Koszul complexes and relative homological algebra of functors over posets

Under certain conditions, Koszul complexes can be used to calculate relative Betti diagrams of vector space-valued functors indexed by a poset, without the explicit computation of global minimal relative resolutions. In relative homological algebra of such functors, free functors are replaced by an arbitrary family of functors. Relative Betti diagrams encode the multiplicities of these functors in minimal relative resolutions. In this article we provide conditions under which grading the chosen family of functors leads to explicit Koszul complexes whose homology dimensions are the relative Betti diagrams, thus giving a scheme for the computation of these numerical descriptors.

1. Introduction 1.1.Relative homological algebra and Koszul complexes.Recently, there has been intense research activity [1,5,6] relating topological data analysis (TDA) with relative homological algebra [3,4,10,11,13] of finite dimensional vector space representations of finite posets.One way of thinking about such representations is as modules over certain finite dimensional algebras, allowing to phrase homological properties of modules in terms of homological properties of the algebra.This is also a common way for expressing relative homological properties of representations.In this case, endomorphism algebras of some chosen modules are often utilized (see e.g.[3,5]).We however take another perspective, central in TDA, and study representations as functors.Local Koszul complexes [9] (see also [18,Sect. 6] and [20,Sect. 1]) are the key reasons why viewing representations as functors is particularly convenient in TDA.In favourable situations, these complexes can be used for calculations of Betti diagrams without explicitly constructing global minimal resolutions, especially if we have some control over the sets of parents of elements in the indexing poset.
The aim of this article is to show that Koszul complexes can also be used to calculate relative Betti diagrams.To do this, we translate from a relative homological algebra of functors indexed by a given poset to the standard homological algebra of functors indexed by a new poset.We are interested in translations for which the homology of the Koszul complexes over the new poset computes the relative Betti diagrams in a way that can be implemented in software (see for example [23]).One of our main contributions in this article is a translation construction which allows us to characterize the set of poset elements (called the degeneracy locus) where local Koszul complexes fail to calculate the relative Betti diagrams.Importantly, in all examples that are currently relevant for TDA the degeneracy loci are negligible.
Aiming at a self-contained presentation, and believing that the tools we introduce may be of interest to a broader audience, we favor an explicit classical formulation of relative homological algebra [10].
1.2.Relative homological approximations.One way of understanding an object is by approximating it.In this article we focus on approximations coming from relative homological algebra (see [13]), where arbitrary objects in an abelian category are approximated by finite direct sums of objects of a chosen collection P. For example, one may ask whether a given object M is isomorphic to a finite direct sum of elements in P. To answer this question we look at the category P ↓ M of morphisms from finite direct sums of elements in P into M .We are interested in situations where this category has a certain distinguished object C 0 → M : for example, a terminal object.However, cases where the category P ↓ M has a terminal object are not that common.More common is the existence of C 0 → M in the category P ↓ M satisfying the following two conditions.The first condition requires that every object in P ↓ M maps to C 0 → M , in which case C 0 → M is called a P-epimorphism (see 2.2).This of course holds if C 0 → M is terminal in P ↓ M , where not only the existence but also the uniqueness of morphisms into C 0 → M is required.The second condition substitutes this uniqueness by a minimality condition: every endomorphism of C 0 → M is an isomorphism.A morphism C 0 → M satisfying these two conditions (being a minimal P-epimorphism) is called a minimal P-cover of M (see 2.4).If it exists, then the minimal P-cover of M is unique up to isomorphism.We regard the minimal P-cover of M as an approximation of M by finite direct sums of elements in P. In particular, M is isomorphic to such a direct sum if, and only if, its minimal P-cover is an isomorphism.
If finite direct sums of elements in P are uniquely determined by the multiplicities of their summands, then we say that P is independent (see 2.1).In this case, there is a unique function β 0 P M : P → N whose value at A in P is the multiplicity of A in the minimal P-cover C 0 of M .The function β 0 P M is called the 0 th P-Betti diagram of M and is a numerical descriptor of the approximation of M by finite direct sums of elements in P.This is only the 0 th step.We can continue by considering the minimal P-cover of the kernel of C 0 → M .By doing this inductively we obtain a chain complex • • • → C n → • • • → C 0 → M called a minimal P-resolution of M .Each object C d in this complex is described by a function β d P M : P → N called the d th P-Betti diagram of M , where, for all A in P, the number β d P M (A) is the multiplicity of A in C d (see 2.6).
Thus every independent collection P, for which all objects admit minimal P-covers, leads to numerical invariants in the form of P-Betti diagrams.The focus of this article is to provide a method of constructing such collections P and to discuss a strategy to effectively compute the associated Betti diagrams in the category Fun(I, vect K ) of functors indexed by a finite poset I with values in finite dimensional K-vector spaces.
1.3.Related work.Since vector space valued functors indexed by posets have played an important role across many areas of mathematics, their categories are well studied, and in particular their homological properties are well understood: see for example [3] and references therein, where such functors are interpreted as modules over finite dimensional algebras.Recently there has also been a growing interest in independent collections in Fun(I, vect K ) [6,2,5] and associated numerical invariants coming from the TDA community, where such functors are commonly called persistence modules, and the poset I is usually R n with the product order.In particular, there are several publications and implementations of various algorithms to calculate numerical homological invariants of such functors (e.g.classical Betti diagrams, rank functions, generalized Euler characteristics) and using them in applications, see for example [15,16,17,21,22,24].
The idea of approximating persistence modules by objects in a chosen collection appears in [14,1,6,2,5].In [14], without relying on homological algebra, approximations in terms of the intervals in the poset are shown to yield a generalization of rank functions, an invariant of persistence modules.In [1], the authors approximate persistence modules in a Grothendieck group generated by interval modules (called spread modules in [5] and in our work, see Example D) using quiver representation theory without resorting to homological algebra methods.Decompositions of rank functions into linear combinations of rank functions of simpler modules, such as rectangles and lower hooks, are explained in [6].Importantly, one such decomposition (called minimal rank decomposition) is defined via relative homological algebra of persistence modules with respect to the collection of lower hooks, which we consider in Example F. In [2], relative homological algebra with respect to interval modules is studied using the language of representations of finite dimensional algebras.For homological algebra results for persistence modules over more general posets, we can also look at [8] and [19].In the former, the authors prove various standard homological algebra results for persistence modules, with a focus on the one-parameter case (where the poset is the real line).In the latter, the author studies presentations of persistence modules, with focus on infinite posets, and concludes with a Hilbert syzygy-type theorem.The authors of [5] present a way of obtaining relative projective resolutions from standard projective resolutions by considering modules over the ring End( P) op and finding a resolution of hom( P, M ).This gives an automatic correspondence of exact structures, but there is no reason for the standard projective resolution of hom( P, M ) to be more easily computed than the relative projective resolution of M .
Our original motivation was to find a homological interpretation of the approximations presented in these articles that would allow for explicit calculations.This article builds on the realization that local Koszul complexes can be used for this purpose.Koszul complexes of modules are well studied, see e.g.[18,Sect. 6].Koszul complexes of graded modules, as presented in [20], have a direct interpretation in terms of persistence modules, based on the equivalence between the category of n-graded modules over the polynomial ring S := K[x 1 , . . ., x n ] and the category Fun(N n , vect K ), where N n is endowed with the product order.Here we sketch this connection, referring to [12,Sect. 3] for more details.The Koszul complex K as defined for example in [20,Def. 1.26] is a minimal free resolution of the field K in the category of n-graded S-modules [20,Prop. 1.28].Given an n-graded S-module M , the Koszul complex of M at a ∈ N n is the grade a part of the n-graded chain complex M ⊗ S K. Its d th homology module is the grade a part of the n-graded S-module Tor S i (M, K), whose dimension is the d th Betti diagram (with respect to the standard projectives) of M at a.This value of the Betti diagram can be computed, for a grade a ∈ N n , by looking only at M restricted to the grades in a small subposet of N n isomorphic to {0 < 1} k for some k ≤ n, whose elements are a together with its parents and all their meets.In this work, we study local Koszul complexes of functors indexed by more general finite posets (see 3.7), more details on which can be found in [9], and use them in the context of relative homological algebra.
The above approach starts with a global resolution of K and considers local parts of M ⊗ S K at different grades.Such a global resolution however may not have an explicit formula, which typically would require global regularity assumptions on the indexing poset.Requiring local regularity only at some grades is a much weaker assumption.In such grades we have an explicit Koszul complex construction which we exploit.For that purpose, the functorial perspective is more advantageous than the language of representations of finite dimensional algebras used in [6,2,5].Our aim is to use Koszul complexes not only as a theoretical tool, but as a computable construction to determine relative Betti diagrams.
1.4.Koszul complexes and thinness.Homological algebra characterizes relevant objects by universal properties, regardless of the category to which they belong, instead of by concrete constructions which may vary greatly with the category.This offers a powerful language in which to express and solve mathematical problems.However, extracting calculable invariants, which is essential in TDA, from universally defined objects is often difficult, costly, and sometimes not even feasible.Therefore the challenge is to introduce a version of homological algebra which retains some conceptual power while allowing for some of the relevant invariants to be described explicitly and algorithmically.
We strike a balance between mathematical and computational effectiveness by using a strategy built on the following two observations.The first is related to the standard independent collection S = {K(a, −) | a ∈ I} consisting of all free functors on one generator (see beginning of Section 3).This collection is always independent and, when I is finite, every functor in Fun(I, vect K ) admits a minimal S-resolution.The first key observation is that under additional assumptions, for example that I is an upper semilattice, for every element a in I and every functor F : I → vect K , there exists an explicit chain complex K a F , called Koszul complex (see 3.7 and discussion in 1.3), whose homology in degree d has dimension equal to the value of the Betti diagram β d S F (K(a, −)) (see Theorem 3.8 and [9, Section 10]).Because of its explicitness, the Koszul complex is an effective tool for computing the values of the Betti diagram with respect to the collection S.
To have an analogous construction for a more general collection P in Fun(I, vect K ), we require a grading of its elements.Thus, instead of an unstructured collection, our starting point is a functor T : J op → Fun(I, vect K ) indexed by a finite poset (J, ).Such a functor leads to a collection P := {T (a) | a ∈ J, T (a) = 0}.This grading allows for a translation between functors indexed by I and functors indexed by J via a pair of standard adjoint functors (see 5.2 and 5.3): where R := Nat I (T , −) assigns to M : I → vect K the functor Nat I (T (−), M ) : J → vect K .The functor T is called thin if the unit of this adjunction η a : K(a, −) → RLK(a, −) is an epimorphism for every a in J.The second key observation is recognizing the importance of the thinness assumption.This assumption enables us to translate between the homological properties of M in Fun(I, vect K ) relative to P and the standard homological properties of RM in Fun(J, vect K ).
1.5.Results.Let I and J be finite posets, and T : J op → Fun(I, vect K ) a thin functor.We give a self-contained proof that the collection P = {T (a) | a ∈ J, T (a) = 0} is independent (5.13).Moreover, we prove that the adjunction between L and R leads to a one-to-one correspondence between minimal covers: for every functor M in Fun(I, vect K ), a natural transformation C 0 → RM is a minimal cover in Fun(J, vect K ) if, and only if, its adjoint LC 0 → M is a minimal P-cover in Fun(I, vect K ) (Theorem 5.14).Although we show that every functor M in Fun(I, vect K ) admits a minimal P-resolution (Corollary 5.16(1)), this resolution cannot in general be obtained via the adjunction.This means that in general the P-Betti diagrams of M differ from the standard Betti diagrams of RM .Elements of J where this happens are called degenerate and form the degeneracy locus (5.18).If J is an upper semilattice, our main result (Corollary 5.20) identifies elements outside the degeneracy locus, for which we can use Koszul complexes to calculate the values of P-Betti diagrams.From a computational perspective, we obtain Algorithm 1 for computing relative Betti diagrams, given a thin functor.// parents defined in 3.7 Furthermore, we produce examples of thin functors T : J op → Fun(I, vect K ), and we apply our results to explicitly calculate the associated relative Betti diagrams.We observe how numerical invariants introduced in other works, such as the rank invariant over lower hooks of [6] (Example F) and the single-source homological spread invariant of [5] (Example E), are included in our general framework.1.6.Organization.We begin in Section 2 by introducing the theory of relative homological algebra, including the notions of independence and acyclicity, covers and resolutions.Next, in Section 3 we recall standard homological algebra for functors indexed by finite posets.In particular, when the poset is an upper semilattice, we show how to compute Betti diagrams explicitly via Koszul complexes.In Section 4, we discuss in more depth resolutions and Betti diagrams of certain functors called filtrations.In particular, we study subfunctors, and discuss several important notions: supports of functors, upsets, and antichains.
Section 5 is the central part of this work, where we introduce gradings of the form T : J op → Fun(I, vect K ).We define the key properties of thinness and flatness, which will allow us to compute relative Betti diagrams over I from standard Betti diagrams over J.
Finally, we connect to other works in Section 6 by exhibiting how rectangles, lower hooks, and spread modules, among others, are handled in our framework.
Acknowledgements.We would like to thank Luis Scoccola for valuable discussions and the anonymous referees for their extensive reports and constructive suggestions.

Relative homological algebra
In this section we recall a setup for relative homological algebra in an abelian category M.This subject, initiated by Hochschild [13], has been extensively developed in the context of the representation theory of finite dimensional algebras (see for example [3,4]).The explicit and self-contained treatment we provide here has the advantage of not assuming familiarity of the reader with such theory.We fix a collection P of objects in M. What follows are fundamental notions of homological algebra relative to P.

2.1.
Freeness.An object in M is called P-free if it is isomorphic to a finite direct sum of elements in P. For example, the zero object is P-free.The collection P is called independent if, for every P-free object F , there is a unique function β : P → N whose support supp β := {A | β(A) = 0} is finite and for which F is isomorphic to A∈P A β(A) .If P is independent, then all of its elements are nonzero, and if two of its elements are isomorphic, then they must be equal.For example, if M is a Krull-Schmidt category and P is a collection of indecomposables, then it is independent if and only if no two elements in P are isomorphic.
2.2.Exactness.Two composable morphisms f : M → N and g : N → L in M are said to form a P-exact sequence if the following sequence of abelian groups is exact for every A in P: of its consecutive morphisms form a P-exact sequence.
If the sequence M f − → N → 0 is P-exact, then f is called a P-epimorphism.A morphism f : M → N is a P-epimorphism if, and only if, hom(A, f ) : hom(A, M ) → hom(A, N ) is surjective for every A in P (i.e.every g : A → N can be expressed as the composition of some morphism A → M with f ).

2.3.
Projectives.An object C 0 in M is called P-projective if, for every f : M → N and g : N → L that form a P-exact sequence, the following is an exact sequence of abelian groups: Note that if M is P-projective, then f : M → N and g : N → L form a P-exact sequence if, and only if, the composition gf is null and the induced morphism M → ker(g) is a P-epimorphism.Every P-free object is P-projective.If every P-projective is P-free, then the collection P is called acyclic.

2.4.
Covers.A P-cover of an object M is a P-epimorphism C 0 → M where C 0 is P-projective.
We say that the category M has enough P-projectives if every object has a P-cover.
A morphism between a P-cover C 0 → M and a P-cover D 0 → M is a morphism f : C 0 → D 0 in M for which the following diagram commutes: There is always a morphism between two P-covers C 0 → M and D 0 → M , since C 0 is P-projective and D 0 → M is a P-epimorphism.
A P-cover C 0 → M is called minimal if all its endomorphisms are isomorphisms.Every two minimal P-covers of M are isomorphic: if C 0 → M ← D 0 are two minimal P-covers, then in any sequence C 0 → D 0 → C 0 → D 0 of morphisms of the P-covers, the first and second ones compose into an isomorphism by minimality, as well as the second and third ones, which implies that all these morphisms are isomorphisms.2.5.Resolutions.A sequence of composable morphisms in M of the form it is P-exact and, for every d ≥ 0, the object C d is Pprojective.Equivalently, the sequence A P-resolution of M is also written as a morphism of chain complexes, by C → M , where M denotes a chain complex concentrated in degree 0 and C = (• • • → C 1 → C 0 ).An object M for which there is a P-resolution C → M is called P-resolvable.If there are enough P-projectives, then all objects are P-resolvable.
A morphism from a P-resolution C → M to a P-resolution D → M is a sequence of morphisms f d : C d → D d for d ≥ 0 for which the following diagram commutes: Such a morphism is an isomorphism if f d is an isomorphism for every d ≥ 0. As with P-covers, there is always a morphism between every two P-resolutions of M .A P-resolution C → M is called minimal if all of its endomorphisms are isomorphisms.A P-resolution C → M is minimal if, and only if, for every d ≥ 0, the morphism ) is a minimal P-cover, where C −1 = M and C −2 = 0. Thus, if every object has a minimal P-cover, then every object has a minimal P-resolution that can be constructed inductively by taking minimal P-covers of successive kernels.Every two minimal P-resolutions of M are isomorphic.
2.6.Betti diagrams.Suppose P is independent (see 2.1) and acyclic (see 2.3).Let C → M be a minimal P-resolution.By acyclicity of P, for every d ≥ 0, the object C d is P-free.Thus, by independence of P, there is a unique function, denoted by β d P M : P → N and called the Since any two minimal P-resolutions of M are isomorphic, the P-Betti diagrams of M do not depend on the choice of a minimal P-resolution, and are invariants of the isomorphism type of M .One should note however that P-Betti diagrams are only defined for objects that have minimal P-resolutions.
To calculate the values of the d th P-Betti diagram of M , a standard strategy is to develop tools to calculate the 0 th P-Betti diagram and then use the fact that, for a minimal resolution C → M , we have the following sequence of equalities: Building a minimal resolution is an inductive procedure involving constructing minimal Pcovers of successive kernels, which typically requires a description of the entire 0 th P-Betti diagram of these kernels.One approach to calculating Betti diagrams that avoids this costly inductive procedure is the construction, given an object M in M and an element A in P, of a chain complex Ideally, this construction is systematic, or even functorial in M .This is the case for standard homological algebra for functors indexed by finite upper semilattices, discussed in the next section, where such chain complexes are given by Koszul complexes.

Standard homological algebra
Fix a field K, choose a finite poset (J, ), and consider the abelian category Fun(J, vect K ) whose objects are functors indexed by J with values in the category vect K of finite dimensional vector spaces.Morphisms in Fun(J, vect K ) are the natural transformations.For an element a in J, denote by K(a, −) : J → vect K the composition of the following functors, where the symbol set denotes the category of finite sets: The functor K(a, −) is called free on one generator in degree a.
In this section we are going to describe the homological algebra of Fun(J, vect K ), as presented in Section 2, relative to the collection S := {K(a, −) | a ∈ J}.This corresponds to standard homological algebra and all the following statements are well known: see for instance [3,26].
3.1.Independence.The collection S is independent since its elements are indecomposable (see 2.1).This can be also shown using radicals, as in [3].Recall that the radical of a functor F : J → vect K is a subfunctor rad(F ) ⊆ F given by rad(F )(a) = s≺a im(F (s ≺ a)) for a in J.The quotient functor H 0 F := F/rad(F ) is semisimple: that is, for every a ≺ b in J, the morphism H 0 F (a ≺ b) is the zero morphism.For a and b in J, (H 0 K(a, −))(b) is 1-dimensional if a = b and 0 otherwise.Thus, if F is free and isomorphic to K(b,−)∈S K(b, −) β(K(b,−)) , then β(K(b, −)) = dimH 0 F (b) and hence the number β(K(b, −)) is uniquely determined by the isomorphism type of F .

3.2.
Exactness.By the Yoneda lemma, for every functor F : J → vect K , the linear map Nat J (K(a, −), F ) → F (a), which assigns to a natural transformation ϕ : K(a, −) → F the element ϕ a (a a) in F (a), is a bijection.Consequently, morphisms f : F → G and g : G → H in Fun(J, vect K ) form an S-exact sequence if, and only if, for every a in J, the linear maps f a : F (a) → G(a) and g a : G(a) → H(a) form an exact sequence of vector spaces.Thus, being S-exact is the same as being exact.In particular, f : F → G is an S-epimorphism if, and only if, it is a standard categorical epimorphism, which means f a : F (a) → G(a) is surjective for all a in J.To detect this we can again use the radical:

and only if, the composition of f with the quotient
Proof.The only if part of this equivalence is clear.Conversely, suppose that the composition is an epimorphism and consider the set of all a in J for which f a is not a surjection.If this set is nonempty, then it contains a minimal element a (since J is finite).Minimality of a means that rad(f ) a : rad(F )(a) → rad(G)(a) is surjective.This, combined with the surjectivity of the composition F (a) → G(a) → H 0 G(a), implies that f a is surjective, which contradicts the assumption.
Since S-epimorphisms are the categorical epimorphisms, S-projectives are the standard projectives, and it turns out that all of them are (S-)free.
3.4.Proposition.Let J be a finite poset.The collection S in Fun(J, vect K ) is acyclic (see 2.3).
Proof.We again use the radical.Let F be projective in Fun(J, vect K ).Consider H 0 F and the free functor a) .Note that H 0 C 0 and H 0 F are isomorphic.We then use projectiveness to construct the following commutative diagram where all of the vertical arrows represent the quotient natural transformations: The vertical natural transformations in this diagram are epimorphisms, so by Lemma 3.3 the horizontal natural transformations are as well.Since the values of the functors are finite dimensional vector spaces, the horizontal natural transformations must therefore be isomorphisms.
3.5.Minimal covers, resolutions, and Betti diagrams.Let F : J → vect K be a functor.As in 3.4, consider H 0 F = F/rad(F ) and the free functor . Since H 0 C 0 and H 0 F are isomorphic, we can use the projectiveness of C 0 to lift the natural transformation C 0 → H 0 F to a natural transformation C 0 → F which makes the following square commute, where the vertical arrows represent the quotient natural transformations: The resulting natural transformation C 0 → F is an epimorphism (see discussion in 3.2) and hence it is an S-cover.It is also minimal by the same reasoning as in 3.4.It follows that there are enough S-projectives and every functor F : J → vect K has a minimal S-cover, and hence also a minimal S-projective resolution.We refer to S-covers and S-projective resolutions simply as covers and resolutions.Since every functor admits a minimal resolution, S-Betti diagrams in all degrees are always defined for every functor.We also refer to them simply as Betti diagrams and denote them as β d F : J → N, omitting the symbol S. A consequence of the above construction of a minimal cover of F is the equality: The Betti diagrams of F are not independent from each other.For example, consider a differential in a minimal resolution of F : ∂ By minimality, each summand K(b, −) of C d+1 must map nontrivially onto at least one summand K(a, −) of C d .This means that every element in supp(β d+1 F ) is bounded below by some element in supp(β d F ). Relations between Betti diagrams are easier to describe when the indexing poset J is an upper semilattice.In this case, to calculate Betti diagrams one can use Koszul complexes.Describing this is the content of the rest of this section.
3.6.Upper semilattices.Recall that the join of a subset S ⊆ J is an element in J, denoted by J S or S when the poset is understood, satisfying the following universal property: s S for every s in S, and, if s a in J for every s in S, then S a.The join is the categorical coproduct in J. Dually, the meet of S is an element in J, denoted by J S or S, satisfying the following universal property: S s for every s in S, and, if a s in J for every s in S, then a S. When they exist, joins and meets are unique.If every nonempty subset of J has a join, then J is called an upper semilattice.If J is an upper semilattice, then every subset S which is bounded below (i.e., for which there exists a in J such that a s for every s in S) also has a meet, given by {a ∈ J | S is bounded below by a}.
A subset L ⊆ J of an upper semilattice is called a sublattice if, for every nonempty subset S ⊆ L, the join S belongs to L. For S ⊆ J, we denote by S := { T | T = ∅, T ⊆ S} ⊆ J the smallest sublattice containing S.

Koszul complexes.
Let a be an element in J. Recall that an element s in J is called a parent of a if s ≺ a and there is no element b in J such that s ≺ b ≺ a.We also say that a covers s.We denote the set of parents of a by U J (a) or U (a) when the ambient poset is understood.
Suppose that a has the following property: every subset S ⊆ U (a) that is bounded below admits a meet S. If J is an upper semilattice, then every element a ∈ J satisfies this property.Fix a total order < on the set of parents U (a).We then assign to every functor F : J → vect K a chain complex of K-vector spaces, called the Koszul complex of F at a, defined for all d ≥ 0 by For example, (K a F ) 1 = s∈U (a) F (s).We define the differentials as follows: as the linear map which on the summand F (s) in s∈U (a) F (s) = (K a F ) 1 is given by F (s ≺ a).
The linear functions ∂ form a chain complex as it is standard to verify that the composition of two consecutive such functions is null.For a natural transformation f : F → G, define: These linear maps form a chain map denoted by K a f : K a F → K a G.The assignment f → K a f is a functor from Fun(I, vect K ) to the category of chain complexes called the Koszul complex at a. Its fundamental property is the following.
3.8.Theorem.Let J be a finite poset.Let a be an element in J such that every nonempty subset S ⊆ U (a) that is bounded below admits a meet S.Then, for every functor F : J → vect K and every d ≥ 0, the following equality holds: The assumption on the element a in Theorem 3.8 is local, depending only on the parents of a.Under this assumption, which is satisfied for example if J is an upper semilattice (see 3.6), in order to calculate the Betti diagram β d F (a) of a functor F at a, we do not need to construct the minimal resolution of F .Instead, it suffices to calculate the homology of the Koszul complex K a F , which can be done for each a and each degree d independently.
Proof of Theorem 3.8.The proof is based on three observations and follows closely what is presented in [9, Section 10].Observation 1.For every natural transformation f : F → G, the linear map H 0 (K a f ) is isomorphic to (H 0 f ) a .In particular, the vector spaces According to Observation 1, β 0 F (a) = dim(H 0 F )(a) (see 3.5) coincides with dim H 0 (K a F ), which is the statement of the theorem in the case d = 0. To prove this statement for d > 0, we need two additional observations: Observation 2. For b in J: Indeed, if b a, then K a K(b, −) = 0 and the statement holds.Otherwise, if b = a, then (K a K(a, −)) 0 = K and (K a K(a, −)) d = 0 for d > 0, and again the statement holds.
Otherwise, b ≺ a. Then (K a K(b, −)) 0 = K and, for every nonempty subset S ⊆ U (a) that is bounded below, K(b, S) is 1-dimensional if S is a subset of (b U (a)) := {s ∈ U (a) | b s}, and 0 otherwise.Consequently the complex K a K(b, −) is isomorphic to which is the augmented chain complex of the standard (|b U (a)| − 1)-dimensional simplex, whose homology is trivial in all degrees.Thus in this case the statement also holds.Observation 3. Since taking direct sums is an exact operation, the Koszul complex at a is an exact functor in the following sense: if f : F → G and g : G → H form an exact sequence, then the following is an exact sequence of chain complexes of vector spaces:

Kaf Kag
According to Observation 3, the Koszul complex functor at a commutes with direct sums, which, combined with Observation 2, implies that for a free functor Let F : J → vect K be a functor and C 0 → F its minimal cover.This minimal cover fits into the following exact sequence: which, according to Observation 3, leads to an exact sequence of chain complexes: which in turn leads to a long exact sequence of vector spaces: According to Observation 1, the function ) is an isomorphism, and and dim H 1 (K a F ), which is the statement of the theorem for d = 1.The theorem for d > 1 follows by induction by applying what we already have proven to the functor Z.
Theorem 3.8, combined with the long exact sequence argument in its proof, implies the following.
3.9.Corollary.Let J be a finite upper semilattice and 0 → F 0 → F 1 → F 2 → 0 an exact sequence in Fun(J, vect K ).Let a be an element in J such that β d F 0 (a) = 0 for every d ≥ 0.

Filtrations, subfunctors, and their standard Betti diagrams
In this section, we discuss in more depth resolutions and Betti diagrams of certain filtrations.Recall that a functor F : J → vect K is called a filtration if, for all a b in J, F (a b) is a monomorphism.For example, free functors and constant functors are filtrations, and in particular the constant functor K J : J → vect K , whose values are the 1-dimensional vector space K, is a filtration.Since subfunctors of a filtration are filtrations, the kernel of a cover C 0 → F is a filtration.Two results of this section are of particular interest to us.One is Corollary 4.3, which describes relations between the supports of Betti diagrams of a filtration whose 0 th Betti diagram has support bounded below.The other one is Corollary 4.9, which describes a similar, but weaker, relation for any subfunctor of the constant functor K J , even if its 0 th Betti diagram does not have a support bounded below.
4.1.Theorem.Let J be a finite upper semilattice and F : J → vect K a functor.Then the following containment holds: Proof.Consider a minimal presentation of F given by an exact sequence: Using the vocabulary of [9], we can discretize the functors of this sequence by the sublattice supp(β 0 F ) ∪ supp(β 1 F ) .By [9, Corollary 10.18.( 2)], we conclude that the Betti diagrams of F indexed by J are isomorphic to those of F restricted to the sublattice supp(β 0 F ) ∪ supp(β 1 F ) ⊂ J, and consequently As a consequence, if β d F (a) = 0, then there is a subset S ⊆ supp(β 0 F ) ∪ supp(β 1 F ) for which a = S.This gives a considerable restriction on what elements of the upper semilattice J can belong to supp(β d F ) for d > 1.
4.2.Corollary.Let J be a finite upper semilattice and F : J → vect K a functor for which supp(β 0 F ) is bounded below (the meet supp(β 0 F ) exists).Then Proof.Let b 0 = supp(β 0 F ).For all a in supp(β 0 F ), we have a b 0 , so there is a monomorphism K(a, −) ⊆ K(b 0 , −).These monomorphisms, for all a in supp(β 0 F ), fit into the following pushout square where the top horizontal arrow represents a minimal cover of F : Then the natural transformation represented by the bottom horizontal arrow is a minimal cover of G. Furthermore, since the kernels of the top and bottom horizontal natural transformations coincide, we get: In particular supp(β 0 G) = {b 0 } and supp(β d G) = supp(β d F ) for every d ≥ 1.Since every element in supp(β 2 G) is bounded by some element in supp(β 1 G), and every element in supp(β 1 G) is bounded by some element in supp(β 0 G) = {b 0 }, we can use Theorem 4.1 to conclude that supp(β 2 G) ⊆ supp(β 1 G) and consequently supp(β 2 G) ⊆ supp(β 1 G) .We then proceed by induction on d.
For a filtration, Corollary 4.2 can be extended to include also the 0 th Betti diagram.

4.3.
Corollary.Let J be a finite upper semilattice and F : J → vect K a filtration such that supp(β 0 F ) is bounded below.Then Proof.The assumptions on F are equivalent to F being a subfunctor of G := K(b, −) n for some b in J and a natural number n.Consider the exact sequence

4.4.
Subfunctors.Next we focus on subfunctors of the constant functor K J : J → vect K .Given a subfunctor F of K J , we denote by F ⊆ K J the inclusion natural transformation.Note that a functor F : J → vect K is isomorphic to a subfunctor of K J if, and only if, F is a filtration and dim F (a) ≤ 1 for every a in J (compare with assumptions in Corollary 4.3).We denote by Sub(J) the set of all subfunctors of K J .We consider two ways of parameterizing this set by posets: via upsets and antichains.

4.5.
Upsets.An upset of a poset (J, ) is a subset U ⊆ J such that, for all a in U and b a, the element b is in U .The set of upsets of J is denoted Up(J) and comes naturally equipped with a distributive lattice structure where the order relation is the inclusion ⊆, joins are unions, and meets are intersections.An example of an upset is the support of a subfunctor F ⊆ K J , defined as the subset supp(F ) := {a ∈ J | F (a) = 0}.In fact, the function F → supp(F ) is a bijection between the collections of subfunctors of K J and upsets in J. Its inverse sends an upset U to the subfunctor K U ⊆ K J defined as, for all a in J, Thus K U ⊆ K J is the unique subfunctor for which supp(K U ) = U .The inclusion of subfunctors F ⊆ G ⊆ K J defines a poset relation that makes this bijection a poset isomorphism, between (Sub(J), ⊆) and (Up(J), ⊆).We are however more interested in the opposite poset to (Sub(J), ⊆).This is because of its relation to antichains.

4.6.
Antichains.An antichain of a poset (J, ) is a subset of J whose elements are pairwise incomparable.In particular, singletons of J are antichains.The set of antichains of J is denoted by Anti(J).Given an upset U of J, the set Min(U ) of its minimal elements is an antichain.In fact, the function U → Min(U ) from upsets to antichains is a bijection, and so we could equip antichains with the induced poset structure from Up(J).However, we would like for the subposet of singletons in Anti(J) to coincide with the poset J.For this reason, we define the relation on antichains by the opposite order on upsets.Thus, given two upsets U and V , we write Min(U ) Min(V ) if V ⊆ U .Explicitly, for two antichains S and T in J, the relation S T holds if, and only if, for every t in T , there is s in S such that s t in J.In this way, (Anti(J), ) is a distributive lattice whose restriction to singletons coincides with J.
To a subfunctor F ⊆ K J we associate the antichain Min supp(F ), which coincides with supp(β 0 F ).The function F → supp(β 0 F ) is a bijection between the collection Sub(J) of subfunctors of K J and the set of antichains Anti(J).Its inverse sends an antichain S to the subfunctor K(S, −) := K (S J) ⊆ K J where (S J) denotes the upset {a ∈ J | ∃s ∈ S, s a}.This induces a distributive lattice structure on subfunctors of K J : given two subfunctors F, G ⊆ K J , we write F G if supp(β 0 F ) supp(β 0 G), which is equivalent to the inclusion supp(F ) ⊇ supp(G).We denote this poset by (Sub(J), ).4.7.Global Koszul complexes.We now discuss free resolutions of subfunctors of K J under the assumption that J is an upper semilattice.Let F ⊆ K J be a subfunctor.Fix a total order < on the antichain supp(β 0 F ).This order is used to construct a chain complex called global Koszul complex of F .For d ≥ 0, define: For example, (KF ) 0 = s∈supp(β 0 F ) K(s, −), which coincides with the minimal cover of F (see 3. 4.8.Proposition.Let J be a finite upper semilattice and F ⊆ K J be a subfunctor.Then KF → F is a free resolution of F . Proof.We need to show that, for every a in J, the complex of vector spaces (KF )(a) → F (a) is exact.Consider the set T := {s ∈ supp(β 0 F ) | s a}.If this set is empty, then the complex (KF )(a) → F (a) is trivial and hence it is exact.Otherwise, the complex (KF )(a) → F (a) is isomorphic to which is the augmented chain complex of the standard (|T | − 1)-dimensional simplex, whose homology is trivial in all degrees.
The global Koszul complex KF → F is a free resolution of F which may fail however to be minimal.A minimal resolution of F is a direct summand of the global Koszul complex, which gives: 4.9.Corollary.Let J be a finite upper semilattice and F ⊆ K J be a subfunctor.Then the following containment holds:

Constructing relative homological algebra
Let (I, ≤) be a finite poset.In this section we present a strategy for defining an independent (see 2.1) and acyclic (see 2.3) collection for relative homological algebra in the category Fun(I, vect K ).Our starting point and the standard assumption throughout the entire section is a functor T : J op → Fun(I, vect K ) where (J, ) is a finite poset.
We define the collection P := {T (a) | a ∈ J, T (a) = 0}.Although no structure on the collection P was needed in our description of the homological algebra relative to P in Section 2, in this section we illustrate the advantage of having the grading on P given by the poset structure on J.The functor T is used to translate between the P-relative homological algebra on Fun(I, vect K ) and the standard homological algebra on Fun(J, vect K ).This translation is done via a pair of adjoint functors.5.1.Example.Throughout this section, we consider the following running example.Let (I, ≤) be the set {0, 1} 2 with the product order, J the set {(u, v) ∈ I 2 | u ≤ v} with the product order, and This example is a specific instance of lower hooks, introduced in [6], which we study in general in Example F.

5.2.
Adjunction.The functor T induces the following pair of adjoint functors: • L assigns to F : J → vect K the following colimit in Fun(I, vect K ): where the summand T (a 1 ) ⊗ F (a 0 ), indexed by a 0 ≺ a 1 , is mapped by d 0 to the summand T (a 1 ) ⊗ F (a 1 ) via the morphism id T (a 1 ) ⊗ F (a 0 ≺ a 1 ); -by d 1 to the summand T (a 0 ) ⊗ F (a 0 ) via the morphism T (a 0 ≺ a 1 ) ⊗ id F (a 0 ) .

Proposition.
The functor L is left adjoint to R.
Proof.Let F : J → vect K and M : I → vect K be two functors.By the universal property of the colimit and the tensor-hom adjunction, the set Nat I (LF, M ) is in natural bijection with the set of sequences of linear maps {f a : F (a) → Nat I (T (a), M )} a∈J making the following square commute for every a 0 ≺ a 1 in J: Thus this set of sequences is in bijection with Nat J (F, Nat I (T , M )) = Nat J (F, RM ).In this way we get a natural isomorphism between Nat I (LF, M ) and Nat J (F, RM ), which gives the desired adjunction.
Let a be in J and consider the free functor K(a, −) : J → vect K .By the adjunction between L and R, for every M : I → vect K , the set Nat I (LK(a, −), M ) is naturally isomorphic to the set Nat J (K(a, −), RM ), which is in natural bijection with RM (a) = Nat I (T (a), M ).Consequently LK(a, −) and T (a) are isomorphic functors for every a in J, and the collection P = {T (a) | a ∈ J, T (a) = 0} can be identified with {LK(a, −) | a ∈ J, LK(a, −) = 0}.Then, using the same representation conventions for functors indexed by J, the functor RM is represented as 5.5.Adjunction unit and counit.For M : I → vect K , the counit ε M : LRM → M is the natural transformation adjoint to id : RM → RM .For F : J → vect K , the unit η F : F → RLF is the natural transformation adjoint to id : LF → LF .If F = K(a, −) for some a in J, then η F is also denoted by η a .The adjunction between L and R implies the commutativity of the following diagrams, which in particular implies that Rε M and ε LF are epimorphisms, and η RM and Lη F are monomorphisms: The functor R can be used to translate between P-exactness in Fun(I, vect K ) and the standard exactness in Fun(J, vect K ): 5.6.Proposition.Let I and J be finite posets.A pair of natural transformations f : M → N and g : N → L in Fun(I, vect K ) form a P-exact sequence (see 2.2) if, and only if, the natural transformations Rf and Rg form an exact sequence in Fun(J, vect K ).
Proof.By definition of the functor R = Nat I (T , −), the pair Rf and Rg forms an exact sequence if, and only if, Nat I (T (a), f ) and Nat I (T (a), g) form an exact sequence for all a in J, which corresponds to P-exactness in Fun(I, vect K ).

5.7.
Corollary.A natural transformation f : M → N in Fun(I, vect K ) is a P-epimorphism if, and only if, Rf is an epimorphism in Fun(J, vect K ).
For example, consider a functor M : I → vect K and the natural transformation ε M : LRM → M .Since Rε M is an epimorphism, ε M is therefore a P-epimorphism.
To translate between minimal P-covers in Fun(I, vect K ) and standard minimal covers in Fun(J, vect K ), an additional assumption on the functor T : J op → Fun(I, vect K ) is required.Here is the key definition of our paper.5.8.Thinness.The functor T : J op → Fun(I, vect K ) is called thin if, for every a in J, the unit natural transformation η a : K(a, −) → RLK(a, −) is an epimorphism.
Our primary examples of thin functors are given by the following result: 5.9.Proposition.Let I and J be finite posets and T : J op → Fun(I, vect K ) a functor.Suppose that • for all a in J, the functor T (a) has at most one generator, i.e. so the unit natural transformation η a : K(a, −) → RLK(a, −) is indeed an epimorphism.
Among thin functors there are functors that satisfy a stronger requirement.
5.11.Flatness.The functor T : J op → Fun(I, vect K ) is called flat if the unit natural transformation η a : K(a, −) → RLK(a, −) is an isomorphism for every a in J for which T (a) = 0.
Every flat functor is thin.Since both left and right adjoints commute with direct sums, if T is thin, then, for every free functor C 0 in Fun(J, vect K ), the unit natural transformation η C 0 : C 0 → RLC 0 is also an epimorphism.If T is flat and C 0 in Fun(J, vect K ) is a free functor, then the unit natural transformation η C 0 : C 0 → RLC 0 is an isomorphism if, and only if, T (a) = 0 for every a in supp(β 0 C 0 ).5.12.Proposition.Let I and J be finite posets.If the functor T : J op → Fun(I, vect K ) is thin (resp.flat), then, for every subposet L ⊆ J, the restriction of T to L op ⊆ J op is also thin (resp.flat).
Proof.For all a in J, the functors RLK(a, −), RT (a), and Nat I (T , T (a)) are isomorphic.Thus T is thin if, and only if, both of the following conditions are satisfied: • for a b in J, Nat I (T (b), T (a)) = 0, • for a b in J, every natural transformation T (b) → T (a) is of the form λT (a b) for some λ in the field K.This characterization implies that if T is thin, then its restriction to L op is also thin.
Similarly, the functor T is flat if, and only if, both of the following conditions are satisfied: As before, this characterization implies that if T is flat, then its restriction to L op is also flat.
Thinness is important because it implies that all the elements in the collection P are indecomposable (as their endomorphism algebras are 1 dimensional), and consequently this collection is independent (see 2.1).Nevertheless we find the following proof insightful.5.13.Proposition.Let I and J be finite posets.If the functor T : J op → Fun(I, vect K ) is thin, then the collection P = {T (a) | a ∈ J, T (a) = 0} is independent (see 2.1).
Proof.Choose a in J.The functor RLK(a, −) = Nat I (T , T (a)) = RT (a) is the zero functor if, and only if, T (a) is the zero functor.Thus, if T (a) is nonzero, then the surjectivity of η a : K(a, −) → RLK(a, −) (T is assumed to be thin) implies that η a is a minimal cover in Fun(J, vect K ), in which case the standard 0 th Betti diagram β 0 RT (a) = β 0 RLK(a, −) : J → N has the following values: Let β : J → N be a function whose support is finite and such that, if β(a) = 0, then T (a) = 0. Consider the functor a∈J T (a) β(a) , which is a finite sum because J is finite.Since R is right adjoint, it commutes with finite direct sums and consequently R a∈J T (a) β(a) is isomorphic to a∈J RT (a) β(a) .According to the calculation above, β is the 0 th Betti diagram of R a∈J T (a) β(a) .The function β is therefore determined uniquely by the isomorphism type of the functor a∈J T (a) β(a) , which gives the independence of T .
If T : J op → Fun(I, vect K ) is thin, then we also have an effective way of constructing (minimal) P-covers in Fun(I, vect K ) using standard (minimal) covers in Fun(J, vect K ).5.14.Theorem.Let I and J be finite posets and T : J op → Fun(I, vect K ) a thin functor (see 5.8).Then, for every M : I → vect K , a natural transformation C 0 → RM is a cover in Fun(J, vect K ) if, and only if, its adjoint LC 0 → M is a P-cover in Fun(I, vect K ).Moreover, if C 0 → RM is a minimal cover, then its adjoint LC 0 → M is a minimal P-cover.
Proof.Since, for every a in J, the functor LK(a, −) is isomorphic to T (a) and L commutes with direct sums, a functor C 0 in Fun(J, vect K ) is free if, and only if, the functor LC 0 in Fun(I, vect K ) is P-free.Recall that all projectives in Fun(J, vect K ) are free.
Choose a functor M : I → vect K , and a natural transformation f : C 0 → RM with C 0 a free functor in Fun(J, vect K ).Let g : LC 0 → M be the adjoint of f .The following commuting diagram describes the relation between f and g: By thinness assumption, the unit η C 0 : C 0 → RLC 0 is an epimorphism.Therefore f is an epimorphism if, and only if, Rg is an epimorphism, which, by Proposition 5.6, happens if, and only if, g is a P-epimorphism.Thus f is a cover if, and only if, g is a P-cover.
Next we discuss minimality.Suppose that f : C 0 → RM is a minimal cover.Since we already showed that g : LC 0 → M is a P-cover, it remains to show its minimality.Let h : LC 0 → LC 0 be an endomorphism of g.This endomorphism fits into the following commutative diagram: As before, the unit η C 0 : C 0 → RLC 0 is an epimorphism.The dashed arrow exists because C 0 is free.By the minimality of f , this dashed arrow is an isomorphism.The natural transformation Rh is therefore an epimorphism.By Corollary 5.7, h : LC 0 → LC 0 is a P-epimorphism.Since LC 0 is P-free, h being a P-epimorphism means that h is an epimorphism.As the values of LC 0 are finite dimensional vector spaces, h is an isomorphism.The natural transformation g is therefore a minimal cover.Theorem 5.14 has several important consequences, including the acyclicity of the collection P: 5.16.Corollary.Let I and J be finite posets.Suppose the functor T : J op → Fun(I, vect K ) is thin.Then (1) every functor in Fun(I, vect K ) admits a minimal P-resolution (see 2.5); (2) the collection Proof.
(1) Every functor in Fun(J, vect K ) admits a minimal cover (see 3.5).Thus according to Theorem 5.14 every functor in Fun(I, vect K ) admits a minimal P-cover, and hence a minimal P-resolution.
(2) Let M be a P-projective object in Fun(I, vect K ).We need to show that it is P-free.Consider a minimal cover C 0 → RM in Fun(J, vect K ) (see 3.5).Its adjoint LC 0 → M , according to Theorem 5.14, is then a minimal P-cover of M in Fun(I, vect K ), and therefore has to be an isomorphism since M is P-projective.Note that LC 0 is P-free, and consequently so is M .
To summarize, if the functor T : J op → Fun(I, vect K ) is thin, then the collection P is independent (see 5.13) and acyclic (see 5.16.( 2)), and every functor M : I → vect K has a minimal P-resolution (see 5.16.( 1)).Thus thinness guarantees that the P-Betti diagram β d P M : P → N is well defined for every d ≥ 0 (see 2.6).The rest of this section is devoted to describing methods to calculate some values of these P-Betti diagrams.We start by looking at what we can prove for a general poset J: 5.17.Proposition.Let I, J be finite posets, and T : J op → Fun(I, vect K ) and M : I → vect K be functors.Proof.Let f : C 0 → RM be a minimal cover.By definition of β 0 (RM ), the functor C 0 is isomorphic to a∈J K(a, −) β 0 (RM ) (a) .Recall that, for every a in J, the functor LK(a, −) is isomorphic to T (a).This, together with the fact that L commutes with direct sums, implies that LC 0 is isomorphic to the functor Suppose T is thin.According to Theorem 5.14, the minimal P-cover of M is given by the adjoint g : LC 0 → M of the minimal cover f : C 0 → RM .We conclude that the equality β 0 P M (T (a)) = β 0 (RM )(a) holds for all a for which T (a) = 0.This shows (1).Suppose T is flat.The natural transformations ker(g) ֒→ LC 0 and g : LC 0 → M form a P-exact sequence in Fun(I, vect K ).According to Proposition 5.6, after applying R, the natural transformations R(ker(g) ֒→ LC 0 ) and Rg form therefore an exact sequence in Fun(J, vect K ).Since R is right adjoint, it preserves monomorphisms and hence R ker(g) is the kernel of Rg.We can then form the following commutative diagram with the indicated arrows representing epimorphisms and monomorphisms, and with the vertical sequences being exact:

Rg
By flatness, RLC 0 is free and hence, by minimality of f , the unit η is an isomorphism.The natural transformation represented by the top horizontal arrow in this diagram is then also an isomorphism.Consequently, for every d ≥ 0 and every a in J, β d (ker(f ))(a) = β d (R ker(g))(a), and we conclude with the following sequence of equalities: This proves (2).
(1) provides an algorithm to calculate the 0 th P-Betti diagram of a functor M : I → vect K when T is thin: first take RM : J → vect K , then calculate its standard 0 th Betti diagram (over the poset J), and finally restrict the obtained 0 th Betti diagram to the a in J for which T (a) = 0.The standard 0 th Betti diagrams can be calculated for example using radicals (see 3.5).This algorithm can be then used inductively in order to calculate the P-Betti diagrams β d P M of M for all d ≥ 0, as explained in 2.6.In every step of this procedure we need to evaluate the functor R on successive kernels (see the second step in the sequence of equalities at the end of the proof of 5.17).We would like to avoid that step, for example by showing beforehand that the P-Betti diagrams over I at T (a) are equal to the corresponding Betti diagrams over J at a.This is the case if, for example, T is flat (see Proposition 5.17.( 2)).However, our key examples (see Section 6) are not flat but thin.For an arbitrary thin functor there could be elements a in J for which the numbers If T is flat, then its degeneracy locus is given by {a ∈ J | T (a) = 0} (see 5.17.( 2)).For an arbitrary thin functor T we do not have an explicit description of its degeneracy locus.However, we can approximate it when J is an upper semilattice.5.19.Theorem (Main result, first half).Let I be a finite poset, J a finite upper semilattice, and T : J op → Fun(I, vect K ) a thin functor.Then the degeneracy locus of T (see 5.18) is contained in the set Proof.Write G := a,d supp(β d ker η a ).Let us prove the contraposition, that for M : I → vect K a functor, a in J \ G, and d ≥ 0, we have T (a) = 0 and β d P M (T (a)) = β d (RM )(a).Note that if T (a) = 0, then supp(β 0 ker η a ) = {a}, so the first statement is true.We proceed by induction on d ≥ 0 for the second statement.The case d = 0 is the statement of Proposition 5.17.Now let d > 0.
We proceed exactly as in the proof of Proposition 5.17.Let M : I → vect K be a functor and consider the following commutative diagram:

Rg
where f is a minimal cover in Fun(J, vect K ), g is its left adjoint, which according to Theorem 5.14 is a minimal P-cover in Fun(I, vect K ), the indicated arrows are epimorphisms and monomorphisms, and the vertical sequences are exact.The thinness assumption implies that the unit natural transformation η (the middle horizontal arrow in the above diagram) is an epimorphism.This, together with the exactness of the vertical sequences, implies that the natural transformation represented by the top horizontal arrow is also an epimorphism.Also by exactness of the vertical sequences, the kernel of this natural transformation is isomorphic to ker(η).In this way we obtain an exact sequence in Fun(J, vect K ) of the form 0 ker(η) ker(f ) R ker(g) 0.
Let us show that for all a in J \ G and d ≥ 0, β d (ker(f ))(a) = β d (R ker(g))(a).Since C 0 is free and both R and L commute with direct sums, ker(η) is isomorphic to a direct sum of functors of the form ker(η a : K(a, −) → RLK(a, −)).By definition of G, we have the containment supp(β d ker(η)) ⊆ G for every d ≥ 0. Thus, for every d ≥ 0 and a in J \ G, β d (ker(η))(a) = 0. Since J is an upper semilattice by assumption, we apply Corollary 3.9 to deduce that We conclude with the following sequence of equalities: With some additional hypotheses on the parameterization, we get an exact description of the degeneracy locus.5.20.Corollary (Main result, second half).Let I be a finite poset, J a finite upper semilattice, and T : J op → Fun(I, vect K ) a thin functor.Suppose that, for all a in J, the sublattice Explicitly, for every functor M : I → vect K , every element a in J such that T (a) = 0, and every d ≥ 0, Proof.For a in J and d ≥ 0, supp(β d ker η a ) is contained in supp(β 0 ker η a ) , which itself is contained in {b ∈ J | T (b) = 0} by hypothesis.By Theorem 5.19, we deduce that the degeneracy locus of T is also contained in {b ∈ J | T (b) = 0}.In fact, they are equal, and they are both equal to ∪ a∈J supp(β 0 ker η a ) .By definition of the degeneracy locus and Theorem 3.8 respectively, for all M : I → vect K , a in J such that T (a) = 0, and d ≥ 0, we get the equalities 5.21.Example.Our running Example 5.1 satisfies the conditions of Corollary 5.20.This is proved in general for lower hooks in Example F, but here we can check this by hand: for instance, for a = (00, 10), we computed the unit natural transformation η a in 5.10.We find that ker η a = , and so supp(β 0 ker η a ) = {(10, 10)} = supp(β 0 ker η a ) , and from 5.1 we know that T ((10, 10)) = 0. We reason similarly for all other a in J.

Examples
This last section is devoted to examples of independent and acyclic collections together with calculations of the induced relative Betti diagrams in the category Fun(I, vect K ), where (I, ≤) is a finite poset.Our strategy for performing these calculations is to find an appropriate parameterization of the collection via an upper semilattice J and a functor T : J op → Fun(I, vect K ) as explained in Section 5.

Example A. Singleton
We start with the simplest nonempty example.Choose a functor P 0 : I → vect K and consider the collection {P 0 } consisting only of this chosen functor.The simplest way to parameterize this collection is to consider the poset [0] with a single element 0 and the functor T : [0] op → Fun(I, vect K ) sending 0 to P 0 .
We do a case analysis by the dimension of Nat I (P 0 , P 0 ).If dim Nat I (P 0 , P 0 ) = 0, then P 0 = 0 and the collection P = {T (a) | a ∈ [0], T (a) = 0} is empty.For every functor M : I → vect K , 0 → M is a minimal P-resolution, and the Betti diagrams of M are therefore given by the empty function ∅ → N.
If dim Nat I (P 0 , P 0 ) = 1, then the functor T is flat (explicitly, K(0, −) → Nat I (P 0 , P 0 ) is an isomorphism).For every functor M : I → vect K , the minimal cover of RM = Nat I (P 0 , M ) is the identity, so by Theorem 5.14, the minimal P-cover of M is its left adjoint, which is the evaluation natural transformation P 0 ⊗ K Nat I (P 0 , M ) → M .By Proposition 5.17: If dim Nat I (P 0 , P 0 ) > 1, then the functor T is not thin, and we observe for instance that the minimal cover of RP 0 is adjoint to the P-cover P 0 ⊗ K Nat I (P 0 , P 0 ) → P 0 , which is not the minimal P-cover of P 0 .
We now proceed with the remaining examples, studying homological algebra relative to various collections of functors in Fun(I, vect K ).In each example, to the extent that is possible, we proceed as follows.We start by defining a collection and its parameterization by a functor T : J op → Fun(I, vect K ).We then check if the functor T is flat or thin.
• If T is flat, then we can apply Proposition 5.17.(2).
• If T is thin, then our aim is to apply Corollary 5.20.To do this we need to verify the other conditions of the corollary, namely that the poset J is an upper semilattice and that T sends the sublattice supp(β 0 ker η a ) to 0 for all a in J.We call this latter assumption the degeneracy condition.
In both cases, we conclude that we can compute Betti diagrams relative to P from Koszul complexes over J.To do this effectively, we characterize parents in J and then give an explicit formula for Koszul complexes.Then, for every functor M : I → vect K , element a in J such that T (a) = 0, and d ≥ 0, we have the equality Finally, we compute an explicit relative projective resolution of the following functor M 0 on a 6 × 6 grid, whose minimal standard resolution is illustrated below: As in Example 5.1, the functor M 0 illustrated here is equal to K on its support (vertices in the shaded area) and 0 elsewhere, with all the transition functions M 0 (v ≤ w) having maximal rank.

Example B. All subfunctors
Collection.We consider the collection of all nonzero elements in Sub(I), the set of all subfunctors of the constant functor K I (see Section 4).
Parameterization.Recall that there is a poset relation on Sub(I) (see 4.6) such that F G if, and only if, supp(F ) ⊇ supp(G), which is equivalent to supp(F ) ⊇ supp(β 0 G).According to 4.5 and 4.6, the poset (Sub(I), ) can be identified with either the opposite of the inclusion poset (Up(I), ⊆) of upsets of I, or with the poset (Anti(I), ) of antichains in I.
In particular (Sub(I), ) is an upper semilattice.
Define T : (Sub(I), ) op → Fun(I, vect K ) to be the identity on objects, and to map a relation F G to the corresponding natural transformation G ⊆ F .The collection P = {T (F ) | F ∈ Sub(I), T (F ) = F = 0} consists of all nonzero subfunctors of K I .
Flatness.This requires the additional assumption of I having a unique maximal element.6.1.Proposition.Let I be a finite poset with a unique maximal element.Then the parameterization T is flat.
Proof.Let F, G ⊆ K I be nonzero subfunctors.By [19,Prop. 3.10(2)], the dimension of the space Nat(G, F ) is equal to the number of connected components of supp(G) fully contained in supp(F ).By hypothesis of I having a unique maximal element, supp(F ) and supp(G) each have only one connected component, so dim(Nat(G, F )) = 1 if F G and 0 otherwise.We deduce that the unit natural transformation η F : K(F, −) → RLK(F, −) is an isomorphism, and so we conclude that T is flat.
According to Proposition 5.17.(2), the flatness of T gives the equality β d P M (F ) = β d (RM )(F ) for every functor M : I → vect K , nonzero subfunctor F ⊆ K I , and d ≥ 0. The P-Betti diagrams over I can be therefore expressed as the standard Betti diagrams over the upper semilattice (Sub(I), ).Thus these standard Betti diagrams can be calculated using Koszul complexes, once we have identified and enumerated parents and their meets in (Sub(I), ) (see 5.20).
Parents.We identify (Sub(I), ) with the opposite poset of upsets Up(I) op via the map F → supp(F ) (see 4.5).For a subset S of I, denote by S c its complement in I and by Max(S) the set of maximal elements of S. The sets of parents in Up(I) op can be described as follows: Proof.Let v be an element of Max(U c ).The set U ∪ {v} is indeed an upset because every element greater than v is not in U c , and hence it is in U .Since U ∪ {v} contains U , the former is indeed a lower bound of the latter in Up(I) op .Finally, since there is only a one-element difference, U ∪ {v} is a parent of U .
Conversely, let V be a parent of U in Up(I) op .Then U is properly contained in V .Let v be a maximal element in V \ U .Since V is an upset, v is also an element of Max(U c ), and consequent V = U ∪ {v}.
Recall that for an upset U of I, the symbol K U denotes the unique subfunctor of K I for which supp(K U ) = U .Thus, in terms of subfunctors, according to Lemma 6.2, the parents of F ⊆ K I in the poset (Sub(I), ) are subfunctors of the form K supp(F )∪{v} ⊆ K I for v in Max(supp(F ) c ).
Koszul complexes.For M : I → vect K and F ⊆ K I , the Koszul complex of RM : Sub(I) → vect K at F is therefore given in degree d ≥ 0 by with the differentials as defined in 3.7.The value β d P M (F ) is then given by the dimension of the d th homology of this chain complex.For example, in the case I is the 6 × 6 grid, the relative projective resolution of M 0 can be illustrated as Since T : Sub(I) op → Fun(I, vect K ) is flat, so are all of its restrictions (see 5.12).Thus one way of producing examples of flat functors for which we can use Koszul complexes to calculate the associated relative Betti diagrams is to find subposets of Sub(I) which are upper semilattices (not necessarily sublattices of (Sub(I), )) and for which we can identify parents of its elements and meets of subsets of parents that are bounded below.The next example is an illustration of this strategy.

Example D. Spread Modules
Collection.We consider the following collection: Functors in this collection have been studied in other works.They are exactly the spread modules as defined in [5].Recall that spread modules are defined using two antichains S and T in I such that, for all s in S there exists t in T such that s ≤ t, and for all t in T there exists s in S such that s ≤ t.The spread with sources S and sinks T is the subset of The spread module with sources S and sinks T is then defined as the functor K [S,T ] : I → vect K where, for all v ≤ w in I, Let Q 0 : Ω op 0 → Fun(I, vect K ) denote this restriction.Thinness.The parameterization Q 0 satisfies the conditions of Proposition 5.9.Indeed, by definition, the functor Q 0 (K(v, −), G) = coker(G ⊆ K(v, −)) has at most one generator, and if the functor is nonzero, then its generator is at v.Moreover, we can check that if Consequently, Q 0 is thin and we can utilize Proposition 5.17.(1) to express the 0 th C 0 -Betti diagrams of functors indexed by I in terms of the standard 0 th Betti diagrams of functors indexed by Ω 0 .
In order to apply Corollary 5.20 to use Koszul complexes to calculate the higher C 0 -Betti diagrams, we need to check that the poset Ω 0 is an upper semilattice and that the functor Q 0 sends supp(β 0 ker η a ) to 0 for all a in Ω 0 .
Upper semilattice.Suppose (I, ≤) is an upper semilattice.Then Ω 0 is also an upper semilattice.To describe the join operation and the parents of elements in Ω 0 , it is convenient to identify it with the following subposet of the product I × Up(I) op Degeneracy.By definition, for all a = (v, U ) in Ω 0 , the support of the kernel of the unit η a : K(a, −) → RLK(a, −) consists of the elements b in Ω 0 such that b a and RLK(a, b) = Nat(Q 0 (b), Q 0 (a)) = 0. Thus the elements in supp(β 0 ker η a ), which are the minimal elements in supp(ker η a ), are of the form (u, (u ≤ I)) where u is a minimal element of U .Note further that Q 0 (v ′ , v ′ ≤ I) = 0 for every v ′ in I. Since the collection of elements in Ω 0 of the form (v ′ , (v ′ ≤ I)) is closed under joins, we conclude that Q 0 sends every element of supp(β 0 ker η a ) to 0.
Parents.The poset Ω 0 has the product order and Lemma 6.2 gives the parents of upsets, so the parents of (v, U ) in Ω 0 are of the form where (v ≤ Max(U c )) is the set of maximal elements of U c bounded below by v. Koszul complex.For M : I → vect K the value of the functor RM : Ω 0 → vect K at (v, U ) in Ω 0 can be identified with where the second equality comes from identifying every element ϕ in Nat(coker The Koszul complex of RM at (v, U ) in degree d ≥ 0 can now be identified with |S|+|T |=d S has lower bound u∈Min(U ∪T ) ker M ( (I≤v) S ≤ u), with differentials as defined in 3.7.Finally, we obtain the relative Betti diagrams of M from the homology of this chain complex.For example, the relative projective resolution of M 0 = coker(K {04,32,40} ⊆ K(00, −)) is itself, since it is already a single-source spread module: −→

Example F. Lower hooks
Collection.Let (I, ≤) be a finite upper semilattice.We now consider the collection of lower hooks, originally defined in [6], which we define as the subcollection of C (see Example D) Parameterization.Let (J, ) be the poset {(v, w) | v, w ∈ I, v ≤ w} equipped with the product order: (v, w) (v ′ , w ′ ) in J if and only if v ≤ v ′ and w ≤ w ′ in I.We identify J with the subposet of the poset Ω 0 (see Example E) consisting of the pairs (v, w ≤ I) for (v, w) in J. Via this identification, we parameterize lower hooks by the restriction of the functor Q 0 : Ω op 0 → Fun(I, vect K ) (see Example E) to J, and denote it by T .Explicitly, for a = (v, w) in J, the functor T (a) is given by coker(K(w, −) ⊆ K(v, −)).
Thinness.The functor T is thin by Proposition 5.12, since it is a restriction of the thin functor Q 0 .
Upper semilattice.The poset J is a finite upper semilattice, inheriting the structure from I. Note however that J is not a sublattice of Ω 0 .
Degeneracy.For all a = (v, w) in J, the sublattice supp(β 0 ker η a ) ⊆ J is sent to 0 by T .Indeed, the support of the kernel of the unit η a : K(a, −) → RLK(a, −) consists of the elements b of J such that b a and RLK(a, b) = Nat(T (a), T (b)) = 0.This support has a single minimal element, (w, w), which is therefore also the only element of supp(β 0 ker η a ).Thus, in this case, supp(β 0 ker η a ) = {(w, w)}, which is sent to 0 by T .By Corollary 5.20, we conclude that the Betti diagrams relative to lower hooks can be computed via Koszul complexes over J.
Parents.First, for all (v, w) in J, we have where (v ≤ U I (w)) is the subset of U I (w) of elements greater or equal to v. Koszul complexes.Next, we identify RM (v, w) = Nat(T (v, w), M ) with ker(M (v ≤ w)).Thus we can compute the Koszul complex of RM at (v, w) in degree d ≥ 0 as |S|+|T |=d S has lower bound ker M ( (I≤v) S ≤ (I≤w) T ), with differentials as defined in 3.7.Finally, we obtain the relative Betti diagrams of M from the homology of this chain complex.

Variation on lower hooks. Now consider the subcollection of C C
We parameterize this by the finite upper semilattice J where ∞ is greater than every element of I.The upper semilattice J is a sublattice of J ∞ .We extend the functor T : J op → Fun(I, vect K ) to J ∞ by sending (v, ∞) to K(v, −) and denote it by the symbol T ∞ .The conditions for Corollary 5.20 also hold for this extension.In addition, C ∞ -exactness implies exactness and rank additivity [6,Prop. 4.3] There is a similar formula for Koszul complexes where we identify RM (v, ∞) with M (v).For both of these examples (T and T ∞ ), the relative projective resolution of M 0 is As observed above, this sequence is rank additive.
Relative projective dimension.By [9, Corollary 10.18] we can deduce an upper bound for the projective dimension of a functor in Fun(J, vect K ), i.e. the maximal length of its minimal projective resolution.By Corollary 5.20, the maximal length of a minimal projective resolution in Fun(I, vect K ) relative to lower hooks will have the same upper bound.Recent work [7] proposes a tighter bound in the case where I is a finite lattice.Here we present a proof of this result using Koszul complexes.
6.5.Proposition (cf.[7,Theorem 5.18]).Let I be a finite lattice such that every its element has at most n parents.Then the projective dimension relative to lower hooks of every functor M : I → vect K is bounded above by 2n − 2.
Proof.By [9, Corollary 10.18], every functor M in Fun(I, vect K ) has projective dimension at most n.For the same reason and by observing that J is also a lattice where every element has at most 2n parents, a functor in Fun(J, vect K ) has projective dimension at most 2n.In particular, this is the case for the functor RM : J → vect K .Let π 1 , π 2 : J → I be the functors defined by π 1 (v, w) = v and π 2 (v, w) = w.Consider the compositions M π 1 and M π 2 in Fun(J, vect K ) and the natural transformation α : M π 1 → M π 2 whose component at any (v, w) in J is M (v ≤ w).Observe that there is an exact sequence of functors 0 → RM = ker(α) ֒→ M π 1 α − → M π 2 .Now let C → M be a minimal free resolution in Fun(I, vect K ).For i = 1, 2, the sequence Cπ i → M π i in Fun(J, vect K ) is also exact because exactness is defined componentwise.Moreover, for u in I and (v, w) in J, the vector space (K(u, −)π 1 )(v, w) is nonzero if, and only if, u ≤ v, or equivalently (u, u) ≤ (v, w).Thus, K(u, −)π 1 is isomorphic to the free functor K((u, u), −).As a consequence, the sequence Cπ 1 → M π 1 is a free resolution in Fun(J, vect K ), and so M π 1 has projective dimension at most n.Analogously, K(u, −)π 2 coincides with the free functor K((e, u), −), where e is the global minimum of the lattice (it exists because I is finite).We have again that the sequence Cπ 2 → M π 2 is a free resolution in Fun(J, vect K ), and M π 2 has projective dimension at most n.
Now consider the exact sequence 0 → RM → M π 1 → im(α) → 0 and let (v, w) be an element in J. Since the functor K (v,w) is exact, the sequence 0 → K (v,w) RM → K (v,w) M π 1 → K (v,w) im(α) → 0 is also exact.As we observed, the elements in the poset J have at most 2n parents, so we obtain the following long exact sequence: 0 H 2n (K (v,w) RM ) H 2n (K (v,w) M π 1 ) H 2n (K (v,w) im(α)) Parameterization.Let (J, ) be the poset {(v, w) | v, w ∈ I, v ≤ w} with the product order, as in the example of lower hooks (see Example F).This time, we identify J with the subposet {(v, ker(K(v, −) → K(−, w) * ) | v ≤ w} of Ω 0 , where the functor K(−, w) * is equal to K at u, when u ≤ w, and 0 otherwise, with transition functions of full rank.Via this identification, we parameterize rectangle functors by the restriction of the functor Q 0 : Ω op 0 → Fun(I, vect K ) (see Example E) to J, and denote it by T .
Thinness.The functor T is thin by Proposition 5.12, since it is a restriction of the thin functor Q 0 .
Upper semilattice.The poset J is a finite upper semilattice just as in the previous example.

Example H. Rectangles on a grid
We can amend the example of rectangles (see Example G) when we have more control over the poset I.In this example, we consider I := {0 < • • • < n} r a finite grid with the product order, denoted by ≤.For v, w in I, we write v + (w − v) i for the element (v 1 , . . ., v i−1 , w i , v i+1 , . . ., v r ).
Collection.We consider the same collection C R as in Example G.
Parameterization.Let (J, ) be the poset {(v, w) | v, w ∈ I, v ≤ w} with the product order, as in the previous example.This time, we identify J with the subposet v, Thinness.The functor T is thin by Proposition 5.12, since it is a restriction of the thin functor Q 0 .
Upper semilattice.The poset J is a finite upper semilattice just as in the previous example.
Degeneracy.For all a = (v, w) in J, T sends the sublattice supp(β 0 ker η a ) to 0. Indeed, the support of the kernel of the unit The sublattice supp(β 0 ker η a ) then consists of the elements (v X , w), for every nonempty subset X of {1, . . ., r}, where, for all i in {1, . . ., r}, the element v X in I is defined by (v X ) i := w i if i ∈ X, v i otherwise.
Note that for (v ′ , w ′ ) in J, T (v ′ , w ′ ) = 0 if, and only if, there exists i ∈ {1, . . ., r} such that v ′ i = w ′ i .As a consequence, T sends the elements of the sublattice supp(β 0 ker η a ) to 0. By Corollary 5.20, we can compute Betti diagrams relative to rectangles via Koszul complexes.
Parents.Parents are exactly the same as for lower hooks (see Example F).
Koszul complexes.For (v, w) in J, we can identify RM (v, w) = Nat(T (v, w), M ) with

Algorithm 1 :
Betti diagrams relative to a parameterization T Input: I finite poset, J finite upper semilattice, T : J op → Fun(I, vect K ) satisfying the conditions of Corollary 5.20, M : I → vect K functor, a ∈ J poset element such that T (a) = 0. Output: the relative Betti diagrams of M at T (a) relative to {T (b) | b ∈ J, T (b) = 0}.U (a) ← {a 1 , . . ., a n } parents of a; the following three conditions are satisfied: (a) for every d ≥ 0, the object C d is P-projective; (b) the composition of every two consecutive morphisms is 0; and (c) for every d ≥ 0, the induced morphism C d → ker(C d−1 → C d−2 ) is a P-epimorphism, where C −2 = 0 and C −1 = M .

,
−)) the translation or parameterization functor.Explicitly, we have the Hasse diagrams (where the order goes up and to the right) where we have represented each nonzero functor T (a) : I → vect K in P by their support: the dotted squares correspond to the poset I, the shaded vertices correspond to the support of the functor, where the functor is equal to K, and the transition functions are all implicitly of maximal rank.The position of each functor T (a) corresponds to the position of a in the Hasse diagram of J.Finally, we have the collection

( 1 )
If T is thin, then β 0 P M (T (a)) = β 0 (RM )(a) for every a in J for which T (a) = 0. (2) If T is flat, then β d P M (T (a)) = β d (RM )(a) for every d ≥ 0 and every a in J for which T (a) = 0.
β d P M (T (a)) = β d (RM )(a) differ.Such elements are called T -degenerate: 5.18.Degeneracy locus.Let T : J op → Fun(I, vect K ) be thin.An element a in J is called T -degenerate if T (a) = 0 or if T (a) = 0 and there exists a functor M : I → vect K and a natural number d ≥ 0, for which β d P M (T (a)) = β d (RM )(a).The collection of T -degenerate elements is called degeneracy locus of T .
Example C. Translated functors Collection.Suppose the poset (I, ≤) is the product {0 < • • • < n} r , which is a distributive lattice.Its elements will be denoted as words w 1 • • • w r .Fix an antichain S in Anti(I) and define I − S := {v ∈ I | S + v ⊆ I}.The set I − S consists of the elements v in I for which all the coordinates of t + v are bounded by n for every s in S. Thus I − S = r i=1 {0, . . ., n − max{s i | s ∈ S}} ⊆ {0, . . ., n} r = I.Moreover, the subposet (I − S) ⊆ I is a sublattice.For v in I − S, the subset S + v ⊆ I is also an antichain in I.The associated subfunctor K(S +v, −) ⊆ K I is called the translation of K(S, −) ⊆ K I by v (see 4.6 for the notation).In this example we consider the collection of all such translations P := {K(S + v, −) | v ∈ I − S}.Parameterization.The lattice I − S provides a natural choice for parameterizing this collection because, for v and w in I − S, the inclusion K(S + v, −) ⊆ K(S + w, −) exists if, and only if, v ≥ w in I.In particular, K(S + v, −) = K(S + w, −) if, and only if, v = w.Set T : (I − S) op → Fun(I, vect K ) to be the functor which maps v in (I − S) op to the translation K(S + v, −).Flatness.This functor is flat by Proposition 5.12, since it can be identified with the restriction of the functor T discussed in Example B to the subposet {K(S + v, −) | v ∈ I − S} ⊆ Sub(I).Although the inclusion of posets may fail to be a sublattice inclusion, the poset I − S is a lattice (since it is a sublattice of I) and consequently the P-Betti diagrams can be calculated via Koszul complexes.Parents.For every element v of (I − S) ⊆ I, the parents of v in I are also in I − S. Thus the parents of v in I − S are the same as those of v in I.Koszul complexes.For M : I → vect K and v in I −S, the Koszul complex of RM : (I −S) → vect K at v is given in degree d ≥ 0 by(K v RM ) d = T ⊆U I−S (v) |T |=dNat(K(S + (I−S) T , −), M ), with differentials as defined in 3.7.Finally, the value β d P M (K(S + v, −)) is given by the dimension of the d th homology of this chain complex.For example, if we choose I = {0 < • • • < 5} 2 and S = {02, 10} ⊂ I, then the projective resolution of M 0 relative to translations of the subfunctor K({02, 10}, −) ⊆ K I can be illustrated as lines indicate summands with multiplicity 2.
(w − v) i ≤ I) v, w ∈ I, v ≤ wof Ω 0 (see Example E).Via this identification, we parameterize rectangle functors by the restriction of the functor T : Ω op 0 → Fun(I, vect K ) (see Example E) to J, and denote it by T .Explicitly, T maps (v, w) to the functor coker( r i=1 K(v + (w − v) i , −) → K(v, −)).

r
i=1 ker M (v ≤ v + (w − v) i ).Thus we can compute the Koszul complex of RM at (v, w) in degree d ≥ 0 as (K (v,w) RM ) d = S⊆U I (v) T ⊆(v≤U I (w)) |S|+|T |=d r i=1 ker M (v S ≤ v S + (w T − v S ) i ) , Parameterization.The restriction of Q : Ω op → Fun(I, vect K ) (see Example D) to the following subposet of Ω is a natural parameterization of the collection C 0 : 6.3.Lemma.The following statements about M : I → vect K are equivalent: