Draft lesson.

Finite-poset sheaves and canonical homological resolutions

Written by GPT-6.1 Sol (OpenAI), October 2026. Public domain (CC0).

A finite poset gives a sheaf category in which every object has explicit finite injective and projective resolutions. Their terms are indexed by strict chains. Their differentials retain the coefficient maps, and their exactness follows from concrete contractions at each point. These resolutions compute higher sections and every Ext group. A commuting diamond, a long chain and a cycle show why the resulting height bound is not always the actual homological dimension.

Prerequisites: Modules and tensor functors over small preadditive categories, §§1–2, proves pointwise exactness and the representable evaluation maps. Augmented finite orders and contracting coface complexes, §4, proves signed face cancellation and the augmented homotopy test. Derived Hom and bounded resolution pairs, §6, proves that the indicated bounded resolutions compute all Ext groups, including their comparison maps. Injective tests and finite homological dimension, §§2–4, gives the dimension criteria. All vector spaces below may be infinite-dimensional.

1. Recover every sheaf and every map from its point diagram

Fix a field \(k\), a universe and choice. Let \(P\) be a finite poset. Give it the upper Alexandrov topology: an open set contains every point above any of its points. Put \(U_x=\{y:x\le y\}\). A sheaf of \(k\)-vector spaces gives a covariant diagram

M(x)=F(U_x),\qquad M_{xy}:M(x)\longrightarrow M(y) \quad(x\le y), \tag{1.1}

by restriction. The equations are \(M_{xx}=1\) and \(M_{yz}M_{xy}=M_{xz}\). In particular maps along two paths through a diamond must agree.

Conversely, for such a diagram define

F_M(O)=\{(m_x)_{x\in O}:M_{xy}m_x=m_y\text{ for }x\le y\text{ in }O\}. \tag{1.2}

Restriction forgets coordinates; the empty value is zero. Compatible sections on an open cover supply a unique value at each point. If \(x\le y\), any cover member containing \(x\) also contains \(y\), so that value satisfies the required equation. This proves existence and uniqueness of gluing for every cover.

Projection to \(x\) identifies \(F_M(U_x)\) with \(M(x)\); its inverse is \(m\mapsto(M_{xy}m)_{y\ge x}\). Both inverse equations follow from the diagram laws. For a sheaf \(F\), restrict a section on \(O\) to every \(U_x\), \(x\in O\). Conversely a tuple in (1.2) gives those local sections. On \(U_x\cap U_y\) their restrictions agree on every \(U_z\) with \(z\ge x,y\); these minimal opens cover the intersection, so separatedness gives agreement there. They therefore glue uniquely on \(O\). These are inverse comparisons with the original sheaf.

A sheaf map gives linear maps on the \(U_x\) commuting with (1.1). A commuting family defines a map on (1.2) coordinatewise. The two constructions are inverse on all maps and natural in the objects. Thus the sheaf category is the full category \(\mathsf D_P=\mathsf{Fun}(P,\mathsf{Vect}_k)\). Kernels, cokernels and exactness in this category are pointwise, either by their direct diagram constructions or by the preadditive-module prerequisite applied to the \(k\)-linearized opposite poset. Global sections are

\Gamma(M)=\{(m_x)_{x\in P}:M_{xy}m_x=m_y\}, \tag{1.3}

and equal \(\operatorname{Hom}_{\mathsf D_P}(k_P,M)\), where \(k_P\) has value \(k\) everywhere and identity transitions. The two maps take a transformation's value at 1 at every point, and send a compatible tuple to scalar multiplication by its coordinates.

2. Both evaluation adjoints, with their Hom inverses

For \(x\in P\) and a vector space \(V\), define diagrams

L_x(V)(y)=\begin{cases}V&x\le y,\\0&\text{otherwise},\end{cases} \qquad I_x(V)(y)=\begin{cases}V&y\le x,\\0&\text{otherwise}.\end{cases} \tag{2.1}

An arrow inside a displayed support acts as identity; every other arrow acts as zero. Transitivity makes these functors. There are inverse natural bijections

\operatorname{Hom}(L_x(V),M)\simeq\operatorname{Hom}_k(V,M(x)), \qquad \operatorname{Hom}(M,I_x(V))\simeq\operatorname{Hom}_k(M(x),V). \tag{2.2}

For the first, evaluate a map at \(x\). Its inverse sends \(a:V\to M(x)\) to \(M_{xy}a\) when \(x\le y\), and to zero elsewhere. For the second, evaluate at \(x\); the inverse for \(b:M(x)\to V\) has component \(bM_{yx}\) when \(y\le x\), and zero elsewhere. Naturality forces those same formulas in the reverse directions, proving both inverse identities. They commute with precomposition, postcomposition and the coefficient maps.

Every vector space is both projective and injective: choose lifts of basis vectors through a surjection, and extend a basis of a subspace to a basis of its ambient space. Therefore (2.2) and pointwise exactness make \(L_x(V)\) projective and \(I_x(V)\) injective. Finite sums or products retain those properties by lifting or extending all their coordinates. Also

\Gamma(I_x(V))\simeq V. \tag{2.3}

The inverse of projection at \(x\) assigns the same vector at every \(y\le x\) and zero elsewhere. The arrow equations hold, and any compatible tuple is forced to have these coordinates.

3. The normalized injective resolution and its actual contraction

Let \(h\) be the largest number of strict arrows in a chain of \(P\). For a nonempty poset, \(0\le h\le |P|-1\). Set

J^n(M)=\prod_{x_0<\cdots<x_n}I_{x_0}(M(x_n))\quad(n\ge0). \tag{3.1}

These are finite products of injectives and are zero for \(n>h\). A map of diagrams acts on each terminal coefficient, defining \(J^n\) on all morphisms. At a point \(y\), a coordinate of \(J^n(M)(y)\) is indexed by \(y\le x_0<\cdots<x_n\) and lies in \(M(x_n)\).

The augmentation \(a:M\to J^0(M)\) sends \(m\in M(y)\) to \((M_{yx}m)_{x\ge y}\). For an output coordinate \(x_0<\cdots<x_{n+1}\), define

(d^ns)_{x_0\ldots x_{n+1}} =\sum_{i=0}^{n}(-1)^i s_{x_0\ldots\widehat{x_i}\ldots x_{n+1}} +(-1)^{n+1}M_{x_nx_{n+1}}s_{x_0\ldots x_n}. \tag{3.2}

The first omission uses the natural map \(I_{x_1}(V)\to I_{x_0}(V)\), identity at points below \(x_0\) and zero elsewhere. The middle omissions keep the first support and last coefficient. The last omission uses the displayed coefficient map. This describes each summand of a genuine diagram morphism, including at points outside its support. The construction commutes with every map \(M\to N\).

The augmentation is killed by \(d^0\): the coordinate at \(x_0<x_1\), evaluated below \(x_0\), is the difference of the two equal composite restriction maps. In \(d^{n+1}d^n\), every double omission occurs twice with opposite signs. The map attached to the two omissions agrees in either order: support inclusions compose by transitivity, terminal coefficient maps compose by the diagram law, and an inclusion commutes with a coefficient map. Thus all terms cancel. This proves the complex equation with the coefficient maps included.

Here is a contraction of the augmented complex at every fixed \(y\). Let \(b_y^0:J^0(M)(y)\to M(y)\) be its coordinate at \(y\). For \(n\ge1\), put

(b_y^ns)_{x_0\ldots x_{n-1}} =\begin{cases}s_{y,x_0,\ldots,x_{n-1}}&y<x_0,\\0&y=x_0.\end{cases} \tag{3.3}

Then \(b_y^0a_y=1\). In degree zero, at \(x>y\), \(b_y^1d^0(s)\) is \(s_x-M_{yx}s_y\), while \(a_yb_y^0(s)\) is \(M_{yx}s_y\). At \(x=y\), the first term is zero and the second is \(s_y\). Their sum is identity.

In higher degree, a coordinate beginning with \(x_0>y\) has in \(b_y^{n+1}d^n(s)\) the term \(s_{x_0\ldots x_n}\) obtained by deleting the prepended \(y\). Every other deletion, including the terminal coefficient term, cancels the corresponding term of \(d^{n-1}b_y^n(s)\); inserting \(y\) shifts that deletion's sign by one. If \(x_0=y\), the first composite is zero, and in the second composite only deletion of that first point survives, giving exactly \(s_{x_0\ldots x_n}\). Hence

b_y^{n+1}d^n+d^{n-1}b_y^n=1\quad(n\ge1). \tag{3.4}

These contractions establish pointwise exactness. They are not asserted to be natural as \(y\) varies; exactness does not require that assertion. We have a functorial finite injective resolution

0\longrightarrow M\xrightarrow a J^0(M)\longrightarrow\cdots \longrightarrow J^h(M)\longrightarrow0. \tag{3.5}

4. A projective resolution with the opposite endpoint

Define

Q_n(M)=\bigoplus_{x_0<\cdots<x_n}L_{x_n}(M(x_0))\quad(n\ge0). \tag{4.1}

Every term is projective and terms above \(h\) are zero. The augmentation \(e:Q_0(M)\to M\), at \(y\), sends the summand \(m\in M(x)\), \(x\le y\), to \(M_{xy}m\).

For \(n\ge1\), the face deleting \(x_0\) uses \(M_{x_0x_1}\), the faces deleting an interior point keep the coefficient and last support, and the face deleting \(x_n\) uses the natural map \(L_{x_n}(V)\to L_{x_{n-1}}(V)\). Sum these faces with signs \((-1)^i\), \(0\le i\le n\), to define \(\partial_n:Q_n\to Q_{n-1}\). On a coordinate at \(y\), this is the rule on a labelled vector

\partial_n[x_0<\cdots<x_n;m] =[x_1<\cdots<x_n;M_{x_0x_1}m] +\sum_{i=1}^{n}(-1)^i[x_0<\cdots<\widehat{x_i}<\cdots<x_n;m]. \tag{4.2}

All chains in this formula end at a point at most \(y\). The support and coefficient maps are natural and their composites commute. Double omission therefore cancels with opposite signs, proving \(\partial_{n-1}\partial_n=0\). The two terms for \(n=1\) have equal images under \(e\), so \(e\partial_1=0\). These constructions commute with all coefficient-diagram maps.

At \(y\), embed \(M(y)\) in the summand \([y;m]\) of \(Q_0(M)(y)\). For \(n\ge0\), define on each labelled vector

t_{n,y}[x_0<\cdots<x_n;m] =\begin{cases}(-1)^{n+1}[x_0<\cdots<x_n<y;m]&x_n<y,\\0&x_n=y.\end{cases} \tag{4.3}

The augmentation composed with its section is identity. In degree zero, if \(x<y\), \(\partial_1t_{0,y}[x;m]=[x;m]-[y;M_{xy}m]\); the augmentation section supplies the missing second term. If \(x=y\), the latter already gives identity. In higher degree with \(x_n<y\), the face deleting the newly appended \(y\) in \(\partial_{n+1}t_{n,y}\) has sign \((-1)^{n+1}(-1)^{n+1}=1\) and gives the original vector. All other faces cancel those in \(t_{n-1,y}\partial_n\), since their outside signs differ by one. If \(x_n=y\), only the face deleting \(x_n\) can be appended back; its two signs are \((-1)^n\), so its composite also gives the original vector. Thus these are the complete augmented contraction equations. Pointwise exactness gives

0\longrightarrow Q_h(M)\longrightarrow\cdots\longrightarrow Q_0(M) \xrightarrow e M\longrightarrow0. \tag{4.4}

Again the pointwise contractions are used for exactness, not as a contraction in the diagram category.

5. The signed complex computing every higher section

Apply global sections to (3.5) and use (2.3). Its actual complex is

C^n(P,M)=\prod_{x_0<\cdots<x_n}M(x_n), \tag{5.1}

with differential (3.2). All comparison maps are the maps on terminal coefficients. In degree zero, its kernel is exactly (1.3), since \((d^0s)_{xy}=s_y-M_{xy}s_x\). The injective-resolution computation in the prerequisite now gives

R^n\Gamma(M)=H^n(C^\bullet(P,M)),\qquad R^n\Gamma(M)=0\quad(n>h). \tag{5.2}

Likewise (4.4), placed in nonpositive cochain degrees, computes every \(\operatorname{Ext}^n(A,B)\) by the complex \(\operatorname{Hom}(Q_n(A),B)\), with differential precomposition by \(\partial_{n+1}\). Alternatively (3.5) for \(B\) computes it by \(\operatorname{Hom}(A,J^n(B))\). The bounded Hom comparison of the prerequisite identifies these computations naturally, including the maps in both inputs. Both complexes are zero above \(h\). Thus \(\mathsf D_P\) has homological dimension at most \(h\), and every object has projective and injective dimension at most \(h\).

If \(P\) is empty, its diagram and sheaf categories are zero, all the displayed terms are zero and all Hom and derived section groups vanish. Set \(h=-1\) in that case. For nonempty \(P\), the bound is only an upper bound: relations and the coefficient diagram determine the actual groups.

6. Four graded exercises with complete solutions

Exercise 1 (grade 2). Let \(P=\{0,a,b,1\}\) be the diamond \(0<a<1\), \(0<b<1\), with \(a,b\) incomparable. Let \(S_x\) be the diagram with value \(k\) only at \(x\). Compute \(\operatorname{Ext}^2(S_0,S_1)\) and the homological dimension. Explain why global sections nevertheless have no positive derived groups.

Solution. Write \(P_x=L_x(k)\). The evaluation quotient \(P_0\to S_0\) is identity at 0 and zero at the other points. Its kernel has value zero at 0 and \(k\) at \(a,b,1\), with both maps to 1 identity. The natural inclusions from \(P_a,P_b\) onto this kernel are identity at \(a,b\) and the sum map \(k^2\to k\) at 1. Its kernel is supported at 1, with vectors \((t,-t)\). Consequently

0\longrightarrow P_1\xrightarrow{(1,-1)}P_a\oplus P_b \xrightarrow{(1,1)}P_0\longrightarrow S_0\longrightarrow0 \tag{6.1}

is exact at every point; all maps commute with the diamond relations. Applying \(\operatorname{Hom}(-,S_1)\) and (2.2) gives zero in degrees zero and one and \(k\) in degree two. Hence \(\operatorname{Ext}^2(S_0,S_1)=k\). The height is 2, so the upper bound is sharp and the homological dimension is exactly 2.

Every compatible tuple for any diagram is determined by its value at the least point 0, with inverse \(m\mapsto(M_{0x}m)_x\). Thus \(\Gamma(M)=M(0)\) naturally. This is an exact evaluation functor and has zero positive derived groups. Equivalently, \(k_P=P_0\) is projective. Nonzero Ext for another source is compatible with exactness of Hom from this particular source.

Exercise 2 (grade 2). For the chain \(0<1<\cdots<m\), build a projective resolution of every diagram of length at most one, even when \(m>1\). Determine its homological dimension.

Solution. Write \(V_i=M(i)\), \(f_i=M_{i,i+1}\) and \(f_{ij}=M_{ij}\). There is the exact row

0\longrightarrow\bigoplus_{i<m}L_{i+1}(V_i) \xrightarrow{\delta}\bigoplus_{i\le m}L_i(V_i) \xrightarrow{\epsilon}M\longrightarrow0. \tag{6.2}

The \(i\)-summand of \(\delta\) has components \(+v\) in the \(i\)-summand and \(-f_i(v)\) in the \((i+1)\)-summand. Their natural support maps define a diagram map. At \(j\), \(\epsilon\) sends \((v_0,\ldots,v_j)\) to \(\sum_{i\le j}f_{ij}v_i\), and is onto using the final summand. It kills \(\delta\) by composition of the \(f_i\).

At \(j\), the coordinates of \(\delta(w_0,\ldots,w_{j-1})\) are \(w_0\), \(w_i-f_{i-1}w_{i-1}\) for \(0<i<j\), and \(-f_{j-1}w_{j-1}\). They determine every \(w_i\) uniquely, so \(\delta\) is injective. Conversely, for a tuple in \(\ker\epsilon\), set \(w_0=v_0\) and recursively \(w_i=v_i+f_{i-1}w_{i-1}\). The final equation \(\sum f_{ij}v_i=0\) is precisely \(v_j=-f_{j-1}w_{j-1}\). This gives the unique preimage and proves exactness, also at \(j=0\).

Both terms are projective by (2.2). If \(m\ge1\), the resolution \(0\to P_1\to P_0\to S_0\to0\), tested against \(S_1\), gives \(\operatorname{Ext}^1(S_0,S_1)=k\). Hence the homological dimension is exactly 1, although the strict-chain height is \(m\). If \(m=0\), the category is vector spaces and has dimension 0. No finite-dimensional value assumption entered the argument.

Exercise 3 (grade 3). Let \(L\) be the poset of nonempty proper subsets of \(\{0,1,2\}\), ordered by inclusion. Compute all higher sections of its constant diagram \(k_L\), with an explicit coordinate on the nonzero positive group.

Solution. Write the three singletons as \(v_0,v_1,v_2\) and the three pairs as \(e_{01},e_{12},e_{20}\). The six strict arrows, in order, are

v_0<e_{01},\quad v_1<e_{01},\quad v_1<e_{12},\quad v_2<e_{12},\quad v_2<e_{20},\quad v_0<e_{20}. \tag{6.3}

There is no strict chain with two arrows, so (5.1) is \(k^6\xrightarrow d k^6\) in degrees zero and one. For each arrow its component is target value minus source value. A tuple in \(\ker d\) has the same value on adjacent points of this six-cycle. Successively following (6.3) forces all six values equal; conversely a constant tuple is killed. Thus \(H^0=k\) and \(\operatorname{rank}d=5\).

For arrow coordinates \((c_0,\ldots,c_5)\), set \(\lambda(c)=c_0-c_1+c_2-c_3+c_4-c_5\). Substituting the target-minus-source formulas cancels all six point values, so \(\lambda d=0\). The functional is onto, since it sends the first coordinate vector to 1. Its kernel has dimension 5 and equals \(\operatorname{im}d\). Hence \(\lambda\) induces an isomorphism \(H^1\to k\), whose inverse sends a scalar to that first coordinate class. Higher groups are zero. The signs also work in characteristic two.

Exercise 4 (grade 4). For \(z\) in a finite poset \(P\), identify all higher sections of \(S_z\) in terms of the strict lower poset \(L_z=\{x:x<z\}\). Apply this to the poset obtained by adjoining a greatest point \(z\) to the six-point poset \(L\) of Exercise 3. Exhibit a nonzero second derived section.

Solution. Define the reduced order-complex cochains of \(L_z\) by one copy of \(k\) for each strict chain of \(r+1\) points in degree \(r\ge0\), with the usual alternating omission differential, and by \(k\) in degree minus one mapping to the constant degree-zero tuple. If \(L_z\) is empty, only this degree-minus-one term remains. This defines \(\widetilde H^r\) algebraically, without a geometric comparison theorem.

In (5.1) for \(S_z\), only chains ending in \(z\) contribute. The chain \((z)\) gives \(k\) in degree zero. In degree \(n\ge1\), remove the final \(z\) to obtain a strict chain of \(n\) points of \(L_z\). This gives inverse coordinate identifications with the reduced complex in degree \(n-1\). The terminal term of (3.2) is zero, since its preceding point has zero \(S_z\)-value. Every other omission is exactly the reduced omission differential; the degree-zero map assigns the same scalar to every chain \(x<z\). Therefore these identifications commute with every differential, and

R^n\Gamma(S_z)\simeq\widetilde H^{n-1}(L_z;k)\quad(n\ge0). \tag{6.4}

For the seven-point example the actual complex is \(k\to k^6\to k^6\), in degrees zero, one and two. The first arrow is the constant tuple inclusion and the second is the map \(d\) of Exercise 3. Their ranks are 1 and 5. Hence degree-zero and degree-one cohomology vanish, while degree two is \(k\), identified by the same \(\lambda\). Its inverse takes the class of the first flag \(v_0<e_{01}<z\). There are no higher terms. Thus the maximal-point sheaf has a genuine nonzero \(R^2\Gamma\), despite having only one nonzero point value. This example reaches the height bound 2 for global sections themselves.