Division properties in exterior algebras of free modules
aa r X i v : . [ m a t h . A C ] F e b DIVISION PROPERTIES IN EXTERIOR ALGEBRAS OF FREE MODULES
B. JAKUBCZYKINSTITUTE OF MATHEMATICS, POLISH ACADEMY OF SCIENCES
Abstract.
Let M be a free module of rank m over a commutative unital ring R and let N beits free submodule. We consider the problem of when a given element of the exterior productΛ p M is divisible, in a sense, over elements of the exterior product Λ r N , where r ≤ p . Precisely,we give conditions under which a given η ∈ Λ p M can be expressed as a finite sum of elementsof Λ r N multiplied (via the exterior product) by elements of Λ p − r M . Necessary and sufficientconditions for such divisibility take a simple form, provided that the submodule is embedded in M with singularities having the depth larger then p − r +1. In the special case where r = rank N the divisibility property means that η = Ω ∧ γ where Ω is the product ω ∧ · · · ∧ ω r of elementsof a basis of N and γ is an element of Λ p − r M .More detailed statements of these results are then used to state criteria for existence anduniqueness of algebraic residua when the “divisor” is defined by elements f , . . . , f k ∈ R . Specialcases are logarithmic multi-residua in complex analysis. Introduction
Let R be a commutative ring with unity and let M be a free R -module of finite rank m . Weshall denote by Λ q M the q th exterior product of M and Λ M = ⊕ mq =0 Λ q M will be the exterioralgebra generated by M . We shall consider division problems in the algebra Λ M related to thefollowing elementary questions.Consider two elements η ∈ Λ p M and α ∈ Λ r M . If p ≥ r then it is natural to ask Question 1. Is η divisible over α , i.e., does it exist γ ∈ Λ p − r M such that η = α ∧ γ ?This question seems to be largely open and is nontrivial even in the case where R is a field and M is a vector space. We will answer it in the special case when α is a product of one-forms. Namely,let k ≤ m and let ω , . . . , ω k be fixed elements of M . We will answer Question 2.
When there exists γ ∈ Λ p − k M such that η = ω ∧ · · · ∧ ω k ∧ γ ?This question, similarly as two other ones stated below, is nontrivial when the submodule of M generated by the elements ω , . . . , ω k is singularly embedded in M which means that the elementsare linearly independent but can not be completed to a basis of M .We can also ask if for a given η ∈ M it is possible to find elements a i ∈ R such that η = P i a i ω i .More generally, Question 3.
Given η ∈ Λ p M , is it possible to represent η as η = X i ω i ∧ γ i , with some γ i ∈ Λ p − M ? An answer to this question was given in the main theorem of [9] for R Noetherian. Questions 2and 3 are particular cases of the following main problem considered in the paper.
Question 4.
Given η ∈ Λ p M and ≤ r ≤ k such that p ≥ r , when one can write η = X ω J ∧ γ J ( R ) with some γ J ∈ Λ p − r M , where ω J ∈ Λ r M are of the form ω J = ω j ∧ · · · ∧ ω j r ? uch questions appear naturally in differential geometry when one considers the module of onedifferential forms or the module of vector fields on N . Singularities of tuples of differential forms, orvector fields, are unevoidable if the topology of the manifolds is not sufficiently trivial. Already for k = 1 a classical result says that on any manifold with nonzero Euler characteristic every smoothvecor field on N has a singular point. As shown recently, every holomorphic one-form on a smoothcomplex projective variety of general type must vanish at some point [8]. Global questions can oftenbe reduced to local ones with the use of basic sheaf theory. It is then of fundamental importanceto understand answers to the above questions for R equal to the ring of germs of functions at fixedpoints of a manifold, at least in the smooh, real analytic, and holomorphic category.Our first main result, statement (i) in Theorem 2.1 in the next section, will give a criterium for η to be representable in the form (R). To have such a representation we will assume that p − r +2 < d ,where d is the depth of the ideal I (Ω) generated by the coefficients of the k-formΩ = ω ∧ · · · ∧ ω k . Without this assumption an analogous representation holds for η replaced by a n η with a an arbi-trary element of the ideal I (Ω) and n sufficiently large (statement (ii) of Theorem 2.1). The proofof Theorem 2.1 is postponed to Section 6. A uniquenss property of the representation is statedin Theorem 2.7. Finally, an equivalent statement of Theorem 2.1, formulated as a property of a(singularly embedded) free submodule of M , is given in 2.8. In the Appendix we recall propertiesof the depth of an ideal used in stating some of the results.In the next two sections we state direct consequences of results presented in Section 2 in thecase where we are given a finitely generated ideal I ⊂ R , in addition to the earlier data. Westate theorems on algebraic residua (Section 3) and on an algebraic version of logarithmic residua(Section 4). In Section 5 we translate the theorem on algebraic logarithmic residua to a theoremon multidimensional residua in complex analysis. In this case the depth condition is replaced by acondition on the codimension of the singularity defined by the form Ω.We do not assume that the ring R and the module M are Noetherian which makes the proofsmore delicate. The reader may notice that some of the results can be stated in a more abstractway using the language of homological algebra.2. Main result
Let R be a commutative ring with unity and let M be a free module over R of finite rank m .We shall denote its q th exterior product by Λ q M , with identifications Λ M = R , Λ M = M , andΛ q M = 0 for q < q > m . Note that, choosing a basis e , . . . , e m in M and q ∈ { , . . . , m } ,we have a natural basis in Λ q M which consists of exterior products e I = e i ∧ · · · ∧ e i q , ≤ i < · · · < i q ≤ m. ( E )Then any element η ∈ Λ q M can be represented as η = P I a I e I , with unique coefficients a I ∈ R , I = i · · · i q .Given elements ω , . . . , ω k ∈ M , we will use the following notation. For a given 0 ≤ r ≤ k wewill denote by J ( r, k ) the set of strictly monotone multiindices J ( r, k ) = { J = j · · · j r : 1 ≤ j < · · · < j r ≤ k } , including the empty one J = ∅ with r = 0. Sometimes, when k is not fixed, it will be convenientto denote the length r of J by h J i .Given J ∈ J ( r, k ), we will denote ω J = ω j ∧ · · · ∧ ω j r . where ω J = 1 when J = ∅ . For example for k = 5, r = 2 and j = 2, j = 3 we will have ω J = ω ∧ ω .Denote Ω = ω ∧ · · · ∧ ω k . Using the basis (E) with q = k we can write Ω = P I a I e I where thesum is taken over I ∈ J ( k, m ). The ideal generated by the coefficients a I , I (Ω) = span R { a I , I ∈ J ( k, m ) } , is independent of the choice of the basis in M and will play a crutial role in our considerations. ecall that the depth of an ideal I ⊂ R , denoted depth( I ), is the largest length of a regularsequence in I (see Appendix for a brief recall of properties of regular sequences). We will denote d = depth I (Ω). In Noetherian rings we have depth I ≤ height I ≤ gen I , where gen I denotes theminimal number of generators of the ideal I . Since the ideal I (Ω) is contained in the ideal generatedby the coefficients of any of the forms ω i , and the latter is generated by m such coefficients, itfollows that d ≤ m in the case where R is Noetherian. Theorem 2.1 (Existence) . Fix positive integers ≤ p, r, s ≤ m = rank M such that p ≥ r and r + s = k + 1 .(i) Assume that p − r ≤ depth I (Ω) − . ( DC ) Then an element η ∈ Λ p M can be represented in the form η = X J ∈J ( r,k ) ω J ∧ γ J , for some γ J ∈ Λ p − r M, ( A ) if and only if ω I ∧ η = 0 , for all I ∈ J ( s, k ) , ( B ) (ii) Without assuming the depth condition (DC) there exists n > such that for any η ∈ Λ p M satisfying (B) and any a ∈ I (Ω) a n η = X J ∈J ( r,k ) ω J ∧ γ J , for some γ J ∈ Λ p − r M. ( A ′ ) Conversely, if (A’) holds for some non-zero-divisor a ∈ R then (B) holds, too. Remark 2.2.
Implication (A) ⇒ (B) is trivial and holds for any 0 ≤ p ≤ m . Namely, multiplyingboth sides of (A) by ω I with I ∈ J ( s, k ) gives zero on the right hand side as each product ω I ∧ ω J ,with J ∈ J ( r, k ), contains twice some ω i due to r + s = k + 1. For the same reason condition (A’)implies that a n η ∧ ω I = 0 for all I ∈ J ( s, k ). Thus, if (A’) holds for a non-zero-divisor a ∈ R (notnecessarily in I (Ω)) then (B) holds, too. Note that both statements are trivial if Ω = 0.For r = 1 the first statement in the theorem reduces to the ensuing known facts. Corollary 2.3. (i) If depth I (Ω) ≥ then any η ∈ M satisfying Ω ∧ η = 0 can be represented as η = P a i ω i with a i ∈ R .(ii) More generally, for any η ∈ Λ p M with p < depth I (Ω) the equality Ω ∧ η = 0 implies that η = P ω i ∧ γ i for some γ i ∈ Λ p − M (answer to Question 3).(iii) Let k = 1 and denote ω = ω . Then for any η ∈ Λ p M , with p < depth I ( ω ) , the equality ω ∧ η = 0 implies that η = ω ∧ γ for some γ ∈ Λ p − M . The second statement was proved in Saito [9] assuming R was Noetherian and in [6] withoutthis assumption. The third statement just means exactness of the Koszul complex M ∂ ω −→ Λ M ∂ ω −→ · · · ∂ ω −→ Λ d − M ∂ ω −→ Λ d M (cf. [4]), where d = depth I ( ω ) and ∂ ω is the operator of exterior multiplication by ω , ∂ ω : α ω ∧ α .Exactness of this complex is used in many contexts but it is usually assumed that R is Noetherian.For r = k Theorem 2.7 gives the following answer to Question 2.
Corollary 2.4.
Assume that k ≤ p ≤ k +depth I (Ω) − and let η ∈ Λ p M . Then η = ω ∧· · ·∧ ω k ∧ γ for some γ ∈ Λ p − k M if and only if η ∧ ω i = 0 for i = 1 , . . . , k . If s + p > m then ω J ∧ η ∈ Λ s + p M = 0 and condition (B) is automatically satisfied. Theinequality s + p = k + 1 − r + p > m and the depth condition p − r ≤ d − m − k ≤ p − r ≤ d −
2. This implies
Corollary 2.5. If s + p > m then condition (A’) holds true and (A) holds under the assumption m − k ≤ p − r ≤ d − . Example 2.6.
Take m = 5 , k = 4 , r = 2 , s = 3 , and p = 3 . Then if elements ω , . . . , ω ∈ M satisfy depth(Ω) ≥ then any η ∈ Λ M can be represented in the form η = P i Let ≤ p, r, s ≤ m be fixed integers such that p ≥ r and r + s = k + 1 . Consideran element η ∈ Λ p M .(i) Assuming that p − r ≤ depth I ( N ) − DC ) we have η ∈ Λ r N (cid:13)∧ Λ p − r M ( A ) if and only if Λ s N ∧ η = 0 . ( B ) (ii) Without assuming (DC) there exists n > such that if η satisfies (B) then, for any a ∈ I ( N ) , a n η ∈ Λ r N (cid:13)∧ Λ p − r M. ( A ′ ) Conversely, if (A’) holds for some non-zero-divisor a ∈ R then (B) holds, too. Note that if for ω , . . . , ω k ∈ M the ideal I (Ω) contains a non-zero-divisor then the submodule N = span R { ω , . . . , ω k } of M is free and ω , . . . , ω k is its basis. (If P i a i ω i = 0 then multiplyingboth sides by ω ∧· · · ˆ ω j · · ·∧ ω k , with ω j omitted, we get a j Ω = 0 and, as there is a non-zero-divisorin I (Ω), we get a j = 0, for all j .) roposition 2.9. Theorem 2.1 implies Theorem 2.8. Conversely, Theorem 2.8 implies Theorem2.1 weakened by the additional assumption in statement (ii) that I (Ω) contains a non-zero-divisor.Proof. To prove that Theorem 2.1 implies Theorem 2.8 note that the assumption that N is free ofrank k means that it has a basis ω , . . . , ω k ∈ M . Any two such bases are related by an invertible k × k matrix over R . This implies that whatever basis ω , . . . , ω k of N we choose the modulesΛ q N , q ≤ k , are spanned by the elements ω I with I ∈ J ( q, k ). In consequence, conditions (A),(A’), and (B) in Theorem 2.8 are equivalent to the corresponding conditions in Theorem 2.1.To show the converse implication one should use the elements ω , . . . , ω k ∈ M in Theorem 2.1to define the submodule N ⊂ M generated by them. Such submodule N is free which followsfrom the fact that I (Ω) contains a non-zero-divisor. In statement (i) of Theorem 2.1 the depthcondition (DC) implies that depth I (Ω) ≥ I (Ω). In statement(ii) the existence of such non-zero-divisor is explicitely assumed. Once N is free the statements ofTheorem 2.8 imply the corresponding statements in Theorem 2.1 since conditions (A), (A’), and(B) in both theorems are equivalent. (cid:3) Algebraic residua We will now state consequences of the results from the preceding section which will be calledtheorems on algebraic residua and on logarithmic algebraic residua. We focus our attension onexact formulations of the theorems so that the statements suggest their proofs as direct corollariesof Theorems 2.1 and 2.7.As before, we fix elements ω , . . . , ω k of the module M and, in addition, elements f , . . . , f ℓ ofthe ring R , where 1 ≤ k ≤ m and ℓ ≥ 1. Consider the quotient ring and the quotient module¯ R = R/f R + · · · + f ℓ R, ¯ M = M/f M + · · · + f ℓ M. We assume that the ideal I = f R + · · · + f ℓ R of R is proper. We can write ¯ R = R/ I , ¯ M = M/ I M and the ¯ R -module ¯ M is again free of rank m having a basis { ¯ e i } induced by the basis { e i } in M .There are canonical homomorphisms R → ¯ R , M → ¯ M and the image of an element of R or M willbe marked with a bar, in particular ω i ¯ ω i = ω i + I M . As the exterior product in Λ M inducesan exterior product in Λ ¯ M , again denoted by ′′ ∧ ′′ , we can introduce the product of all ¯ ω i ¯Ω = ¯ ω ∧ · · · ∧ ¯ ω k which is an element of Λ k ¯ M . We will denote by I ( ¯Ω) the ideal generated by the coefficients of ¯Ωwritten in the natural basis induced in Λ k ¯ M by the basis ¯ e , . . . , ¯ e m .Additionally, we introduce the ideal I ( f , . . . , f ℓ , Ω) := span R {I , I (Ω) } generated by the elements f , . . . , f ℓ and by the coefficients of the k -form Ω. Then I ⊂ I ( f , . . . , f ℓ , Ω)and I (Ω) ⊂ I ( f , . . . , f ℓ , Ω). The depth of the ideal I ( ¯Ω), to be used later, can be computed fromthe ensuing relation. Proposition 3.1. If R is Noetherian and the sequence f , . . . , f ℓ is regular then depth I = ℓ and depth I ( f , . . . , f ℓ , Ω) = depth I ( ¯Ω) + ℓ. Proof. The proof is an exercise on regular sequences and depth. The equality depth I = ℓ is aconsequence of regularity of the sequence f , . . . , f ℓ and the inequality depth I ≤ gen I true for anyproper ideal in Noetherian rings (Property 9 in Appendix). The displayed equality follows fromthe following property: if I is a proper ideal of a Noetherian ring R then any regular sequence in I can be completed to a maximal regular sequence in I of length equal to depth I (Property 7 inAppendix). Taking I = I ( f , . . . , f ℓ , Ω) and the regular sequence f · · · , f ℓ in I , we can completeit to a maximal regular sequence f · · · , f ℓ , · · · , f q with q = depth I . Then f ℓ +1 + I , . . . , f q + I isa regular sequence in I ( ¯Ω) and thus we have depth I ≤ ℓ + depth I ( ¯Ω).Vice versa, let ¯ d = depth I ( ¯Ω) and consider a regular sequence f ℓ +1 + I , . . . , f ℓ + ¯ d + I in I ( ¯Ω)with f i ∈ I (Ω). Then the sequence f , . . . , f ℓ , . . . , f ¯ d of elements of I is regular and we get theconverse inequality depth I ≥ ℓ + depth I ( ¯Ω). (cid:3) efore formulating the main result of this section (Theorem 3.4) we first state its particular caserelated to geometric residua. Theorem 3.2. (i) Assume that p ≥ k and p ≤ depth I ( ¯Ω) + k − . ( DC ) Then an element η ∈ Λ p M can be represented in the form η = ω ∧ · · · ∧ ω k ∧ γ + X j f j ξ j , ( A ) for some γ ∈ Λ p − k M and ξ j ∈ Λ p M , if and only if ω i ∧ η = X j f j β i,j ∀ i ∈ { , . . . , k } ( B ) for some β i,j ∈ Λ p +1 M .(ii) Let p ≥ k . Then without assuming the depth condition (DC) there exists n > such thatfor any η ∈ Λ p M satisfying (B) and any a ∈ I (Ω) a n η = ω ∧ · · · ∧ ω k ∧ γ + X j f j ξ j , ( A ′ ) for some γ ∈ Λ p − k M and ξ j ∈ Λ p M . Conversely, if (A’) holds for some non-zero-divisor a ∈ R then (B) holds, too.(iii) If, in the representation (A’), a is a non-zero-divisor in R or a = 1 as in (A) then the ( p − k ) -form γ on M ∗ is unique, modulo P j f j Λ p − k M , when restricted to K = T i ker ω i ⊂ M ∗ . In the last statement we do not assume that a ∈ I (Ω). There uniqueness means that for tworepresentations of the same a n η with γ ′ , ξ ′ j and γ ′′ , ξ ′′ j on the right hand side we will have( γ ′ − γ ′′ ) | K = X j f j γ j | K for some γ j ∈ Λ p − k M . The element¯ γ | K := γ | K + X j f j Λ p − k M | K , with γ appearing in the formula (A), can be called algebraic residue of η with respect to theelements ω , . . . , ω k ∈ M , modulo the ideal I ⊂ R generated by f , . . . , f ℓ . Example 3.3. Let R = O ( C n ) be the ring of holomorphic function germs at 0 ∈ C n and let f , . . . , f ℓ be its elements. Consider elements ω , . . . , ω k ∈ M = R m which can be treated as germsat 0 of holomorphic maps C n → C m . More geometrically, if C n is replaced by a complex manifold X of dimension n and C n × C m is replaced by a holomorphic vector bundle E → X with fiber F ≃ C m then we can think of R as the ring of holomorphic fuction germs at a point x ∈ X andof ω i as germs at x of holomorphic sections of E . Then M is isomorphic to the module of germsat x of holomorphic sections of the bundle E and Λ q M is isomorphic to the R -module of germs at x of holomorphic sections of the bundle Λ q E which is the q th exterior product of the bundle E .Assume additionally that the ideal I generated by the germs f j is radical or, equivalently, that anyfunction germ f vanishing on the common zeros Z = { f = · · · = f ℓ = 0 } of f j belongs to I . Inthis case the algebraic residue γ | K + P j f j Λ p − k M | K has the geometric interpretation of the field,defined on the set germ Z , of the skew-symmetric ( p − k )-form γ ( x ) ∈ Λ p − k E x defined pointwiseon the common kernel K x = ∩ j ker ω j ( x ) ⊂ Λ p − k E ∗ x , where x are regular points in Z of maximaldimension dim x Z = n − ℓ . A detailed discussion of the case where ℓ = k and ω j = df j will givenin Section 5.The following general result, including the above theorem when one takes r = k , is nothing elsebut a restatement of Theorems 2.1 and 2.7 in the quotient ring ¯ R and the quotient modules ¯ M and Λ q ¯ M ≃ Λ q M/ I Λ q M . heorem 3.4. (i) Fix integers ≤ p, r, s ≤ m such that p ≥ r and r + s = k + 1 . Assume that p − r ≤ depth I ( ¯Ω) − . ( DC ) Then an element η ∈ Λ p M can be represented in the form η = X J ∈J ( r,k ) ω J ∧ γ J + X i f i ξ i , ( A ) for some γ J ∈ Λ p − r M and ξ i ∈ Λ p M , if and only if for all I ∈ J ( s, k ) ω I ∧ η = X i f i β I,i ( B ) for some β I,i ∈ Λ p M .(ii) Fix ≤ r, s ≤ k satisfying r + s = k + 1 and let p ≥ r . Then, without assuming the depthcondition (DC), there exists n > such that for any η ∈ Λ p M satisfying (B) and any a ∈ I (Ω) a n η = X J ∈J ( r,k ) ω J ∧ γ J + X i f i ξ i , ( A ′ ) for some γ J ∈ Λ p − r M and ξ i ∈ Λ p M . Conversely, if (A’) holds for some non-zero-divisor a ∈ R then (B) holds, too.(iii) If a in the representation (A’) is a non-zero-divisor in R , or a = 1 as in (A), then theelements γ J are unique, modulo P i f i Λ p − r M , when treated as skew-symmetric ( p − r ) -linear formson M ∗ restricted to K = T i ker ω i . In statement (iii) it is not necessary that a ∈ I (Ω). The uniqueness there means that for tworepresentations of the same a n η , with γ ′ J , ξ ′ i and γ ′′ J , ξ ′′ i , respectively, we will have( γ ′ J − γ ′′ J ) | K = X i f i γ J,i | K for some γ J,i ∈ Λ p − r M and all J ∈ J ( r, k ), where we interpret γ ′ J − γ ′′ J as skew-symmetric ( p − r )-linear forms on M ∗ . Proof. Statements (i) and (ii) in the theorem follow from the corresponding statements in Theorem2.1 where the ring R should be replaced by the quotient ring ¯ R = R/ I and the modules M andΛ q M by the quotient modules ¯ M = M/ I M and Λ q ¯ M ≃ Λ q M/ I Λ q M . Namely, the reader caneasily verify that conditions (A), (A’) and (B) in Theorem 2.1 stated for the quotient modules areequivalent to the corresponding conditions in Theorem 3.4 and thus the first two statements inTheorem 3.4 are direct consequences of Theorem 2.1.Statement (iii) follows analogously from Theorem 2.7 where the original modules should bereplaced with the quotient ones. (cid:3) We will adapt Theorem 3.2 to a form which resembles theorems on multidimensional logarithmicresidua playing important role in complex analysis. Let, as earlier, R be a unital commutativering. Recall that a derivation on R is a map X : R → R which has the properties X ( f + g ) = X ( f ) + X ( g ) ,X ( f g ) = X ( f ) g + f X ( g ) , for all f, g ∈ R . Such derivations can be added, ( X + Y )( f ) = X ( f ) + Y ( f ), and multiplied byelements of R , ( aX )( f ) = aX ( f ), so that the set of all derivations Der( R ) is a module over R .Denote M = Der( R ) and assume that it is a free module of finite rank m . Any element f ∈ R defines a homomorphism df : M → R defined by df ( X ) = X ( f )and called here algebraic differential of f . We shall fix elements f , . . . , f k ∈ R and consider thecorresponding homomorphisms df , . . . df k : M → R which are elements of the dual free module, df , . . . , df k ∈ M ∗ = (Der( R )) ∗ . We will apply Theorem 3.2 to the dual module M ∗ , instead of M .Consider the quotient ring ¯ R = R/ I and the quotient module¯ M ∗ = M ∗ /f M ∗ + · · · + f k M ∗ . he ¯ R -module ¯ M ∗ is again free and has rank m . Let ¯ f i = f i + I ∈ ¯ R and d ¯ f i = df i + I M ∗ ∈ ¯ M ∗ denote the quotient counterparts (equivalence classes) of the elements f i and df i . Consider theproduct ¯Ω = d ¯ f ∧ · · · ∧ d ¯ f k which is an element of Λ k ¯ M ∗ . Let I ( ¯Ω) denote the ideal in ¯ R generated by the coefficients of ¯Ωwritten in the basis ¯ e i ∧ · · · ∧ ¯ e i k , i < · · · < i k , where ¯ e , . . . , ¯ e m is a basis in M ∗ . Theorem 3.5 ( Algebraic logarithmic residua). Assume the set M = Der( R ) of derivationsof the ring R is a free module of finite rank m .(i) Let p ≥ k and assume that p ≤ depth I ( ¯Ω) + k − . ( DC ) Then an element η ∈ Λ p M ∗ can be represented in the form η = df ∧ · · · ∧ df k ∧ γ + X j f j ξ j , ( A ) for some γ ∈ Λ p − k M ∗ and ξ j ∈ Λ p M ∗ , if and only if for all i ∈ { , . . . , k } df i ∧ η = X j f j β i,j ( B ) for some β i,j ∈ Λ p M ∗ .(ii) Let p ≥ k . Then, without assuming the depth condition (DC), there exists n > such thatfor any η ∈ Λ p M ∗ satisfying (B) and any a ∈ I (Ω) a n η = df ∧ · · · ∧ df k ∧ γ + X j f j ξ j ( A ′ ) for some γ ∈ Λ p − k M ∗ and ξ j ∈ Λ p M ∗ . Conversely, if (A’) holds for some non-zero-divisor a ∈ R then (B) holds, too.(iii) If, in the representation (A’), a is a non-zero-divisor in R or a = 1 as in (A), then theskew-symmetric ( p − k ) -linear form γ J on M is unique, modulo P j f j Λ p − k M ∗ , when restricted to K = T i ker df i ⊂ M . The theorem is an immediate corollary to Theorem 3.2. The reader may formulate an analogouscorollary to Theorem 3.4. The element¯ γ | K := γ | K + X j f j Λ p − k M ∗ | K , with γ appearing in (A), can be called logarithmic residue of η with respect to f , . . . , f k ∈ R .Note that the assumptions (DC) and p ≥ k imply that depth I ( ¯Ω) ≥ 2, thus the product f · · · f k has no repeated factors since then ¯Ω = 0.Suppose that R is an integral domain and let K be its field of fractions. Then for the R -module M = R m we have the embeddings R ⊂ K , R m ⊂ K m , Λ q R m ⊂ Λ q K m . Consider an element¯ η = ηf · · · f k ∈ Λ p K m , η ∈ Λ p R m , where the product f · · · f k has no repeated factors. A direct consequence to Theorem 3.5 is Corollary 3.6. Assume that R is an integral domain and M = Der( R ) is a free module of rank m . Then Theorem 3.5 applied for η implies equivalent statements for ¯ η with formulas (A) and (A)replaced by ¯ η = df f ∧ · · · ∧ df k f k ∧ γ + X j ξ j ˆ f j , γ ∈ Λ p − k M ∗ , ξ j ∈ Λ p M ∗ ,a n ¯ η = df f ∧ · · · ∧ df k f k ∧ γ + X j ξ j ˆ f j , γ ∈ Λ p − k M ∗ , ξ j ∈ Λ p M ∗ , where ˆ f j denotes the product f · · · f k with omitted f j . . Multidimensional logarithmic residua in complex analysis One consequence of Theorem 3.5 from the preceding section is a result in complex geometrystated below. Let N denote a complex manifold of dimension m which, without loosing generality,can be taken C m . We denote by R = O x ( N ) the ring of holomorphic function germs at a point x ∈ N (e.g. x = 0 ∈ C m ). The ring O x ( N ) is a Noetherian unique factorization domain.Consider the R -module Vec x ( N ) of germs at x of holomorphic vector fields on the manifold N and denote by Λ x ( N ) the R -module of germs at x of holomorphic one forms on N . Both modulesare free of rank m and are dual to each other by the natural pairing h ω, X i = ω ( X ) of ω ∈ Λ x N and X ∈ Vec x ( N ). By Λ qx ( N ) we denote the R -module of germs at x of holomorphic q -forms on N . Germs at x of considered objects can be replaced by their representatives defined in a smallneighbourhood of x and, if convenient, we will do this without mentioning.Fix x ∈ N and consider function germs f , . . . , f k in O x ( N ). The set of their common zeros Z f = { z ∈ N : f ( z ) = · · · = f k ( z ) = 0 } is an analytic set germ at x . Let df i denote holomorphic differentials of f i .Consider the holomorphic k-form germ Ω ∈ Λ kx ( N ) defined pointwise asΩ( z ) = df ( z ) ∧ · · · ∧ df k ( z ) . We will also use the set germ of common zeros of f = ( f , . . . , f k ) and Ω, Z f, Ω = { z ∈ N : f ( z ) = 0 , Ω( z ) = 0 } . Recall that, given an open subset U ⊂ N and an analytic subset Z ⊂ U , a point z ∈ Z is called regular if the intersection of Z with a small neighbourhood V ⊂ U of z is a complex submanifoldof U , otherwise such point is called singular . The subset of regular (resp. singular) points in Z willbe denoted Z reg (resp. S ). The set S is nowhere dense in Z . The dimension of Z at a regular point z ∈ Z is just the dimension of the submanifold Z ∩ V , denoted dim z Z . The dimension of Z at asingular point z ∈ S ⊂ Z , also denoted dim z Z , is defined as the supremum of dimensions of Z atregular points in Z in a small neighbourhood of z (exactly, dim z Z = lim sup w → z, w ∈ Z reg dim w Z ).The codimension of Z at z ∈ Z is defined as codim z Z = dim N − dim z Z .Points in Z which satisfy Ω( z ) = 0 will be called strongly regular and the set of such points will bedenoted by Z rreg . At points z ∈ Z rreg the differentials df ( z ) , . . . , df k ( z ) are linearly independent.It then follows from the holomorphic version of implicit function theorem that in a neighbourhood U of any such point z ∈ Z rreg the set of zeros Z is a submanifold of codimension k . The inclusions Z rreg ⊂ Z reg ⊂ Z become equalities in each such neighbourhood. Moreover, we have T z Z reg = ker df ( z ) ∩ · · · ∩ ker df k ( z ) , z ∈ Z rreg . ( T S ) Theorem 4.1. (i) Let p ≥ k . Assume that the sequence of germs f , . . . , f k is regular and p ≤ codim x Z f, Ω − . ( CD ) Then a holomorphic p-form η ∈ Λ px ( N ) can be written as η = df ∧ · · · ∧ df k ∧ γ + X j f j ξ j , ( A ) for some γ ∈ Λ p − kx ( N ) and ξ j ∈ Λ px ( N ) , if and only if for all i ∈ { , . . . , k } df i ∧ η = X j f j β i,j ( B ) for some β i,j ∈ Λ px ( N ) .(ii) Let p ≥ k . Then, without assuming the codimension condition (CD), there exists n > suchthat for any holomorphic p-form η ∈ Λ px ( N ) satisfying (B) and any holomorphic function germ a ∈ I (Ω) a n η = df ∧ · · · ∧ df k ∧ γ + X j f j ξ j , ( A ′ ) for some γ ∈ Λ p − kx ( N ) and ξ j ∈ Λ px ( N ) . Conversely, if (A’) holds for some nonzero a ∈ O x ( N ) then (B) holds, too. iii) The (p-k)-form γ ( z ) in (A’), as well as in (A), restricted to the complex tangent subspaces T z S reg , z ∈ Z rreg , is unique. Remark 4.2. It can be seen that statement (ii) implies Theorem 1 in Alexandrov [1] which isbasic in his exposition of fundamentals of multidimensional complex residua theory. Statement (i)seems new. Proof. Statements (i) and (ii) of the theorem follow from the corresponding statements in Theorem3.5 if we take there R = O x ( N ) - the ring of holomorphic function germs, and M = Λ x ( N ) - themodule of germs at x holomorphic one forms. Then Λ q M coincides with the R module of germsat x of holomorphic q -forms on N , Λ q M = Λ qx ( N ), and M ∗ can be identified with the moduleVec x ( M ) of germs at x of holomorphic vector fields on N . The ring O x ( N ) is Noetherian and hasno non-zero-divisors. Thus, the assumption that a in statements (ii) and (iii) of Theorem 3.5 isa non-zero-divisor is equivalent to a being nonzero. The depth condition (DC) in Theorem 3.5 isequivalent to the codimension condition (CD) above. This follows from the known fact in analyticgeometry that the codimension of the set of zeros of a family of holomorphic functions germs in O x ( N ) is equal to the depth of the ideal generated by these functions germs in O x ( N ), see e.g.Chapter 5.3 in [5]. We then obtain codim x Z f, Ω = depth I ( f , . . . , f k , Ω) = k + depth I ( ¯Ω) where inthe second equality we use Proposition 3.1. Thus p ≤ codim x Z f, Ω − p − k ≤ depth I ( ¯Ω) − a n η with γ ′ and γ ′′ on the right side. Subtracting one from the other gives zero on the leftside. Additionally, at points z ∈ Z the functions f i vanish thus, denoting ∆ γ = γ ′ − γ ′′ , we findthat df ( z ) ∧ · · · ∧ df k ( z ) ∧ ∆ γ ( z ) = 0 , z ∈ Z. At strongly regular points z ∈ Z rreg the one forms df ( z ) , . . . , df k ( z ) are linearly independent.Thus, on their common kernel K ( z ) = ker df ( z ) ∩ · · · ∩ ker df k ( z ), the ( m − k )-form ∆( z ) vanishes(linear algebra). This proves that γ ′ and γ ′′ restricted to the subspace K ( z ) coincide at points z ∈ Z rreg . By formula (TS) we have K ( z ) = T z S reg , thus γ ′ and γ ′′ coincide on T z S reg . (cid:3) Proof of Theorem 2.1 Throughout the proof R denotes a commutative ring with a unit an M is a free module over R of rank m . The starting point is the following elementary version of statement (i) in Theorem 2.7. Lemma 5.1. If the elements ω , . . . , ω k ∈ M can be completed to a basis in M then for arbitrary p ≥ and any η ∈ Λ p M condition (B) implies condition (A).Proof. Fix p , r and s as in Theorem 2.1. Let elements α , . . . , α m − k ∈ M complete ω , . . . .ω k toa basis of M . Then we have a basis of Λ p M which consists of exterior products of the form ω J ∧ α K = ω j ∧ · · · ∧ ω j h J i ∧ α k ∧ · · · ∧ α k h K i , with strictly monotone (possibly empty) multiindices J and K such that their lengths h J i and h K i satisfy h J i + h K i = p (see notation introduced in Section 2). Suppose that η ∈ Λ p M is written asa linear combination of the elements of this basis, η = X J,K a J,K ω J ∧ α K . ( E )Condition (B) states that ω I ∧ η = 0 for all strictly monotone multiindices I of length h I i = s . Wehave ω I ∧ η = X J,K a J,K ω I ∧ ω J ∧ α K . ( E )The products ω I ∧ ω J ∧ α K are elements of a basis in Λ p + s M whenever ω I and ω J do not containthe same ω i , i.e., when I T J = ∅ with I and J treated as sets. Thus, condition ω I ∧ η = 0 impliesthat a J,K = 0 for J such that I T J = ∅ . Given J with h J i < r , we can always find I with h I i = s , s + r = k + 1, such that J ∪ I = ∅ , thus a J,K = 0 for any ( J, K ) with h J i < r . This means thatall nonzero coefficients a J,K in the expansion (E) have h J i ≥ r which is equivalent to the fact thatcondition (A) holds. (cid:3) n the general proof we will use the ensuing special case of r = 1 in statement (i) of Theorem2.1 which was proved in [9] for R Noetherian and in [6], Theorem 2.3, without this assumption. Lemma 5.2. If p < depth I (Ω) and η ∈ Λ p M satisfies Ω ∧ η = 0 then η = P ω i ∧ γ i for some γ , . . . , γ k ∈ Λ p − M . We shall also need the following known fact (for an easy proof see e.g. [6], Lemma 2.6) Lemma 5.3. If a sequence a , . . . , a r ∈ R is regular then, for any n ≥ , the sequence a n , a , . . . , a r is also regular. Proof of Theorem 2.1. Statement (ii). We first prove the second statement which will beused for proving the first one. It is enough to prove that condition (A’) holds for the coefficientsof Ω. Indeed, if it holds for any such coefficient with some power n then it holds for an arbitraryelement of I (Ω) (which is a finite linear combinations of the coefficients) with a sufficiently largepower depending on n and on the length of the linear combination.Let a be a coefficient of Ω. We may assume that a is not nilpotent, otherwise condition (A’)trivially holds. Consider the multiplicative set of nonnegative powers of a , A = { a i } i ≥ where a = 1. Let R [ a ] denote the localization of R with respect to A . Elements of the ring R [ a ] canbe represented as sums of “fractions” of the form b/a i , b ∈ R . Similarly, let M [ a ] denote thelocalization of M with respect to A , with elements represented as sums of “fractions” of the form m/a i , m ∈ M . Then M [ a ] is a module over R [ a ] . Analogously, the modules Λ p M can be localizedwith respect to the multiplicative set A and these localizations are isomorphic to Λ p M [ a ] . We havecanonical homomorphisms R → R [ a ] and M → M [ a ] given by the transformations b [ b ] := b/ ω [ ω ] := ω/ 1. In particular, the basis e , . . . , e m of M is transformed into the basis[ e ] , . . . , [ e m ] of M [ a ] and the basis e I , I ∈ J ( p, m ) of Λ p M is transformed into the basis [ e I ] ofΛ p M [ a ] , where [ e I ] = [ e i ] ∧ · · · ∧ [ e i p ].The image [ a ] of a coefficient a of Ω under the homomorphism R → R [ a ] is a coefficient of[Ω] = [ ω ] ∧ · · · ∧ [ ω k ] and is a unit in R [ a ] . This implies that the elements [ ω ] , . . . , [ ω k ] can becompleted to a basis in M [ a ] . Therefore, we can use Lemma 5.1 for the elements [ ω ] , . . . , [ ω k ]of M [ a ] and [ η ] ∈ Λ p M [ a ] . Condition (B) satisfied for the elements ω , . . . , ω k of M , η ∈ Λ p M and ω I ∈ Λ s M implies that it is satisfied for the corresponding elements [ ω ] , . . . , [ ω k ] of M [ a ] ,[ η ] ∈ Λ p M [ a ] , and [ ω I ] = [ ω i ] ∧ · · · ∧ [ ω i s ]. It follows from the lemma that[ η ] = X J ∈J ( r,k ) [ ω J ] ∧ ˜ γ J , ˜ γ J ∈ Λ r M [ a ] , where [ ω J ] = [ ω j ] ∧ · · · ∧ [ ω j r ]. One can write ˜ γ J = γ J /a n for some γ J ∈ Λ r M and n ≥ 0, thesame for all J . This implies that equality (A’) in Theorem 2.7 holds. Statement (i). By Remark 2.2 it is enough to prove that condition (B) implies (A). We willuse induction with respect to 1 ≤ r ≤ min { k, p } , in decreasing order. The case r = k ≤ p. In this case we have s = 1, thus condition (B) means that ω i ∧ η = 0for all 1 ≤ i ≤ k . Let a , . . . , a d ∈ I (Ω) be a regular sequence, d = depth I (Ω), then a n , a , . . . , a k is also regular for any n > 0, by Lemma 5.3. From statment (ii) it follows that the exists n suchthat, for a = a n , aη = Ω ∧ γ, for some γ ∈ Λ p − k M. Consider the quotient ring ¯ R = R/aR , the quotient module ¯ M = M/aM , and the modules Λ q ¯ M ≃ Λ q M/a Λ q M , for any q . Then we have natural homomorphisms b ∈ R ¯ b ∈ ¯ R , ω ∈ M ¯ ω ∈ ¯ M ,and η ∈ Λ p M ¯ η ∈ Λ p ¯ M . Applying these homomorphisms to the above equality gives0 = ¯Ω ∧ ¯ γ, with ¯ γ ∈ Λ p − k ¯ M where ¯Ω = ¯ ω ∧ · · · ∧ ¯ ω k . The sequence ¯ a , . . . , ¯ a d ∈ I ( ¯Ω) is regular in ¯ R , thus depth I ( ¯Ω) ≥ d − d − > p − r by the depth assumption (DC). It follows then that depth I ( ¯Ω) >p − r = p − k . We can now apply Lemma 5.2 for the quotient module, with ¯ γ playing the role of η , to deduce that ¯ γ = X ¯ ω i ∧ ¯ γ i with some ¯ γ i ∈ Λ p − k − ¯ M . ifting this equality to the original module gives γ = X ω i ∧ γ i + aξ, with some γ i ∈ Λ p − k − and ξ ∈ Λ p − k M . Plugging such γ to the above formula for aη gives aη − a Ω ∧ ξ = Ω ∧ (cid:16)X ω i ∧ γ i (cid:17) and then a ( η − Ω ∧ ξ ) = 0 , since Ω ∧ ω i = 0. Using the fact that a = a n was a non-zero-divisor we deduce that η = Ω ∧ ξ ,which was to be proved. The case r = p ≤ k. By condition (DC) we have that depth I (Ω) ≥ 2, thus there is a regularsequence a , a ∈ I (Ω). It follows fom Lemma 5.3 that a n , a is also a regular sequence, for any n ≥ 1, and this fact will be used below.By statement (ii) of the theorem condition (B) implies condition (A’) which means, in particular,that there exist n ≥ b J ∈ R such that a n η = X J b J ω J , where J ∈ J ( r, k ). Denote a = a n and consider the quotient ring ¯ R = R/aR and the quotientmodule ¯ M = M/aM . After passing to the quotients the above equality reads0 = X J ¯ b J ¯ ω J , ¯ b J ∈ ¯ R, ¯ ω J ∈ Λ r ¯ M , with ¯ ω J = ¯ ω j ∧ · · · ∧ ¯ ω j r . For a given J ∈ J ( r, k ) let J ′ ∈ J ( k − r, k ) denote the multiindex whichis complementary to J , i.e., J T J ′ = ∅ . Then multiplying both sides of the above equality by ¯ ω J ′ gives 0 = ¯ b J ¯ ω J ∧ ¯ ω J ′ = ± ¯ b J ¯Ωwhere ¯Ω = ¯ ω ∧ · · · ∧ ¯ ω k . Since the ideal I ( ¯Ω) contains the non-zero-divisor ¯ a (the sequence a n , a was regular), we deduce that all ¯ b J = 0. This means that b J = ac J = a n c J , for some c J ∈ R .Plugging such b J to the formula for a n η gives a n η = a n X J c J ω J , c J ∈ R. The fact that a n is a non-zero-divisor in R implies that η = P J c J ω J , which was to be shown. The general case. To prove statement (i) for all r < min { k, p } assume it is true for r + 1.Denote d = depth I (Ω) and let a , . . . , a d be a regular sequence in I (Ω) where, by our assumption, d ≥ p − r + 2. By Lemma 5.3 the sequence a n , a , . . . , a d is also regular. Condition (B) impliescondition (A’), by statement (ii), thus for η satisfying (B) we have a n η = X J ∈J ( r,k ) ω J ∧ γ J for some n ≥ γ J ∈ Λ p − r ( M ).Denote a = a n . As earlier, we introduce the quotient ring ¯ R = R/aR and the quotient modules¯ M = M/aM , Λ i ¯ M ≃ Λ i M/a Λ i M . Under canonical homomorphisms elements of the original ringand modules have their canonical images, denotesd with bars, in the quotient ring and modules.For ¯Ω = ¯ ω ∧ · · · ∧ ¯ ω k we have depth I ( ¯Ω) ≥ d − a , . . . , ¯ a d ∈ I ( ¯Ω) is regular.When replacing the elements in the above equality with their counterparts in the quotient objectswe get zero on the left side, thus 0 = X J ∈J ( r,k ) ¯ ω J ∧ ¯ γ J where ¯ ω J = ¯ ω j ∧· · · ∧ ¯ ω j r and ¯ γ J ∈ Λ p − r ¯ M . Pick a multiindex J ∈ J ( r, k ) and let J ′ ∈ J ( r ′ , k ) beits complement satisfying r ′ + r = k and J T J ′ = ∅ . Multiplying both sides of the above equalityby ¯ ω J ′ we obtain 0 = ¯ ω J ′ ∧ ¯ ω J ∧ ¯ γ J = ± ¯Ω ∧ ¯ γ J , ince all other products in the sum vanish as they contain a repeated ¯ ω i . We can now use Lemma5.2 with ¯ γ J ∈ Λ p − r ¯ M playing the role of η and Ω replaced with ¯Ω. Namely, p − r < depth I (Ω) − d = depth I (Ω) ≤ depth I ( ¯Ω) + 1, by the inequality mentionedearlier. Therefore the asumption p − r < depth I ( ¯Ω) required in the lemma is satisfied and wededuce that ¯ γ J = X i ¯ ω i ∧ ¯ γ J,i for some ¯ γ J,i ∈ Λ p − r − ¯ M .The above equaliy can be written in the original modules as γ J = X i ω i ∧ γ J,i + aξ J , for some γ J,i ∈ Λ p − r − M and ξ J ∈ Λ p − r M . Plugging such γ J to the formula for a n η and takinginto account that a = a n we obtain a n ( η − X J ω J ∧ ξ J ) = X J X i ω J ∧ ω i ∧ γ J,i . Multiplying both sides by ω I ′ , with arbitrary I ′ ∈ J ( k − r, k ), we find that a n ( η − X J ω J ∧ ξ J ) ∧ ω I ′ = 0since the product ω J ∧ ω i ∧ ω I ′ vanishes as it has a repeated ω j for some j ∈ { , . . . , k } . Since a n is a non-zero-divisor we deduce that( η − X J ω J ∧ ξ J ) ∧ ω I ′ = 0 , for any I ′ ∈ J ( k − r, k ). This means that the form η − P J ω J ∧ ξ J satisfies condition (B) with s ′ = k − r . Take r ′ = r + 1, then r ′ + s ′ = k + 1 and we can use the induction assumption on r as weassumed that statement (i) holds for r + 1 = r ′ . We conclude that the element η ′ = η − P ω J ∧ ξ J has a representation η − X J ω J ∧ ξ J = X ˆ J ω ˆ J ∧ ˆ γ ˆ J with ˆ J in J ( r + 1 , k ) and some ˆ γ ˆ J ∈ Λ p − r − ¯ M . Since ω ˆ J are products of r + 1 forms among ω , . . . , ω k , the sum on the right side can be rearranged to a sum P J ω J ∧ e γ J , with J ∈ J ( r, k )and e γ J ∈ Λ p − r M . This implies that η can be represented in the form (A) which ends the proof. ✷ Appendix: properties of the depth Let R be a commutative ring with unit. Recall that a sequence of elements a , . . . , a q of R iscalled regular if ( a , . . . , a q ) = R and a i is a non-zero-divisor on R/ ( a , . . . , a i − ) for i = 1 , . . . , q (in particular, a is a non-zero-divisor on R ). Here ( a , . . . , a i ) denotes the ideal in R generatedby the elements a , . . . , a i .Given a proper ideal I ⊂ R , the depth of I denoted depth I is the supremum of lengths of regularsequences in I . Additionally, one defines depth R = ∞ . Below we list several properties which canbe useful when verifying the depth condition (DC).(1) If I , I are ideals in R and I ⊂ I then depth I ≤ depth I (trivial).(2) If a , . . . , a r ∈ I is a regular sequence then depth I ≥ depth I/ ( a , . . . , a q ) + q .(3) If for a fixed i the sequences a , . . . , a i − , b, a i +1 , . . . , a r and a , . . . , a i − , c, a i +1 , . . . , a r are regular, and a i = bc , then a , . . . , a i − , a i , a i +1 , . . . , a r is regular. Vice versa, if a , . . . , a i , . . . , a r is regular, with a i = bc , and ( a , . . . , a i − , b, a i +1 , . . . , a r ) R = R then a , . . . , a i − , b, a i +1 , . . . , a r is regular.(4) A sequence a , . . . , a r is regular if and only if a i , . . . , a i r r is regular for given i , . . . , i r ≥ I is equal to the depth of the radical of I .For Propertites (3) and (4), see e.g. [3], Exercises in Chapter 7.1.For R Noetherian we have the following additional properties (cf. e.g. [4] or [3], Chapter 7.1).(5) If a , . . . , a r is a maximal (with respect to inclusion) regular sequence in I then r = depth I . 6) Any regular sequence a , . . . , a q in a proper ideal I ⊂ R can be completed to a maximalregular sequence a , . . . , a q , . . . , a r in I .(7) If a , . . . , a q ∈ I is a regular sequence then depth I = depth( I/ ( a , . . . , a q )) + q (followsfrom (5) and (6)).(8) If R is local then any permutation of a regular sequence is regular.(9) The depth, the hight and the minimal number of generators of I , denoted gen I , satisfythe inequalities depth I ≤ height I ≤ gen I .(10) depth I = gen I if and only if I is generated by a regular sequence.(11) If R is a Cohen-Macauley ring then depth I = height I for any ideal I ∈ R (and vice versa).(12) If ω = ( a . . . , a r ) ∈ R r and the ideal I in R generated by a , . . . , a r is proper and nonzerothen depth I is equal to the maximal p such that i th cohomology H i : Im ∂ i − / ker ∂ i inthe Koszul complex ∂ i : Λ i R r → Λ i +1 R r defined by η ω ∧ η vanishes for all i < p . References [1] A.G. Aleksandrov, Multidimensional residue theory and the logarithmic de Rham complex, J. of Singularities Vol. 5 (2012), 1-18.[2] A.G. Aleksandrov, Residues of Logarithmic Differential Forms in Complex Analysis and Geometry, Anal.Theory Appl. , Vol. 30, No. 1 (2014), 34-50.[3] S. Balcerzyk, T. J´ozefiak, Commutative Rings; Dimension, Multiplicity, and Homological Methods, PolishScientific Publishers, Warsaw, 1989.[4] D. Eisenbud, Commutative Algebra, Springer Verlag 1994.[5] P. Griffiths, J. Harris, Principles of Algebraic Geometry, Wiley and Sons, New York, 1978.[6] B. Jakubczyk, Exterior multiplication with singularities: a Saito theorem in vector bundles, Ann. Polon.Math. Vol. 125, No.2 (2020), 117-138.[7] B. Jakubczyk, M. Zhitomirskii, Local reduction theorems and invariants for singular contact structures, Ann. Inst. Fourier , Vol. 51 (2001), 237–295.[8] Mihnea Popa, Christian Schnell, Kodaira dimension and zeros of holomorphic one-forms, Ann. of Math. Ann. Inst. Fourier , Vol. 26, No.2 (1976), 165-170., Vol. 26, No.2 (1976), 165-170.