Draft lesson.

Local systems, tensor duals and global projectivity on finite posets

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

For arbitrary topological spaces, see the locally constant module and finite-generation morphism comparison. Finite point values do not by themselves give a tensor dual: the transition maps must also be invertible. Once they are, their inverses transport a diagram around paths and produce a representation of a groupoid. These local systems are categorically compact. Their global projectivity has a separate, exact criterion involving least points, and a local system can be acyclic for global sections while still admitting a nonsplit extension.

Use Finite-poset sheaves and canonical homological resolutions, §§1–5, for the sheaf/diagram equivalence, evaluation projectives and complete cochain resolutions. Its Exercise 4 proves the lower-link calculation for simple sheaves. Tensor and Hom for twisted module stacks, §1 and Theorem 6.1, gives the general internal-Hom and dual-basis maps. Dualizable sheaves, finite covers and global projectivity, §§1–2 and 4, proves the global Hom factor, both tensor-adjoint orders and the unit criterion. The direct proofs here specialize those constructions and check every point transition.

1. Pointwise tensor and finite-rank local systems

Fix a field \(k\), a universe and choice, and let \(P\) be a finite poset with upper Alexandrov topology. Write \(\mathsf D_P=\mathsf{Fun}(P,\mathsf{Vect}_k)\). Its tensor product has value \(M(x)\otimes_kN(x)\) and transition \(M_{xy}\otimes N_{xy}\). Its unit \(k_P\) has constant value \(k\) and identity transitions. These are the sheaf tensor and unit under the prerequisite's equivalence: the minimal neighbourhood value is the stalk, and tensor sheafification has the tensor of the stalk values. Equivalently, the bilinear universal maps on those neighbourhoods glue uniquely. Associativity, symmetry and units are the natural pointwise vector-space maps.

A finite-rank local system means a diagram \(L\) with every \(L(x)\) finite-dimensional and every \(L_{xy}\) invertible. Its rank is constant on each component of the undirected comparability graph. On \(U_x=\{y:x\le y\}\), the maps \(L_{xy}\) give a natural isomorphism from the constant diagram with value \(L(x)\) to \(L|_{U_x}\). Thus these are locally finite free sheaves. Projectivity and Ext below always mean those in the full category \(\mathsf D_P\); the local-system subcategory is specified separately.

2. Classify every tensor dual, including the transitions

Theorem 2.1. A diagram is dualizable if and only if it is a finite-rank local system. Its canonical dual is

Evaluation is the ordinary pairing at each point; coevaluation is the tensor corresponding to identity on that point value. These maps satisfy both triangles and identify any supplied dual with (2.1).

Proof. Suppose \(D\) is a dual, with evaluation \(D\otimes L\to k_P\) and coevaluation \(k_P\to L\otimes D\). At \(x\), write its coevaluation of 1 as a finite tensor sum \(\sum_i l_i\otimes d_i\). The first triangle says

Thus \(L(x)\) is finite-dimensional. The second triangle similarly makes \(D(x)\) finite-dimensional. The map \(D(x)\to L(x)^*\), \(d\mapsto\epsilon_x(d,-)\), has inverse \(\phi\mapsto\sum_i\phi(l_i)d_i\): (2.2) proves one inverse equation, and the second triangle proves the other. Consequently the pairing is nondegenerate on both sides.

For \(x\le y\), put \(T=L_{xy}\), \(S=D_{xy}\). Naturality of evaluation says \(\epsilon_y(Sd,Tl)=\epsilon_x(d,l)\). If \(Tl=0\), nondegeneracy at \(x\) forces \(l=0\), so \(T\) is injective. Naturality of coevaluation says \(\eta_y(1)=\sum_iTl_i\otimes Sd_i\). The triangle at \(y\) then writes every \(u\in L(y)\) as \(\sum_i\epsilon_y(Sd_i,u)Tl_i\), so \(T\) is surjective. Hence it is invertible. The evaluation equation under the just-proved dual identifications forces \(S=(T^{-1})^*\). This proves necessity and naturality of the identification of \(D\) with the canonical dual.

Conversely, (2.1) is a functor: for composable arrows, inversion reverses their order and linear dual reverses it back. Evaluation is natural because \(((T^{-1})^*\phi)(Tl)=\phi(l)\). For a basis \((e_i)\) of \(L(x)\), set \(\eta_x(1)=\sum_i e_i\otimes e_i^*\). This is basis independent: under the map \(l\otimes\phi\mapsto(u\mapsto\phi(u)l)\) to \(\operatorname{End}_k(L(x))\), it is identity. That map is invertible, with inverse \(A\mapsto\sum_i A(e_i)\otimes e_i^*\); both inverse identities are the coordinate expansion in a basis. Transport by \(T\otimes(T^{-1})^*\) sends the endomorphism identity to \(T1T^{-1}=1\), so coevaluation is natural. The two triangles at each point are \(l=\sum_i e_i^*(l)e_i\) and \(\phi=\sum_i\phi(e_i)e_i^*\). They therefore hold as diagram maps. This proves sufficiency and all claims, including zero-dimensional values. \(\square\)

For any diagram \(N\), the canonical internal-Hom map has inverse at every minimal neighbourhood:

Indeed a map on \(U_x\) is determined by \(a_x:L(x)\to N(x)\), with inverse components \(a_y=N_{xy}a_xL_{xy}^{-1}\). These commute with every further arrow by the two functor laws, and evaluating at \(x\) gives the other inverse. Restriction along \(x\le y\) sends \(a_x\) to that same formula for \(a_y\). The vector-space tensor/Hom comparison sends \(\phi\otimes n\) to \((l\mapsto\phi(l)n)\); its inverse is \(a\mapsto\sum_i e_i^*\otimes a(e_i)\). Both inverse identities are the two coordinate expansions, and its restriction is exactly \((T^{-1})^*\otimes N_{xy}\). Thus (2.3) includes all restrictions. In particular

on all maps, naturally in both inputs.

3. Universal path inverses and every loop representation

Construct a groupoid \(\Pi(P)\) with objects the points. A morphism is a composable finite word of arrows \(x\le y\) and formal reverse arrows, modulo the smallest congruence imposing identity relations, composition of two original arrows, and both cancellations with a reverse arrow. Concatenation respects that congruence and defines composition. A word followed by its reversed, oppositely oriented word cancels one pair at a time to identity, in either order. Thus its reversed word is a two-sided inverse after quotienting. Equal words have equal inverses by uniqueness of a two-sided inverse, which makes inversion well defined. This gives a groupoid. Its Hom sets are small: finite words form a small set and the quotient remains a set.

A functor on \(P\) whose arrow images are invertible extends uniquely to \(\Pi(P)\), by sending each reverse arrow to the inverse map. Every relation is preserved, proving existence; the word generators force uniqueness. A natural transformation on \(P\) extends too: from \(a_yT=T'a_x\), multiply by the inverses to get \(a_xT^{-1}=(T')^{-1}a_y\). The naturality squares therefore hold for reverse arrows and their words. Restriction and extension are inverse on objects and on all natural transformations. Consequently the local-system subcategory is equivalent to

Choose one base point \(b\) in each comparability component, and choose a path \(\gamma_x:b\to x\), with \(\gamma_b=1\). Paths exist by the definition of that component. Put \(G_b=\operatorname{Aut}_{\Pi(P)}(b)\). Evaluation of a functor at \(b\), including its loop maps, is a finite-dimensional representation of \(G_b\).

Here is its inverse. From \(\rho:G_b\to\operatorname{GL}(V_b)\), give each point of that component value \(V_b\); on \(a:x\to y\), use

The concatenated word is a based loop. The intervening \(\gamma_y\gamma_y^{-1}\) cancels in two successive arrows, proving composition; identities give identity maps. For an original functor \(L\), the components \(L(\gamma_x):L(b)\to L(x)\) give the natural isomorphism from this reconstructed representation to \(L\), since multiplying (3.2) by \(L(\gamma_y)\) gives \(L(a)L(\gamma_x)\). Conversely its value and loop action at \(b\) recover \(V_b,\rho\) exactly.

A representation map intertwining all based loops supplies a natural transformation by (3.2). For the original functors, its component at \(x\) is \(L'(\gamma_x)a_bL(\gamma_x)^{-1}\). Naturality on each path shows that any natural transformation is forced to have these components. These are both inverse maps on every morphism. Thus (3.1) is also equivalent to the finite product of the representation categories of the \(G_b\). The construction here is the word groupoid; no identification with a topological fundamental group is needed.

4. Compactness and the exact global projectivity criterion

Every pointwise finite-dimensional diagram on finite \(P\) is categorically compact, even if it is not a local system. Its Hom is the kernel of the linear map

Both products are finite. Each component Hom preserves filtered colimits using a finite basis of \(M(x)\). Filtered vector-space colimits commute with the finite products and the kernel: choose common representatives for the finitely many coordinates, then a common stage witnessing their finitely many equations or equalities. Diagram colimits are pointwise. Hence the comparison for (4.1) is the actual invertible filtered Hom comparison. In particular every local system of §1 is compact.

Call a component active for \(L\) if its rank is nonzero. Write \(P_m=L_m(k)\) for the projective evaluation diagram of the resolution prerequisite, with value \(k\) at points above \(m\) and zero elsewhere.

Theorem 4.1. A finite-rank local system \(L\) is projective in \(\mathsf D_P\) if and only if each active component has a least point. In that case it is compact projective.

Proof. Let \(\mathcal M\) be the set of minimal points. Every point of a finite poset lies above one. There is an actual epimorphism

whose \(m\)-summand at \(x\ge m\) uses \(L_{mx}\). These maps commute with transitions. At \(x\), at least one such map is an isomorphism, so \(e_x\) is onto. The source terms are projective by the evaluation adjunction, since all vector spaces are projective.

If \(L\) is projective, choose a section \(s:L\to E(L)\). At a minimal point \(m\), its only possible summand is the \(m\)-summand, and \(e_m=1\), so \(s_m=1\). If an active point \(x\) lies above two different minimal points \(m,n\), naturality gives \(s_xL_{mx}\) equal to the inclusion of the \(m\)-summand, and \(s_xL_{nx}\) equal to the inclusion of the \(n\)-summand. Both \(L_{mx}\) and \(L_{nx}\) are onto, so the image of \(s_x\) would be each of these distinct nonzero summands. Their intersection is zero, a contradiction. Thus every point in an active component has a unique minimal ancestor.

For \(x\le y\), that unique ancestor is the same for \(x,y\): an ancestor of \(x\) is also one of \(y\). Along any comparability path the label is consequently constant. On each active component there is therefore just one minimal point, and it lies below every point. It is the least point.

Conversely, on a component with least point \(m\), the components \(L_{mx}\) identify \(L\) with \(P_m\otimes L(m)\). This is a natural isomorphism by the functor law and is projective by the evaluation adjunction. Inactive components are zero. The finite direct sum over components gives a projective \(L\). Its already-proved compactness gives the last assertion. \(\square\)

It follows that \(\Gamma\) is exact if and only if every component of \(P\) has a least point. Indeed \(\Gamma=\operatorname{Hom}(k_P,-)\) is left exact; exactness is exactly projectivity of the unit, which is active on every component. Thus these same components characterize when every dualizable sheaf is compact projective. If \(P\) is empty, the category is zero and all the statements have their vacuous interpretations.

5. All ambient Ext groups from the dual tensor

For a finite-rank local system \(L\) and arbitrary diagram \(N\), there are natural isomorphisms

To prove the resolution claim, tensor by \(L^\vee\) is right adjoint to tensor by \(L\). At a point, the adjunction takes \(f:L\otimes A\to B\) to \(a\mapsto\sum_i e_i^*\otimes f(e_i\otimes a)\); its inverse evaluates \(\phi\otimes b\) on the \(L\)-input. The coordinate expansions prove both inverse identities. The natural coevaluation and evaluation make the maps natural on the entire diagram. Applying the same construction to \(L^\vee\), with its proved bidual identification, gives the other adjoint order, as in the duality prerequisite.

Tensor is exact here because vector-space tensor is exact and diagram exactness is pointwise. Therefore tensor by \(L^\vee\), as a right adjoint to an exact functor, preserves injectives. If \(N\to J\) is an injective resolution, \(L^\vee\otimes J\) is an injective resolution of \(L^\vee\otimes N\). Equation (2.4) identifies the entire complexes \(\operatorname{Hom}(L,J)\) and \(\Gamma(L^\vee\otimes J)\), including every differential. Taking cohomology and then the complete resolution formula of the prerequisite proves (5.1). Precomposition by a map between local systems induces the pointwise dual map, so the same equations give naturality in that input as well. This computation takes place in the full diagram category, not only among representations.

6. Four graded exercises with complete solutions

Exercise 1 (grade 1). On \(0<1\), let \(M=(k^2\xrightarrow{(u,v)\mapsto u}k)\). Show directly that it has no tensor dual, although it has finite point values, is categorically compact, and its tensor functor is exact.

Solution. Suppose \(D\) were a dual. The vector-space triangles make its pairing at 0 nondegenerate, as in (2.2). For \(l=(0,1)\), naturality of evaluation gives \(\epsilon_0(d,l)=\epsilon_1(D_{01}d,M_{01}l)=0\) for every \(d\), since \(M_{01}l=0\). Nondegeneracy would make \(l=0\), contradicting its second coordinate. Hence there is no dual. Formula (4.1) has finite-dimensional source point values and proves compactness. Tensor is exact at both vector spaces, hence exact on the diagram category. These are distinct properties: finite point values and categorical finite presentation do not force invertible transitions or a tensor dual.

Exercise 2 (grade 3). Classify every finite-rank local system on the six-point cycle poset with arrows, in order,

Describe all morphisms, tensor and dual maps, the based loop group, and every global-section cohomology group with both explicit comparison maps.

Solution. The first five arrows form a spanning tree. Transport a chosen value \(V\) at \(v_0\) through them, using inverse maps when crossing an arrow backwards. This identifies every point value with \(V\) and makes those five maps identity. The last map is then one \(T\in\operatorname{GL}(V)\). Write this diagram \(L_T\). These transports give a natural isomorphism from \(L_T\) to the original diagram.

A map \(L_T\to L_U\) has the same component \(A:V\to W\) everywhere, by naturality on the five tree edges, and its remaining condition is \(AT=UA\). Conversely this condition defines such a map. Thus isomorphisms are exactly the invertible intertwining maps. Tensor gives \(L_{T\otimes U}\); the dual is \(L_{(T^{-1})^*}\), by (2.1).

Choose the tree paths \(\gamma_x\) of §3. The loop using the last edge and the inverse tree path acts as \(T\). Every word can be factored into based edge loops by inserting \(\gamma_x\gamma_x^{-1}\) between its steps. Tree edge loops cancel, and the remaining edge contributes this loop or its inverse. Therefore every based loop is its integer power. Conversely the functor sending every tree edge to 0 and the last edge to 1 in the one-object group \(\mathbb Z\) respects all relations: there are no strict two-arrow chains, and reverse cancellation is respected. It distinguishes every power. Thus the based word loop group is exactly \(\mathbb Z\), and the classification includes all its finite-dimensional representations.

The cochain complex is \(V^6\xrightarrow d V^6\); its arrow values are target minus source for the first five edges and target minus \(T\) times source for the last. Its kernel is the constant six-tuple with value \(v\in\ker(T-1)\), by successively solving the five tree equations. Projection at \(v_0\) is the inverse, so \(H^0=\ker(T-1)\).

For arrow coordinates \(c=(c_0,\ldots,c_5)\), put \(\Lambda(c)=c_0-c_1+c_2-c_3+c_4-c_5\). Substitution gives \(\Lambda d=(T-1)v_0\), so it induces \(H^1\to V/(T-1)V\). Its inverse sends the class of \(v\) to the class of \((v,0,0,0,0,0)\). To verify both identities, suppose \(\Lambda(c)=(T-1)w\). Set the source value at \(v_0\) to \(w\), then solve the first five equations consecutively for \(e_{01},v_1,e_{12},v_2,e_{20}\). The sixth equation is precisely that displayed condition, so the result has coboundary \(c\). This proves its kernel and the inverse identities. There are no higher cochain terms, hence

Intertwining maps commute with both these comparisons, since their point and arrow components are all \(A\).

Exercise 3 (grade 3). On the diamond \(0<a,b<1\), put \(V=k^2\) at each point and give its four side maps

in order \(0\to a,0\to b,a\to1,b\to1\). Verify the diagram and give its actual constant model, dual and global projectivity.

Solution. Direct multiplication gives \(CA=DB=AB=H=\left(\begin{smallmatrix}2&1\\1&1\end{smallmatrix}\right)\). All four side matrices and \(H\) have determinant 1 over every field, including characteristics two and three. Use \(H\) for the transition \(0\to1\). This gives all diamond functor laws. The components \((1,A,B,H)\) define the natural isomorphism from the constant diagram \(V\) to this diagram; the two top naturality squares are exactly \(CA=DB=H\).

Its dual has the inverse transpose on each side:

The direct top map is \(H^{-t}\). The inverse-transpose laws give its functor equations, and (2.1) supplies natural evaluation, coevaluation and both triangles. The constant model is \(P_0\otimes V\), so it is compact projective. Its sections are naturally the value \(V\) at the least point 0; both inverse maps are projection there and transport by \((1,A,B,H)\). Thus all positive global-section cohomology vanishes. Commutativity of the diamond, not an arbitrary choice of four invertible matrices, permitted this trivialization.

Exercise 4 (grade 4). Choose \(\lambda\in k^\times\setminus\{1\}\), for example \(k=\mathbb F_3\), \(\lambda=2\). Use the rank-one cycle local system \(L_\lambda\) of Exercise 2. Prove that all its global-section cohomology vanishes but it is not projective in \(\mathsf D_P\). Construct a nonsplit short exact sequence and compute its ambient \(\operatorname{Ext}^1\) with the indicated kernel.

Solution. Since \(\lambda-1\) is invertible, (6.2) gives \(H^0=H^1=0\), and the height-one complex gives all higher groups zero. The diagram is nevertheless dualizable and compact by §§2 and 4. Its connected component has three minimal vertices and no least point, so Theorem 4.1 makes it nonprojective.

Here is an explicit witnessing sequence. Put \(N=S_{e_{01}}\). Let \(E\) agree with \(L_\lambda\) at every other point and have \(E(e_{01})=k^2\). Use the maps \(u\mapsto(u,0)\) from \(v_0\) and \(u\mapsto(u,u)\) from \(v_1\); all four other cycle arrows retain their maps, including the final \(\lambda\). There are no strict two-arrow chains, so these define a diagram. Inject \(N\) into the second coordinate at \(e_{01}\), and quotient there by \((a,b)\mapsto a\); at other points the quotient is identity. The maps commute with every arrow and the row

is pointwise exact. A section is forced to be identity at both \(v_0,v_1\). Naturality would then make its value on 1 at \(e_{01}\) both \((1,0)\) and \((1,1)\), a contradiction. Thus the sequence does not split.

Tensoring \(N\) by \(L_\lambda^\vee\) leaves a one-dimensional simple diagram supported at \(e_{01}\). Formula (5.1) computes \(\operatorname{Ext}^1(L_\lambda,N)\) as its first global-section cohomology. Its explicit complex is \(k\to k^2\), with scalar sent to the diagonal, from the two lower vertices \(v_0,v_1\); it has zero kernel and one-dimensional cokernel. The map \((a,b)\mapsto b-a\) identifies that cokernel with \(k\), with inverse the class of \((0,t)\). Therefore \(\operatorname{Ext}^1(L_\lambda,N)=k\), while its degree-zero and higher-than-one groups are zero. Acyclicity of \(L_\lambda\) itself tests Hom from the unit, not Hom from \(L_\lambda\) into every coefficient diagram.

Further reading

The Stacks Project, Duals, describes the general ringed-site dualizability criterion. The vector-space transition proof, both dual triangles, word transport and projectivity tests are given above. Original lesson text: CC0-1.0.