Quadratic Gorenstein algebras with many surprising properties
aa r X i v : . [ m a t h . A C ] J un QUADRATIC GORENSTEIN ALGEBRAS WITH MANYSURPRISING PROPERTIES
JASON MCCULLOUGH AND ALEXANDRA SECELEANU
Abstract.
Let k be a field of characteristic 0. Using the method ofidealization, we show that there is a non-Koszul, quadratic, Artinian,Gorenstein, standard graded k -algebra of regularity 3 and codimension8, answering a question of Mastroeni, Schenck, and Stillman. We alsoshow that this example is minimal in the sense that no other idealiza-tion that is non-Koszul, quadratic, Artinian, Gorenstein algebra, withregularity 3 has smaller codimension.We also construct an infinite family of graded, quadratic, Artinian,Gorenstein algebras A m , indexed by an integer m ≥
2, with the followingproperties: (1) there are minimal first syzygies of the defining ideal indegree m + 2, (2) for m ≥ A m is not Koszul, (3) for m ≥
7, theHilbert function of A m is not unimodal, and thus (4) for m ≥ A m does not satisfy the weak or strong Lefschetz properties. In particular,the subadditivity property fails for quadratic Gorenstein ideals.Finally, we show that the idealization of a construction of Roos yieldsnon-Koszul quadratic Gorenstein algebras such that the residue field k has a linear resolution for precisely α steps for any integer α ≥ Introduction
Let k be a field and let S = k [ x , . . . , x c ] be a standard graded polynomialring over k . Consider R = S/I an Artinian standard graded quotient of S .A recent problem that has attracted some attention is to identify what con-ditions on a quadratic Gorenstein algebra R force R to be Koszul. Whileevery such R with reg( R ) ≤ r = reg( R ) = 3 andcodim( R ) = c for all c ≥
9; this negatively answered a question of Conca,Rossi, and Valla [9]. Mastroeni et. al. pose the following question: For whichpositive integers ( r, c ) does there exist a non-Koszul, quadratic, Gorensteinalgebra with regularity r and codimension c ? In a second paper [26], theysettle the question in all cases except three, namely ( r, c ) = (3 , , (3 , , and(3 , r, c ) = (3 ,
8) by finding a non-Koszul, quadratic, Gorenstein algebra withthese parameters. We do so by applying Nagata’s idealization construc-tion to a non-Koszul Artinian algebra of codimension 4, which comes from
Key words and phrases. free resolution, regularity, Gorenstein, Koszul, idealization.
Roos’ list of quadratic algebras in four variables [32]. Moreover, we showthat the other two cases ( r, c ) = (3 , , (3 ,
7) cannot be similarly settled viaidealizations.Our second construction addresses the subadditivity property of Goren-stein ideals. Set t i ( R ) = sup { j | Tor Si ( R, k ) j = 0 } . The numbers t i ( R )measure the maximal degree of a minimal generator of the i -th syzygy mod-ule of R as an S -module. They are primarily of interest because of theirrelation to regularity as reg( S/I ) = max i ≥ { t i ( S/I ) − i } . The ring R issaid to satisfy the subadditivity property if t a ( R ) + t b ( R ) ≥ t a + b ( R ) for all a, b ≥
1. It is easy to see that complete intersections satisfy subadditivity(Proposition 4.1) while general Cohen-Macaulay ideals do not (cf. [12, Ex-ample 4.4]). Several recent papers have studied the subadditivity propertyfor various classes of ideals [1, 14, 22]. It is conjectured that monomial idealsand Koszul ideals satisfy subadditivity [2, Conjecture 6.4]. To our knowl-edge, there were no known counterexamples to subadditivity for Gorensteinideals; some positive results for Gorenstein ideals are proved in [13]. InSection 4, we show that subadditivity fails in a strong way for quadraticGorenstein ideals. As a consequence of our methods, we obtain an infinitefamily of quadratic Gorenstein ideals that are non-Koszul, have arbitrarilyhigh degree first syzygies, have non-unimodal Hilbert function, and do notsatisfy the strong or weak Lefschetz properties. This provides a counterex-ample to a conjecture of Migliore and Nagel [29, Conjecture 4.5]; an earliercounterexample was given by Gondim and Zappala [16].The third construction modifies a separate example of Roos [31] to showthat there is no finite test of the Koszul property even for quadratic Goren-stein algebras. In Section 5, we show that for any integer α ≥
2, thereis a quadratic, Artinian, Gorenstein k -algebra B α with codim( B α ) = 14and reg( B α ) = 3 such that the resolution of k as a B -module is linear forprecisely α many steps. 2. Background
Here we collect notation and results needed in the rest of the paper.Let k be a field, S = k [ x , . . . , x n ] a standard graded polynomial ring overk, and R = S/I , where I is a homogeneous ideal of S . Then R inher-its a decomposition R = ⊕ i ≥ R i as K -vector spaces with the propertythat R i · R j ⊆ R i + j . The Hilbert function of R is HF R ( i ) = dim k ( R i ).If HF R ( i ) = 0 for i ≫ R is Artinian; this is equivalent to requiringdim k ( R ) < ∞ or that R satisfies the descending chain condition on ideals.The generating function for the Hilbert function is the Hilbert series of R defined as HS R ( t ) = P i dim k ( R i ) t i and similarly for a graded R -module.For a graded Artinian ring R , the h -vector records the nonzero values of theHilbert function of R . The syzygy modules of R are denoted Syz Si ( R ). The regularity of R is reg( R ) = max { j | β Sij ( R ) = 0 } , where β Sij ( R ) = Tor Si ( R, k ) j are the graded Betti numbers of R over S . Regularity is one of the most UADRATIC GORENSTEIN ALGEBRAS 3 well-studied invariants of graded k -algebras and has connections to sheafcohomology and computational complexity [4]. In particular, if R = S/I asabove, reg S ( S/I ) + 1 is an upper bound on the degrees of a minimal gener-ating set of I ; however, there are examples showing that reg S ( S/I ) can bedoubly exponential in the degrees of the generators of I [24].The ring R is called Koszul if k has a linear free resolution over R ; that is, β Rij ( k ) = 0 for all j > i . It is well-known that Koszul algebras are defined byquadratic ideals and that ideals having a Gr¨obner basis of quadrics defineKoszul algebras, but both of these implications are irreversible [8, Remark1.10 and Example 1.20]. Every quadratic complete intersection (that is, ringsof the form S/ ( f , . . . , f m ), where f , . . . , f m is a graded regular sequenceon S ) is Koszul by a result of Tate [34]. There are many examples ofKoszul algebras in algebraic geometry and these algebras enjoy a rich dualitytheory. The article [7] contains a modern introduction to the theory of(commutative) Koszul algebras.A graded Artinian k -algebra R is said to satisfy the weak Lefschetz prop-erty if there is a linear form ℓ ∈ R such that for each non negative integer i the k -linear map R i → R i +1 , r ℓr is either injective or surjective. Sim-ilarly, R is said to satisfy the strong Lefschetz property if there is a linearform ℓ ∈ R such that for each pair of non negative integers i, j the k -linearmap R i → R i + j , r ℓ j r is either injective or surjective. Lefshetz propertiesof Artinian k -algebras have been well-studied and we refer the reader to [21]or [30] for an overview of the area.If R is graded Artinian, then the canonical module of R is given by ω R =Ext nS ( R, S )( − n ) and the canonical module of an R -module M is given by ω M = Ext nS ( M, S )( − n ). In this case R is called level if ω R is generated ina single degree. The minimal number of generators of ω R is called the type of R and denoted throughout this paper by type( R ). If type( R ) = 1, i.e. if ω R is isomorphic to R , up to a shift in the grading, then R is Gorenstein.Equivalently, R is Artinian and Gorenstein if it is injective as an R -module.Gorenstein ideals have symmetric Betti tables and thus Gorenstein ringshave palindromic h -vectors. There are many examples of Gorenstein ringsof interest in algebraic geometry, such as coordinate rings of many canonicalcurves, rings of invariants, and monomial curves. We refer the reader to [23]for a history of Gorenstein rings.Following [25], we say that R is superlevel if R is level and ω R is linearlypresented over R . Note that for R to be superlevel, it is sufficient for R tobe level and ω R be linearly presented over S . The idealization (sometimescalled the Nagata idealization or trivial extension) of R with respect to itscanonical module is the ring e R := R ⋉ ω R ( − reg( R ) − , with multiplication given by ( r , z ) · ( r , z ) = ( r r , r z + r z ). When R islevel, e R is a standard graded ring. It is well-known that when R is Artinian, J. MCCULLOUGH AND A. SECELEANU e R is Artinian and Gorenstein; see [5, proof of Theorem 3.3.6] or [18, Theorem2.76]. Mastroeni, Schenck, and Stillman observed the following: Theorem 2.1 ([25, Proposition 2.2, Lemma 2.3, Theorem 2.5]) . Let R = S/I be a standard graded, Artinian k -algebra. (1) If R is level, then e R is a standard graded, Artinian, Gorenstein k -algebra. In this case codim( e R ) = codim( R ) + type( R ) and reg( e R ) = reg( R ) + 1 . (2) If R is quadratic and superlevel, then e R is quadratic. (3) If R is not Koszul, then e R is not Koszul. (4) e R ∼ = S [ y , . . . , y t ] / (( I ) + L + ( y , . . . , y t ) ) , where t = type( R ) and L = t X i =1 f i y i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( f , . . . , f t ) ∈ Syz S ( ω R ) ! . Thus idealizations of superlevel, Artinian, quadratic algebras are a conve-nient way of constructing quadratic Gorenstein algebras. All three of theconstructions in this paper use this idea.3.
A non-Koszul, quadratic, Gorenstein ring with codimension8 and regularity 3
A construction of Matsuda [27] shows that not every quadratic, Goren-stein ideal is Koszul. Matsuda’s example had regularity 4. Conca, Rossi,and Valla showed that every quadratic, Gorenstein algebra with regularity 2was Koszul [9, Proposition 2.12] and asked asked whether every such algebrawith regularity 3 was Koszul [9, Question 6.10]. It is known that all qua-dratic, Gorenstein algebras of regularity 3 and codimension at most 5 areKoszul [6, 9]. Mastroeni, Schenck, and Stillman [25] constructed counterex-amples in all codimensions c ≥
9. They then posed the following question:
Question 3.1 ([25, Question 1.3]) . For which positive integers c and r is every quadratic Gorenstein ring R with codim( R ) = c and reg( R ) = r Koszul?
In a second paper [26] Mastroeni, Schenck and Stillman settle this questionfor all ordered pairs ( r, c ) except for (3 , , , r, c ) = (3 , k -algebra of regularity 2. ApplyingTheorem 2.1 to it, we obtain the following result. Theorem 3.2.
Let k be a field of characteristic and let S = k [ u, x, y, z ] .Let I = ( x + yz + u , xu, x + xy, xz + yu, zu + u , y + z ) . Then R = S/I is non-Koszul, Artinian, superlevel, with reg( R ) = 2 and type( R ) = 4 . UADRATIC GORENSTEIN ALGEBRAS 5
Consequently, its idealization e R = R ⋉ ω R ( − is a non-Koszul, quadratic,Gorenstein, Artinian, graded k -algebra with reg( e R ) = 3 and codim( e R ) = 8 .Proof. That
S/I is not Koszul follows from computations done by Roos [32].A Macaulay2 [20] calculation shows that
S/I has graded Betti table0 1 2 3 40: 1 - - - -1: - 6 4 - -2: - - 9 12 4.In particular,
S/I is superlevel and type( R ) = 4. Therefore by Theorem 2.1, e R = R ⋉ ω R (1) is a non-Koszul, quadratic, Gorenstein, standard graded k -algebra with reg( e R ) = 3 and codim( e R ) = 8. (cid:3) The h -vector of e R is (1 , , , R to settle the two remaining cases ( r, c ) = (3 , , e R is codim( R ) + type( R ), we would need to find aquadratic, superlevel, Artinian algebra with codim( R ) + type( R ) ≤
7. Thisis impossible in view of the following result.
Proposition 3.3. If R is a level, Artinian, quadratic non-Koszul algebrawith reg( R ) = 2 , then codim( R ) + type( R ) ≥ . Hence e R = R ⋉ ω R hascodimension at least .Proof. Assume towards a contradiction that codim( R ) + type( R ) ≤
7. Since R is level with reg( R ) = 2, type( R ) = Hilb R (2). If Hilb R (2) ≤
2, then R is Koszul by [3, 4.8]. If Hilb R (2) = 3, we may appeal to [7, Theorem 1.1]to conclude that R is Koszul. If Hilb R (2) ≥
4, then c = codim( R ) ≤ R (2) ≤ (cid:0) c +12 (cid:1) − ≤ R is Artinian andquadratic. (cid:3) Subadditivity fails for Gorenstein ideals
To place our second result in context, we begin by showing that homoge-neous complete intersection rings R enjoy the subadditivity property, thatis t a ( R ) + t b ( R ) ≥ t a + b ( R ) for all a, b ≥ Proposition 4.1.
Subadditivity holds for homogeneous complete intersec-tions.Proof.
Let I = ( f , . . . , f c ) be a homogeneous complete intersection idealwith deg( f i ) = d i . Then S/I is resolved by a Koszul complex. We order thegenerators so that d ≥ d ≥ · · · ≥ d c . By the construction of the Koszul J. MCCULLOUGH AND A. SECELEANU complex, it follows that t i ( S/I ) = P ij =1 d j . Hence for any positive integers a, b with a + b ≤ c we have t a + b ( S/I ) = a + b X j =1 d j = a X j =1 d j + a + b X j = a +1 d j ≤ a X j =1 d j + b X j =1 d j = t a ( S/I ) + t b ( S/I ) . (cid:3) On the other hand, subadditivity fails in general, even for Cohen-Macaulayideals cf. [12, Example 4.4]. Note that to study subadditivity for Cohen-Macaulay, and in particular, for Gorenstein rings, it suffices to considerArtinian rings R = S/I , since the graded Betti numbers of R over S are thesame as those of R/ ( ℓ ) over S/ ( ℓ ) for any linear form ℓ ∈ S regular on R .The following Lemma is similar to an example due to Caviglia [12, Ex-ample 4.4]. We use the notation [ − ] : Z → N , [ x ] = max { x, } . Lemma 4.2.
Fix a natural number m and a field k with char( k ) = 0 or char( k ) > m + 1 . Let S = k [ x , . . . , x m ] and consider the ideals C =( x , . . . , x m ) and I = C + (cid:0) ( x + · · · + x m ) (cid:1) . Then for m ≥ , R := S/I is an Artinian, quadratic algebra that has the following properties. (1)
The Hilbert function of R is Hilb R ( i ) = h(cid:0) mi (cid:1) − (cid:0) mi − (cid:1)i . (2) reg( R ) = m . (3) β S ,m +2 ( R ) = 0 , and moreover, t ( R ) = m + 2 . (4) R is superlevel. (5) R is not Koszul.Proof. Tensoring with a field extension of k , if necessary, one may assumethat k is infinite. Set L = C : I and note that L is directly linked to I . Let ℓ = x + · · · + x m and consider the homomorphism µ : ( S/C ) → ( S/C ),where µ ( x ) = ℓ x for which Ker( µ ) = L/C and Coker( µ ) = S/I . Since ℓ is astrong Lefschetz element for S/C (see [21, Theorem 3.35] and the referencestherein for the characteristic zero case and [10, Theorem 7.2] for positivecharacteristics), the k -linear functions µ i : ( S/C ) i ℓ → ( S/C ) i +2 , obtainedby restricting µ to each of the graded components of S/C , are injective for i ≤ m − i ≥ m −
1. It follows that the Hilbert functionof R = S/I isHilb R ( i ) = [dim k (( S/C ) i ) − dim k (( S/C ) i − )] = (cid:20)(cid:18) mi (cid:19) − (cid:18) mi − (cid:19)(cid:21) . In particular, R is Artinian with Hilb R ( m ) = 0, while Hilb R ( m + 1) = 0 andso reg( R ) = m .Moreover, the injectivity of the maps µ i above shows dim k ( L i /C i ) = 0for i ≤ m −
1, so L has no minimal generators below degree m besidesthe quadratic generators of C . The non-injectivity of µ m shows that L hasminimal generators in degree m . UADRATIC GORENSTEIN ALGEBRAS 7
Consider the graded short exact sequence0 → S/L ( − ℓ −→ S/C −→ S/I → . Since β ,m +2 ( S/C ) = 0 and β ,m +2 ( S/L ( − β ,m ( S/L ) = 0, it followsfrom the long exact sequence of Tor that β ,m +2 ( S/I ) = 0. Since the onlyminimal generators for both C and L below degree m are those in thecomplete intersection C , it follows that β i,j ( S/C ) = 0 for j < i and β i,j ( S/L ( − β i,j +2 ( S/L ) = 0 for j < min { m − i, i − } . Again by the long exact sequence of Tor we obtain that β i,j ( S/I ) = 0 for j < min { i, m + i } .On the other hand, since R = S/I is Artinian and its Hilbert functionsatisfies Hilb R ( i ) = (h(cid:0) mi (cid:1) − (cid:0) mi − (cid:1)i = 0 for i > m and (cid:0) mm (cid:1) − (cid:0) mm − (cid:1) = 0 , for i = m. it follows that reg( R ) = m ; so the vanishing of Betti numbers β i,j ( R ) = 0for j < min { i, m + i } forces β i,m + i ( R ) = 0 for m < i ≤ m . In particular, R is superlevel.Finally to see that R = S/I is not Koszul, note that if m ≥
3, then t ( R ) = m + 2 > t ( R ) + t ( R ). If R were Koszul, then this wouldcontradict [2, Theorem 6.2]. For m = 2 the non Koszul property can bechecked by direct computation in Macaulay2 [20]. (cid:3) We now construct a family of quadratic, Artinian, Gorenstein graded ringsthat have several bad properties.
Theorem 4.3.
Fix an integer m ≥ and let k be a field with char( k ) = 0 or char( k ) > m + 1 . There exists a quadratic, Artinian, Gorenstein, graded k -algebra A with the following properties (1) codim( A ) = 2 m + h(cid:0) mm (cid:1) − (cid:0) mm − (cid:1)i . (2) A is not Koszul. (3) t ( A ) = m + 2 . (4) reg( A ) = m + 1 . (5) The Hilbert function of A is HF A ( i ) = (cid:20)(cid:18) mi (cid:19) − (cid:18) mi − (cid:19)(cid:21) + (cid:20)(cid:18) mm − i + 1 (cid:19) − (cid:18) mm − i − (cid:19)(cid:21) . In particular, • A does not satisfy the subadditivity property if m ≥ , and • A has a non-unimodal Hilbert function if m ≥ and thus does notsatisfy the weak or strong Lefschetz properties.Proof. Set A = e R = R ⋉ ω R ( − m − R = S/I is the Artinianalgebra introduced in the statement of Lemma 4.2 for S = k [ x , . . . , x m ].That A is quadratic, Artinian, non-Koszul, and Gorenstein with the claimed J. MCCULLOUGH AND A. SECELEANU regularity, codimension and Hilbert function follows from Theorem 2.1 andLemma 4.2.Also by Theorem 2.1 we can write a presentation for A as A = T /M with M = (( I ) + L + ( y , . . . , y t ) ) , where t = type( R ) = h(cid:0) mm (cid:1) − (cid:0) mm − (cid:1)i and T = S [ y , . . . , y t ]. It will beuseful at this time to view A as a bigraded ring with respect to the gradingobtained by assigning degree (1 ,
0) to the variables of S and degree (0 ,
1) tothe variables y i . The short exact sequence of bigraded T-modules0 → L + ( y , . . . , y t ) → T /IT → A → · · · → Tor T ( L + ( y , . . . , y t ) , k ) → Tor T ( T /IT, k ) → Tor T ( A, k ) → · · · . Since ( L + ( y , . . . , y t ) ) ( ∗ , = 0, also (Tor T ( L + ( y , . . . , y t ) , k )) ( ∗ , = 0 forall ∗ ∈ N . By contrast, since y , . . . , y t is a regular sequence on T /IT , itfollows that Tor T ( T /IT, k ) = Tor S ( S/I, k ) ⊗ S T is concentrated in degrees( ∗ , T ( T /IT, k ) ֒ → Tor T ( A, k ). Hence t ( A ) ≥ t ( S/I ) = m .On the other hand, since reg( S/I ) = m , we get from Theorem 2.1 thatreg( A ) = m + 1; i.e. t i ( A ) ≤ m + 1 + i for all i ≥
0. Moreover, since A is Gorenstein and quadratic, the symmetry of the Betti table of A over T forces t i ( A ) ≤ m + i for all i < m + t . In particular, this implies that t ( A ) = m + 2 > t ( A ) + t ( A ) and thus A fails the subadditivityproperty when m ≥ m ≥
7. If m = 7, then HF A (3) = 1988 < A (2). If m = 8, then HF A (3) = 6732 < A (2). If m = 9 thenHF A (3) = 24054 < A (2). We now show HF A (1) > HF A ( (cid:4) m (cid:5) )for m ≥
10. By (5) we haveHF A (1) = 2 m + (cid:18) mm (cid:19) − (cid:18) mm − (cid:19) = 2 m + 2 · (2 m + 1)! m !( m + 2)!and thus it suffices to show that · (2 m +1)! m !( m +2)! / HF A (cid:0)(cid:4) m (cid:5)(cid:1) >
1. Consider firstthe case m = 2 n , whence by use of Pascal’s formula one computesHF A ( n ) = (cid:18) nn (cid:19) − (cid:18) nn − (cid:19) + (cid:18) nn + 1 (cid:19) − (cid:18) nn − (cid:19) = (cid:18) n + 1 n + 1 (cid:19) − (cid:18) n + 1 n − (cid:19) = (4 n + 1)! · n + 1) (3 n + 2)!( n + 1)! . UADRATIC GORENSTEIN ALGEBRAS 9
We deduce the desired inequality by considering the function · (2 m +1)! m !( m +2)! HF A ( n ) = 2 · (4 n + 1)!(2 n )!(2 n + 2)! · (3 n + 2)!( n + 1)!(4 n + 1)! · n + 1) = (2 n + 3)(2 n + 4) · · · (3 n + 1)(3 n + 2)( n + 2)( n + 3) · · · (2 n )(2 n + 1) = n +1 Y i = n +2 (cid:18) n + 1 i (cid:19) · n + 1 > (cid:18) n + 12 n + 1 (cid:19) n · n + 1 ≥ (cid:18) (cid:19) n · n + 1 . Clearly the last function above attains arbitrarily large values asymptoti-cally and one can check that its values surpass 1 for n ≥
6. In the remainingcase, n = 5, the claim 58806 = HF A (1) > HF A ( n ) = 48279 can be checkedby direct computation. The case when m is odd is similar and we omit thedetails. (cid:3) That the family of Artinian algebras in Theorem 4.3 fails to satisfy thestrong Lefschetz property can be deduced from [15, Proposition 2.1], howeverthe stronger statement regarding the failure of the weak Lefschetz propertyis to our knowledge new.
Example 4.4.
Take m = 3 and consider S = k [ x , . . . , x ]. ApplyingLemma 4.2, we set C = ( x , . . . , x ) and I = C + ( x + · · · + x ) . Then R = S/I has the following Betti table over S :0 1 2 3 4 5 60: 1 - - - - - -1: - 7 - - - - -2: - - 21 - - - -3: - - 14 105 132 70 14In particular, t ( S/I ) = 2 and t ( S/I ) = 5, so R is an Artinian algebra thatfails subadditivity; i.e. t ( R ) > t ( R ) + t ( R ). Moreover, R is superlevelwith reg( R ) = 3 and h -vector (1 , , , e R = R ⋉ ω R ( − e R is Artinian,Gorenstein with h -vector (1 , , , ,
1) and reg( e R ) = 4. As e R is qua-dratic, t ( e R ) = 2 while t ( e R ) = 5 by Theorem 4.3. Thus e R is an Artinian,Gorenstein algebra for which subadditivity fails.It is worth noting that while we know the Hilbert function of e R fromTheorem 4.3, the full Betti table of e R , as a quotient of a polynomial ringin 20 variables, is not so clear. It would be very interesting to have a fulldescription of the resolution of the idealization e R in terms that of R . Example 4.5.
When m = 7, the Gorenstein k -algebra e R from Theorem 4.3has non-unimodal h -vector (1 , , , , , , , , . Thus e R is a quadratic Gorenstein algebra with codim( e R ) = 1444 and reg( e R ) = 8and for which both the weak and strong Lefshetz properties fail. The first ex-ample of a (non-quadratic) Gorenstein algebra with non-unimodal h -vectorwas famously constructed by Stanley [33]. Quadratic non-Koszul Gorenstein algebras with linearresolutions of arbitrarily high order
Let k be a field of characteristic 0. In [31, Thereom 1’] Roos gave ex-amples of graded Artinian non-Koszul quadratic k -algebras A α for integers α ≥ k has a linear resolution for precisely the first α steps beforea minimal non-linear syzygy. Here we note that these algebras are super-level and that their Gorenstein idealizations have the same property. Thisremoves any hope of a ‘finite test’ for the Koszul property in the context ofquadratic Gorenstein algebras.To state the result, we first recall the definition of the graded Poincareseries of a module. Fix a graded module M over a graded ring R . The gen-erating function for the bigraded k -vector space Tor R ∗ ( M, k ) ∗ is P MR ( x, y ) := P i,j β Rx,y ( M ) x i y j . Thus the graded Poincare series of R is P kR ( x, y ), whichencodes the resolution of k over R . Theorem 5.1.
Let k be a field of characteristic and fix a positive integer α ≥ . Let S = k [ u, v, w, x, y, z ] and I = ( x , xy, y , yz, z , zu, u , uv, v , vw, w , xz + αzw − uw, zw + xu +( α − uw ) . Then the idealization e R of R = S/I is quadratic, Gorenstein, non-Koszul,and k has linear resolution for precisely α steps in the resolution over e R .Proof. By [31, Theorem 1’], R has Hilbert series HS R ( t ) = 1 + 6 t + 8 t ,whence reg( R ) = 2, and type( R ) = 8, and it is easy to check with Macaulay2[20] that R is superlevel. Thus e R = R ⋉ ω R ( −
3) is Artinian, quadratic, andGorenstein with reg( e R ) = 3 and codim( e R ) = 6 + 8 = 14.We need only argue that the resolution of k over e R is linear for exactly α steps. To achieve this, we recall the details of the construction of R from[31]. It is shown therein that R itself is an idealization R = A ⋉ M ( − A = k [ x, y, u, v ] / ( x , xy, y , v , vw, w ) = B ⊗ k C , B = k [ x, y ] / ( x, y ) ,and C = k [ v, w ] / ( v, w ) and M is an A -module with Hilbert series HS M ( t ) =2 + 4 t . By a result of Gulliksen [17, Theorem 2] combined with the fact that M and ω R are linearly presented, the relevant graded Poincar´e series arerelated by P k e R ( x, y ) = P kR ( x, y )(1 − xyP ω R ( − R ( x, y )) − (1) P ω R R ( x, y ) = P ω R A ( x, y )(1 − xyP MA ( x, y )) − . (2) Stanley’s construction was also given via idealization for the ring A = k [ x, y, z ] / ( x, y, z ) . Since A has h -vector (1 , , , e A = A ⋉ ω A has h -vector (1 , , , , e A is clearly not quadratic. UADRATIC GORENSTEIN ALGEBRAS 11
Since [31] shows that both the resolution of k over R and the resolution of M over A are linear for exactly α steps, it suffices to show that the resolutionof ω R over A is linear.By [19, proof of Claim 2] there is an isomorphism of R -modules ω R ( − ∼ = ω M ( − ⋉ ω A ( − R = A ⋉ M ( − a, m ) · ( s, t ) = ( as, at + s ( m )), where we view s ∈ Hom A ( M, ω A ) ∼ = ω M .This induces an isomorphism of A -modules ω R ( − ∼ = ω M ( − ⊕ ω A ( − ω M ( −
1) and ω A ( −
2) are generated in degree 0. Thusit suffices to show that both ω A ( −
2) and ω M ( −
1) have linear resolutionsover A . The former module decomposes as a tensor product ω A ( − ∼ = ω B ( − ⊗ k ω C ( − ω A ( −
2) over A isin turn the tensor product of the resolutions of ω B ( −
1) over B and ω C ( − C . Since the rings B and C contain no elements of degree greater thanone, the differentials in the resolutions of ω B ( −
1) and ω C ( −
1) are linear. As B and C are level of regularity 1, ω B ( −
1) and ω C ( −
1) are each generatedin degree zero, hence their resolutions are linear and so is the resolution of ω A ( −
2) over A .Finally, in order to analyze the resolution of ω M ( −
1) over A one mustdig deeper into the structure of the module M . Since A is Artinian, [11,Proposition 21.1] shows that ω M ( − ∼ = Hom k ( M (+1) , k ) as an A -module.Consequently the Hilbert series of this module is HS ω M ( − ( t ) = 4 + 2 t . Fix k -bases { f ∗ , f ∗ , f ∗ , f ∗ } for ( ω M ( − and { e ∗ , e ∗ } for ( ω M ( − , where e i , f i refer to the dual elements in M and M respectively. The A -modulestructures of M and ω M can be completely described by the following dualtables, the first of which can be deduced from [32, Equation (3)] f f f f e v w x + αw e α − w − x w y f ∗ f ∗ f ∗ f ∗ e ∗ v w x + αw e ∗ α − w − x w y The leftmost table should be interpreted to mean that, for example, ve = f in M , while the rightmost table should be interpreted to mean that, dually, vf ∗ = e ∗ in ω M . (That for instance ye = 0 is thus implicit in the table.)Equipped with this information, we implement a strategy for determiningTor A ( ω M ( − , k ) inspired by Roos’s original approach: we compute thehomology of the tensor product of the bar resolutions for k over B and C further tensored with ω M . Since ω M is concentrated in only two degrees,the only potential nonzero components of Tor Ai ( ω M ( − , k ) are in degrees i and i + 1. Furthermore Tor Ai ( ω M ( − , k ) i +1 is the cokernel of( ω M ( − ⊗ k i M q =0 B ⊗ i − q ⊗ k C ⊗ i → ( ω M ( − ⊗ k i − M q =0 B ⊗ i − q − ⊗ k C ⊗ i s ⊗ b ⊗ · · · ⊗ b i − q ⊗ c ⊗ · · · ⊗ c q sb ⊗ · · · ⊗ b i − q ⊗ c ⊗ · · · ⊗ c q + ( − i − q sc ⊗ b ⊗ · · · ⊗ b i − q ⊗ c ⊗ · · · ⊗ c q . It is easily verified that this map is surjective as for arbitrary µ, λ ∈ k, b j ∈ B , c j ∈ C we have λf ∗ ⊗ y ⊗ b ⊗· · ·⊗ b i − q ⊗ c ⊗· · ·⊗ c q +( − i − q µf ∗ ⊗ b ⊗· · ·⊗ b i − q ⊗ v ⊗ c · · ·⊗ c q ( µe ∗ + λe ∗ ) ⊗ b ⊗ · · · ⊗ b i − q ⊗ c ⊗ · · · ⊗ c q . Thus Tor Ai ( ω M ( − , k ) i +1 = 0 for i >
0, which concludes the proof. (cid:3)
Acknowledgment.
We thank Matt Mastroeni, Hal Schenck, and MikeStillman for useful conversations and bringing Question 3.1 to our atten-tion.
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