Connections between a local ring and its associated graded ring

JOURNAL OF ALGEBRA 111, 300-305 (1987) Connections between a Local Ring and Its Associated Graded Ring RALF FRGBERG Department of Mathematics, University of Stockholm, Box 6703, Stockholm S-l 1385, Sweden Communicated by D. A. Burhsbaum Received December 28, 1984 It is a general phenomenon that a local ring is “at least as nice” as its graded associative. We start by giving some results of a homological nature in this direction (see Example 1). These first results are not new; see, e.g., [ 11. But since we need the results later on, we give a proof. We use the spectral sequence TorgrR(gr M, gr N) * TorR(M, N). (In [ 11, a “dual” spectral sequence is used.) The main results of this article concern some extremal situations, where we can prove that the local ring is “just as nice” as its associated graded; cf. Theorem 1 and its corollary. Let the local ring (S, n, k) be a factor of the regular local ring (R, m, k). Then TorzR(gr S, k) has a grading, TorfrR (gr S, k) = @jai (ToryR(gr S, k)),, induced from the grading of the rings. Conditions on the set of (i,j) for which (ToryR(gr S, k)), vanishes yield E, = E, in he spectral sequence, which is interpreted as equalities for Betti numbers in the graded and ungraded cases.If the local ring S is complete we even get a close relation between the finite resolutions of S over R and of gr S over gr R, respectively. This generalizes, e.g., results in [lo]. Finally, we consider resolutions of k and find that in some situations we have equality for Poincare series, xi dim,(TorS(k, k)) z’= xi dim,(ToryS(k, k)) zi. 1. FINITE RESOLLJTIONS Let (R, m, k) be a local ring with maximal ideal m and residue class field k = R/m and let A4 and N be finitely generated R-modules. Let grR (and gr M, gr N, resp.) be the associated graded objects by means of the m-filtrations. Then there is a spectral sequence TorgrR(gr M, gr N) => TorR(M, N); cf. [9, Chap. 11.61.This spectral sequence will be one of our ingredients in a comparison of R and gr R. Just the existence of the spectral 300 0021-8693/87 $3.00 Copyright 0 1987 by Academic Press, Inc. All rights of reproduction in any form reserved. LOCAL RING AND ASSOCIATED GRADED RING 301 sequencegives some connection which often can be interpreted in terms of inequalities among Betti numbers. EXAMPLE 1. The case R regular, A4 = R/I, and N = k. Let (S, n, k) be a local ring. The completion S of S in the m-adic topology is then a factor ring of a regular ring, S = R/Z, where I is in the square of the maximal ideal of R. It is well known that S is Cohen-Macaulay (and Gorenstein, resp.) if and only if S is Cohen-Macaulay (and Gorenstein, resp.). We define the Betti numbers of S by hi(S) = dim, TorR(S, k). We define the complete intersection defect of S to be c.i.d. (S) = 6, (S) - dim R + dim S; here 6,(S) is the minimal number of generators of Z. Then S is a complete intersection (i.e., I is generated by a regular sequencein R) if and only if c.i.d.(S) = 0. If S is Cohen-Macaulay, we define the type of S to be type (S) = b,,(S), where c = dim R - dim S = maxi i; b,(S) # O}. The analogue definitions of those above apply for the case of a graded ring A = k[X,, .... X,,]/Z, in particular if A = gr S for some local ring S. The spectral sequencegives the inequality for Betti numbers b,(gr S) b b;(S). This gives (a) depth gr S<depth S. (b) If gr S is Cohen-Macaulay, then S is Cohen-Macaulay and furthermore type (gr S) 2 type (S). (c) If gr S is Gorenstein, then S is Gorenstein. (d) c.i.d. (gr S) > c.i.d. (S). In particular, if gr S is a complete intersection, then S is a complete intersection. (e) If gr S is regular, then S is regular. ProojY Since all the above-mentioned qualities are preserved by completion S to S and since gr S = gr S, we can assumethat S is complete, thus a factor ring of a regular ring R. We apply the spectral sequence for R regular, M = S, and N = k. Thus we have bi(gr S) 2 bi( S). (a) depth S = dim R - max{ i; hi(S) # 0) and correspondingly for gr S. Since bi(gr S) 3 hi(S) we have max{i; bi(gr S) # 0} > max(i; hi(S) ZO}. S’mce dim R = dim gr R we have depth gr S 6 depth S. (b) Suppose gr S to be Cohen-Macaulay. Then dim S = dim gr S = depth gr S < depth S, thus dim S = depth S (since dim S 2 depth S always); hence S is Cohen-Macaulay. Type (S) is the highest non-vanishing hi(S) and correspondingly for gr S. (c) Gorenstein is Cohen-Macaulay of type 1. (d) We have b,(grS)bbl(S), dimgrR=dimR and dimS= dim gr S. (e) We have that S (and gr S, resp.) is regular if and only if b,(S) = 0 (and b,(gr S) = 0, resp.). 302 RALF FRijBERG We now describe the spectral sequencein more detail in order to be able to draw conclusions which do not follow just from the existence. Let K be the Koszul complex S( T,, .... T,,; dT,= xi), where x,, .... x, constitutes a minimal system of generators for the maximal ideal n in S. We filter K by FpK=(x,, .... x,, T,, .... T,,)” K. The grading on K is defined by deg T, = -1 for all i. Put Kp.y = Fpk n K,. Then we have Ef.4 = (Tor?,R(gr S, k)), and d?“: EfY + E,P+ r,q+ I. From this construction of the spectral sequence it follows directly that E, = E, is guaranteed by the condition max{j; (ToryR(gr S, k)),#O} <min{j; (TorB;R,(gr S, k)),#O} (*) for all i. The following theorem follows. THEOREM 1. If E, = E, in the spectral sequence TorgrR(gr S, k) * TorR(S, k), in particular if the condition (* ) holds, then h,(gr S) = b,(S). COROLLARY. (a) (b) Suppose E, = E, in the spectral sequence. Then depth gr S = depth S. gr S is Cohen-Macaulay they are Cohen-Macaulay, if and only if S is Cohen-Macaulay. they have the same type. rf gr S is Gorenstein if and only zj’S is Gorenstein. (d) c.i.d.(gr S) = c.i.d.(S). In particular gr S is a complete intersection (c) if and only if S is a complete intersection. We now give examples of classesof rings satisfying the conditon (*). EXAMPLE 2. If gr S has a pure resolution, i.e., if for each i there is at most one j such that (ToryR(gr S, k)), ~0, then the condition (*) is satisfied. EXAMPLE 3. The notion of compressed algebras was introduced in [6] for artinian rings and extended in [4] to Cohen-Macaulay rings. We do not give the technical definition here, but only enumerate some examples. The following classes of graded algebras are compressed; cf. [4]. Compressed algebras satisfy (*) [4, Proposition 163. Thus, Theorem 1 applies to local rings S whose associated graded algebra is any of the following. (i) The extremal Cohen-Macaulay rings are rings from [S]. Examples of such 303 LOCAL RING AND ASSOCIATED GRADED RING (a) k[X,]/Z, 1 d i < m, 1~ j < n, and Z generated by all maximal minors of (X,); (b) k[X,]/Z, 1 d i, j 6 n, X, = Xii, and Z generated by all submaximal minors of (X,); (c) tangent cones of local Cohen-Macaulay rings of maximal embedding dimensions; (d) tangent cones of a rational surface singularity. (ii) The extremal Gorenstein rings from [S]. Examples of such rings are (a) k[ X,]/Z, 1 < i, j < 2n + 1, XV = -X,, Xii = 0, and Z generated by all (2n x 2n)-Pfaffians; (b) k[X,,]/Z, 1 6 i, j Q n, and Z generated by all submaximal minors; (c) tangent cones of Gorenstein rings of maximal embedding dimension; (d) tangent cones of an elliptic surface singularity. (e) If A is a triangulation of a sphere with n vertices and with a maximal number of faces in each dimension, then the associated Stanley-Reisner ring k[A] is an extremal Gorenstein ring. (iii) If A =k[X,, .... X,]/(fi, .... f,) is a graded complete intersection, then A is a compressed algebra if and only if r = 1 or r = 2 and or r=3 and degf,=degf,=degf,=2. Idegf,-degf,l<l 4. Suppose that (1 - Z)’ Hilbs(Z) = (1 - Z)k CizO dim, ) 2’ = Cj=A (‘L”) Z’ + sZ’ for some k, n, t, and s and furthermore Wlmi + ’ that gr S is Cohen-Macaulay. Then the condition (*) is satisfied. Examples of such rings are coordinate rings for points in “uniform position” in P”; cf. [S-j. EXAMPLE Proof. It follows by (Torp’R(grS,k))j=Oforj#t+i, the technique of [4, Sect. 41 that tfi-1. If S is complete we can get a close relation even on the resolution level for S and gr S, not only on the homology level. We first need a general lemma. LEMMA 2. Let S be a complete local ring and R a complete regular local ring mapping on S. Let (Fe, d) be a minimal graded gr R-resolution of gr S. Then there exists an R-resolution (FL, d’) of S such that (gr Fi, gr d’) = (F,, 4. 304 RALF FRiiBERG ProoJ: Suppose is constructed such that: (1) (2) (3) FPRh is (4) It is exact. grd,‘=d, forj=O, 1, .... i. dJFpRbj) =dj(Rh)n FpRb-l for j=O, 1, .... i and all p, where the filtration part induced by the m-filtration of R. bj=rank,,,Fi forj=O, 1, .... i. We shall construct a map 4, I : Rbtil -+ Rb’ preserving these conditions. Let Fb,+,= gr Re, 0 gr Re, @ . . . 0 gr Reb,+,be the next step in F, and let di+ ,(e,) = uj so that {u,> is a minimal graded set of generators for Ker d, and let a,i be a lifting of a1to RbJ+‘.Then, since di(aj) = 0, we seethat d,!(ai) lies higher in the filtration that ai. Hence by (3) we can adjust ai with an element 6; of higher filtration degree than a,! so that a,!- b,’E Ker di and so that a; - bi has initial form aj in gr Rbl. Let Fb,,, = Re; @ Re; @ . . . @ Reb,,, and define d: + ,(ei) = a; - bl. Then d:, , d: = 0. Let z’ E Ker dj and let z be the initial form of z’. Then z E Ker di, so z is a linear combination of the a;s. We can lift this linear combination to a cycle U’ with the same initial form as z’, so that z’ - U’ has higher filtration degree than z’. Continuing like this and using the completeness of R we can make F’ exact at RbJ.A very similar reasoning (again using the completeness of R) shows that d;+ ,(FpRb”‘) = di+ l(Rb”‘) n FpRb’, which proves the lemma by induction on i, the induction start being trivial. THEOREM 3. Let S be a complete local ring such that hi(S) = bi(gr S). Then a minimal graded gr R-resolution of gr S can be liffed to a minimal R-resolution qf S. Proof Everything but the minimality follows from the lemma. But if b((S) = b,(gr S), then Fi is minimal. 2. RESOLUTIONS OF k We apply the spectral sequence TorgR(gr M, gr N) * TorR(M, N) for R = arbitrary local ring and M = N = k. Let PR(Z) = Ciao dim, Tor”(k, k) . Z’ denote the Poincare series for R and correspondingly for gr R. As in the preceding section we get LOCAL RING AND ASSOCIATED GRADED RING 305 THEOREM 4. If E,=E, in the spectral sequence, in particular if max(j; (ToryR(k, k)),# 0} < min(j: (TorcR,(k, k)),#O} for all i, then &dZ) = PR(Z). COROLLARY. If R is a local ring with gr R a Koszul algebra, i.e., i#j, then PgrR(Z) = PR(Z). (ToryR(k, k)),=Ofor Remark. The corollary extends results in [3,7]. We enumerate some examples of Koszul algebras. EXAMPLE 5. The following classes of rings are Koszul algebras; cf. [2, Sect. 1.171. (i) kCX,, .... ~,I/~, where I is generated by an arbitrary set of monomials of degree two; (ii) “most” algebras k[X, , X,, X,]/Z, where I is generated by elements of degree two; 111) graded k-algebras A = @ iz0 Ai with dim, A, < 2, where A = k[X(-’I, ...>~,ll(fI 7.... f,) and deg fi = 2 for all i; (iv) k[X,, .... X,1/1, where Z is generated by a regular sequence of elements of degree two. 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