Fitzgerald, Robert W.

Robert W. Fitzgerald

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Bass Series for Small Witt Rings

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Southern Illinois University CarbondaleOpenSIUCArticles and Preprints Department of Mathematics2006Bass Series for Small Witt RingsRobert W. FitzgeraldSouthern Illinois University Carbondale, rfitzg@math.siu.eduFollow this and additional works at: http://opensiuc.lib.siu.edu/math_articlesPart of the Number Theory CommonsPublished in Communications in Algebra, 34, 1753-1762.This Article is brought to you for free and open access by the Department of Mathematics at OpenSIUC. It has been accepted for inclusion in Articlesand Preprints by an authorized administrator of OpenSIUC. For more information, please contact opensiuc@lib.siu.edu.Recommended CitationFitzgerald, Robert W. "Bass Series for Small Witt Rings." ( Jan 2006).BASS SERIES FOR SMALL WITT RINGSRobert W. FitzgeraldSouthern Illinois UniversityThroughout R is a finitely generated (abstract) Witt ring . We will usually assumeI3R = 0.Our interest in Ext arises from a desire to examine combinatorial techniques coming fromring theory. The two principal objects of study for a local ring (A,m, k), are ExtA(k, k) andExtA(k,A). The dimension of ExtnA(k, k) is the rank of the nth free module in a minimalfree resolution of k. If A is also Artinian then every finitely generated injective A-moduleI is, by [8], the direct sum of µ(I) many copies of E(k), the injective hull of k . Thedimension of ExtnA(k,A) is µ of the nth injective module in a minimal injective resolutionof A.We often work with the generating functions for these dimensions. Specifically, set:PA(t) =∑i≥0(dimExtiA(k, k))tiIA(t) =∑i≥0(dimExtiA(k,A))tiHA(t) =∑i≥0dim(mi/mi+1)ti.Here PA(t), IA(t) and HA(t) are respectively the Poincare´ series, the Bass series and theHilbert series of A. We note that for Artinian A, the Hilbert series is in fact a polynomial.But it is not the Hilbert polynomial. Also, to avoid confusion with the Bass series we willwrite the fundamental ideal of a Witt ring R as IR instead of the usual IR.1. Elementary type case.Lemma 1.1.(1) Suppose I3S = 0, I3T = 0 and R = S u T . Then:IR(t) =IS(t)HS(−t) + IT (t)HT (−t) + tHR(−t)(2) Let S be any local Artinian Witt ring and let R = S[E1]. Then IR(t) = IS(t).Typeset by AMS-TEX12 ROBERT W. FITZGERALDProof. (1) Lescot [7] gives:IR(t)PR(t)=IS(t)PS(t)+IT (t)PT (t)+ tand [4] gives PW (t)−1 = HW (−t) for any Witt ring with I3W = 0.(2) We apply Foxby-Thorup [5], since S is a free (hence flat)R-module. In their notation,Q = IS, C = S/(IS ∩ R)S = S/IR · S ∼= k[E1] and Q′ = QC = IS/IR · S ∼= {0, 1 + e},where E1 = {1, e}. We obtain:µnS(IS, S) =∑i+j=nµiC(E1, C)µjR(IR,R)IS(t) = IC(t)IR(t).Now C is Gorenstein (by Bass’ criterion [2]) so IC(t) = 1 and IS(t) = IR(t). ¤We note that (1.1)(1) also holds for S and T local Witt rings of elementary type, sinceagain P (t)−1 = H(−t), by [4]. In fact, (1.1)(1) should be stated for any Fro¨berg Wittring and (1.1)(2) for any Witt ring. Here I assume always that R is non-real and finitely-generated (equivalently, local Artinian).Lemma 1.2.(1) If R is of local type then IR(t) = 1.(2) If R is of quasi-local type, say R = Dn[E1], then:IR(t) =n− t1− nt .Proof. (1) R is Gorenstein by [3], hence self-injective. Thus dimExt0(k,R) = 1 anddimExtn(k,R) = 0, for all n > 0.(2) Let S = Z4 or Z2[E1]. Then S is Gorenstein and again IS(t) = 1. Now Dn is aproduct of n copies of S, so an easy induction argument using (1.1)(1) shows IDn(t) =(n− t)/(1− nt). The result then follows from (1.1)(2). ¤We introduce the following notation:g = dim IR/I2Rh = dim I2R.For an infinite series p =∑aiti, q =∑biti we write p ≥ q if ai ≥ bi for all i.Set en = dimExtnR(k, k). When I3R = 0 the Fro¨berg relation yields:en =[n/2]∑i=0(−1)i(n− ii)gn−2ihi.For convenience, set en = 0 if n < 0. Also note that e0 = 1 and e1 = g.BASS SERIES FOR SMALL WITT RINGS 3Theorem 1.3. Suppose I3R = 0 and R is non-degenerate of elementary type. Let kdenote Z/2Z.(1) dimExt0(k,R) = h.(2) dimExtn(k,R) ≥ (h− 1)(gen−1 − (h+ 1)en−2), for all n ≥ 1.(3)IR(t) ≥ h− gt+ t21− gt+ ht2 .(4) The following are equivalent:(a) R is indecomposable.(b) Equality holds in (3).(c) Equality holds in (2) for all n ≥ 1.(d) Equality holds in (2) for some n ≥ 1.Proof. (1) Ext0(k,R) = Hom(k,R) ∼= ann IR = I2R, since R is non-degenerate. (2) and(3) are equivalent, as a simple computation shows. We prove (3). If R is of local typethen:IR(t) = 1 =h− gt+ t21− gt+ ht2 ,since h = 1. If R is quasi-local, say R = Dn[E1], then:IR(t) =n− t1− nt=n− t1− nt ·1− t1− t=n− (n+ 1)t+ t21− (n+ 1) + nt2 =h− gt+ t21− gt+ ht2 ,as desired. We note that this computation also proves (a)→(b) in (4) also. Finally, ifR = S u T then by induction:IR(t) =IS(t)HS(−t) + IT (t)HT (−t) + tHR(−t)≥ (hS − gSt+ t2) + (hT − gT t+ t2) + tHR(−t)=h− (g − 1)t+ 2t2HR(−t)≥ h− gt+ t21− gt+ ht2For (4), we have already shown (a)→(b) and that (b)↔(c). Clearly (c)→(d). Thus itsuffices to prove (d) implies (a). We will show the contrapositive. Suppose R = S u T .Then as above:IR(t) ≥ h− (g − 1)t+ 2t2HR(−t) .4 ROBERT W. FITZGERALDOnce again using the Fro¨berg relation 1/HR(−t) = PR(t) we get for n > 0:dimExtn(k,R) ≥ hen − (g − 1)en−1 + 2en−2.Now comparing coefficients in (1− gt+ ht2)PR(t) = 1 yields:en − gen−1 + hen−2 = 0 for n > 0en = gen−1 − hen−2.Thus:dimExtn(k,R) ≥ hgen−1 − (g − 1)en−1 − h2en−2 + 2en−2= (h− 1)gen−1 − (h2 − 1)en−2 + gen−1 + en−2= (h− 1)(gen−1 − (h+ 1)en−2) + gen−1 + en−2Thus we are done if we show gen−1+en−2 > 0 for every n > 0. If not, then for some n > 0we have en−1 = en−2 = 0. Since em = gem−1+hem−2 for all m > 0 we see that em = 0 forallm ≥ n−2. But then PR(t) is a polynomial which is impossible as PR(t)HR(−t) = 1. ¤We note that (1.3)(1) holds for any non-degenerate Witt ring with I3R = 0 since theproof used only these two facts. The next section will show (1.3)(2) also holds for such Ras does part of (4).We also note that since (1.1)(1) holds for all elementary type Witt rings (even if I3R 6=0), it is possible to compute IR(t) for any R of elementary type. In place of (1.3)(4) onecan show that if In+1R = 0:[n−1/2]∑i=0∑j+k=2i+1(−1)k(dim IkR/Ik+1R)(dimExtj(k,R)) ≥ 0with equality iff R is indecomposable. I suggest replacing (1.3) with this result for Fro¨bergWitt rings.2. General case.Supposedn−→ Ren −→ · · · d1−→ Re1 d0−→ R ε−→ k −→ 0is a minimal free resolution (here again en = dimExtnR(k, k)). For n ≥ 1 set Fn = Renand set F0 = R and F−1 = k. Also for n ≥ 1 set Zn = ker dn−1 ⊂ Fn with Z0 = IR andZ−1 = k. We note that, by the construction of a minimal resolution, that en+1 is the sizeof a minimal generating set for Zn. The following is standard.Lemma 2.1. Extn+2(k,R) = Ext1(Zn, R), for all n ≥ 0.Proof. We show by induction that Ext1(Zn, R) = Extj+1(Zn−j , R) for 0 ≤ j ≤ n+1. Thisis clear for j = 0. For j > 0 we have the exact sequence:0 −→ Zn−j −→ Ren−j dn−j−1−→ Zn−j−1 −→ 0.Ren is free so Extm(Ren , R) = 0 for all m > 0. The induced long exact sequence is:0 −→ Extj+1(Zn−j , R) δ−→ Extj+2(Zn−(j+1), R) −→ 0.¤BASS SERIES FOR SMALL WITT RINGS 5Proposition 2.2. Suppose that I3R = 0 and R is non-degenerate. Then:(1) dimExtn(k,R) ≥ (h− 1)(gen−1 − (h+ 1)en−2) for all n ≥ 1.(2)IR(t) ≥ h− gt+ t21− gt+ ht2 .Proof. A simple computation shows that (1) and (2) are equivalent. We again have theshort exact sequence:0 −→ Zn−1 −→ Ren−1 −→ Zn−2 −→ 0which yields:0 −→ Hom(Zn−2, R) −→ Hom(Ren−1 , R) −→ Hom(Zn−1, R)−→ Ext1(Zn−2, R) −→ Ext1(Ren−1 , R) ≡ 0We get:(2.3) dimExt1(Zn−2, R) = `(Hom(Zn−2, R))− en−1(1 + g + h) + `(Hom(Zn−1, R)).Claim. `(Hom(Zm, R)) ≥ hem+1 + em.When m = −1, Zm = k so that `(Hom(Zm, R)) = h. Since e−1 = 0 and e0 = 1,the Claim is proven in this case. Suppose m ≥ 0. Now Zm has em+1 elements in aminimal generating set. So dim(Hom(Zm, I2R)) = hem+1. Further, let p(Zm, R) denotethe subgroup generated by the em many projections Rem → R restricted to Zm. We assertthat Hom(Zm, I2R) ∩ p(Zm, R) = 0. Suppose instead that p is a non-zero member of theintersection. Write p =∑pii|Zm , where pii denotes the projection of Rem onto its ithcoordinate. We may assume pi1 is one of the summands of p. Now I2R · Fm ⊂ IR ·Zm by[4, 3.9,2.5]. Let σ be a non-zero element of I2R, and set x = (σ, 0, . . . , 0) ∈ I2R ·Fm. Thenp(x) = σ 6= 0. But since p ∈ Hom(Zm, I2R), we have p(x) ∈ p(I2R · Fm) ⊂ p(IR · Zm) ⊂I3R = 0, a contradiction. This proves the assertion.We thus have:Hom(Zm, I2R)⊕ p(Zm, R) ⊂ Hom(Zm, R).So `(Hom(Zm, R) ≥ hem+1 + em,proving the Claim. Plugging into (2.3) yields:dimExtn(k,R) = dimExt1(Zn−2, R)≥ hen−1 + en−2 − en−1(1 + g + h) + hen + en−1= en−2 − gen−1 + hen.Now from the Fro¨berg relation [4,3.9] PR(t)HR(−t) = 1 we have en = gen−1 − hen−2. So:dimExtn(k,R) ≥ en−2 − gen−1 + ghen−1 − h2en−2= (h− 1)(gen−1 − (h+ 1)en−2).6 ROBERT W. FITZGERALD¤We remark that when n = 1 (2.2) says:dimExt1(k,R) ≥ (h− 1)g,since e0 = 1 and e−1 = 0. We are unable to show equality holds in (2.2) iff R is indecom-posable, except in this case of n = 1.Probably should state that, by the proof, equality holds in (2.2)(1) iff(Em) Hom(Zm, R) = Hom(Zm, I2R)⊕ p(Zm, R)for m = n− 1, n− 2.Corollary 2.4. Suppose that I3R = 0. Let K be the Koszul complex on a minimalgenerating set for IR and let ci = dimk Homi(K). Then:h− gt+ t21− gt+ ht2 ≤∑g−1i=1 cg−iti − tg+11−∑ i = 1gciti+1 .Moreover, equality never occurs.Proof. This inequality is (2.2) combined with the inequality of [1]. If equality holds thenIR(t) is the common value, hence R is a Golod ring. Golod’s result [6] says:PR(t) =(1 + t)g1−∑gi=1 citi+1 .But by [4, 3.9] R is a Fro¨berg ring, that is,PR(t) =11− gt+ ht2 .Thus:1− c1t2 − · · · − cgtg+1 = (1 + t)g(1− gt+ ht2),which is impossible by a comparison of degrees. ¤Remark. This is optional. We can give the first four terms of the inequality in (2.4)(assuming g ≥ 4).h ≤ cgg(h− 1) ≤ cg−1(h− 1)(g2 − h− 1) ≤ cg + cg−2g(h− 1)(g2 − 2h− 1) ≤ cgc2 + c1cg−1 + cg−3.The first is always equality but already the second is not (ever?).BASS SERIES FOR SMALL WITT RINGS 7Lemma 2.5. Suppose D〈1,−a〉 ⊂ D〈1,−c〉, D〈1,−b〉 ⊂ D〈1,−d〉 and D〈1,−ab〉 ⊂D〈1,−cd〉. Then c ∈ D〈1,−bd〉 and d ∈ D〈1,−ac〉.Proof. Note that ac ∈ D〈1,−c〉,bd ∈ D〈1,−d〉 and abcd ∈ D〈1,−cd〉. So:〈〈−c,−bd〉〉 ' 〈〈−c,−abcd〉〉 ' 〈〈−d,−abcd〉〉 ' 〈〈−d,−ac〉〉.Thus c ∈ D〈1,−bd〉 iff d ∈ D〈1,−ac〉.Now since −a ∈ D〈1,−c〉,−ab ∈ D〈1,−cd〉 and −bD〈1,−d〉 we have:〈〈−c,−b〉〉 ' 〈〈−c, ab〉〉 ' 〈〈−d, ab〉〉 ' 〈〈−d,−a〉〉.By linkage, there exists a t such that:〈〈−b,−c〉〉 ' 〈〈−b,−t〉〉 ' 〈〈−d,−t〉〉 ' 〈〈−d,−a〉〉.In particular, ct ∈ D〈1,−b〉 and t ∈ D〈1,−bd〉. Now D〈1,−b〉 ⊂ D〈1,−b〉 ∩ D〈1,−d〉 ⊂D〈1,−bd〉. Thus c ∈ D〈1,−bd〉. ¤Theorem 2.6. Suppose I3R = 0 and R is non-degenerate. Then R is indecomposable iffdimExt1(k,R) = (h− 1)g.Proof. We need only Em for m = −1, 0 by ? and the case m = −1 is clear as Z−1 = kimplies Hom(Z−1, R) = Hom(Z−1, I2R). Moreover, Z0 = IR so p(Z0, R) = idIR. SodimExt1(k,R) = (h− 1)g iff Hom(IR,R) = {0, idIR}+Hom(IR, I2R).Suppose first that R = S u T . Then IR = IS ⊕ IT so that the projection IR → IS isin Hom(IR,R) but not in {0, idIR}+Hom(IR, I2R). Thus dimExt1(k,R) > (h− 1)g.Now suppose that dimExt1(k,R) > (h − 1)g. Then there exists an α ∈ Hom(IR,R)such that α /∈ {0, idIR} + Hom(IR, I2R). Fix a basis a1, . . . , ag of G. If σ ∈ I2R then0 = α(σ〈1,−ai〉) = σ · α(〈1,−ai〉). Thus α(〈1,−ai〉) ∈ IR. Write:α(〈1,−ai〉) = 〈1,−bi〉+ σi,for some bi ∈ G and σi ∈ I2R. Now β : IR → I2R by β(〈1,−ai〉) = σi is a well-definedhomomorphism and α− β is still not in {0, idIR}+Hom(IR, I2R). We may thus assumethat α(〈1,−ai〉) = 〈1,−bi〉, for 1 ≤ i ≤ g. Note that some bi 6= 1 as α 6= 0. Let H be thesubgroup of G generated by the bi; then H 6= 1. Let K be the subgroup of G generatedby the aibi. Note that as α 6= idIR, for some i we have ai 6= bi and so K 6= 1. Also, clearlyeach ai ∈ HK so that G = HK.Claim. If x ∈ H and y ∈ K then x ∈ D〈1,−y〉.For 1 ≤ i ≤ g, if c ∈ D〈1,−ai〉 then 0 = α(〈〈−ai,−c〉〉) = 〈1,−c〉α(〈1,−ai〉) =〈〈−c,−bi〉〉. Thus D〈1,−ai〉 ⊂ D〈1,−bi〉. Let j 6= i and 1 ≤ j ≤ g. Then D〈1,−aj〉 ⊂D〈1,−bj〉 also.If c ∈ D〈1,−aiaj〉 then 〈〈−c,−ai〉〉 ' 〈〈−c,−aj〉〉. Applying α yields 〈〈−c,−bi〉〉 '〈〈−c,−bj〉〉. Thus c ∈ D〈1,−bibj〉 and we have D〈1,−aiaj〉 ⊂ D〈1,−bibj〉. Then (2.5)gives bi ∈ D〈1,−ajbj〉 for all j 6= i. Also D〈1,−ai〉 ⊂ D〈1,−bi〉 implies that bi ∈ D〈1, ai〉∩8 ROBERT W. FITZGERALDD〈1, bi〉 ⊂ D〈1,−aibi〉. Hence every bi is in D〈1,−y〉 for every y ∈ K. So if x ∈ H andy ∈ K then x ∈ D〈1,−y〉, proving Claim.To show we have a decomposition of R, we need to show D〈1,−xy〉 = D〈1,−x〉 ∩D〈1,−y〉 for all x ∈ H and y ∈ K. Let t = ∏ aij ∈ D〈1,−xy〉. Set t′ = ∏ bij , over thesame indices as t. Then 0 = α(〈〈−t,−xy〉〉) = 〈〈−t′,−xy〉〉. Now t′ ∈ H so y ∈ D〈1,−t′〉.Thus x = xy · y ∈ D〈1,−t′〉. Further, tt′ ∈ K and so x ∈ D〈1,−tt′〉. Thus x ∈ D〈1,−t〉,t ∈ D〈1,−x〉 and, as t ∈ D〈1,−xy〉, t ∈ D〈1,−y〉 also. Hence R decomposes. ¤Remark. Here is a proof that if R decomposes thendimExtn(k,R) > en−2 − gen−1 + hen,for all n ≥ 0. Namely, let pi be the projection of Zn−1 onto the first coordinate. IfR = S uT then IR = IS⊕ IT . Let σ be the projection of IR onto IS. Set ϕ = σpi. Thenϕ ∈ Hom(Zn−1, R). We check that ϕ /∈ Hom(Zn−1, I2R) ⊕ p(Zn−1, R). Pick non-zeroα ∈ I2S and β ∈ I2T and let γ = (α, β) ∈ I2R. As in (2.2) γv1 ∈ Zn−1, where v1 =(1, 0, . . . , 0). Now ϕ(γv1) = α. But if ψ ∈ Hom(Zn−1, I2R) then ψ(γv1) ∈ IRψ(Zn−1) = 0.And if ψ ∈ p(Zn−1, R) is a combination of projections then ψ(γv1) = γ or 0. Henceϕ /∈ Hom(Zn−1, I2R)⊕p(Zn−1, R). The result follows from the, as yet unwritten, corollaryto (2.2).References1. L. Avramov and J. Lescot, Bass numbers and Golod rings, Math. Scand. 51 (1982), 199–211.2. H. Bass, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8–28.3. R. Fitzgerald, Gorenstein Witt rings, Canad. J. Math. 40 (1988), 1186–1202.4. R. Fitzgerald, Local artinian rings and the Fro¨berg relation, Rocky Mtn. J. Math. 26 (1966), 1351–1369.5. H.-B. Foxby and A. Thorup, Minimal injective resolutions under flat base change, Proc. Amer. Math.Soc. 67 (1977), 27–31.6. E.S Golod, Homomorphisms of some local rings, Dokl. Akad. Nauk SSSR 144 (1962), 479–482. (Rus-sian)7. J. Lescot, La se´rie de Bass d’un produit fibre´ d’anneaux locaux, P. Dubreil and M.-P. Malliavin AlgebraSeminar, 35th year (Paris, 1982), Lecture Notres in Math., vol. 1029, Springer, 1983, pp. 218–239.8. E. Matlis, Injective modules over Noetherian rings, Pac. J. Math. 8 (1958), 511–528.Carbondale, IL 69201-4408E-mail address: rfitzg@math.siu.edu

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Bass Series for Small Witt Rings

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