s .

u,


MONOMIAL ALMOST COMPLETE INTERSECTIONS

2

4089

Notation, Definitions and Background results

As above, k is a field,

S = k [ x l , . . . ,x,],

m = ( x l , . . . ,x,), R = S I I .

Definition: The Poincard series lll$'(z) of

M

is defined to be

The Poincare series of R, Pn(z), is the Poincard series of its residue field k. That is, 03

PR(z) =

C dimk(Tor?(k, k))zi. 1=0

When I is a monomial ideal, the resolution of any multigraded R-module M over R is again multigraded. To each free module in the multigraded resolution with multidegree j = ( j l , . . . ,j,) we associate the monomial yJ = y)'I . . . Yl .; The multigraded Poincari series P/(y,, . . . ,y,, z ) is defined to be

such that P ~ ( z = ) PA4(z,1 , . . . , I ) .

Note: For simplicity of notation we will use mi to denote monomials in

S

as well as the monomials involving y;'s appearing in t h e Poincark series. For example m, = x:' . . . xj,. E S will represent yi' . . . yi;" if it appears in

P R ( Y I , . . , Yn, 2). Next we state Tate's Theorem [1_4, Theorem 4) for monomial ideals generated by S-sequences, and we give the multigraded Poincard series for ease of reference. T h e o r e m 1 Let t l , . . . , t , and s , , . . . , s , be S-regular sequences of n o n o mials such that the ideal J = ( s l , . . . , s , ) is contained in the ideal It' = ( t l , . . . , t,), (r 5 w r ) . W e reorder the monomials so that s , = q t ; . Assume that c, # 1 l o r i = 1 , . . . , t and c; = 1 for i = t + 1 , . . . , r . Lef A = S f J , li' = K I J , and lel c,, I, denoie the images of c,, ti in A. Let K denote the Koszul complex on Ti over A with i = 1 , . . . , w , and let B. be the "divided power algebraJ' on the degree 2 vniiables S,willl i = 1 , . . . , t . Then the algebra

CHARALAMBOUS AND REEVES

4090

is acyclic and therefore yields a minimal free resolution of the A-module A I R . 1\4oreover Y i s a multigraded resolution und d is a multigraded map of multidegree zero whenever Ti, S;arc given Ihe multidegrees of t , and s ; . In A/K . 1s: this case the multigraded Poincare' series PA

Y is an A-algebra, meaning that Y is an associative, graded, unitary, strictly skew commutative algebra over A and the differential d is a skew derivation of degree (-1): d ( x y ) = ( d x ) y ( - l ) ' x ( d y ) , where x E l'i. Hence forth the notation in (*) will always denote Tate's resolution. Under certain conditions Gulliksen [8, 226-2281 gives the construction of a minimal resolution of k over a quotient R of S. We translate his result to the particular case in which we are interested.


+

Theorem 2 Let A be a complete intersection ring and suppose that the m i n imal resolution Y o f the residue field k of A i s a n A-algebra. Let K be a proper graded ideal of A. Let R = A I K , Y = Y @ A R and E be a graded R-module such that for i 1, E' is a trivial module while for i 2 2, E' is a free R-module of rank equal to the dimension of the k-vector space H ; - I ( Y ) . Finally, assume Y has trivial Massey operations. T h e n the diflerential on Y m a y be extended to a differential on the graded R-module X = Y @R T ( E ) , turning A' into a m i n i m a l resolution of k over R. T ( E ) denotes the tensor algebra of the graded R-module E , while the tensor product with Y i s to be regarded as a tensor product of graded algebras.