DG Structure on Length 3 Trimming Complexes and Applications to Tor Algebras
aa r X i v : . [ m a t h . A C ] N ov DG STRUCTURE ON LENGTH 3 TRIMMING COMPLEXESAND APPLICATIONS TO TOR ALGEBRAS
KELLER VANDEBOGERT
Abstract.
In this paper, we consider the iterated trimming complex associ-ated to data yielding a complex of length 3. We compute an explicit algebrastructure in this complex in terms of the algebra structures of the associatedinput data. Moreover, it is shown that many of these products become trivialafter descending to homology. We apply these results to the problem of real-izability for Tor-algebras of grade 3 perfect ideals, and show that under mildhypotheses, the process of “trimming” an ideal preserves Tor-algebra class. Inparticular, we construct new classes of ideals in arbitrary regular local ringsdefining rings realizing Tor-algebra classes G ( r ) and H ( p, q ) for a prescribedset of homological data. Introduction
Let ( R, m , k ) be a regular local ring with maximal ideal m and residue field k . Aresult of Buchsbaum and Eisenbud (see [3]) established that any quotient R/I of R with projective dimension 3 admits the structure of an associative commutativedifferental graded (DG) algebra. Later, a complete classification of the multiplica-tive structure of the Tor algebra Tor R • ( R/I, k ) for such quotients was establishedby Weyman in [11] and Avramov, Kustin, and Miller in [2].Absent from this classification was a complete description of which Tor-algebrastructures actually arise as the Tor-algebra of some quotient
R/I with some pre-scribed homological data. More precisely, let
R/I have a length 3 DG-algebraminimal free resolution: F • : 0 → F → F → F → R, with m = rank( F ), n = rank( F ). Let · := · ⊗ k , and define p = rank( F ) , q = rank( F · F ) ,r = rank (cid:16) F → Hom k ( F , F ) (cid:17) . Then, Avramov posed the question (see [1, Question 3.8]):Which tuples ( m, n, p, q, r ) are realized by some quotient ring
R/I ?This is often referred to as the realizability question , and Avramov gives bounds onthe possible tuples that can occur in this same paper along with some conjectureson the tuples associated to certain Tor-algebra classes. One such conjecture wasrelated to the Tor-algebra class G , where Avramov had posed: Question . If R/I is class G ( r ) for some r >
2, then is
R/I
Gorenstein?
Date : November 26, 2020.
The counterexamples to Question 1.1 were constructed by Christensen, Veliche,and Weyman using a remarkably simple construction. Given an m -primary ideal I = ( φ , . . . , φ n ) ⊆ R , one can “trim” the ideal I by, for instance, forming theideal ( φ , . . . , φ n − ) + m φ n . It turns out that this will yield an ideal that defines a non-Gorenstein quotient ring which is also of class G ( r ) (see [5]).More generally, computational evidence suggests that the process of trimming anideal tends to preserve Tor-algebra class. In this paper, we set out to answer whythis is true. In practice, there are two ways of computing multiplication in the Tor-algebra Tor • ( R/I, k ) for a given ideal I . First, let K • denote the Koszul complexresolving R/ m . Then, one can descend to the homology algebra H • ( K • ⊗ R/I ), withmultiplication induced by the exterior algebra K • . Alternatively, one can producean explicit DG-algebra free resolution F • of R/I , tensor with R/ m , and descend tohomology with product induced by the algebra structure on F • .We take the latter approach in this paper. Luckily, an explicit free resolutionof trimmed ideals is constructed in [10]. More generally, we construct an explicitalgebra structure on arbitrary iterated trimming complexes of length 3 (see Theorem3.3). In the case that the ambient ring is local, we are then able to show thatthe possible nontrivial multiplications in the Tor-algebra are rather restricted (seeCorollary 3.7).This algebra structure is then applied to the previously mentioned case wherethe ideal I is obtained by trimming. We focus on ideals defining rings of Tor-algebra G and H , and show that under very mild assumptions, trimming an idealpreserves these Tor-algebra classes. This allows us to construct novel examples ofrings of class G ( r ) and H ( p, q ) obtained as quotients of arbitrary regular local rings( R, m , k ) of dimension 3, and we further add to the realizability question posed byAvramov (see Corollary 5.8 and 5.14).The paper is organized as follows. In Section 2, we set the stage with conven-tions and notation to be used throughout the rest of the paper along with somebackground. In Section 3, an explicit product on the length 3 iterated trimmingcomplex is constructed. In the case that the complexes involved further admit thestructure of DG-modules over each other, then this product may be made evenmore explicit (see Proposition 3.4). As corollaries, we find that only a subset ofthe products on the iterated trimming complex are nontrivial after descending tohomology.In Section 4, we focus on the case of trimming an ideal (in the sense of Chris-tensen, Veliche, and Weyman [5]). Assuming that certain products on the minimalfree resolution of an ideal sit in sufficiently high powers of the maximal ideal, weshow that trimming an ideal will either preserve the Tor-algebra class or yield aGolod ring (see Lemma 4.10 and 4.13). In the case that the minimal presenting ma-trix for these quotient rings has entries in m , the restrictions become even tighterand we can say precisely which Tor-algebra class these new ideals will occupy (seeCorollary 4.11 and 4.14).Finally, in Section 5, we begin to construct explicit quotient rings realizing tu-ples of the form ( m, n, p, q, r ). In particular, we construct an infinite class of newexamples of class G ( r ), and can say in general that there are rings of arbitrarilylarge type with Tor-algebra class G ( r ), for any r > k [ x, y, z ], see [8]). Likewise, we construct aninfinite class of rings of Tor-algebra class H ( p, q ) that are not hyperplane sections, G STRUCTURE ON LENGTH 3 TRIMMING COMPLEXES 3 which, combined with the process of linkage, can be used to conclusively show theexistence of rings realizing many of the tuples falling within the bounds imposedby Christensen, Veliche, and Weyman in [6].2.
Background, Notation, and Conventions
In this section, we first introduce some of the notation and conventions that willbe in play throughout the paper. We will introduce iterated trimming complexes(see Definition 2.7), the algebra structure on which is the main subject of Section 3.We also discuss the realizability question posed originally by Avramov, and discussthe progress on this question due to Christensen, Veliche, and Weyman in [6].Throughout the paper, all complexes will be assumed to have nontrivial termsappearing only in nonnegative homological degrees.
Notation 2.1.
The notation ( F • , d • ) will denote a complex F • with differentials d • . When no confusion may occur, F or F • may be written instead.Given a complex F • as above, elements of F n will often be denoted f n , withoutspecifying that f n ∈ F n . Definition 2.2. A differential graded algebra ( F • , d • ) (DG-algebra) over a com-mutative Noetherian ring R is a complex of finitely generated free R -modules withdifferential d and with a unitary, associative multiplication F ⊗ R F → F satisfying(a) F i · F j ⊆ F i + j ,(b) d i + j ( x i x j ) = d i ( x i ) x j + ( − i x i d j ( x j ),(c) x i x j = ( − ij x j x i , and(d) x i = 0 if i is odd,where x k ∈ F k .The following setup will be used to introduce iterated trimming complexes . Formore details and proofs of the accompanying statements, see [10]. Setup 2.3.
Let R be a regular local ring or a standard graded polynomial ring overa field k . Let I ⊆ R be a (homogeneous) ideal and ( F • , d • ) denote a (homogeneous)free resolution of R/I .Write F = F ′ ⊕ (cid:16) L ti =1 Re i (cid:17) , where, for each i = 1 , . . . , t , e i generates a freedirect summand of F . Using the isomorphism Hom R ( F , F ) = Hom R ( F , F ′ ) ⊕ (cid:16) t M i =1 Hom R ( F , Re i ) (cid:17) write d = d ′ + d + · · · + d t , where d ′ ∈ Hom R ( F , F ′ ) and d i ∈ Hom R ( F , Re i ) .For each i = 1 , . . . , t , let a i denote any homogeneous ideal with d i ( F ) ⊆ a i e i , and ( G i • , m i • ) be a homogeneous free resolution of R/ a i .Use the notation K ′ := im( d | F ′ : F ′ → R ) , K i := im( d | Re i : Re i → R ) , andlet J := K ′ + a · K + · · · + a t · K t . KELLER VANDEBOGERT
Proposition 2.4.
Adopt notation and hypotheses of Setup 2.3. Then for each i =1 , . . . , t there exist maps q i : F → G i such that the following diagram commutes: F q i ~ ~ ⑥⑥⑥⑥⑥⑥⑥ d i ′ (cid:15) (cid:15) G i m i / / a i , where d i ′ : F → R is the composition F d i / / Re i / / R , and where the second map sends e i . Proposition 2.5.
Adopt notation and hypotheses as in Setup 2.3. Then for each i = 1 , . . . , t there exist maps q ik : F k +1 → G ik for all k > such that the followingdiagram commutes: F k +1 q ik (cid:15) (cid:15) d k +1 / / F kq ik − (cid:15) (cid:15) G ik m ik / / G ik − Theorem 2.6 ([10]) . Adopt notation and hypotheses as in Setup 2.3. Then themapping cone of the morphism of complexes (2.1) · · · d k +1 / / F k q k − ... q tk − (cid:15) (cid:15) d k / / · · · d / / F d ′ / / q ... q t (cid:15) (cid:15) F ′ d (cid:15) (cid:15) · · · L m ik / / L ti =1 G ik − L m ik − / / · · · L m i / / L ti =1 G i − P ti =1 m i ( − ) · d ( e i ) / / R is a free resolution of R/J . Definition 2.7.
The iterated trimming complex associated to the data of Setup2.3 is the complex of Theorem 2.6.The following setup will be used for the rest of this section, and will be used todiscuss results related to Question 2.9.
Setup 2.8.
Let ( R, m , k ) denote a local ring. Let R/I have a length DG-algebraminimal free resolution: F • : 0 → F → F → F → R, with m = rank( F ) , n = rank( F ) . Let · := · ⊗ k , and define p = rank( F ) , q = rank( F · F ) , G STRUCTURE ON LENGTH 3 TRIMMING COMPLEXES 5 r = rank (cid:16) F → Hom k ( F , F ) (cid:17) . Question . Which tuples ( m, n, p, q, r ) are realized by the data of Setup 2.8 forsome quotient ring
R/I ?For the definition of the Tor-algebra class H ( p, q ) appearing in the followingTheorem, see Theorem 4.1. Theorem 2.10 ([6], Theorem 1 . . Adopt notation and hypotheses as in Setup 2.8.Let Q = R/I be a Cohen-Macaulay local ring of codimension and class H ( p, q ) .Then the following inequalities hold: p m − and q n. Moreover, the following are equivalent:(i) p = n + 1 .(ii) q = m − .(iii) p = m − and q = n .If the conditions ( i ) − ( iii ) are not satisfied, then there are inequalities p n − and q m − with p = n − ⇐⇒ q ≡ m − and q = m − ⇐⇒ p ≡ n − . Christensen, Veliche, and Weyman further conjecture that within the boundssupplied by Theorem 2.10, there are ideals defining rings of class H ( p, q ) realizingthe tuples ( m, n, p, q ) (that is, these bounds are sharp; see the conjectures of 7 . J ⊆ R is directly linked to the ideal I if there existsa grade 3 complete intersection a ⊆ I such that J = ( a : I ). Two ideals I and J are linked if there exists a sequence of direct links connecting I and J . In [6], itis carefully studied how the Tor-algebra of an ideal relates to that of any directlylinked ideal. In particular, one has the following Proposition. Proposition 2.11 ([6]) . Adopt notation and hypotheses as in Setup 2.8. Assume I ⊆ R is a grade perfect ideal.(a) If I defines a ring of Tor-algebra class G ( r ) , then it is directly linked to agrade perfect ideal J defining a ring of Tor-algebra class H ( p ′ , q ′ ) realizingthe tuple ( n + 3 , m − , p ′ , q ′ ) , where p ′ > min { r, } .(b) If I defines a ring of Tor-algebra class H ( p, q ) with p > , q > , and m − > p , then it is directly linked to a grade perfect ideal J of class H ( q, p ) realizing the tuple ( n + 3 , m − , q, p ) . (c) If I defines a ring of Tor-algebra class H ( p, q ) with p > , q > , and m − > p > , then it is directly linked to a grade perfect ideal J of class H ( q, p − realizing the tuple ( n + 2 , m − , q, p − . KELLER VANDEBOGERT (d) If I defines a ring of Tor-algebra class H ( p, q ) with p > , q > , and m − > p > , then it is directly linked to a grade perfect ideal J of class H ( q, p − realizing the tuple ( n + 1 , m − , q, p − , where m > or n > . Combining Proposition 2.11 with the explicit examples given in Section 5 willallow us to combine the processes of linkage and trimming to compute explicitexamples realizing the tuples falling within the bounds of Theorem 2.10.3.
Algebra Structure on Length 3 Iterated Trimming Complexes
In this section, we show that if the complexes associated to the input data ofSetup 2.3 are length 3 DG-algebras, then the product on the resulting iteratedtrimming complex of Theorem 2.6 may be computed in terms of the products onthe aforementioned complexes. The proof of this fact is a long and rather tediouscomputation; moreover, in full generality, the products have certain componentsthat are only defined implicitly. In the case that the complexes involved admitadditional module structures over one another, these products may be made moreexplicit (see Proposition 3.4). However, after descending to homology, many ofthese products either vanish completely or become considerably more simple. Thisfact is made explicit in the corollaries at the end of this section, and will be takenadvantage of in Section 4.The following is essentially the proof of Proposition 1 . Proposition 3.1.
Let ( F • , d • ) denote a length resolution of a cyclic module M admitting a product satisfying axioms ( a ) − ( d ) of Definition 2.2. Then, the productis associative.Proof. Since F • has length 3, the only nontrivial triple product can occur between3 elements e , e , and e of homological degree 1. One computes: d (cid:0) ( e · e ) · e − e · ( e · e ) (cid:1) = d ( e · e ) · e + d ( e ) e · e − d ( e ) e · e + e · d ( e · e )= d ( e ) e · e − d ( e ) e · e + d ( e ) e · e − d ( e ) e · e + d ( e ) e · e − d ( e ) e · e = 0 . Since d is injective, the result follows. (cid:3) Notation 3.2.
Given a DG-algebra F • , the notation · F denotes the product on F • . Given two free modules F and G , elements of the direct sum F ⊕ G will bedenoted f + g ∈ F ⊕ G . Theorem 3.3.
Adopt notation and hypotheses as in Setup 2.3, and assume thatthe complexes F • and G i • ( i t ) are length DG-algebras. Then the length
G STRUCTURE ON LENGTH 3 TRIMMING COMPLEXES 7 F ′ ⊗ F ′ → F ⊕ (cid:16) t M i =1 G i (cid:17) f · T f ′ := f · F f ′ + t X i =1 g i , where m i ( g i ) = q i ( f · F f ′ ) ,F ′ ⊗ G i → F ⊕ (cid:16) t M j =1 G j (cid:17) f · T g i := m i ( g i ) e i · F f + t X j =1 g j , where m i ( g i ) = d ( f ) g i + m i ( g i ) q i ( e i · F f ) ,m j ( g j ) = m i ( g i ) q j ( e i · F f ) for j = iG i ⊗ G i → F ⊕ (cid:16) t M j =1 G j (cid:17) g i · T g ′ i := − g i · G i g ′ i d ( e i ) ,G i ⊗ G j → F ⊕ (cid:16) t M k =1 G k (cid:17) i = jg i · T g j := m i ( g i ) m j ( g j ) e i · F e j + t X k =1 g k , where m i ( g i ) = m i ( g i ) m j ( g j ) q i ( e i · F e j ) + m j ( g j ) d ( e j ) g i ,m j ( g j ) = m i ( g i ) m j ( g j ) q j ( e i · F e j ) − m i ( g i ) d ( e i ) g j ,m k ( g k ) = m i ( g i ) m j ( g j ) q k ( e i · F e j ) for k = i, j,F ′ ⊗ F → F ⊕ (cid:16) t M i =1 G i (cid:17) f · T f := f · F f + t X i =1 g i , for some g i ∈ G i ,F ′ ⊗ G i → F ⊕ (cid:16) t M j =1 G j (cid:17) f · T g i := t X j =1 g j , for some g j ∈ G j G i ⊗ G i → F ⊕ (cid:16) t M j =1 G j (cid:17) g i · T g i = − g i · G i g i d ( e i ) , KELLER VANDEBOGERT G i ⊗ G j → F ⊕ (cid:16) t M k =1 G k (cid:17) g i · T g j := t X k =1 g k , i = j, for some g k ∈ G k ,G i ⊗ F → F ⊕ (cid:16) t M j =1 G j (cid:17) g i · T f := − m i ( g i ) e i · F f + t X j =1 g j , for some g j ∈ G j . Proof.
Observe that, by Proposition 3.1, it suffices to show that the contendedproducts satisfy axiom ( b ) of Definition 2.2. The proof thus becomes a straightfor-ward verification of this identity, and will be split accordingly into all of the cases.For convenience, let π i : F → R denote the composition F −−−−−−→ Re i → R, where the second map sends e i
1. Observe that m i ◦ q i = π i ◦ d =: d ′ i . Likewise,let p : F → F ′ denote the natural projection. Observe that d ′ = p ◦ d . Case 1: F ′ ⊗ F ′ → F ⊕ (cid:16) L ti =1 G i (cid:17) . We first need to verify the existence ofeach g i ; by exactness of each G i • , it suffices to show that q i ( f · F f ′ ) is a cycle. Onecomputes: m i ◦ q i ( f · F f ′ ) = π i (cid:16) d ( f · F f ′ ) (cid:17) = π i (cid:16) d ( f ) f ′ − d ( f ′ ) f (cid:17) = 0 , since π i ( F ′ ) = 0 for all i. Thus the desired g i exists for all i . It remains to verify the DG axiom: ℓ ( f · T f ′ ) = d ′ ( f · F f ′ ) + t X i =1 (cid:16) − q i ( f · F f ′ ) + m i ( g i ) (cid:17) = d ( f ) f ′ − d ( f ′ ) f = ℓ ( f ) f ′ − ℓ ( f ′ ) f . Case 2: F ′ ⊗ G i → F ⊕ (cid:16) L tj =1 G j (cid:17) . We first verify the existence of the desired g j . One computes: m i (cid:16) d ( f ) g i + m i ( g i ) q i ( e i · F f ) (cid:17) = d ( f ) m i ( g i ) + m i ( g i ) π i ( d ( e i · F f ))= d ( f ) m i ( g i ) − m i ( g i ) d ( f )= 0 ,m j (cid:16) m i ( g i ) q j ( e i · F f ) (cid:17) = m i ( g i ) π j ( d ( e i · F f ))= 0 . G STRUCTURE ON LENGTH 3 TRIMMING COMPLEXES 9
It remains to verify the DG axioms: ℓ ( f · T g i ) = m i ( g i ) d ′ ( e i · F f ) − t X j =1 m i ( g i ) q j ( e i · F f ) + t X j =1 m i ( g i )= m i ( g i ) d ( e ) f + d ( f ) g i = ℓ ( f ) g i − ℓ ( g i ) f . The above uses that d ′ = p ◦ d , implying d ′ ( e i · F f ) = d ( e i ) f . Case 3: G i ⊗ G i → F ⊕ (cid:16) L tj =1 G j (cid:17) . One computes directly: ℓ ( g i · T g ′ i ) = m i ( − g i · G i g ′ i d ( e i ))= − m ( g i ) g ′ i d ( e i ) + m ( g ′ i ) g i d ( e i )= ℓ ( g i ) g ′ i − ℓ ( g ′ i ) g i Case 4: G i ⊗ G j → F ⊕ (cid:16) L tk =1 G k (cid:17) , i = j . We verify the existence of g i ; theproof of the existence of g j is identical. One computes: m i (cid:16) m i ( g i ) m j ( g j ) q i ( e i · F e j ) + m j ( g j ) d ( e j ) g i (cid:17) = m i ( g i ) m j ( g j ) π i ◦ d ( e i · F e j ) + m j ( g j ) d ( e j ) m i ( g i )= − m i ( g i ) m j ( g j ) d ( e j ) + m j ( g j ) d ( e j ) m i ( g i )=0 . It remains to show the DG axiom: ℓ ( g i · T g j ) = m i ( g i ) m j ( g j ) d ′ ( e i · F e j )+ t X k =1 (cid:16) − m i ( g i ) m j ( g j ) q i ( e i · F e j ) + m k ( g k ) (cid:17) = m j ( g j ) d ( e j ) g i − m i ( g i ) d ( e i ) g j = ℓ ( g i ) g j − ℓ ( g j ) g i . In the above, notice that d ′ ( e i · F e j ) = p ( d ( e i ) e j − d ( e j ) e i ) = 0. Case 5: F ′ ⊗ F → F ⊕ (cid:16) L ti =1 G i (cid:17) . Observe that f · T ( d ′ ( f ) − t X i =1 q i ( f )) = f · F d ′ ( f ) + t X i =1 g i − t X i =1 (cid:16) m i ◦ q i ( f ) e · F f + t X j =1 g i,j (cid:17) , where m i (cid:16) g i − t X j =1 g j,i (cid:17) = q i ( f · F d ′ ( f )) − d ( f ) q i ( f ) − t X j =1 q i ( d j ( f ) · F f )= q i ( f · F d ( f ) − d ( f ) f )= − m i ◦ q i ( f · F f ) , for each i = 1 , . . . , t. This implies that g i − P tj =1 g j,i + q i ( f · F f ) is a cycle, so that there exist g i ∈ G i such that m i ( g i ) = g i − t X j =1 g j,i + q i ( f · F f ) . Using this along with the fact that t X i =1 (cid:16) g i − t X j =1 g i,j (cid:17) = t X i =1 (cid:16) g i − t X j =1 g j,i (cid:17) , one obtains: f · T ( d ′ ( f ) − t X i =1 q i ( f )) = f · F d ( f ) − t X i =1 q i ( f · F f ) + t X i =1 g i , whence upon choosing f · T f := f · F f + t X i =1 g i , one immediately obtains ℓ ( f · T f ) = d ( f ) f − f · F d ( f ) + t X i =1 q i ( f · F f ) + t X i =1 g i = d ( f ) f − f · T ( d ′ ( f ) − t X i =1 q i ( f ))= ℓ ( f ) f − f · T ℓ ( f ) . Case 6: F ′ ⊗ G i → F ⊕ (cid:16) L tj =1 G j (cid:17) . One computes: ℓ ( f ) g i − f · T m i ( g i ) = d ( f ) g i − t X j =1 g ′ j , where m j ( g ′ j ) = ( d ( f ) m i ( g i ) if i = j, . G STRUCTURE ON LENGTH 3 TRIMMING COMPLEXES 11
This implies that ( d ( f ) g i − g ′ i if i = j, and g ′ j otherwiseare cycles. By exactness of each G j • , there exist g j ∈ G j such that(3.1) m j ( g j ) = ( d ( f ) g i − g ′ i if i = j, − g ′ j otherwise . Case 7: G i ⊗ G i → F ⊕ (cid:16) L tj =1 G j (cid:17) . One computes: ℓ ( g i · T g i ) = m i ( − g i · G i g i d ( e i ))= − m i ( g i ) g i d ( e i ) + g i · G i m i ( g i ) d ( e i )= ℓ ( g i ) g i − g i · T ℓ ( g i ) . Case 8: G i ⊗ G j → F ⊕ (cid:16) L tk =1 G k (cid:17) , i = j . One computes: ℓ ( g i ) g j − g i · T ℓ ( g j ) = − m i ( g i ) d ( e i ) g j − t X k =1 g ′ k where m k ( g ′ k ) = ( − m i ( g i ) d ( e i ) m j ( g j ) if k = j , whence ( g ′ i + m i ( g i ) d ( e i ) g j if k = jg ′ k otherwise , are cycles, implying there exists g k ∈ G k such that(3.2) m k ( g k ) = ( g ′ i + m i ( g i ) d ( e i ) g j if k = jg ′ k otherwise . Thus, one may define g i · T g j := t X k =1 g k , and this product will satisfy the Leibniz rule. Case 9: G i ⊗ F → F ⊕ (cid:16) L tj =1 G j (cid:17) . One computes: ℓ ( g i ) f − g i · T ℓ ( f )= − m i ( g i ) d ( e i ) f − g i · T ( d ′ ( f ) − t X j =1 q j ( f ))= − m i ( g i ) d ( e i ) f + d ′ ( f ) · T g i + t X j =1 g i · T q j ( f ) = − m i ( g i ) d ( e i ) f + m i ( g i ) e i · F d ′ ( f ) + t X j =1 g j − g i · G i q i ( f ) d ( e i )+ t X j = i (cid:16) m i ( g i ) m j ( q j ( f )) e i · F e j + t X k =1 g ′ j,k (cid:17) = − m i ( g i ) d ( e · F f ) − g i · G i q i ( f ) d ( e i ) + g i + t X j = i (cid:16) g j + t X k =1 g ′ j,k (cid:17) . Observe that P tj = i (cid:16) g j + P tk =1 g ′ j,k (cid:17) = P tj = i (cid:16) g j + P tk = i g ′ k,j (cid:17) + P tk = i g ′ k,i , and m j (cid:16) g j + t X k = i g ′ k,j (cid:17) = m i ( g i ) q j ( e i · F d ′ ( f ))+ X k = i m i ( g i ) d ′ k ( f ) q j ( e i · F e k ) − m i ( g i ) d ( e i ) q j ( f )= m ( g i ) q j ( e i · F d ( f )) − m i ( g i ) d ( e i ) q j ( f )= − m i ( g i ) m j ( q j ( e i · F f )) ,m i (cid:16) − g i · q i ( f ) d ( e i ) + g i + t X k = i g ′ k,i (cid:17) = − m i ( g i ) q i ( f ) d ( e i )+ d ′ i ( f ) d ( e i ) + d ( d ′ ( f )) g i + m i ( g i ) q i ( e i · F d ′ ( f ))+ X k = i (cid:16) m i ( g i ) d ′ k ( f ) q i ( e i · F e k ) + d ′ k ( f ) d ( e k ) g i (cid:17) = d ( d ( f )) g i − m i ( g i ) q i ( f ) d ( e i )+ m i ( g i ) q i ( e i · F d ( f ))= − m i ( g i ) m i ( q i ( e i · F f )) . Thus, g j + t X k = i g ′ k,j + m i ( g i ) q j ( e i · F f ) , and(3.3) − g i · q i ( f ) d ( e i ) + g i + t X k = i g ′ k,i + m i ( g i ) q i ( e i · F f )(3.4)are both cycles, implying there exist g j ∈ G j such that m j ( g j ) = ( − g i · q i ( f ) d ( e i ) + g i + P tk = i g ′ k,i + q i ( e i · F f ) if i = j,g j + P tk = i g ′ k,j + q j ( e i · F f ) otherwise . G STRUCTURE ON LENGTH 3 TRIMMING COMPLEXES 13
Defining g i · T f := − m i ( g i ) e i · F f + t X j =1 g j , one combines this with the first computation of this case to find: ℓ ( g i · T f ) = ℓ ( − m i ( g i ) e i · F f + t X j =1 g j )= − m i ( g i ) d ( e i · F f ) + t X j =1 − m i ( g i ) q j ( e i · F f ) + t X j =1 m j ( g j )= ℓ ( g i ) f − g i · T ℓ ( f ) . (cid:3) As previously mentioned, in the case that each G i • has an additional DG-modulestructure, some of the products of Theorem 3.3 may be made more explicit. Proposition 3.4.
Adopt notation and hypotheses as in the statement of Theorem3.3, and assume that G i • admits the structure of a DG-module over each G j • for all j t . Then, the following products can be extended to a DG-algebra structureon T • : F ′ ⊗ G i → F ⊕ (cid:16) t M j =1 G j (cid:17) f · T g i := m i ( g i ) e i · F f + t X j =1 g i · G j q j ( e i · F f ) ,G i ⊗ G j → F ⊕ (cid:16) t M k =1 G k (cid:17) i < jg i · T g j := m i ( g i ) m j ( g j ) e i · F e j + t X k =1 g k , where g k = m j ( g j ) g i · G i q i ( e i · F e j ) if k = im i ( g i ) g j · G j q j ( e i · F e j ) if k = jm i ( g i ) g j · G k q k ( e i · F e j ) otherwise ,F ′ ⊗ G i → F ⊕ (cid:16) t M j =1 G j (cid:17) f · T g i := − t X j =1 g i · G j q j ( e i · F f ) ,G i ⊗ G j → F ⊕ (cid:16) t M k =1 G k (cid:17) g i · T g j := ( − P k = i m i ( g i ) g j · G k q k ( e i · F e j ) if i < j − m i ( g i ) g j · G j q j ( e i · F e j ) if j < i, Remark . Observe that the assumption that G • is a DG-module over each G j • is satisfied if G i • = G j • for all 1 i, j t . Proof.
In order to show that the products in the statement of the Proposition arewell-defined and may be extended to a product on all of T • , one only needs to verifythe identities in the statement and proof of Theorem 3.3. The verification is splitinto all 4 cases: Case 1: F ′ ⊗ G i → F ⊕ (cid:16) L tj =1 G j (cid:17) . One has: m j ( g i · G j q j ( e i · F f )) = m i ( g i ) q j ( e i · F f ) − g i · π j ( d ( e · F f ))= m i ( g i ) q j ( e i · F f ) + ( d ( f ) g i if i = j . Case 2: G i ⊗ G j → F ⊕ (cid:16) L tk =1 G k (cid:17) . One computes: m i ( g i ) = m j ( g j ) m i ( g i ) q i ( e i · F e j ) − m j ( g j ) g i · π i ( d ( e i · F e j ))= m j ( g j ) m i ( g i ) q i ( e i · F e j ) + m j ( g j ) d ( e j ) g i ,m j ( g j ) = m i ( g i ) m j ( g j ) q j ( e i · F e j ) − m i ( g i ) g j · π j ( d ( e i · F e j ))= m i ( g i ) m j ( g j ) q j ( e i · F e j ) − m i ( g i ) d ( e i ) g j ,m k ( g k ) = m i ( g i ) m j ( g j ) q k ( e i · F e j ) − m i ( g i ) g j · π k ( d ( e i · F e j ))= m i ( g i ) m j ( g j ) q j ( e i · F e j ) . Case 3: F ′ ⊗ G i → F ⊕ (cid:16) L tj =1 G j (cid:17) . The identity 3.1 must be verified. Onecomputes: m j ( − g j ) = − m i ( g i ) · G j q j ( e i · F f ) − g i · π j ( d ( e i · F f ))= d ( f ) g i − m i ( g i ) · G j q j ( e i · F f ) if i = j, = − m i ( g i ) · G j q j ( e i · F f ) , otherwise. Case 4: G i ⊗ G j → F ⊕ (cid:16) L tk =1 G k (cid:17) . The identities of 3.2 must be verified.One computes: m k ( m i ( g i ) g j · G k q k ( e i · F e j )) = m i ( g i ) m j ( g j ) · G k q k ( e i · F e j )+ ( m i ( g i ) d ( e i ) g j if k = j i < jm j ( m i ( g i ) g j · G j q j ( e i · F e j )) = m i ( g i ) m j ( g j ) · G j q j ( e i · F e j )+ m i ( g i ) d ( e j ) g j
2G STRUCTURE ON LENGTH 3 TRIMMING COMPLEXES 15 = g i · T m j ( g j ) + m i ( g i ) d ( e j ) g j if j < i. This completes the proof. (cid:3)
Notation 3.6.
Let ( R, m , k ) denote a regular local ring. Let · denote the functor · ⊗ R k .The following corollaries are immediate consequences of the statement and proofof Theorem 3.3. Corollary 3.7.
Let ( R, m , k ) denote a regular local ring. Assume that the complexes F • and G i • (for i t ) are minimal. Then the only possible nontrivial productsin the algebra T • are F ′ · T F ′ , F ′ · T F , G i · T F , and F ′ · T G i . Corollary 3.8.
Let ( R, m , k ) denote a regular local ring. Assume that the complexes F • and G i • (for i t ) are minimal. Then the map T • → F • f i f i , ( f i ∈ F i ) ,g ji , ( g ji ∈ G ji ) , is a homomorphism of k -algebras. Consequences for Tor Algebra Structures
In this section, we take advantage of Corollary 3.7 and study how the process oftrimming an ideal affects the Tor-algebra class. As in turns out, if the multiplicationbetween certain homological degrees has coefficients appearing in sufficiently highpowers of the maximal ideal, then the Tor-algebra class will be preserved. Wealso give explicit examples showing that if this assumption is not satisfied, thenit is possible to obtain new nontrivial multiplication in the associated trimmingcomplex.We begin with the Tor-algebra classification provided by Avramov, Kustin, andMiller in [2].
Theorem 4.1 ([2], Theorem 2 . . There are nonnegative integers p , q , and r and bases { f i } , { f i } , and { f i } for Tor R ( R/I, k ) , Tor R ( R/I, k ) , and Tor R ( R/I, k ) ,respectively, such that the multiplication in Tor R + ( R/I, k ) is given by one of thefollowing: CI : f = f f , f = f f , f = f f f i f j = δ ij f for i, j TE : f = f f , f = f f , f = f f B : f f = f , f f = f , f f = f G ( r ) : f i f i = f , i r H ( p, q ) : f p +11 f i = f i , i pf p +11 f p + i = f i , i q Remark . In terms of the tuples presented in the Realizability Question 2.9, theclasses G ( r ) and H ( p, q ) have:G( r ) : ( m, n, , , r )H( p, q ) : ( m, n, p, q, q ) . For this reason, a tuple coming from a class G ( r ) ring will often be shortenedto ( m, n, r ), and a tuple coming from a class H ( p, q ) ring will be shortened to( m, n, p, q ). These tuples will be referred to as the associated tuple .The following definition is introduced out of convenience for stating the resultsappearing later in this section. Definition 4.3.
A Tor-algebra is in standard form if the basis elements have beenchosen such that the multiplication is given by one of the possibilities of Theorem4.1. A DG-algebra free resolution F • is in standard form if the multiplicationdescends to a Tor-algebra in standard form after applying − ⊗ R k . Example 4.4.
Let R = k [ x ij | i < j ] and X = ( x ij ) be a generic n × n skewsymmetric matrix, with n odd. Given an indexing set I = ( i < · · · < i k ), letPf I ( X ) := pfaffian of X with rows and columns from I removed.Define d := (Pf ( X ) , − Pf ( X ) , . . . , ( − i +1 Pf i ( X ) , . . . , Pf n ( X )) , and consider the complex F • : 0 / / R d ∗ / / R n X / / R n d / / R / / . Then F • admits the structure of an associative DG-algebra with the following prod-ucts: f i · F f j = n X k =1 ( − i + j + k Pf ijk ( X ) f k ,f i · F f j = δ ij f . If n >
5, then F • is in standard form since it descends to a Tor-algebra of class G ( n ) in standard form. Notation 4.5. If A = A ⊕ A ⊕ A ⊕ k is a finite dimensional graded-commutative k -algebra, then A ⊥ := { a ∈ A | a · A = 0 } . The following Lemma is a coordinate free characterization for length 3 k -algebrasrealizing certain types of algebra classes. Lemma 4.6 ([2], Lemma 2 . . Suppose A = A ⊕ A ⊕ A ⊕ k is a finite dimensionalgraded-commutative k -algebra with A = 0 . Then:(a) A has form H (0 , if and only if A · A = 0 .(b) A has form H (0 , q ) for some q > if and only if codim A ⊥ = 1 and dim A · A = q .(c) A has form G ( r ) for some r > if and only if dim A · A = 1 and codim A ⊥ = r . G STRUCTURE ON LENGTH 3 TRIMMING COMPLEXES 17
Notation 4.7.
Let I = ( φ , . . . , φ n ) ⊆ R be an m -primary ideal and F • a DG-algebra free resolution of R/I in standard form. Given an indexing set σ = { σ < · · · < σ t n } , definetm σ ( I ) := ( φ i | i / ∈ σ ) + m ( φ j | j ∈ σ ) . The transformation I tm σ ( I ) will be referred as trimming the ideal I .The following setup will be in effect for the remainder of this section. Setup 4.8.
Let ( R, m , k ) denote a regular local ring of dimension . Let I =( φ , . . . , φ n ) ⊆ R be an ideal and σ = (1 σ < · · · < σ t n ) be an indexing set.Let ( F • , d • ) and ( K • , m • ) be minimal DG-algebra free resolutions and R/I and k ,respectively. By Theorem 2.6, a free resolution of R/ tm σ ( I ) may be obtained asthe mapping cone of a morphism of complexes of the form: (4.1) 0 / / F Q (cid:15) (cid:15) d / / F Q (cid:15) (cid:15) d ′ / / F ′ d (cid:15) (cid:15) L ti =1 K L ti =1 m / / L ti =1 K L ti =1 m / / L ti =1 K − P ti =1 m ( − ) d ( e σi ) / / R, where F ′ := M j / ∈ σ Re j and d ′ : F d −→ F proj −−→ F ′ . Let T • denote the mapping cone of 4.1. The following Lemma says that in the context of Setup 4.8, the possible non-trivial multiplications in the homology algebra are even more restricted than thatof Corollary 3.7.
Lemma 4.9.
Adopt notation and hypotheses as in Setup 4.8 and assume that I ⊆ m . Then, F · F F ⊆ Ker( Q ⊗ k ) F · F F ⊆ Ker( Q ⊗ k ) In particular, the only possible nontrivial products in the algebra T • are given by F ′ · T F ′ and F ′ · T F . Proof.
One has: d ( F · F F ) ⊆ m F = ⇒ m i ◦ q ( F · F F ) = d ′ i ( F · F F ) ⊆ m = ⇒ q i ( F · F F ) ⊆ m K for all i. Likewise, q i ( d ( F · F F )) ⊆ m · q ( F · F F ) ⊆ m G i = ⇒ m ( q i ( F · F F )) = q ( d ( F · F F )) ⊆ m G i = ⇒ q i ( F · F F ) ⊆ m K for all i. The latter claim about the triviality of the products K · T F and F ′ · T K followsimmediately from the definition of the products given in Theorem 3.3 along withthe identities 3.1 and 3.3. (cid:3) The following lemma makes precise the previously mentioned fact that, undermild hypotheses, the Tor-algebra class G ( r ) is preserved by trimming. Lemma 4.10.
Adopt notation and hypotheses as in Setup 4.8 and assume that F • is in standard form with F · F F ⊆ m F . If R/I defines a ring of class G ( r ) , thenfor all indexing sets σ , the ideal tm σ ( I ) defines either a Golod ring or a ring of Toralgebra class G ( r ′ ) , for some r − | σ ∩ [ r ] | − rank( Q ⊗ k ) r ′ r − | σ ∩ [ r ] | .Proof. By Lemma 4.9, f may be chosen as part of a basis for Ker( Q ⊗ k ); theparameter r ′ arises from counting the rank of the induced mapKer( Q ⊗ k ) → Hom k ( F ′ , F ) . By definition of the product on T • , the assumption F · F F ⊆ m F implies that f · T f ′ = f · F f ′ = 0 for all f , f ′ ∈ F ′ . Thus, the induced map F → Hom k ( F ′ , F ) has rank r − | σ ∩ [ r ] | . But this mapmay be written as the compositionKer( Q ⊗ k ) ֒ → F → Hom k ( F ′ , F ) , whence one finds that r − | σ ∩ [ r ] | − rank( Q ⊗ k ) r ′ r − | σ ∩ [ r ] | . (cid:3) Corollary 4.11.
Adopt notation and hypotheses as in the statement of Lemma4.10. If d ( F ) ⊆ m F , then tm σ ( I ) defines a ring of Tor-algebra class G ( r − | σ ∩ [ r ] | ) .Proof. The assumption d ( F ) ⊆ m F implies that Q ⊗ k = 0. (cid:3) The following example shows that the assumption F · F F ⊆ m F in Lemma4.10 is necessary. Example 4.12.
Let R = k [ x , x , x ] and X := x x x x x − x x − x − x − x − x − x . Let I = Pf( X ) = ( x , − x x , x − x x , − x x , x ), the ideal of 4 × X . The ring R/I has Tor-algebra class G (5) and the multiplication satisfies F · F ⊆ m F and F · F m F . It may be shown using Macaulay2 [7] thattm ( I ) defines a ring of Tor-algebra class B and tm ( I ) defines a ring of Tor-algebra class H (3 , T • cannotpossibly agree with the multiplication on F • .It turns out that trimming also tends to preserve the Tor-algebra class H ( p, q ): G STRUCTURE ON LENGTH 3 TRIMMING COMPLEXES 19
Lemma 4.13.
Adopt notation and hypotheses as in Setup 4.8 and assume that F • is in standard form of class H ( p, q ) with the property that f i · F f j ∈ m F for all i, j = p + 1 . Then,(i) if p + 1 ∈ σ , tm σ ( I ) defines a Golod ring, and(ii) if p + 1 / ∈ σ , then tm σ ( I ) defines either a Golod ring or a ring of class H ( p − | σ ∩ [ p ] | , q ′ ) , where q − rank( Q ⊗ k ) q ′ q .Proof. In an identical manner to the proof of Lemma 4.10, the assumption f i · F f j ∈ m F implies f i · T f j = f i · F f j = 0 for all i, j = p + 1 . Case 1: p + 1 ∈ σ . Since f p +11 / ∈ F ′ , it follows that F ′ · T F ′ = 0. For identicalreasons, F ′ · T F = 0. Thus, the multiplication in the Tor-algebra is trivial, so R/I is a Golod ring.
Case 2: p + 1 / ∈ σ . By Lemma 4.9, f , . . . , f p may be chosen as part of a basisof Ker( Q ⊗ k ). This immediately implies that the only nontrivial products in T • are of the form f i · T f p +11 = f i for 1 i p, i / ∈ σ. Thus, dim k F ′ · T F ′ = p − | σ ∩ [ p ] | . Likewise, f p +13 , . . . , f p + q may be chosen as partof a basis for Ker( Q ⊗ k ). Moreover, the rank of the induced mapKer( Q ⊗ k ) → Hom k ( F ′ , F )is at least q − rank( Q ⊗ k ), and it is evidently at most q . (cid:3) Corollary 4.14.
Adopt notation and hypotheses as in the statement of Lemma4.13. If d ( F ) ⊆ m F , then tm σ ( I ) is either Golod or defines a ring of Tor-algebra class H ( p − | σ ∩ [ p ] | , q ) . Again, the assumption that f i · F f j ∈ m F for i, j = p + 1 in Lemma 4.13 isnecessary, as the following example shows. Example 4.15.
Let R = k [ x , x , x ] and I = ( x − x x , − x x , x , x ). Thering R/I has Tor-algebra class H (3 , F • of R/I satisfies f i · f j ∈ m F , f i · f j / ∈ m F , where i, j = 5. However,it can be shown using Macaulay2 [7] that tm ( I ) has Tor-algebra class TE.5. Examples
In this section, we employ the theory developed in Section 4 for the constructionof explicit examples of rings realizing Tor-algebra classes G ( r ) and H ( p, q ) for agiven set of parameters ( m, n, p, q, r ) (as in Setup 2.8 and Question 2.9). Theseexamples are constructed in an arbitrary regular local ring ( R, m , k ); in particular,we will construct explicit novel examples of ideals defining rings of Tor-algebra class G ( r ) (and arbitrarily large type). One can combine the examples of this sectionwith the results of Proposition 2.11 and Section 4 to obtain an even larger class oftuples.We begin by first adopting the following simple setup. Setup 5.1.
Let ( R, m , k ) denote a regular local ring of dimension (or a standardgraded polynomial ring over a field). Let m = ( x , x , x ) , where x , x , x is aregular sequence. Let K := Re ⊕ Re ⊕ Re and K • denote the Koszul complexinduced by the map sending e i x i . The matrices appearing in the following two definitions were inspired by matricesconstructed in [5] and further generalized in [8]. Here, we extend this definition toarbitrary local rings for the construction of our examples.
Definition 5.2.
Adopt notation and hypotheses as in Setup 5.1. Let U jm (for j m ) denote the m × m matrix with entries from R defined by:( U jm ) i,m − i = x , ( U jm ) i,m − i +1 = x , ( U jm ) i,m − i +2 = x for i m − j ( U jm ) i,m − i = x , ( U jm ) i,m − i +1 = x , ( U jm ) i,m − i +2 = x for i > m − j and all other entries are defined to be 0.To see the pattern, observe that: U = (cid:18) x x x x (cid:19) , U = x x x x x x x , U = x x x x x x x Definition 5.3.
Define V jm (for j < m ) to be the (2 m +1) × (2 m +1) skew symmetricmatrix V jm := O O x ( U jm ) T − ( O x ) T x O − U jm − ( x O ) T O . If j = m , then V mm is the skew symmetric matrix V mm := O O x ( U mm ) T − ( O x ) T x O − U mm − ( x O ) T O . Lastly, if j = m + 1, then V m +1 m is the skew symmetric matrix V m +1 m := O O x ( U mm ) T − ( O x ) T x O − U mm − ( x O ) T O . Definition 5.4.
Let m > I jm (for 0 j m + 1)by I jm := Pf( V jm ) , where Pf( V jm ) denotes the ideal of 2 m × m pfaffians of V jm . Setup 5.5.
Adopt notation and hypotheses as in Setup 5.1. Define d := ( Pf ( V jm ) , − Pf ( V jm ) , . . . , ( − i +1 Pf i ( V jm ) , . . . , Pf n ( V jm )) , (for m > and j m + 1 ) and consider the complex F • : 0 / / R d ∗ / / R n V jm / / R n d / / R / / . Recall that F • is a minimal free resolution of R/I jm in standard form of class G (2 m +1) with product as in Example 4.4. G STRUCTURE ON LENGTH 3 TRIMMING COMPLEXES 21
Proposition 5.6.
Adopt notation and hypotheses as in Setup 5.5. Define q i : F → K by sending: f m +3 − i e if < i j + 1 m + 1 x e if j + 1 < i m + 1 − x e if m + 1 < i m + 1 − j − e if m + 1 − j < i m + 1 ,f m +2 − i e if i j, i < m + 1 x e if j < i < m + 10 if i = m + 1 − x e if m + 1 < i m + 1 − j − e if m + 1 − j < i m + 1 ,f m +1 − i e if i j − x e if j − < i < m + 1 − x e if i = m + 1 , j < m + 1 − e if i = m + 1 , j = m + 1 − x e if m + 1 < i m + 1 − j − e if m + 1 − j < i < m + 1 , and all other basis elements are sent to . Then the following diagram commutes: F d ′ i ❇❇❇❇❇❇❇❇ q i (cid:15) (cid:15) K m / / m . Notation 5.7.
Let a and b be positive integers with a < b . The notation [ a ]will denote the set { , , . . . , a − , a } and the notation [ a, b ] will denote the set { a, a + 1 , . . . , b − , b } . Corollary 5.8.
Adopt notation and hypotheses as in Setup 5.5. Let σ = (1 σ < · · · < σ t m + 1) denote an indexing and assume:(a) m > , or(b) m = 2 and j = 0 .Then:(1) The ideal tm i ( I jm ) defines either a Golod ring or a ring of Tor algebra class G (2 m + 1 − t − rank( q i ⊗ k )) .(2) If t m + 1 − j , then the ideal tm [ t ] ( I jm ) defines either a Golod ring or aring of Tor-algebra class G (2 m + 1 − t − min { t, j } ) .(3) More generally, the ideal tm σ ( I jm ) defines either a Golod ring or a ring ofTor-algebra class G (2 m + 1 − t − rank( Q ⊗ k ) + | σ ∩ { j | Q ( f j ) = 0 }| ) Proof.
The assumptions ( a ) and ( b ) ensure that F · F ⊆ m F , so that by Lemma4.10, tm σ ( I jm ) defines either a Golod ring or a ring of class G ( r ′ ), where 2 m − rank( q i ⊗ k ) r ′ m . By construction, Ker( Q ⊗ k ) for each i has basis given by { f j ∈ F | Q ( f j ) = 0 } . Case 1:
By the above, one immediately has that rank (cid:0)
Ker( Q ⊗ k ) → Hom k ( F ′ , F ) (cid:1) =2 m − rank( q i ⊗ k ). This is because (by Proposition 5.6) i / ∈ { j | Q ( f j ) = 0 } . Case 2:
Using Proposition 5.6, one finds that rank( Q ⊗ k ) = min { r, j } .Moreover, since t < m + 1 − j ,[ t ] ∩ { j | Q ( f j ) = 0 } = ∅ , whence rank (cid:0) Ker( Q ⊗ k ) → Hom k ( F ′ , F ) (cid:1) m + 1 − t − min { t, j } . ByLemma 4.10, one has equality. Case 3:
Define S := σ ∩ { j | Q ( f j ) = 0 } . If j ∈ S , then in T • , the directsummand Rf j has been omitted from F . Thus, removal of the direct summandgenerated by f j has no effect on the rank of the induced map δ : Ker( Q ⊗ k ) → Hom k ( F ′ , F ) . By inclusion-exclusion, this implies that δ has rank2 m + 1 − t − rank( Q ⊗ k ) + | S | . (cid:3) Remark . Let S := σ ∩ { j | Q ( f j ) = 0 } . Then, in terms of the associatedtuple (see Remark 4.2), the transformation I jm tm σ ( I jm ) transforms the tuple(2 m + 1 , , m + 1) as so: I jm tm σ ( I jm )(2 m + 1 , , m + 1) (2 m + 2 t + 1 − rank( Q ⊗ k ) , t, m + 1 − t − rank( Q ⊗ k ) + | S | )Corollary 5.8 immediately allows us to fill in a large class of tuples: Corollary 5.10.
Adopt notation and hypotheses as in Setup 4.8. Let ( m, n, r ) bea tuple of positive integers satisfying either:(1) m − r = 3( n − , n > , and n + r > ,(2) m − r = 3( n − − , n > , and n + r > ,(3) m − r = 3( n − , n > , and n + r > .Then there exists an ideal J defining a ring of Tor-algebra class G ( r ) realizing thistuple.Proof. Case 1(a): n + r is even. Write n + r = 2 k + 2 for some integer k > [ n − ( I k ). By Corollary 5.8, this has the effect of transformingthe associated tuple in the following way:(2 k + 1 , , k + 1) (2 k + 1 + 2( n − , n − , k + 1 − ( n − n + r + 2 n − , n, n + r − − n + 1)= ( m, n, r ) . G STRUCTURE ON LENGTH 3 TRIMMING COMPLEXES 23
Case 1(b): n + r is odd. Write n + r = 2 k + 1 for some integer k >
3. Considerthe ideal tm [ n − ( I k ). By Corollary 5.8, this has the effect of transforming theassociated tuple in the following way:(2 k + 1 , , k + 1) (2 k + 1 + 2( n − − , n − , k + 1 − ( n − − n + r + 2 n − , n, n + r − n + 1 − m, n, r ) . Case 2(a): n + r is even. Write n + r = 2 k for some k >
3. Consider the ideal ( tm , k +1 ( I k ) if n = 3 , tm { , k +1 }∪ [3 ,n − ( I k ) if n > . By Proposition 5.6, rank( Q ⊗ k ) = 4 and σ ∩ { j | Q ( f j ) = 0 } = { , k + 1 } so | σ ∩ { j | Q ( f j ) = 0 }| = 2. By Corollary 5.8, this has the effect of transformingthe associated tuple as so:(2 k + 1 , , k + 1) (2 k + 1 + 2( n − − , n, k + 1 − ( n − − n + r + 2 n − , n, n + r − − ( n − m, n, r ) . Case 2(b): n + r is odd. Write n + r = 2 k + 1 for some k >
3. Consider the ideal ( tm , k +1 ( I k ) if n = 3 , tm { , k }∪ [3 ,n − ( I k ) if n > . By Proposition 5.6, rank( Q ⊗ k ) = 3, and exactly as in Case 2( a ), | σ ∩ { j | Q ( f j ) = 0 }| = |{ , k + 1 }| = 2 . By Corollary 5.8, this has the effect of transforming the associated tuple as so:(2 k + 1 , , k + 1) (2 k + 1 + 2( n − − , n, k + 1 − ( n − − n + r + 2 n − , n, n + r − − ( n − m, n, r ) . Case 3(a): n + r is odd. Write n + r = 2 k + 1 for some k >
3. Consider the idealtm [ n − , k +1 ( I k ). By Proposition 5.6, rank( Q ⊗ k ) = 4, and (recalling that n > σ ∩ { j | Q ( f j ) = 0 } = { , , k + 1 } , so | σ ∩ { j | Q ( f j ) = 0 }| = 3. By Corollary 5.8, this has the effect of transformingthe associated tuple as so:(2 k + 1 , , k + 1) (2 k + 1 + 2( n − − , n, k + 1 − ( n − − n + r + 2 n − , n, n + r − − ( n − m, n, r ) . Case 3(b): n + r is even. Write n + r = 2 k + 2 for some k >
3. Consider theideal tm [ n − , k +1 ( I k ). By Proposition 5.6, rank( Q ⊗ k ) = 3, and (recalling that n > | σ ∩ { j | Q ( f j ) = 0 }| = |{ , , k + 1 }| = 3 By Corollary 5.8, this has the effect of transforming the associated tuple as so:(2 k + 1 , , k + 1) (2 k + 1 + 2( n − − , n, k + 1 − ( n − − n + r + 2 n − , n, n + r − − ( n − m, n, r ) . (cid:3) Example 5.11.
Corollary 5.10 is far from being an exhaustive list of the possibletuples ( m, n, r ). For example, let R = k [ x , x , x ] and consider the ideal J := ( x x , x x − x , x x , x x x , x , x x , x ) . One may verify using the TorAlgebras package in Macaulay2 that J defines a ringof Tor-algebra class G (2) and realizes the tuple (7 , , G ( r ) en masse besides trimming; because of this, the realizableclasses covered by Corollary 5.10 are bound to be rather restricted. Next, weconsider rings of class H ( p, q ). Setup 5.12.
Adopt notation and hypotheses as in Setup 5.1. Let X p denote the p × ( p − matrix X p := x · · · x x · · · x x x · · · x x · · ·
00 0 x ...... . . . . . . . . .
00 0 · · · · · · x · · · · · · x and define J p := I p − ( X p ) + ( x p − ) = (∆ , . . . , ∆ p , x p − ) . Let H • denote the Hilbert-Burch resolution of R/I p − ( X p ) and G • := 0 → R x p − −−−→ R . The minimal free resolution of J p may be obtained as the tensor product F • :=( H ⊗ G ) • : F • : 0 / / H ⊗ G / / (cid:0) H ⊗ G (cid:1) ⊕ H / / H ⊕ G / / R. The following multiplication makes F • into an algebra resolution in standard formof Tor-algebra class H ( p, p − : h · F h ′ = h · H h ′ ,h · F g = h ⊗ g ,h · F g = h ⊗ g , where h , h ′ ∈ H , h ∈ H , g ∈ G . G STRUCTURE ON LENGTH 3 TRIMMING COMPLEXES 25
In an identical manner, let F ′• be a minimal algebra resolution of J ′ p := I p − ( X ′ p ) +( x n − ) in standard from of Tor-algebra class H ( p, p − , where X ′ p := x · · · x x · · · x x x · · · x x · · ·
00 0 x ...... . . . . . . . . .
00 0 · · · · · · x · · · · · · x . Proposition 5.13.
Adopt notation and hypotheses as in Setup 5.12. Assume H = L pi =1 Rh i , where h i ∆ i . Define q i : F → K for i < p by sending: h i − e ( i > ,h i − e ( i > ,h i e ( i < p ) ,h i ⊗ g
7→ − x p − e , and all other basis elements to . If i = p + 1 , write each ∆ j = x ∆ ,j + x ∆ ,j + x ∆ ,j . Define q p +11 : F → K by sending: h j ⊗ g ∆ ,j e + ∆ ,j e + ∆ ,j e , (1 j p ) and all other basis elements to . Then the following diagram commutes: F d ′ i ❇❇❇❇❇❇❇❇ q i (cid:15) (cid:15) K m / / m . Corollary 5.14.
Adopt notation and hypotheses as in Setup 5.12 with p > . Then(1) if p + 1 / ∈ σ , the ideal tm σ ( J p ) defines either a Golod ring or a ring of Tor-algebra class H ( p − t, p − − rank( Q ⊗ k ) . In particular, if σ = [ t ] for some t < p , the ideal tm [ t ] ( J p ) defines a ring of Tor-algebra class H ( p − t, p − − t ) .(2) if p + 1 ∈ σ , the ideal tm σ ( I ) defines a Golod ring.(3) if p + 1 / ∈ σ , the ideal tm σ ( J ′ p ) defines a ring of Tor-algebra class H ( p − t, p − .(4) if p + 1 ∈ σ , the ideal tm σ ( J ′ p ) defines a Golod ring.Proof. As in the proof of Corollary 5.8, one hasKer( Q ⊗ k ) = Span k { f j ∈ F | Q ( f j ) = 0 } . Moreover, the assumption that p > Case 1:
Since Ker( Q ⊗ k ) is obtained by simply deleting basis elements withnonzero image under Q ⊗ k , it follows that rank(Ker( Q ⊗ k ) → Hom k ( F ′ , F ) r − rank( Q ⊗ k ). By Lemma 4.13, the result follows. Case 2:
This is simply case ( i ) of Lemma 4.13. Cases 3 and 4:
Observe that F ′• has the property that d ( F ) ⊆ m F . Thusthe conclusion follows by Corollary 4.11. (cid:3) Remark . In terms of the tuples of the associated tuple (see Remark 4.2), thetransformation J p tm σ ( J p ) transforms the tuple ( p + 1 , p − , p, p −
1) as so: J p tm σ ( J p )( p + 1 , p − , p, p − ( p + 1 + 2 t − rank( Q ⊗ k ) , p − t, p − t, p − − rank( Q ⊗ k )) ,J ′ p tm σ ( J ′ p )( p + 1 , p − , p, p − ( p + 1 + 2 t, p − t, p − t, p − . We conclude with some discussion on the problem of realizability. The results ofSection 4 are stated for arbitrary rings of a given Tor-algebra class. However, onemust start with a ring of a given Tor-algebra class and then apply the trimmingprocess to obtain a new ideal with some new set of parameters. The only simplecandidates for “initial” ideals of Tor-algebra class G ( r ) and H ( p, q ) are grade 3Gorenstein ideals and grade 3 hyperplane sections, respectively. Even though usinga combination of linkage and trimming can obtain many of the tuples falling withinthe bounds of Theorem 2.10, one is tempted to ask: Question . Are there other “canonical” sources of rings of Tor-algebra class G ( r ) and H ( p, q ), distinct from grade 3 Gorenstein ideals or hyperplane sections?Enlarging the set of starting ideals from which one can begin the process of link-age/trimming would immediately allow one to add to the question of realizability.As it turns out, rings of Tor-algebra class G ( r ) arise generically when working in apolynomial ring. The examples arising in [9] are already obtained by trimming aGorenstein ideal, but it is shown more generally in [4] that generically obtained ringsof type 2 are of class G ( r ) under appropriate hypotheses. To the author’s knowl-edge, there are fewer results of this flavor for rings of Tor-algebra class H ( p, q ), eventhough these rings seem to be ubiquitous. References
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