A MATRIX OF LINEAR FORMS WHICH IS ANNIHILATED BY A VECTOR OF INDETERMINATES

Transcription

1 A MATRIX OF LINEAR FORMS WHICH IS ANNIHILATED BY A VECTOR OF INDETERMINATES ANDREW R KUSTIN, CLAUDIA POLINI, AND BERND ULRICH ABSTRACT Let R = k[t 1,,T f ] be a standard graded poynomia ring over the fied k and Ψ be an f g matrix of inear forms from R, where 1 g < f Assume [ T 1 T f ] Ψ is 0 and that gradei g (Ψ is exacty one short of the maximum possibe grade We resove R = R/I g (Ψ, prove that R has a g-inear resoution, record expicit formuas for the h-vector and mutipicity of R, and prove that if f g is even, then the idea I g (Ψ is unmixed Furthermore, if f g is odd, then we identify an expicit generating set for the unmixed part, I g (Ψ unm, of I g (Ψ, resove R/I g (Ψ unm, and record expicit formuas for the h-vector of R/I g (Ψ unm (The rings R/I g (Ψ and R/I g (Ψ unm automaticay have the same mutipicity These resuts have appications to the study of the bow-up agebras associated to ineary presented grade three Gorenstein ideas CONTENTS 1 Introduction Main resuts 5 3 Conventions, notation, and preiminary resuts 10 4 The maps and modues of M ε 14 5 The doube compexes T ε and B 6 6 The map of compexes ξ ε : Tot(T ε Tot(B 34 7 The h-vector of H 0 (M ε 46 8 Appication to bowup agebras 57 References 61 Date: January 5, 018 AMS 010 Mathematics Subject Cassification Primary 13D0; Secondary 13C40, 13A30 The first author was partiay supported by the Simons Foundation The second and third authors were partiay supported by the NSF Keywords: Buchsbaum-Rim compex, depth sensitivity, determinanta idea, divided power agebra, generaized Eagon-Northcott compexes, Gorenstein idea, grade unmixed part of an idea, Hibert series, h-vector, matrix of inear forms, mutipicity, Pfaffians, Rees agebra, resiient ideas, specia fiber ring, unmixed part of an idea, tota compex 1

2 ANDREW R KUSTIN, CLAUDIA POLINI, AND BERND ULRICH 1 INTRODUCTION It is shown in [4] that, Determinanta ideas associated to sufficienty genera matrices of inear forms are resiient in the sense that they remain of the expected codimension, or prime, even moduo a certain number of inear forms The cases where the origina matrix is generic, symmetric generic, cataecticant, or 1-generic have been particuary we-studied; see for exampe, [5, 8, 11, 19, 0] We study a famiy of resiient determinanta ideas of matrices of inear forms The ideas we consider are described by a structure condition and a grade condition Each of these ideas has a inear resoution and defines a ring which is Cohen-Macauay on the punctured spectrum, but is not Cohen-Macauay The previousy studied resiient ideas associated to a matrix of inear forms have a defined rings which are Cohen-Macauay For the time being, et R be a Noetherian ring, T 1,,T f an R-reguar sequence, and Ψ an f g matrix with entries in R, where g < f (Throughout most of the paper, g is aowed to take the vaue 1; but to simpify the exposition, in the beginning of the introduction, we insist that g be at east Assume that the matrix Ψ has two properties First of a, assume that [ T1 T f ] Ψ = 0 It is cear from this property that the idea I g (Ψ generated by the maxima minors of Ψ does not have the maximum possibe grade, f g+1 The second property of Ψ is that gradei g (Ψ is ony one short of the maximum possibe grade, that is, gradei g (Ψ = f g Let R be the R-agebra R/I g (Ψ and et δ = f g For each integer ε in the set { δ 1, δ, δ+1 }, we construct a compex Mε When ε is either δ δ+1 or, that is when ε = δ, then Mε is a resoution of R by free R-modues Though the compex M ε may not be minima, it can be used to see that the projective dimension of R over R is f 1 or f depending on whether δ is even or odd We deduce that R is never perfect as an R-modue, but is grade unmixed if δ is even, indeed depth R p = δ for every associated prime p of R In particuar, I g (Ψ is unmixed whenever δ is even and R is Cohen-Macauay Despite the faiure of perfection, the resoution M ε speciaizes, in the sense that S R M ε is a resoution of S R R by free S-modues whenever S is a Noetherian R-agebra, T 1,,T f forms an S-reguar sequence, and δ gradesi g (Ψ We can say more if the ring R is non-negativey graded with R 0 a fied and T 1,,T f as we as the entries of Ψ are inear forms In this case, the minima homogeneous free R- resoution of R is g-inear The ring R is not Cohen-Macauay; so the Betti numbers in the minima resoution of R and the mutipicity of R are not an obvious consequence of the fact that the minima homogeneous free R-resoution of R is g-inear Nonetheess, we provide expicit formuas for the h-vector and the mutipicity of R; these resuts do not depend on the parity of δ

3 A MATRIX OF LINEAR FORMS WHICH IS ANNIHILATED BY A VECTOR OF INDETERMINATES 3 The mutipicity of R and the unmixedness of I g (Ψ pay critica roes in [16], where we describe expicity, in terms of generators and reations, the specia fiber ring and the Rees agebra of ineary presented grade three Gorenstein ideas in poynomia rings of odd Kru dimension On the other hand, if δ is odd, then the idea I g (Ψ is mixed We identify the grade unmixed part, I g (Ψ gunm, of I g (Ψ (see (314, prove that M ε, with ε = δ 1, is a resoution of R/I g (Ψ gunm, and provide expicit formuas for the h-vector of R/I g (Ψ gunm (The rings R = R/I g (Ψ and R/I g (Ψ gunm automaticay have the same mutipicity Utimatey, the parameter ε takes the vaue δ δ 1 or If ε = δ, then Mε resoves R; and if ε = δ 1, then M ε resoves R/I g (Ψ gunm Of course, if δ is even, then δ 1 and δ are equa as are I g (Ψ and I g (Ψ gunm Take ε = δ The resoution Mε was obtained in stages First we identified a fairy standard compex Tot(B with H 0 (Tot(B = R The doube compex B is a quotient of of a arger doube compex V: the coumns of V are Koszu compexes and the rows of V are truncations of generaized Eagon-Northcott compexes Aas, Tot(B has higher homoogy Indeed, when δ is even, H 1 (Tot(B 0 A ong ook at H 1 (Tot(B tod us that this homoogy came from Pfaffians Macauay experimentation ead us to a compex Tot(T ε whose homoogy is equa to H 1 (Tot(B Curiousy, T ε is aso a quotient of V We became convinced that there exists a map of compexes Tot(T ε ξ ε Tot(B which induces an isomorphism on a of the higher homoogy Eventuay, we were abe to record a formua for ξ ε The mapping cone, L ε, of ξ ε is an infinite resoution of R The resoution M ε is a finite subcompex of L ε of the proper ength which has the same homoogy as L ε After we had found the resoution M δ of R, and reaized that I g (Ψ is mixed when δ is odd, we ooked for the generators of I g (Ψ gunm /I g (Ψ In this search, we were inspired by the work of Mark Johnson [10] and Susan Morey [18] in the case f = g+1 We iustrate our answer in the context of the motivating situation from [16] Let R 0 be a fied, R 1 be the poynomia ring R 0 [T 1,,T f ], and R be the bi-graded poynomia ring R = R 0 [X 1,,X g,t 1,,T f ] where each X i has bi-degree (1,0 and each T i has bi-degree (0,1 Let φ be an f f aternating matrix with bi-homogeneous entries of degree (1,0 and Ψ be an f g matrix with bi-homogeneous entries of degree (0,1 Assume that f + g is odd and T 1 X 1 φ = Ψ T f X g

4 4 ANDREW R KUSTIN, CLAUDIA POLINI, AND BERND ULRICH Consider the (f + g (f + g aternating matrix [ ] φ Ψ B = Ψ t 0 Let B i be ( 1 i+1 times the Pfaffian of B with row and coumn i removed View B f+g as a poynomia in R 1 [X 1,,X g ] and et C = c R1 (B f+g be the content of B f+g It foows that C is a homogeneous idea in R 1 = R 0 [T 1,,T f ] generated by forms of degree g 1 At this point, we use eementary techniques to show that (101 C + I g (Ψ I g (Ψ : (T 1,,T f In this paper we prove that if δ gradei g (Ψ, then the ideas C + I g (Ψ, I g (Ψ : (T 1,,T f, I g (Ψ : (T 1,,T f, and I g (Ψ gunm of R 1 a are equa These facts are used in [16] where we describe expicity, in terms of generators and reations, the specia fiber ring and the Rees agebra of ineary presented grade three Gorenstein ideas in poynomia rings of even Kru dimension To prove (101, first observe first that the coumn vector [ T1 T f X 1 X g ] t is in the nu space of B; then use ower order Pfaffians of B to see that for each index i, with 1 i f, the row vector [0,,0,B f+g,0,,0, B i ], with B f+g in position i, is in the row space of B It foows that B f+g T i +B i X g = 0 for 1 i f The definition of B i shows that B i is in the idea I g (Ψ; hence T i B f+g I g (Ψ and T i c R1 (B f+g = c R1 (T i B f+g I g (Ψ This competes the proof of (101 The generating set for I g (Ψ gunm /I g (Ψ that we use throughout most of the paper is amost coordinate-free; consequenty, at first gance, it ooks much different than the content of B f+g ; however the generating set that we use has the advantage that ony sma modifications of the resoution M δ of R produces the resoution M δ 1 of R/I g (Ψ gunm We return to a coordinate-dependent matrix version of the resuts in Section 8 There are precedents for the resoution of J gunm to be obtained from the resoution of J using ony sma modifications: the modues are changed sighty, the maps are changed sighty, but the form of the resoution is not reay atered Reca, for exampe, the type two amost compete intersection ideas and the deviation two Gorenstein ideas of Huneke- Urich [9] The Gorenstein ideas are the unmixed part of ideas which have the same form as the amost compete intersection ideas Furthermore, the resoutions of the Gorenstein ideas and the amost compete intersections have the same form; see [13, 14, 15] Indeed,

5 A MATRIX OF LINEAR FORMS WHICH IS ANNIHILATED BY A VECTOR OF INDETERMINATES 5 these ideas are the case g = 1 of the ideas I g (Ψ and I g (Ψ gunm which are resoved in the present paper The main resuts of the paper, Theorems 5 and 7, are stated in Section Section 3 consists of conventions, notation, and preiminary resuts The maps and modues of M ε, a of the numerica information about M ε, and exampes of M ε are given in Section 4 The doube compexes B, T ε, and V are introduced in Section 5, where we aso compute the homoogy of Tot(B and Tot(T ε The fina piece of the puzze, ξ ε, is defined in Section 6 The proofs of Theorem 5 and part (b of Theorem 7 appear near the end of Section 6 The Hibert series part of Theorem 7, assertion (a, is estabished in Section 7 In Section 8 we return to the situation of bowup agebras and organize our concusions in the exact anguage of [16] We aso give the compete proof that equaity hods in (101 MAIN RESULTS The word matrix does not appear in the officia data of the paper (Data 1 We expain the transition Each coumn of Ψ, from Section 1, is annihiated by the row vector [ T1 T f ] It foows that each coumn of Ψ is equa to an aternating matrix times T f In order to understand Ψ one must consider the g reevant aternating matrices We dea with Pfaffian identities invoving mutipe aternating matrices by making use of the divided power structure on the subagebra n n F of an exterior agebra F In other words, we work with an R-modue homomorphism µ : G F, where G and F are free R-modue of rank g and f, respectivey (Notice that if one picks bases for G and F, then µ seects g aternating matrices, one for each coumn of Ψ If Ψ ony has one coumn, then the situation has been studied extensivey; see, for exampe [9, 13, 1, 14, 15] Curiousy, many of the ideas invoved in the present project are aready present in the case where Ψ ony has one coumn Every construction in sections through 7 is buit using Data 1 or Data 3 (We return to a coordinate-dependent matrix version of the resuts in Section 8 The interesting resuts are estabished after we impose the hypotheses of Data 1 Let R be a commutative Noetherian ring, F and G be free R-modues of rank f and g, respectivey, with 1 g < f, be an eement of F, and µ : G F be an R-modue homomorphism Define Ψ : G F by Ψ(γ = (µ(γ for γ in G Let δ represent f g, ε represent an integer with δ 1 ε δ, and R be the R-agebra R/I g(ψ T 1

6 6 ANDREW R KUSTIN, CLAUDIA POLINI, AND BERND ULRICH Hypotheses Adopt Data 1 Assume that (a δ gradei g (Ψ, (this bound is one ess than the maxima possibe grade for I g (Ψ, and (b f gradei 1 ( Data 3 is more compicated than Data 1, is meaningfu ony when δ is odd, and pertains ony to the generating set for I g (Ψ gunm and the first differentia in M ε when ε = δ 1 (Reca that I g (Ψ gunm is the grade unmixed part of I g (Ψ; see 314 for more detais It is possibe to ignore Data 3 and sti read most of the paper The data of 1 is coordinatefree; however the data of 3 is not competey coordinate-free; it requires that one pair of dua basis eements (Y 1,X 1, in G G be distinguished The idea C of R, as described in Data 3, does not depend on the choice of (Y 1,X 1 (See the discussion surrounding (101 for a competey different description of I g (Ψ gunm /I g (Ψ in a specia case; but do notice that in (101 one coumn of Ψ is treated in a distinguished manner and any other coumn of Ψ woud work just as we Data 3 Adopt Data 1 (a Assume δ odd Decompose G and G as (31 G = RY 1 G + and G = RX 1 G + with Y 1 (X 1 = 1, Y 1 (G + = 0, and G + (X 1 = 0, where Y 1 G, X 1 G, G + and G + are free submodues of G and G, respectivey, of rank g 1 Define an R-modue (integration homomorphism X 1 : D i G D i+1 G by way of the decomposition (31; that is, with γ i j,+ D i j (G + Notice that ( i X 1 X ( j 1 γ i j,+ = i X ( j+1 1 γ i j,+, j=0 j=0 (3 Y 1 ( X 1 γ i = γ i for a γ i D i (G Define (33 c : D δ 1 (G f F g G to be the R-modue homomorphism (34 c(γ δ 1 = [D(µ]( X 1 γ δ 1 ( g 1 Ψ(ω G+ ω G+ Y 1 (b Define C to be the foowing idea of R { ann R (cokerc if δ is odd (35 C = 0 if δ is even and define R = R/(I g (Ψ + C Remarks 4 (a The homomorphism c of (33 and is defined ony when δ is odd

7 A MATRIX OF LINEAR FORMS WHICH IS ANNIHILATED BY A VECTOR OF INDETERMINATES 7 (b If δ is odd, then the target of c is a free R-modue of rank one; so our definition of C is a coordinate-free way of saying that C is the image of c (c We show in (6 that C I g (Ψ : I 1 ( (d Some hints about why the generators for I g (Ψ gunm /I g (Ψ as cacuated in (101 are the same as the generators of C are given in (4411 and (441 A compete proof may be found in Theorem 83a Theorems 5 and 7 are the main resuts of the paper The short version of these theorems is that we resove R/I g (Ψ and R/I g (Ψ gunm, where I g (Ψ gunm is the grade unmixed part of the idea I g (Ψ (see, for exampe, 314, and we record the Hibert series, mutipicity, and h-vectors of these rings Theorem 5 Adopt Data 1 and 3 The foowing statements hod (a The maps and modues of M ε, given in Definition 41, form a compex Assume that Hypotheses are in effect for the rest of the statements (b If ε = δ, then the compex Mε is a resoution of R by free R-modues (c If ε = δ 1, then the compex Mε is a resoution of R by free R-modues Assume that I 1 ( is a proper idea of R for the rest of the statements (d The projective dimension of the R-modue R is equa to { f 1, if δ is even, and f, if δ is odd, and the projective dimension of the R-modue R is equa to f 1, if δ is even, f, if δ is odd, 3 δ, and g, f 1, if δ is odd, g = 1, and gradei g (Ψ δ, and 1, if δ = 1 and g (e If δ is even, then depthr p = δ for every p Ass R (R (f Assume that one of the foowing three hypotheses is in effect: (i δ is even, or (ii δ is odd and g, or (iii δ is odd, g = 1, and gradei g (Ψ δ Then depthr p = δ for every p Ass R ( R (g If one of the three hypotheses of (f is in effect, then the ideas C + I g (Ψ, I g (Ψ : I 1 (, I g (Ψ : I 1 (, and I g (Ψ gunm of R are equa; and in particuar, R = R/I g (Ψ gunm

8 8 ANDREW R KUSTIN, CLAUDIA POLINI, AND BERND ULRICH (h If one of the three hypotheses of (f is in effect and ε = δ 1, then the compex Mε is a resoution of R/I g (Ψ gunm by free R-modues Remark 6 We comment on Hypothesis 5fiii We note that if g = 1 and δ is odd, then it is possibe for I g (Ψ to have grade δ + 1; for exampe if [ ] 0 I 1 f Ψ = T I f 0 T f In this situation, R is not of any interest A compete characterization of this situation is given in Observation 68 Theorem 7 gives the Hibert series of three famiies of rings because Theorem 5 shows that R/I g (Ψ in the case that I g (Ψ is not a grade unmixed idea, if ε = δ+1 H 0 (M ε, = R/I g (Ψ in the case that I g (Ψ is a grade unmixed idea, if ε = δ, R/I g (Ψ gunm in the case that I g (Ψ is not a grade unmixed idea, if ε = δ 1 Theorem 7 Adopt Data 1 and 3 Assume that Hypotheses are in effect, that R is a non-negativey graded ring, and that : R( 1 f R and µ : R g R (f are homogeneous (degree preserving R-modue homomorphisms Then the foowing statements aso hod (a Assume that g, or ese, that g = 1 and f is odd Then the Hibert series of H 0 (M ε is equa to HS H0 (M ε (s = HS R (s HN H0 (M ε (s for HN H0 (M ε (s = (1 s f g hn H0 (M ε (s and (1 s g j ε 1 hn H0 (M ε (s = + g s where = ( 1 δ+1( g+ j 1 j ( +f g 1 =0 + f g ( 1 +δ( g+ 1 =0 g 1 =0 ( +f g 1 s χ(ε = δ 1 ( g+ε 1 ε s g f+ s g 1 s j+g f + q(g,f ( 1 +g+1( ( g g+ j 1 g+f j j =g j ε 1 g 3, if ε = δ 1, q(g, f = g, if ε = δ, and g 1, if ε = δ+1 s,

9 A MATRIX OF LINEAR FORMS WHICH IS ANNIHILATED BY A VECTOR OF INDETERMINATES 9 In particuar, if R is a standard graded poynomia ring of Kru dimension dimr over a fied, then (i HS H0 (M ε (s = hn H 0 (M ε (s (1 s, dimr f+g (ii the h-vector of H 0 (M ε is with h = ( +f g 1 ( +f g 1 j ε 1 hv(h 0 (M ε = (h 0,,h q(g,f,, if 0 g, χ(ε = δ 1 and (iii the mutipicity of H 0 (M ε is ( g+ε 1 ε ( 1 +g+1( ( g g+ j 1 g+f j j e(h 0 (M ε = hn H0 (M ε (1 =, if = g 1, and δ/ i=0, if g q(g, f, ( f i δ i which is equa to { the number of monomias of even degree at most δ in g 1 variabes, the number of monomias of odd degree at most δ in g 1 variabes,, if δ is even, or if δ is odd (b If R is a standard graded poynomia ring over a fied, then the minima resoution of R by free R-modues is g-inear Remarks 8 (a In item (a of Theorem 7, we use the notation of [1, 541] to denote the Hibert series HS H0 (M ε (s, the numerator of the Hibert series HN H0 (M ε (s, the simpified Hibert numerator hn H0 (M ε (s, and the h-vector hv(h 0 (M ε, of H 0 (M ε We gave two formuations for the simpified Hibert numerator hn H0 (M ε (s: one yieds the h-vector hv(h 0 (M ε quicky and the other yieds the mutipicity e(h 0 (M ε quicky Reca that the Hibert series of a Noetherian graded ring S = 0 i S i, (with S 0 an Artinian oca ring is the forma power series HS S (z = i λ S0 (S i z i, where λ S0 ( represents the ength of an S 0 -modue, and the mutipicity of S is e(s = (dims! im i λ S (S/m i S i dims, where m is the maxima homogeneous idea of S and dim represents Kru dimension

10 10 ANDREW R KUSTIN, CLAUDIA POLINI, AND BERND ULRICH (b If (81 g = 1 and f is even, then the correct statement which is anaogous to (a of Theorem 7 is HN R (s = (1 s f and hn R (s = 1 Indeed, the form for HN R (S (1 s f g, as given in (a, continues to be correct, even in the presence of the hypotheses (81; however, in the presence of (81, HN R (S = 1 s; (1 s f g consequenty, it is not appropriate to ca the quotient hn R (s See Remark 7 for more detais On the other hand, when the hypothesis of (81 are in effect, then there is no meaningfu statement about HN H0 (M ε (s, with ε = δ 1, which is anaogous to (a of Theorem 7 In this situation, I g (Ψ gunm = (I g (Ψ,α for some homogeneous eement α in R of degree 0; see, for exampe, Exampe 44g Thus, the Hibert series of H 0 (M ε, computed using this grading, is the same as the Hibert series of 0 (c Assertion 7b is not true, in genera, for R/I g (Ψ gunm, when I g (Ψ gunm I g (Ψ because, for exampe, in genera, I g (Ψ gunm has generators of different degrees See Exampe 44c or Exampe 44d 3 CONVENTIONS, NOTATION, AND PRELIMINARY RESULTS Data 1 and 3 are in effect throughout this section 31 Uness otherwise noted, a functors wi be functors of R-modues; that is,, Hom, (, Sym i, D i, i, and : mean R, Hom R, Hom R (,R, Sym R i, DR i, i R, and : R respectivey 3 If I and J are ideas in a ring R, then the saturation of J by I in R is J : I = J : I n = {r R ri n J for some n} n=1 33 We denote the tota compex of the doube compex X by Tot(X 34 If z is a cyce in a compex, then we denote the corresponding eement of homoogy by z 35 If δ is odd, then the decomposition of (31 is used in the description of c from (33 and the description of C from (35; otherwise, our compexes are described in a coordinatefree manner We make much use of the divided power structures on the agebras D (G and

11 A MATRIX OF LINEAR FORMS WHICH IS ANNIHILATED BY A VECTOR OF INDETERMINATES 11 F; in particuar, the R-modue homomorphism µ : G F automaticay induces a homomorphism D(µ : D (G F of divided power agebras and the composition D (G D(µ F incusion F, which we aso denote by D(µ, is used extensivey in our cacuations Let Y 1,,Y g be a basis for G and X 1,,X g be the corresponding dua basis for G ( Y (351 Let represent the set of monomias of degree i in Y 1,,Y g i If m = Y a 1 1 Y a g g is in ( Y i, then et m represent the eement X (a 1 1 X (a g g of D i (G Observe that {m m ( Y i } is the basis for Di (G which is dua to the basis ( Y i of Symi G Consider the evauation map ev : Sym i G D i (G R and et ev : R D i (G Sym i G be the dua of ev Both of these R-modue homomorphisms are competey independent of coordinates; and therefore the eement ev (1 = m m D i (G Sym i G m ( Y i is competey independent of coordinates; this eement wi aso be used extensivey in our cacuations 36 In a simiar manner, if ω G is a basis for g (G and ω G is the corresponding dua basis for g G, then the eement ω G ω G is a canonica eement of g (G g G This eement is aso used in our cacuations 37 We reca some of the properties of the divided power structure on the subagebra F of the exterior agebra F Suppose that e 1,,e f is a basis for the free R-modue F and f = a i1,i e i1 e i 1 i 1 <i f is an eement of F, for some a i1,i in R Let A be the f f aternating matrix with a i, j, if i < j, A i, j = 0, if i = j, and a i, j, if j < i For each positive integer, the -th divided power of f is f ( = A I e I F, I

12 1 ANDREW R KUSTIN, CLAUDIA POLINI, AND BERND ULRICH where the -tupe I = (i 1,,i roams over a increasing sequences of integers with 1 i 1 and i f, e I = e i1 e i, and A I is the Pfaffian of the submatrix of A which consists of rows and coumns {i 1,,i }, in the given order Furthermore, F is a DGΓ-modue over F In particuar, if F and v 1,,v s are homogeneous eements of F, then ( v ( 1 1 v ( s s = s (v j v ( 1 1 v ( j 1 j v ( s s j=1 For more detais see, for exampe, [6, Appendix A4] or [3, Appendix and Sect ] 38 If I and J are integers, then V I,J, VI,J T and V I,J B a represent the free R-modue I F D J (G Of course the rank of V I,J is ( ( f g+j 1 for any choice of ; that is might be T, B, or empty I J 39 We aways use f i for an arbitrary eement of i F and γ j for an arbitrary eement of D j (G 310 The notation ε J I+J δ If S is a statement then means χ(s = {(I,J ε J and I + J δ 1} { 1, if S is true, 0, if S is fase 31 If M is a matrix (or a homomorphism of free R-modues, then I r (M is the idea generated by the r r minors of M (or any matrix representation of M We denote the transpose of a matrix M by M t 313 The grade of a proper idea I in a Noetherian ring R is the ength of a maxima R-reguar sequence in I The unit idea R of R is regarded as an idea of infinite grade 314 Let I be a proper idea in a Noetherian ring R The idea I is grade unmixed if gradep = gradei for a associated prime ideas p of R/I The grade unmixed part of I is the idea I gunm which satisfies either of the foowing two equivaent conditions: (a I gunm is the smaest idea K with I K, gradek = gradei, and K is grade unmixed, or (b I gunm is the argest idea K with I K and gradei < grade(i : K Furthermore, if K is any grade unmixed idea of R with I K and gradek = gradei < grade(i : K, then K = I gunm In particuar, if I = i Q i is a primary decomposition of I, with each Q i a p i -primary idea of R, then I gunm is the intersection of the primary components Q i of I which correspond to prime ideas p i with gradep i = gradei

13 A MATRIX OF LINEAR FORMS WHICH IS ANNIHILATED BY A VECTOR OF INDETERMINATES 13 Of course, if R is Cohen-Macauay, then I gunm is the usua unmixed part of I We reca that if I p are ideas with p prime, then gradei gradep depthr p ; consequenty, if depthr p = gradei for a p Ass R I, then I is automaticay grade unmixed Proof of the assertions in 314 If K 1 and K are ideas which both satisfy gradei < grade(i : K i, then gradei < grade(i : (K 1 +K ; hence the hypothesis that R is Noetherian guarantees that an idea K which satisfies (b exists Fix this K We show that K aso has the property of (a Let x be a maxima R-reguar sequence in I A of the cacuations may be made in R/(x So no harm is done if we prove the statement when gradei = 0 We first show K is grade unmixed of grade 0 Use the primary decomposition of K to write K = A 1 A, where every associated prime idea of A R 1 has grade 0, and, either A = R, or A is a proper idea and every associated prime idea of A R has positive grade Observe that (I : KA 1 A (I : KK I; hence, (I : KA (I : A 1 Observe that (I : KA has positive grade It foows that K A 1 and 0 < grade(i : A 1 The defining property of K now guarantees that K = A 1 and therefore, K is grade unmixed Now we show that K has property (a We have aready shown that K has grade 0 and is grade unmixed We prove that K is the smaest such idea Let J be any grade unmixed idea of R with I J and gradej = 0 We prove that K J It suffices to show K p J p for a p in Ass( R J Let p be in Ass( R J The hypotheses on J guarantees that gradep = 0 and therefore, (I : K p On the other hand, (I : KK I J; hence, K p = (I : K p K p I p J p With respect to the furthermore assertion, I gunm K by (a because K is grade unmixed and K I gunm by (b because gradei < grade(i : K The assertion about primary decomposition is now obvious 315 Let pd R (M represent the projective dimension of an R-modue M 316 Let I be a proper idea in a Noetherian ring R Since one can compute Ext R (R/I,R from a projective resoution of R/I, one obviousy has (3161 gradei pd R R/I; if equaity hods, then I is caed a perfect idea Reca, for exampe, that if I is a proper homogeneous idea in a poynomia ring R over a fied, then I is a perfect idea if and ony if R/I is a Cohen-Macauay ring (This is not the fu story For more information, see, for

14 14 ANDREW R KUSTIN, CLAUDIA POLINI, AND BERND ULRICH exampe, [, Prop 1619] or [1, Thm 15] A perfect idea I of grade g is a Gorenstein idea if Ext g R (R/I,R is a cycic R-modue Lemma 317 Adopt the notation of 35 If A and B are non-negative integers and Γ is an eement of D B (G, then ( B m m(γ = Γ A m ( Y A Proof It suffices to prove the resut for Γ = X (b 1 1 X (b g g where each b i is a nonnegative integer and i b i = B Let m = Y a 1 1 Y a g g, where each a i is a non-negative integer and i a i = A Observe that ( ( m b1 bg m(γ = Γ, a 1 because X (a i i X (b i a i i = ( b i (b a i X i i (The most recent equation hods for a non-negative integers a i and b i At this point, we have shown that On the other hand, m m(γ = m ( Y A (3171 a 1 + +a g =A a 1 + +a n =A ( b1 a 1 ( bg a g a g ( b1 a 1 = ( bg ( B A a g Γ Indeed the eft of (3171 is the coefficient of x A y B A in the eft side of the poynomia (317 (x + y b1 (x + y b g = (x + yb, and the right side of (3171 is the coefficient of x A y B A in the right side of (317 4 THE MAPS AND MODULES OF M ε The object M ε is the foca point of this paper We introduce the maps and modues of M ε in Definition 41; a of the numerica information about M ε is contained in Remarks 4; and some exampes of M ε are given in Exampes 43 and 44 We prove in Remark 64 that M ε is a compex and in Lemma 66 that M ε is a resoution when Hypotheses are in effect Utimatey, M ε is a subcompex of L ε, L ε is the mapping cone of Tot(T ε ξ ε Tot(B, and T ε and B are quotient sub-doube compexes of the doube compex V We use L ε, T ε, B, and V to prove Lemma 66; however, M ε is the object of interest in this paper; and therefore, we introduce it, in compete detai, first

15 A MATRIX OF LINEAR FORMS WHICH IS ANNIHILATED BY A VECTOR OF INDETERMINATES 15 The conventions and notation of Section 3 are used throughout this section; in particuar, the modues V i, j are defined in 38, the symbos X, Y and ( Y i are defined in 35, the notation ε J I+J δ 1 is expained in 310, the function χ is expained in 311, and the bases ω G,ω G are expained in 36 As one reads Definition 41, it might be hepfu to simutaneousy foow Exampe 43 where we record M ε, a of its constituent pieces, and a of its forms, when (g, f = (3,9 and ε = δ Definition 41 Adopt Data 1 and 3 The maps and modues M ε are described as foows: (a As a graded R-modue M ε = ( ε J I+J δ 1 V Tε I,J ( j ε 1 δ i+ j V B i, j ( f F g G, with (i VI,J Tε in position I + J δ +, (ii Vi, B j in position i + j δ + 1, and (iii ( f F g G in position 0 (b The R-modue homomorphisms of M ε are described beow (i If I +J δ+ = N, N, 0 I, ε J, and I +J δ 1, then VI,J Tε is a summand of M ε N and χ(ε J 1 g Ψ(X f I Y (γ J VI+1,J 1 Tε Mε N 1 =1 (411 d( f I γ J V Tε I,J = +( f I γ J V Tε + i+ j=i+j δ i+ j I 1,J Mε N 1 ( 1 I+ j( J 1 j J ε V B i, j Mε N 1 [D(µ](m f I m(γ J m ( Y J j (ii If i+ j δ+1 = N, N, δ i+ j, 0 i, j, and j ε 1, then Vi, B j is a summand of M ε N and d( f i γ j V B i, j = (iii The R-modue homomorphism is d : M ε 1 = V B δ,0 g Ψ(X f i Y (γ j Vi+1, B j 1 Mε N 1 =1 +χ(δ i + j 1( f i γ j V B i 1, j Mε N 1 δ 1 Tε χ(ε = V0,ε Mε 0 = ( f F g G d( f δ V B δ,0 = f δ ( g Ψ(ω G ω G,

16 16 ANDREW R KUSTIN, CLAUDIA POLINI, AND BERND ULRICH and, if ε = δ 1, then (41 d(γ ε V Tε 0,ε = c(γ ε for c as defined in (33 Remarks 4 (a If δ = 1 and ε = δ 1, then Mε does not have any non-zero summands of the form Vi, B j Otherwise, the modue V B f,ε 1 is a non-zero summand of Mε g+ε 1 ; furthermore, if g + ε 1 < N, then M ε N does not contain any non-zero summands of the form Vi, B j The vaue of g + ε 1 is f, if ε = δ 1, f 1, if ε = δ, and f, if ε = δ+1 (b If δ = 1 and ε = δ+1, then Mε does not have any non-zero summands of the form VI,J Tε Otherwise, V0,δ 1 Tε is a non-zero summand of Mε δ; furthermore, if δ + 1 N, then M ε N does not contain any non-zero summands of the form VI,J Tε (c The argest index N with M ε N 0 is 1, if ε = δ 1 and δ = 1, f 1, if ε = δ 1 and 1 = g, (41 N max = f, if ε = δ 1, δ, and g, f 1, if ε = δ, and f, if ε = δ+1 furthermore, if N max is the parameter of (41, then M ε N max is equa to V0,δ 1 Tε, χ(ε = δ 1 χ( = gv 0,δ 1 T χ(ε = δχ(1 = gv 0,δ 1 T V B ; f,ε 1, if ε = δ 1 and 1 = g or ε = δ 1 and 1 = δ, otherwise (d As noted in (b, M ε does not have any non-zero summands of the form VI,J Tε ε = δ+1 Tε Otherwise, the modue V0,ε is a non-zero summand of Mε N 0 for 1, if ε = δ 1, (4 N 0 = ε δ + =, if ε = δ, and 3, if ε = δ+1 and δ; if δ = 1 and furthermore, if N < N 0, then M ε N does not contain any non-zero summands of the form VI,J Tε (e If R is a bi-graded ring and : R( 1,0 f R and µ : R(0, 1 g R (f

17 A MATRIX OF LINEAR FORMS WHICH IS ANNIHILATED BY A VECTOR OF INDETERMINATES 17 are bi-homogeneous R-modue homomorphisms, then the maps and modues of M ε are bi-homogeneous with VI,J Tε rankv Tε R(f g I J, g J I,J, V B i, j R(f g i j, g j rankv B i, j, and ( f F g G R Indeed, under the given hypotheses, Ψ : R( 1, 1 g R f and D(µ : R(0, J (J+g 1 J R ( J f are aso bi-homogeneous R-modue homomorphisms (f The hypotheses of (e are in effect in the generic case where R = R 0 [T 1,,T f,{a i, j,k 1 i < j f and 1 k g}], degt i = (1,0, dega i, j,k = (0,1, (e i = T i, for 1 i f, and µ(x k = A i, j,k e i e j, 1 i< j f for 1 k g, with e 1,,e f a basis for F and X 1,,X g a basis for G (g If R is a graded ring and : R( 1 f R and µ : R g R (f are homogeneous R-modue homomorphisms, then the maps and modues of M ε are homogeneous with { M ε R if N = 0 N R( g + N β N R( g + 1 N β N if 1 N N max, as given in (41, where (43 β N = I+J δ+=n ε J I+J δ 1 rankv Tε I,J and β N = i+ j δ+1=n j ε 1 δ i+ j rankv B i, j, for 1 N N max (h The hypotheses of (g are in effect in the specia case where R is the poynomia ring R = R 0 [T 1,,T f ], degt i = 1, degα i, j,k = 0, (e i = T i, for 1 i f, and µ(x k = α i, j,k e i e j, 1 i< j f for 1 k g, with e 1,,e f a basis for F, and X 1,,X g a basis for G, and α i, j,k R 0 (i If ε is equa to δ δ+1 or, Mε is a resoution, and the hypotheses of (h are in effect with R 0 a fied, then there is a quasi-isomorphism from M ε to the g-inear minima resoution 0 R( g N max + 1 b Nmax R( g 1 b R( g b 1 R,

18 18 ANDREW R KUSTIN, CLAUDIA POLINI, AND BERND ULRICH with b N = β N β N+1 for β N and β N as defined in (43 (This remark is a consequence of three facts First of a, the generators of M ε 1 a have degree g; secondy, every homogeneous R-modue homomorphism R( a R( a 1 is necessariy zero; and thirdy, the image of a homogeneous R-modue homomorphism R( a A R( a B is a free summand of the target However, the anaogous statement is not true when ε = δ 1 See Exampes 44c and 44d (j In Definition 41bi, it is not necessary to impose the condition j ε 1 in the V B component of d(v Tε I,J ; because, if the expression ( J 1 j (44 ( 1 I+ j i+ j=i+j J ε δ i+ j [D(µ](m f I m(γ J Vi, B j m ( Y J j is non-zero, then the inequaity j ε 1 is automaticay satisfied Indeed, if (44 is non-zero, then j J However j can not equa J; because, if j = J, then i = I and δ i + j = I + J δ 1; which is impossibe Thus, 0 J j 1 On the other hand, the binomia coefficient ( J 1 j J ε is not zero; so 0 J ε J 1 j, and j ε 1, as caimed (This remark, which ooks technica, is actuay the proof of the assertion that M ε is a subcompex of L ε ; see Remark 64 Exampe 43 Let (g, f = (3,9 The parameter δ (which equas f g is even, and therefore the constraint δ 1 ε δ forces ε to equa δ = 3 In this exampe, we record Mε, a of its constituent pieces, and a of its forms Definition 41a says that M ε = with (431 V Tε I,J (43 V B i, j ( 9 F 3 G, i, j - (431 (I,J {(I,3 0 I } {(I,4 0 I 1} {(0,5} and (43 (i, j {(i,0 6 i 9} {(i,1 5 i 9} {(i, 4 i 9} It is usefu to consider the doube compexes of Tabe 431 (The doube compexes T ε and B are officiay introduced in Definition 51; the compex Tot(433 is a shift (see (63 of a subcompex of Tot(T ε and the compex Tot(434 is a subcompex of Tot(B It is shown in Lemma 6 that there is a map of compexes ξ from a shift of Tot(433 to Tot(434,

19 A MATRIX OF LINEAR FORMS WHICH IS ANNIHILATED BY A VECTOR OF INDETERMINATES 19 (433 V Tε 0,5 Ψ V Tε 1,4 Ψ V Tε,3 V Tε 0,4 Ψ V Tε 1,3 V0,3 Tε and (434 V B 9, V B 8, Ψ V B 9,1 V B 7, Ψ V B 8,1 Ψ V B 9,0 V B 6, Ψ V B 7,1 Ψ V B 8,0 V B 5, Ψ V B 6,1 Ψ V B 7,0 V B 4, Ψ V B 5,1 Ψ V B 6,0 3 Ψ 9 F 3 G TABLE 431 Doube compexes which are used in the construction of the first version (435 of M ε in Exampe 43 with ξ(v Tε 0,3 V B 6,0, so that Mε is the mapping cone of ξ As a graded modue, M ε is (435 V0,4 Tε V1,3 Tε V0,5 Tε V1,4 Tε V,3 Tε V V B 0,3 Tε 0 V9, B V8, B V7, B V6, B V5, B 4, V V B 5,1 B V6,0 B 9 F 3 G, V9,1 B V8,1 B V7,1 B 6,1 V B V B 7,0 V9,0 B 8,0

20 0 ANDREW R KUSTIN, CLAUDIA POLINI, AND BERND ULRICH 0 R( 10, 5 6 R( 9, 5 54 R( 7, 8 1 R( 6, R( 8, 5 16 R( 7, R( 8, 4 3 R( 7, 4 7 R( 5, 7 15 R( 5, R( 6, R( 6, R( 6, 3 1 R( 4, 6 90 R( 5, R( 5, 4 5 R( 5, 3 9 R( 3, 6 10 R( 4, R( 3, 3 84 R R( 4, 3 36 TABLE 43 The compex M ε from Exampe 43 when the data is bihomogeneous, as described in Remark 4e with 9 F 3 G in position zero (One can use the formuas given in Definition 41a to cacuate the position of each summand VI,J Tε and V i, B j of Mε ; however, if the doube compexes (433 and (434 are avaiabe, then it is easy to read the position of each summand of M ε from the mapping cone construction If the data is bi-homogeneous, as described in Remark 4e, then M ε is given in Tabe 43 The rank of V I,J is given in 38 The bi-homogeneous twists in M ε are given in Remark 4e or may be read from the doube compexes (433 and (434 as soon as one knows that 9 F 3 G = R and V0,3 T = R( 3, 610 If the hypotheses of Remark 4g are in effect, then M ε is 0 R( 10 6 R( 9 54 R( 7 1 R( R( R( 8 19 R( R( R( 4 90 R( 3 10 R( 3 84 R R( R( If the hypotheses of Remark 4i are in effect, then M ε is quasi-isomorphic to (436 0 R( 10 6 R( 9 54 R( 8 19 R( R( R( 5 64 R( 4 34 R( 3 74 R Exampes 44 These exampes are presented more quicky than Exampe 43

21 A MATRIX OF LINEAR FORMS WHICH IS ANNIHILATED BY A VECTOR OF INDETERMINATES 1 (a Let (g, f = (,6 and ε = δ = The modues of Mε are 0 V B 6,1 V Tε 0,3 V B 5,1 V1, Tε V0, Tε V4,1 B V3,1 B V4,0 B 6 F G V6,0 B V5,0 B If the hypotheses of Remark 4e are in effect, then M ε is 0 R( 6, 3 R( 4, 5 4 R( 5, 3 1 R( 3, 4 18 R( 4, 3 30 R( 4, 1 If the hypotheses of Remark 4g are in effect, then M ε is 0 R( 6 R(, 4 3 R( 3, 3 40 R(, 15 R R( 3, 6 R( 4 4 R( 3 18 R( 3 R( 15 R R( 5 1 R( 4 31 R( 3 46 If the hypotheses of Remark 4i are in effect, then M ε is quasi-isomorphic to (441 0 R( 6 R( 5 1 R( 4 7 R( 3 8 R( 1 R (b Let (g, f = (3,6 and ε = δ+1 = The modues of M ε are V B 0 V6,1 B V 5,1 B 4,1 V6,0 B V Tε 0, V B 3,1 V B 5,0 V,1 B V B V4,0 B 3,0 6 F 3 G If the hypotheses of Remark 4e are in effect, then M ε is 0 R( 8, 4 3 R( 7, 4 18 R( 4, 5 6 R( 6, 4 45 R( 5, 4 60 R( 6, 3 1 R( 5, 3 6 (44 R( 4, 4 45 R( 4, 3 15 R( 3, 3 0 R

22 ANDREW R KUSTIN, CLAUDIA POLINI, AND BERND ULRICH If the hypotheses of Remark 4g are in effect, then M ε is 0 R( 8 3 R( 7 18 R( 6 46 R( 4 6 R( 5 66 R( 4 60 R( 3 0 R If the hypotheses of Remark 4i are in effect, then M ε is quasi-isomorphic to (443 0 R( 8 3 R( 7 18 R( 6 46 R( 5 66 R( 4 54 R( 3 0 R (c Let (g, f = (3,6 and ε = δ 1 = 1 The modues of M ε are 0 V B 6,0 V T1 0, V B 5,0 V T1 1,1 V B 4,0 V T1 0,1 V B 3,0 6 F 3 G If the hypotheses of Remark 4e are in effect, then M ε is (444 0 R( 6, 3 1 R( 4, 5 6 R( 3, 4 18 R(, 4 3 R R( 5, 3 6 R( 4, 3 15 R( 3, 3 0 If the hypotheses of Remark 4g are in effect, then M ε is 0 R( 6 1 R( 4 6 R( 3 18 R( 3 R R( 5 6 R( 4 15 R( 3 0 In the present exampe, ε = δ 1 and ε δ If the rest of the hypotheses of Remark 4i, other than the hypothesis ε = δ, are in effect, then we do not know the graded Betti numbers in a minima homogeneous resoution of H 0 (M ε ; indeed, we do not know if these Betti numbers can be determined from the data (ε, f, g or if more information about the R-modue µ (of Data 1 is required (d Let (g, f = (4,7 and ε = δ 1 = 1 The modues of M ε are 0 V B 7,0 V B 6,0 V T1 0, V B 5,0 V T1 1,1 V B 4,0 V T1 0,1 V B 3,0 If the hypotheses of Remark 4e are in effect, then M ε is 0 R( 8, 4 1 R( 7, 4 7 If the hypotheses of Remark 4g are in effect, then M ε is (445 0 R( 8 1 R( F 4 G R( 5, 6 10 R( 4, 5 8 R( 3, 5 4 R R( 6, 4 1 R( 5, 4 35 R( 4, 4 35 R( 5 10 R( 4 8 R( 6 1 R( 5 35 R( 3 4 R( 4 35 d 1 R

23 A MATRIX OF LINEAR FORMS WHICH IS ANNIHILATED BY A VECTOR OF INDETERMINATES 3 Notice that the graded Betti numbers show that it is not possibe for (445 to be quasiisomorphic to a pure compex; indeed, the image of d 1 is generated in two different degrees (e One can use Macauay [7] to verify the graded Betti numbers of (436, (441, (443, and (444, when the Hypotheses are in effect (f When f = n + 1 is odd, g = 1, and ε = δ, then Mε is isomorphic to the compex (446 M = i F h ( j i+ j n 0 j p+q n 1 0 q p Fλ (q of [15, Def 15] In M, i F h ( j is in position i + j and p Fλ (q in position p + q + The modue f F g G in M ε corresponds to 0 F h (0 in M, the modue Vi, B j in Mε corresponds to f i F h (i+ j δ in M, and the modue VI,J Tε in Mε corresponds to I Fλ (J n in M (g When f = n is even, g = 1, and ε = δ 1, then Mε is isomorphic to the compex (447 M = i F h ( j p Fλ (q i+ j n 1 0 j p+q n 1 0 q of [14, Thm 4] (Earier versions of the compex may be found in [13, 1] In M, i F h ( j is in position i + j and p Fλ (q in position p + q + 1 The modue f F g G in M ε corresponds to 0 F h (0 in M, the modue V B i, j in Mε corresponds to f i F h (i+ j δ in M, and the modue V Tε I,J in Mε corresponds to I Fλ (J+1 n in M (h Let δ = 1 If ε = δ+1 = 1, then M ε is (448 0 V B f,0 V B f 1,0 V B,0 V B 1,0 (411 f F g G The subcompex 0 M ε f M ε 1 of M ε is a truncation of the Koszu compex associated to If ε = δ 1 = 1, then M ε is (449 0 V Tε 0,0 (41 f F g G Once we prove Theorem 5, then the exactness of (448 gives that I g (Ψ = αi 1 ( for some eement α in R and the exactness of (449 identifies α as a generator of the image of (4410 c(1 = µ(x 1 ( g 1 Ψ(X X g Y g Y 1 under any isomorphism from f F g G to R (We use the notation of 35 These ideas, but not this phrasing, are aready known by Mark Johnson [10] and Susan Morey [18]

24 4 ANDREW R KUSTIN, CLAUDIA POLINI, AND BERND ULRICH A more concrete version of (4410 is obtained as foows Let X 1,,X g be a basis for G, e 1,,e f a basis for F, µ(x k = a i, j,k e i e j F, T be the coumn vector [(e 1,,(e f ] t, 1 i< j f and, for 1 k g, et A k be the f f aternating matrix with a i, j,k in position i, j when i < j In R, c(1 is a unit times the Pfaffian of A 1 A T A g T T (4411 t A 0 T t A g It is usefu to observe that (4411 is aso the coefficient of Z 1 in the Pfaffian of Z 1 A 1 + Z A + + Z g A g A T A g T T t A 0 T t A g In this discussion, Z 1,,Z g are indeterminates over R; they pay the roe of pace hoders (i Take f to be odd and g = If the hypotheses of Theorem 5 are in effect, then the idea I g (Ψ gumn is perfect of grade f = δ and R/I g (Ψ gumn is resoved by M ε, for ε = δ 1 To describe a generating set for this idea, et A 1 and A be f f aternating matrices with entries in R and et T be a f 1 coumn vector, again with entries in R The idea I g (Ψ gumn is generated by (441 (the coefficients of Z 1 and Z 1Z in the Pfaffian of +I ( [ A 1 T A T ], [ ] Z1 A 1 + Z A A T T transpose A 0 provided (441 has grade at east δ and I 1 ( has grade at east f Once again, Z 1 and Z are indeterminates over R; they pay the roe of pace hoders 45 The fina non-zero map of M is of considerabe interest For most choices of (g, f, this map is : Vf,ε 1 B Vf 1,ε 1 B If this map is written as a matrix, then it ooks ike the matrix of Tabe 451, where e 1,e f is a basis for F and T i = (e i The matrix has ( g+ε ( f g 1 rows and g+ε g 1 coumns For sma vaues of g or δ this genera form is perturbed sighty The compete form of the ast map is given in Observation 46

25 A MATRIX OF LINEAR FORMS WHICH IS ANNIHILATED BY A VECTOR OF INDETERMINATES 5 T 1 T T f T 1 T T f T 1 T T f TABLE 451 The ast non-zero map of M ε for most (g, f See 45, Observation 46, and (466 Observation 46 Adopt Data 1 and 3 Retain the description of M ε as given in Definition 41 and Remarks 4; in particuar, the vaue of N max is given in (41 Then the fina non-zero map of M ε is and this map is d Nmax : M ε N max M ε N max 1 (461 (46 (463 (464 (465 V Tε 0,0 V B f,0 (41 f F g G, [ ] V Tε 0,δ 1 V Tε 0,δ 1 V B f,ε 1 V B f,ε 1 V B [ Ψ ξ ] f 1,0 if δ = 1 and ε = δ 1,, if δ = 1 and ε = δ+1 V Tε 1,δ Vf,ε 1 B, ] [ Ψ 0 ξ [ 0 ] χ(3 δv Tε 1,δ V B f 1,ε 1, V0,δ 1 Tε Vf 1,ε 1 B,, if δ and (ε, g = ( δ 1,1, if δ and (ε, g equas ( δ 1, or ( δ,1, if δ and (ε, g equas ( δ 1,3, or ( δ,, or ( δ+1,1

26 6 ANDREW R KUSTIN, CLAUDIA POLINI, AND BERND ULRICH (466 V B f,ε 1 [ ] V B f 1,ε 1, if δ and (ε, g satisfy one of the foowing: ε = δ 1 and 4 g, or ε = δ and 3 g, or ε = δ+1 and g The map ξ : V0,δ 1 Tε V (δ ε,ε 1 B, which appears in (463 and (464, is given by (467 ξ(γ δ 1 = ( 1 ε 1 [D(µ](m m(γ δ 1 V(δ ε,ε 1 B Mε δ 1; m ( δ ε Y furthermore, { f, if (ε, g = ( δ 1 (δ ε =,1, and f 1, if (ε, g = ( δ 1, or ( δ,1 Proof If δ = 1, then M ε is recorded in (448 and (449 Henceforth, we assume δ The differentia d Nmax is given in Definition 41b We need ony dea with the modues The modue M ε N max is given in Remark 4c One easiy cacuates the summands of M ε N max 1 For exampe, if ε = δ+1 and Vi, B j is a summand of Mε N max 1 = M ε f 1, then hence, i + j δ + 1 = f 1 and j ε 1 = δ+1 1; f i + δ = j δ 1 It foows that f 1 i On the other hand, f i and δ have the same parity; so i = f 1 A simiar argument works in the remaining cases 5 THE DOUBLE COMPLEXES T ε AND B Adopt Data 1 and 3 The doube compexes T ε and B, which are used in the construction of M ε, both are quotients of the doube compex V In Definition 51, we introduce a three doube compexes V, T ε, and B; we aso introduce the sub-doube-compex U of V with V/U equa to B (We have no need for the subcompex of V which defines T ε Pease keep in mind that, for our purposes, T ε and B are the important compexes We have introduced V and U in order to cacuate the homoogy of B; see Lemma 57 (It is not difficut to compute the homoogy of T ε ; see Lemma 53 Definition 51 Adopt Data 1 and 3 Define V, T ε, B and U to be the doube compexes which are given in Tabes 511 and 51 The doube compexes T ε and B are quotients of V and U is the subcompex of V with V/U = B The modues are indexed in such a way that Tot(V 0 = ( f F g G V i, j i+ j=δ 1 (a The modue V i, j is the free R-modue i F D j (G, see 38

27 A MATRIX OF LINEAR FORMS WHICH IS ANNIHILATED BY A VECTOR OF INDETERMINATES 7 V : Ψ Ψ V δ,1 V δ+1,0 Ψ Ψ Vδ 1,1 V δ,0 Ψ Ψ Vδ,1 V δ 1,0 g Ψ g Ψ 0 f F g G f 1 F g G V 0,1 Ψ V 1,0 g Ψ g+1 F g G 0 V 0,0 g Ψ g F g G T ε : V 0,ε+ Ψ V 1,ε+1 Ψ V,ε 0 V Ψ 0,ε+1 V 1,ε 0 0 V 0,ε, TABLE 511 The doube compexes V and T ε from Definition 51 (b The R-modue homomorphism V i, j V i 1, j sends f i γ j to ( f i γ j (Reca the convention 39

28 8 ANDREW R KUSTIN, CLAUDIA POLINI, AND BERND ULRICH B : Ψ Ψ V Ψ δ, V δ+1,1 V δ+,0 Ψ Ψ Vδ 1, Ψ V δ,1 V δ+1,0 0 0 Ψ Ψ Vδ, Ψ Vδ 1,1 Vδ,0 f F g G, U : Ψ Ψ Vδ,1 V δ 1,0 g Ψ f 1 F g G V 0,1 Ψ V 1,0 g Ψ g+1 F g G 0 V 0,0 g Ψ g F g G TABLE 51 The doube compexes B and U from Definition 51 Ψ (c The R-modue homomorphism V i, j Vi+1, j 1 sends f i γ j to Ψ(X f i Y (γ j g (The eement X Y of G G is expained in 35 =1 g Ψ (d The R-modue homomorphism V i,0 g+i F g G is f i f i ( g Ψ(ω G ω G (The symbos ω G and ω G are expained in 36 Remarks 5 (a We aways use f i for an arbitrary eement of i F and γ j for an arbitrary eement of D j (G ; see 39

29 A MATRIX OF LINEAR FORMS WHICH IS ANNIHILATED BY A VECTOR OF INDETERMINATES 9 (b Reca from the definition of Ψ in Data 1 that Ψ = 0 Observe that the squares V i, j V i+1, j 1 V i 1, j V i, j 1 from the compexes of Definition 51 anti-commute and the squares V i,0 i+g F g G V i 1,0 i 1+g F g G commute on-the-nose (c Let N be an integer (i The modue Tot(V N is equa to ( f+n F g G (51 V i, j, if δ N, and 0, if N < δ, where the sum is taken over (51 {(i, j i + j δ + 1 = N, and 0 i, j} (ii The modue Tot(T ε N is equa to 0, if N 0, and V i, j, if 1 N, (5 where the sum is taken over (5 {(i, j i + j δ + 1 = N, 0 i, and ε j} (iii The modue Tot(B N is equa to 0, if N 1, f F g G, if N = 0, and V i, j, if 1 N, (53 where the sum is taken over (53 {(i, j i + j δ + 1 = N, δ i + j, and 0 i, j}

30 30 ANDREW R KUSTIN, CLAUDIA POLINI, AND BERND ULRICH (iv The modue Tot(U N is equa to V i, j, if 0 N, (54 ( f+n F g G V i, j, if δ N 1, and (54 0, if N < δ where the sum is taken over (54 {(i, j i + j δ + 1 = N, i + j δ 1 and 0 i, j} Lemma 53 Adopt Data 1 Assume Hypothesis b hods Reca the doube compexes T ε and V of Definition 51 (a If N is an integer, then H N (Tot(T ε { R I 1 ( D δ+n 1 (G, if δ + N 1 is even and ε δ + 1 N, and 0, otherwise Furthermore, if δ + N 1 is even and ε δ + 1 N, then (531 {z m m ( Y δ+n 1 } is a basis for the free R/I 1 (-modue H N (Tot(T ε (See Remark 54 and (351 (b If 1 δ and 1 N, then { R/I1 ( D δ+n 1 (G, if δ + N 1 is even, and H N (Tot(V 0, otherwise Furthermore, if δ + N 1 is even and 1 N, then (53 {Z m m ( Y δ+n 1 } is a basis for the free R/I 1 (-modue H N (Tot(V (See Remark 54 and (351 Remarks 54 (a We expain the notation in (531 and (53 For each eement γ of D δ+n 1 (G, cyces z ε γ in Tot(T ε δ+n 1 and Z γ in Tot(V δ+n 1 are defined in Lemma 55 In particuar, the homoogy cass z m of (531 refers to the homoogy cass of the cyce z m in Tot(T ε δ+n 1 Lemma 55 which corresponds to the eement m of D δ+n 1 (G as described in (b A ess technica statement of Lemma 53a is that the non-zero homoogy of Tot(T ε is caused by the externa corners V 0, j0 in T ε Furthermore, the corner V 0, j0 contributes R/I 1 ( D j0 (G to H j0 δ+1(tot(t ε We use (5 to read that V 0, j0 sits in position j 0 δ+1 of Tot(T ε Aso, it is immediate that if N is j 0 δ + 1, then j 0 = (N + δ 1/

31 A MATRIX OF LINEAR FORMS WHICH IS ANNIHILATED BY A VECTOR OF INDETERMINATES 31 Proof of Lemma 53 (a Each coumn of T ε is the tensor product of a free R-modue with the Koszu compex associated to a reguar sequence Consequenty, the homoogy of each coumn of T ε is we understood We view Tot(T ε as the imit of a sequence of mapping cones; each mapping cone adjoins one more coumn from T ε The coumns are arranged in such a way that the homoogy of the new coumn does not interfere with the homoogy of the tota compex of the previous coumns We provide some detais Let j 0 be an integer with ε j 0 ; and A and P be the subcompexes of Tot(T ε We assume by induction that A = Tot( V, j0 1 and P = Tot( V, j0 H N (A 0 ony if δ + N 1 is even, and ε δ + 1 N j 0 δ 1 We see that P/A is the coumn iv i, j0 of T ε ; thus P/A is the tensor product of the Koszu compex associated to with D j0 (G Hypothesis b ensures that { R/I 1 ( D j0 (G, if N = j 0 δ + 1, and H N (P/A 0, otherwise The ong exact sequence of homoogy which is induced by the short exact sequence of compexes yieds that 0 A P P/A 0 0, if j 0 δ + N, H N (P R/I 1 ( D j0 (G, if N = j 0 δ + 1, H N (A, if N j 0 δ Furthermore, the cyces that represent H N (A continue to represent H N (P for N j 0 δ Aso, it is not difficut to identify a basis for the free R/I 1 (-modue H j0 δ+1(p One takes a basis for D j0 (G Each eement of this basis is a cyce in the compex P/A One ifts the cyce in P/A to a cyce in P and then takes the homoogy cass In 35, we identified the basis {m m ( Y j 0 } for D j0 (G In Lemma 55 we ift m to the cyce z m in P Tot(T ε (b The proof is simiar to the proof of (a The ony difference is that the stair-case pattern invoving exterior ower eft-hand corners starts with the second coumn instead of the first coumn That is, H δ+1 (Tot(V and H δ (Tot(V are not cacuated using the method of the proof of (a On the other hand, the statement of the resut makes no caim about these homoogies because δ < δ < N Once 1 N, then the modue H N (Tot(V is non-zero if and ony if Tot(V N contains one of the corners V 0, j0 of V Furthermore, (51 shows that V 0, j0 is a summand of Tot(V N if and ony if j 0 δ + 1 = N The rest of the proof is identica to the proof of (a