1 Introduction↩︎

1.1 Background↩︎

Let \(S\) be a finitely generated (commutative) algebra over a field \(\Bbbk\) of characteristic zero. The cotangent cohomology modules \(T^i_{S/\Bbbk}\) for \(i\in\mathbb{Z}_{\geq 0}\) introduced in [1] are finitely generated \(S\)-modules that encode aspects of the deformation theory of \(\mathop{\mathrm{Spec}}S\): \(T^0_{S/\Bbbk}\) is just the set of \(\Bbbk\)-linear derivations from \(S\) to itself, \(T^1_{S/\Bbbk}\) is the space of isomorphism classes of first order deformations of \(\mathop{\mathrm{Spec}}S\) over \(\Bbbk\), \(T^2_{S/\Bbbk}\) contains the obstructions to lifting infinitesimal deformations of \(\mathop{\mathrm{Spec}}S\), and the higher cotangent cohomology \(T^i_{S/\Bbbk}\) for \(i>2\) is relevant for deformations of complete intersections in \(\mathop{\mathrm{Spec}}S\), see e.g. [2], [3]. Likewise, the higher cotangent cohomology modules \(T^i_{S/\Bbbk}\) are important in derived deformation theory, see e.g. [4]. For an example geometric application in our setting, see Corollary 1 in §9.5.

While the modules \(T^i_{S/\Bbbk}\) for \(i=0,1,2\) have relatively concrete descriptions, they can still be challenging to explicitly compute in practice. For the higher cotangent cohomology modules the situation is even more challenging: in characteristic zero they can be described via Harrison cohomology [5], but this is not particularly amenable to explicit computations. It is thus an interesting problem to give more explicit descriptions of the \(T^i_{S/\Bbbk}\) for special classes of rings \(S\).

Along these lines, Altmann and Christophersen [6] studied the cotangent cohomology modules \(T^1_{S/\Bbbk}\) and \(T^2_{S/\Bbbk}\) when \(S\) is a Stanley-Reisner ring, that is, a quotient of a polynomial ring by a square-free¹ monomial ideal. Recall that to any simplicial complex \(\mathcal{K}\) on vertex set \(\mathcal{V}\), there is an associated Stanley-Reisner ideal \(I_\mathcal{K}\subseteq \Bbbk[x_v \;| \;v\in\mathcal{V}]\), see §3. The ideal \(I_\mathcal{K}\) is precisely the square-free monomial ideal generated by products of variables whose corresponding vertices do not belong to a common face, and all square-free monomial ideals arise in this way. Hence for a fixed finite set \(\mathcal{V}\), this establishes a one-to-one correspondence between simplicial complexes with vertex set contained in \(\mathcal{V}\) and square-free monomial ideals in the polynomial ring \(\Bbbk[x_v \;| \;v \in \mathcal{V}]\).

Denote by \(S_\mathcal{K}\) the Stanley-Reisner ring \[S_\mathcal{K}:=\Bbbk[x_v\;| \;v\in\mathcal{V}]/I_\mathcal{K}.\] We set \[T^i(\mathcal{K}) := T^i_{S_\mathcal{K}/\Bbbk}\] for every \(i \in \mathbb{Z}_{\geq 0}\). The ring \(S_\mathcal{K}\) has a natural \(\mathbb{Z}^{\mathcal{V}}\)-grading; this descends to a \(\mathbb{Z}^{\mathcal{V}}\)-grading on the cotangent cohomology modules as well. For any \(\mathbf{c}\in\mathbb{Z}^{\mathcal{V}}\), we denote by \(T^i_{\mathbf{c}}(\mathcal{K})\) the weight² \(\mathbf{c}\) piece of \(T^i(\mathcal{K})\). The main contribution of [6] is a description for \(i=1,2\) of \(T^i_{\mathbf{c}}(\mathcal{K})\) in terms of the \((i-1)\)st relative cohomology groups of some subsets determined by \(\mathbf{c}\) of the cone over the topological realization of \(\mathcal{K}\) (cf. [6].

Inspired by the results of Altmann and Christophersen, the goal of this paper is to provide general arguments that also allow us to study the higher cotangent cohomology modules \(T^i_{S/\Bbbk}\), \(i>2\) when \(S\) is a Stanley-Reisner ring.

1.2 General results↩︎

To summarize our results, we introduce a little more notation.

If \(\mathcal{K}\) is a simplicial complex, recall that the link of a face \(f\in \mathcal{K}\) is \[\mathop{\mathrm{lk}}(f,\mathcal{K})=\{g\in \mathcal{K}\;|\;f\cap g=\emptyset\;\textrm{and}\;f\cup g\in \mathcal{K}\}.\] Our first result is the following localization formula, generalizing [6] to the cases \(i>2\):

In [6], Altmann and Christophersen show that \(T^i_{\mathbf{a}-\mathbf{b}}(\mathcal{K})\) for \(i=1,2\) vanishes unless \(\mathbf{b}\) only has \(0,1\)-entries. This is not true for higher cotangent cohomology (see Example 10), but we are able to show the following bound:

For any simplicial complex \(\mathcal{K}\), Theorem 1 and Proposition 1 allow us to reduce the computation of \(T^i_{S_\mathcal{K}/\Bbbk}\) to the computation of \(T^i_{-\mathbf{b}}(\mathcal{K}')\) as \(\mathcal{K}'\) ranges over the links of \(\mathcal{K}\) and \(\mathbf{b}\) ranges over finitely many possibilities. Such \(T^i_{-\mathbf{b}}(\mathcal{K}')\) can be computed as the cohomology of an explicit complex of finite dimensional vector spaces (see §2.6), but the terms in this complex are often too large for computations to be practical.

One might hope to realize \(T^i_\mathbf{c}(\mathcal{K})\) for \(i>2\) as the \((i-1)\)st relative cohomology of some subsets of the cone over the topological realization of \(\mathcal{K}\), generalizing the results of [6] for \(i=1,2\). However, this cannot be possible, as there are simplicial complexes \(\mathcal{K}\) such that \(T^i_{S_\mathcal{K}/\Bbbk}\neq 0\) for \(i\) arbitrarily large (see Example 7). Instead, we will have to settle for describing \(T^i_{\mathbf{a}-\mathbf{b}}(\mathcal{K})\) for special choices of \(\mathbf{b}\). By the localization formula, we can and will always reduce to the case \(\mathbf{a}=0\).

For \(\mathbf{b}\in\mathbb{Z}_{\geq 0}^\mathcal{V}\) with non-zero entries \(j_1,\ldots,j_k\) listed in non-increasing order, we say \(\mathbf{b}\) has type \((j_1,j_2,\ldots,j_k)\). We define \(|\mathbf{b}|\) to be the sum \(j_1+\ldots+j_k\). Given any subset \(b\subseteq\mathcal{V}\), set \[\mathcal{K}_b=\left\{f\in \mathcal{K}\;\Big|\;f\cap \big(b \cup \mathcal{V}(\bigcap_{v\in b}\mathop{\mathrm{lk}}(v,\mathcal{K}))\big)=\emptyset\right\}.\] Here, \(\mathcal{V}(\bigcap_{v\in b}\mathop{\mathrm{lk}}(v,\mathcal{K}))\) denotes the set of vertices of \(\bigcap_{v\in b}\mathop{\mathrm{lk}}(v,\mathcal{K})\). The set \(\mathcal{K}_b\) is a subcomplex of \(\mathcal{K}\). When considering cohomology groups of complexes such as \(\mathcal{K}_b\), we will implicitly identify \(\mathcal{K}_b\) with its topological realization, see §3.1.

Our first result along these lines is the following:

1.3 Results for flag complexes↩︎

For our further results, we will restrict to the case that \(I_\mathcal{K}\) is generated by quadrics. This is equivalent to \(\mathcal{K}\) being a flag complex, see §3.2. In this situation, we have a significant improvement on Proposition 1.

We also show that \(T^i_{\mathbf{a}-\mathbf{b}}(\mathcal{K})=0\) if \(|\mathbf{b}|= i+1\) for \(i\leq 4\) and conjecture that this holds for all \(i\) (see Remark 1, and Propositions 1 and 1).

We likewise obtain:

To describe our results for \(\mathbf{b}\) of type \((1,1,1)\), we need a different simplicial complex \(\widehat\mathcal{K}_b\). Recall that the join \(\mathcal{K}_1*\mathcal{K}_2\) of simplicial complexes \(\mathcal{K}_1\) and \(\mathcal{K}_2\) on disjoint vertex sets is the complex whose faces are of the form \(f_1\cup f_2\), where \(f_1\in \mathcal{K}_1\) and \(f_2\in \mathcal{K}_2\). Let \(\Delta(\mathcal{E}_b)\) be the full simplex whose vertices correspond to \(2\)-element subsets of \(b\) not contained in \(\mathcal{K}\). We define \(\widehat\mathcal{K}_b\) to be the subcomplex of \(\Delta(\mathcal{E}_b)*\mathcal{K}_b\) arising as the union of \[\Delta(\mathcal{E}_b)^{(0)}*\mathcal{K}_b\] with \[\Delta(f)*(\mathop{\mathrm{lk}}(\{w_f\}, \mathcal{K})\cap \mathcal{K}_b)\] as \(f\) ranges over all edges of \(\Delta(\mathcal{E}_b)\). Here, \(\Delta(\mathcal{E}_b)^{(0)}\) is the \(0\)-skeleton of \(\Delta(\mathcal{E}_b)\), \(\Delta(f)\) is the simplicial complex consisting of \(f\) and all its subsets, and \(w_f\) is the element of \(b\) contained in both vertices of \(f\).

The above results combine to give a complete topological description of \(T^3_{\mathbf{c}}(\mathcal{K})\) when \(\mathcal{K}\) is a flag complex. In fact, similar statements also apply when \(\mathcal{K}\) is not a flag complex as long as one chooses the weight \(\mathbf{c}\) carefully, see Remarks 1 and 1.

For describing \(T^4(\mathcal{K})\) for a flag complex \(\mathcal{K}\), it turns out that the interesting remaining case of \(\mathbf{b}\) to deal with are types \((1,1,1,1)\) and \((2,1,1)\), see Proposition 1. We deal with these in §8.5 and §8.6, also obtaining a complete topological description of \(T^4_{\mathbf{c}}(\mathcal{K})\) when \(\mathcal{K}\) is a flag complex.

We illustrate these results with a simple example.

1.4 Applications↩︎

For applications in deformation theory, we are especially interested in situations where cotangent cohomology vanishes. Christophersen and the first author give concise sufficient criteria for \(T^2\)-vanishing when \(\mathcal{K}\) is a flag complex that is topologically a sphere, see [7] . Using our results, we obtain similar criteria for \(T^3\)-vanishing:

In [8] there is a classification of all triangulations of the \(2\)-sphere with vanishing \(T^2\). We obtain here classifications of triangulations of the \(2\)-sphere with vanishing \(T^3\).

For \(n\geq4\), let \(\mathcal{A}_n\) denote the simplicial complex whose set of vertices are the diagonals of a regular \(n\)-gon, and whose faces consist of collections of non-crossing diagonals, see §9.3. This is the boundary of the polytope dual to the \((n-4)\)-dimensional associahedron. Christophersen and the first author [7] show that \(T^2(\mathcal{A}_n)=0\). We may extend this:

In Corollary 1 in §9.5 we illustrate one geometric consequence of the above vanishing.

We note that since \(S_{\mathcal{A}_4}\) is a complete intersection, \(T^i(\mathcal{A}_4)=0\) for all \(i\geq 2\). Using our results, we may compute that \(T^4(\mathcal{A}_5)=0\), see Example 13. On the other hand, \(T^4(\mathcal{A}_n)\neq 0\) for \(n\geq 6\), see Remark 1.

1.5 Method↩︎

The cotangent cohomology modules \(T^i_{S/\Bbbk}\) may be defined as follows. Let \(P_\bullet\) be a Koszul-Tate resolution of \(S\), that is, a semifree commutative differential (non-negatively) graded algebra (cdga) that is quasi-isomorphic to \(S\), see §2.1. Then \(T^i_{S/\Bbbk}\) is the \(i\)th cohomology of the complex \(\mathop{\mathrm{Der}}^\bullet(P_\bullet,S)\) of graded derivations.

The first challenge is finding an explicit Koszul-Tate resolution \(P_\bullet\) of \(S\). Viewing \(S\) as a quotient of a polynomial ring \(R\), Hancharuk, Laurent-Gengoux, and Strobl [9] show how to construct such \(P_\bullet\) by starting with a free resolution of \(S\) as an \(R\)-module only making finitely many choices along the way. They call the resulting resolution an arborescent resolution of \(S\). In §2.2, we give an alternative inductive construction of arborescent resolutions that avoids some of the technicalities of [9].

When \(S\) is the quotient of a polynomial ring \(R\) by a monomial ideal \(I\), \(S\) has a canonical free resolution as an \(R\)-module, the so-called Taylor resolution (see §2.4). In fact, the Taylor resolution has the structure of a cdga, but is not semifree. The results of [9] imply that there is a canonical arborescent resolution of \(S\) in this case, see §2.3 for a self-contained treatment. We make this resolution as explicit as we can in §2.5. In §2.6, we use this resolution to define explicit complexes \(W_\bullet^\mathbf{c}\) of finite-dimensional vector spaces whose \(i\)th homology is dual to the weight \(\mathbf{c}\) graded piece of \(T^i_{S/\Bbbk}\).

Some results, such as the localization formula in the Stanley-Reisner case (Theorem 1), follow directly from the above discussion. For our topological interpretations of \(T^i_\mathbf{c}(\mathcal{K})\), more work is needed. Under appropriate assumptions on \(\mathbf{c}\) and homological degree, we show in §6 that one may replace the complex \(W_\bullet^\mathbf{c}\) by a simpler complex we call the non-splitting complex. We then show in §8 that these resulting complexes are homotopic to the relevant complexes of (reduced) simplicial chains arising in the topological setting.

Notation and conventions↩︎

Throughout, \(\Bbbk\) will always denote a field of characteristic zero. All rings are commutative with identity unless otherwise stated. Any weight vector \(\mathbf{c}\in \mathbb{Z}^\mathcal{V}\) will always be written as \(\mathbf{c}= \mathbf{a}- \mathbf{b}\) where \(\mathbf{a}, \mathbf{b}\in \mathbb{Z}_{\geq 0}^\mathcal{V}\) are as in Notation 1, i.e.they have disjoint supports.

Acknowledgements↩︎

We are grateful to Klaus Altmann, Adam Boocher, Jan Christophersen, Enrico Fatighenti and Frank Moore for helpful conversations.

All three authors were partially supported by an NSERC Alliance Catalyst grant. NI was partially supported by an NSERC Discovery grant. FM and AP are members of INdAM-GNSAGA. AP acknowledges funding from the European Union - NextGenerationEU under the National Recovery and Resilience Plan (PNRR) - Mission 4 Education and research - Component 2 From research to business - Investment 1.1, Prin 2022, “Geometry of algebraic structures: moduli, invariants, deformations”, DD N. 104, 2/2/2022, proposal code 2022BTA242 - CUP J53D23003720006.

2 Arborescent resolutions↩︎

2.1 Koszul-Tate resolutions and cotangent cohomology↩︎

Let \(S\) be a finitely-generated \(\Bbbk\)-algebra. A Koszul-Tate resolution of \(S\) is a semifree⁴ commutative differential graded algebra (cdga) \(P=P_\bullet\) with \(P_i=0\) for \(i<0\), along with a quasi-isomorphism \(P \to S\). Here we use a homological grading for \(P\): the differential \(\partial\) of \(P_\bullet\) decreases the homological degree by one. We consider \(S\) as a cdga concentrated in homological degree \(0\). Such a Koszul-Tate resolution may be constructed inductively as in [10] but we note that at each step (of which there are potentially infinitely many), there are choices that must be made.

The cotangent cohomology \(T^i_{S/\Bbbk}\) may be defined in terms of \(P_\bullet\). For \(i\in\mathbb{Z}\), let \(\mathop{\mathrm{Hom}}_\Bbbk^i(P,S)\) consist of those \(\Bbbk\)-linear maps \(\phi\) from \(P_\bullet\) to \(S\) satisfying \(\phi(P_j)=0\) for \(j\neq i\). We let \(\mathop{\mathrm{Der}}_\Bbbk^i(P,S)\) consist of those \(\phi\in\mathop{\mathrm{Hom}}_\Bbbk^i(P,S)\) satisfying the graded Leibniz rule \[\phi(yz)=\phi(y)z+(-1)^{ij}y\phi(z)\;\qquad\forall y\in P_j,z \in P.\] The vector spaces \(\mathop{\mathrm{Der}}_\Bbbk^i(P,S)\) form a cochain complex \(\mathop{\mathrm{Der}}_\Bbbk^\bullet(P,S)\) with differential \(\partial^*\) given by \[\phi\mapsto (-1)^{\deg \phi+1}\phi\circ \partial.\] The \(i\)th cotangent cohomology of \(S\) over \(\Bbbk\) is the \(i\)th cohomology of this complex: \[\label{eq46cotangentcohomology} T^i_{S/\Bbbk}=H^i(\mathop{\mathrm{Der}}_\Bbbk^\bullet(P,S)),\tag{1}\] see e.g. [11].

2.2 Inductive construction of arborescent resolutions↩︎

Here we give an alternate and self-contained description of arborescent resolutions [9].⁵ Let \(R\) be a polynomial ring over \(\Bbbk\), and \(S\) a quotient of \(R\). Let \(F_\bullet\) be a free resolution of \(S\) as an \(R\)-module with differential \(d\).⁶ We may assume that \(F_0=R\); moreover, by Hilbert’s syzygy theorem, we may assume that \(F_\bullet\) has only finitely many non-zero terms. Our goal is to use \(F_\bullet\) to construct a Koszul-Tate resolution \(P_\bullet\to S\).

We give an inductive description of \(P_\bullet=\bigoplus_{i\geq 0} P_i\), first as a commutative graded algebra (without differential). Let \(P(0)=R\), viewed as a commutative graded algebra concentrated in degree \(0\). Set \(P_0^\mathrm{old}=0\) and \(P_0^\mathrm{new}=0\). We now proceed inductively on \(i \geq 0\), assuming we have defined \(P(i)\) and \(P_j^\mathrm{old}\) and \(P_j^\mathrm{new}\) for all \(0 \leq j\leq i\). We set \[P_{i+1}^\mathrm{new}:=F_{i+1}\oplus P^{\mathrm{old}}_i\] and we define \(P(i+1)\) to be the (graded) symmetric algebra over \(R\) of the graded free \(R\)-module \[\bigoplus_{1 \leq j \leq i+1} P_j^\mathrm{new},\] where the elements of \(P_j^\mathrm{new}\) have homological degree \(j\). Set \(P_{i+1}^{\mathrm{old}}=P(i)_{i+1}\).

We have inclusions \(P(i)\subseteq P(i+1)\), and \[P(i)_j=P(i+1)_j=P_j^\mathrm{new}\oplus P_j^\mathrm{old}\] for any \(j\leq i\). Finally, we set \[P_\bullet=\bigcup_{i\geq 0} P(i)=\bigoplus_{i\geq 0} \left(P_i^\mathrm{new}\oplus P_i^\mathrm{old}\right),\] with algebra structure induced by those of the \(P(i)\).⁷

For instance, the homogeneous summands of low degree of \(P_\bullet\) are: \[\begin{align} P_0 &= R, \\ P_1 &= F_1, \\ P_2 &= \underbrace{F_2}_{P_2^\mathrm{new}} \oplus \underbrace{\wedge^2_R F_1}_{P_2^\mathrm{old}}, \\[.2cm] P_3 &= F_3 \oplus \underbrace{\wedge^2_R F_1}_{P_2^\mathrm{old}} \oplus \underbrace{\wedge^3_R F_1 \oplus (F_1 \otimes_R F_2)}_{P^\mathrm{old}_3}, \\[.2cm] P_4 &= F_4 \oplus \underbrace{\wedge^3_R F_1 \oplus (F_1 \otimes_R F_2)}_{P_3^\mathrm{old}} \oplus \underbrace{ (F_1 \otimes_R \wedge^2_R F_1) \oplus (\wedge^2_R F_1 \otimes_R F_2) \oplus \wedge^4_R F_1 \oplus \mathrm{S}^2_R F_2}_{P_4^\mathrm{old}}. \end{align}\]

We now wish to endow \(P_\bullet\) with a differential \(\partial\), which we will also inductively define. With respect to the direct sum decomposition \[P_j=F_j\oplus P_{j-1}^\mathrm{old}\oplus P_j^\mathrm{old},\] the differential \(\partial_j\) we construct will have the form \[\label{eqn:diffform} \begin{array}{l} \begin{array}{c c c} F_j& P_{j-1}^\mathrm{old}& P_j^\mathrm{old}\end{array}\\ \partial_j= \left(\begin{array}{c c c} d_j & -\rho_{j} & \alpha_j\\ \\ 0 & -\gamma_{j-1} & \beta _j\\ \\ 0 & \mathop{\mathrm{Id}}& \gamma_j \end{array}\right) \begin{array}{c} F_{j-1}\\ \\ P_{j-2}^\mathrm{old}\\ \\ P_{j-1}^\mathrm{old} \end{array} \end{array}\tag{2}\] where \(d\) is the differential of \(F_\bullet\) and \(\mathop{\mathrm{Id}}\) is the identity map. The differential must satisfy the graded Leibniz rule: \[\label{eqn:gl} \partial_j(yz)=\partial_k(y)z+(-1)^ky\partial_{k-j}(z)\qquad \forall y\in P_k,\;z\in P_{k-j}.\tag{3}\] Likewise, it must square to zero: \[\label{eqn:closed} \partial_{j-1} \partial_j=0.\tag{4}\]

Since \(P_0^\mathrm{old}=P_1^\mathrm{old}=0\), we begin with \(\partial_0=0\), \(\partial_1=d_1\).

2.3 Arborescent resolutions for cdgas↩︎

Suppose that the free resolution \(F_\bullet\) in fact has the additional structure of a cdga. In this case, there is a canonical choice for the differential \(\partial\) of \(P_\bullet\), and the map \(\pi_F\) of Remark 1 is a map of cdgas.

Indeed, we will define maps \(\rho_{i+1} \colon P_{i}^\mathrm{old}\to F_{i}\) as follows. For any \(y\in P_j^\mathrm{new}\), let \(\mathrm{pr}(y)\) denote its image under the projection \(\mathrm{pr} \colon F_j\oplus P_{j-1}^\mathrm{old}\to F_{j}\) to the first factor. We then set \[\rho_{i+1}(y_1\cdots y_k)=\mathrm{pr}(y_1)\cdots \mathrm{pr}(y_k)\] for any \(y_j\in P_{\delta_j}^\mathrm{new}\), \(j=1,\ldots,k\) satisfying \(\delta_1+\ldots+\delta_k=i\). Here, the product on the left hand side is in \(P_\bullet\), and the product on the right hand side is in \(F_\bullet\). This extends linearly to give an \(R\)-linear map \(\rho_{i+1} \colon P_{i}^\mathrm{old}\to F_{i}\).

2.4 Monomial ideals and the Taylor resolution↩︎

In the remainder of this paper, we will always be considering the following situation: \(R=\Bbbk[x_1,\ldots,x_n]\) will be a polynomial ring over \(\Bbbk\), \(I\subsetneq R\) will be a proper ideal generated by monomials, and \(S=R/I\) will be the quotient. The ring \(R\) has a natural \(\mathbb{Z}^n\)-grading. Since \(I\) is a monomial ideal, \(R/I\) also has a \(\mathbb{Z}^n\)-grading. In \(R\) we will use multi-index notation, e.g. for any \(\mathbf{c}= (\mathbf{c}_1,\dots,\mathbf{c}_n) \in\mathbb{Z}^n\), we set \[x^\mathbf{c}=\prod_{i=1}^n x_i^{\mathbf{c}_i}.\]

The Taylor resolution \(F_\bullet\) is a \(\mathbb{Z}^n\)-graded free resolution of \(S\), as we will now recall (cf. [12]). For input data, we fix \(\mathcal{G}\) to be any set and \(\lambda \colon \mathcal{G}\to I\) to be a map whose image consists of monomials generating \(I\) as an ideal. Typically, we will take \(\mathcal{G}\) to the minimal monomial generators of \(I\), and \(\lambda\) simply to be the inclusion.

We now take \(F_0=S\), and let \(F_1\) to be the free \(\mathbb{Z}^n\)-graded \(R\)-module with basis \(e_f\) indexed by the elements \(f \in \mathcal{G}\). Here, the weight \(\mathbf{wt}(e_f)\) is simply the weight \(\mathbf{wt}(f) \in \mathbb{Z}^n\) of \(\lambda(f)\), and the differential \(d_1\) sends \(e_f\) to \(\lambda(f)\).

For \(i>1\), we set \(F_i=\bigwedge^i_R F_1\), but with a modified \(\mathbb{Z}^n\)-grading. Denote \[e_{f_1}\wedge e_{f_2}\wedge \cdots \wedge e_{f_i}\in F_i\] by \[\{f_1 f_2\ldots f_i\}\] for any \(f_1,\ldots,f_i\in \mathcal{G}\). Then we set \[\mathbf{wt}(\{f_1\ldots f_i\})=\mathbf{wt}(\mathop{\mathrm{lcm}}(\lambda(f_1),\ldots,\lambda(f_m))).\]

The differential \(d_i \colon F_i\to F_{i-1}\) is defined via \[\label{eq46differentialfree} d_i(\{f_1\ldots f_i\})=\sum_{j=1}^i (-1)^j x^{\mathbf{wt}(\{f_1\ldots f_i\})-\mathbf{wt}(\{f_1\ldots \widehat{f_j}\ldots f_i\})} \{f_1\ldots \widehat{f_j}\ldots f_i\}.\tag{11}\] Here, \(\widehat{f_j}\) indicates we are removing the \(f_j\) term. The differential \(d\) is similar to the differential for the exterior algebra on \(F_1\), but its terms are multiplied by monomials to make \(d\) homogeneous with respect to the \(\mathbb{Z}^n\)-grading.

As described above, \((F_\bullet,d)\) is a \(\mathbb{Z}^n\)-graded free resolution of \(S\) as an \(R\)-module. Moreover, it has a canonical cdga structure [12]: the product of \(\{f_1\ldots f_i\}\in F_i\) and \(\{f_1'\ldots f_j'\}\in F_j\) is defined to be \[x^{\mathbf{wt}(\{f_1\ldots f_i\})+\mathbf{wt}(\{f_1'\ldots f_j'\})-\mathbf{wt}(\{f_1\ldots f_i f_1'\ldots f_j'\})} \{f_1\ldots f_i f_1'\ldots f_j'\}\in F_{i+j}.\] Again, this is essentially the product on the exterior algebra on \(F_1\), but modified so as to preserve the \(\mathbb{Z}^n\)-grading.

We now find ourselves in the setting of §2.3: the Taylor resolution \(F_\bullet\) of \(S\) is itself a cdga, so from it we obtain a canonical Koszul-Tate resolution \(P_\bullet \to S\) (Proposition 1). See also [9]. We will call \(P_\bullet \to S\) a Koszul-Tate-Taylor resolution of \(S\).

2.5 Explicit differential in monomial case↩︎

We now wish to explicitly describe \(R\)-module generators and the differential for a Koszul-Tate-Taylor resolution \(P_\bullet\) of \(S\). As in §2.4, we have \(R\)-module generators for \(F_i\) of the form \(\{f_1\ldots f_i\}\). Given \(p\in P_i\) and \(q\in P_j\), we denote their product in \(P_{i+j}^\mathrm{old}\subseteq P_{i+j}\) simply by \(pq\). On the other hand, given \(y\in P_i^\mathrm{old}\), we may view it as an element of \(P_{i+1}^\mathrm{new}\subseteq P_{i+1}\); we denote this element by \([y]\).

Hence, we obtain \(R\)-module generators of \(P_\bullet\) from strings involving elements \(f\in \mathcal{G}\), and matched pairs of \(\{ \;\}\) and \([ \;]\). Note that the (graded) commutativity of \(P_\bullet\) induces relations among these generators, e.g. \([\{f_1\}\{f_2\}]=-[\{f_2\}\{f_1\}]\).

Notice that all generators of homological degree at most \(4\) are primitive or quasiprimitive; in homological degree \(5\) this is no longer the case, e.g.\([[\{f_1\}\{f_2\}]\{f_3\}]\).

In order to describe the differential \(\partial\), we need a little more notation.

2.6 Computing graded pieces of cotangent cohomology↩︎

Continuing with \(S=\Bbbk[x_1,\ldots,x_n]/I\) for a monomial ideal \(I\), let \(P=P_{\bullet}\to S\) be a Koszul-Tate resolution respecting the \(\mathbb{Z}^n\)-grading of \(S\), for instance, a Koszul-Tate-Taylor resolution. As noted in Remark 1, the derivation complex \(\mathop{\mathrm{Der}}_{\Bbbk}^\bullet(P,S)\) is also \(\mathbb{Z}^n\)-graded, and so the \(T^i_{S/\Bbbk}\) are as well. For any weight \(\mathbf{c}\in \mathbb{Z}^n\), we will describe an explicit complex of finite dimensional \(\Bbbk\)-vector spaces whose homology is dual to the weight \(\mathbf{c}\) piece of \(T^i_{S/\Bbbk}\).

To that end, for each \(i\geq 0\), fix a \(\mathbb{Z}^n\)-homogeneous basis for \(P_i^\mathrm{new}\) as an \(R\)-module. For any \(\mathbf{u}\in \mathbb{Z}_{\geq 0}^n\), let \(W_{i,\mathbf{u}}\subseteq P_i^\mathrm{new}\) be the \(\Bbbk\)-vector space generated by homogeneous basis elements of weight \(\mathbf{u}\). Each \(W_{i,\mathbf{u}}\) is finite dimensional. Notice that \[P_i^\mathrm{new}=\bigoplus_{\mathbf{u}\in \mathbb{Z}_{\geq 0}^n} W_{i,\mathbf{u}}\otimes_\Bbbk R.\] Typically, we will take the homogeneous basis of \(P_i^\mathrm{new}\) to consist of generators as described in §2.5. The differential \(\partial\) of \(P_\bullet\) induces a \(\Bbbk\)-linear map \[\overline{\partial}_i \colon \bigoplus_{\mathbf{u}\in\mathbb{Z}_{\geq 0}^n} W_{i,\mathbf{u}} \to \bigoplus_{ \mathbf{u}\in \mathbb{Z}_{\geq 0}^n} W_{i-1,\mathbf{u}}\] defined via the composition \[\bigoplus_{\mathbf{u}\in\mathbb{Z}_{\geq 0}} W_{i,\mathbf{u}} \xrightarrow{\iota} P_i\xrightarrow{\partial_i} P_{i-1} \xrightarrow{\pi_R} P_{i-1}^\mathrm{new}\xrightarrow{\pi_{\Bbbk}} \bigoplus_{\mathbf{u}\in\mathbb{Z}_{\geq 0}^n} W_{i-1,\mathbf{u}}\] where \(\iota\) denotes the inclusion, \(\pi_R\) the natural projection of \(R\)-modules coming from the decomposition \(P_{i-1}=P_{i-1}^\mathrm{new}\oplus P_{i-1}^\mathrm{old}\), and \(\pi_\Bbbk\) is obtained by tensoring with the map \(R=\Bbbk[x_1,\ldots,x_n]\to \Bbbk\) defined by \(x_j\mapsto1\) for every \(j\in\{1,\ldots,n\}\).

We give an easy example where Theorem 7 allows us to compute the cotangent cohomology.

\[\begin{array}{c c c c} \textrm{Degree} &y & \textrm{Weight} & \overline\partial(y)\\ \hline 3 & [\{f_1\}\{f_2\}] & (3,1) & \{f_1f_2\}\\ 4 & [\{f_1f_2\}\{f_1\}] & (3,2) & [\{f_2\}\{f_1\}]=-[\{f_1\}\{f_2\}]\\ 4 & [\{f_1f_2\}\{f_2\}] & (4,1) & -[\{f_1\}\{f_2\}]\\ 5 & [\{f_1f_2\}\{f_1\}\{f_2\}] & (5,2) & [\{f_1f_2\}\{f_2\}]-[\{f_1f_2\}\{f_1\}]\\ 5 & [[\{f_1\}\{f_2\}]\{f_1\}] & (5,1) & [\{f_1f_2\}\{f_1\}]\\ 5 & [[\{f_1\}\{f_2\}]\{f_2\}] & (4,2) &[\{f_1f_2\}\{f_2\}]\\ 5 & [\{f_1f_2\}\{f_1f_2\}] & (4,2) &2[\{f_1f_2\}\{f_2\}]-2[\{f_1f_2\}\{f_1\}]\\ \end{array}\]

3 Simplicial complexes and Stanley-Reisner rings↩︎

3.1 Basic notions↩︎

Let \(\mathcal{V}\) be a finite non-empty set. An abstract simplicial complex on the vertex set \(\mathcal{V}\) is a collection \(\mathcal{K}\) of subsets of \(\mathcal{V}\) such that if \(f\in \mathcal{K}\), then any subset of \(f\) is also in \(\mathcal{K}\). Any simplicial complex on \(\mathcal{V}\) is always a subcomplex of the complex \(\Delta(\mathcal{V})\), the set of all subsets of \(\mathcal{V}\).

The faces of \(\mathcal{K}\) are simply the elements of \(\mathcal{K}\). The dimension of a face is one less than its cardinality. Zero-dimensional faces of \(\mathcal{K}\) are called vertices. One-dimensional faces of \(\mathcal{K}\) are called edges. We let \(\mathcal{V}(\mathcal{K})\) consist of all those \(v\in \mathcal{V}\) such that \(\{v\}\) is a vertex of \(\mathcal{K}\).

To any simplicial complex \(\mathcal{K}\), we may construct an associated Stanley-Reisner ideal \(I_\mathcal{K}\) and ring \(S_\mathcal{K}\), cf. [13]. Let \(R=\Bbbk[ x_v\;|\;v\in \mathcal{V}]\). The Stanley-Reisner ideal of \(\mathcal{K}\) is the monomial ideal \[I_\mathcal{K}=\left\langle \prod_{v\in f} x_v\;\middle|\;f\subseteq \mathcal{V},\;f\notin \mathcal{K}\right\rangle.\] Note that to generate \(I_\mathcal{K}\), it suffices to consider those monomials corresponding to minimal non-faces \(f\) of \(\mathcal{K}\). The Stanley-Reisner ring \(S_\mathcal{K}\) is the quotient \(R/I_\mathcal{K}\). Since \(I_\mathcal{K}\) is a monomial ideal, it is homogeneous with respect to the natural \(\mathbb{Z}^\mathcal{V}\)-grading on \(R\), and hence \(S_\mathcal{K}\) is also \(\mathbb{Z}^\mathcal{V}\)-graded.

The topological realization of a simplicial complex \(\mathcal{K}\) is the union in \(\mathbb{R}^\mathcal{V}\) of the simplices \[\mathop{\mathrm{conv}}\{e_v\;|\;v\in f\}\subseteq \mathbb{R}^\mathcal{V}\] where \(\mathop{\mathrm{conv}}\) denotes convex hull, \(e_v\) is the basis vector of \(\mathbb{R}^\mathcal{V}\) corresponding to \(v\in\mathcal{V}\), and \(f\) ranges over all faces of \(\mathcal{K}\). In this paper, when considering topological chain complexes, homology, and the like, we will abuse notation by identifying \(\mathcal{K}\) with its topological realization.

We recall from §1.2 than for any face \(f\in\mathcal{K}\), its link in \(\mathcal{K}\) is \[\mathop{\mathrm{lk}}(f,\mathcal{K})=\{g\in \mathcal{K}\;|\;f\cap g=\emptyset\;\textrm{and}\;f\cup g\in \mathcal{K}\}.\] Given simplicial complexes \(\mathcal{K}_1\) and \(\mathcal{K}_2\) on \(\mathcal{V}_1\) and \(\mathcal{V}_2\), their join \(\mathcal{K}_1*\mathcal{K}_2\) is the simplicial complex on the disjoint union of \(\mathcal{V}_1\) and \(\mathcal{V}_2\) whose faces are of the form \[f_1\cup f_2\qquad\mathrm{where}\;f_1\in \mathcal{K},f_2\in \mathcal{K}'.\]

3.2 Flag complexes↩︎

A flag complex is an abstract simplicial complex \(\mathcal{K}\) such that the minimal non-faces have cardinality \(2\). This is equivalent to \(I_\mathcal{K}\) being generated by quadratic monomials. Likewise, it is equivalent to the condition that \(\mathcal{V}=\mathcal{V}(\mathcal{K})\), and \(f\subseteq \mathcal{V}\) belongs to \(\mathcal{K}\) if and only if every \(2\)-element subset of \(f\) belongs to \(\mathcal{K}\). These ideals \(I_\mathcal{K}\) also arise as edge ideals of graphs, see e.g. [16] for details. It is straightforward to see that any link of a flag complex remains a flag complex.

When considering a flag complex \(\mathcal{K}\), it will be useful to consider the minimal non-faces of \(\mathcal{K}\): we let \(G(I_\mathcal{K})\) be the graph with vertices \(\mathcal{V}\) and edges consisting of all \(2\)-element subsets of \(\mathcal{V}\) not belonging to \(\mathcal{K}\). The simplicial complex \(\mathcal{K}\) is determined by \(G(I_\mathcal{K})\). To avoid confusion between \(G(I_\mathcal{K})\) and \(\mathcal{K}\), when drawing figures, we will denote edges as dashed lines. For example, in Figure 4, we depict \(G(I_\mathcal{K})\) for the complex \(\mathcal{K}\) from Example 7.

Given any generator \(y\in P_\bullet\) in a Koszul-Tate-Taylor resolution of \(S_\mathcal{K}\), we let \(G(y)\) be the subgraph of \(G(I_\mathcal{K})\) whose edges are the edges of \(G(I_\mathcal{K})\) appearing in any string \(s\) representing \(y\), and whose vertices are exactly the vertices of the aforementioned edges. We note that a primitive generator is determined up to sign by \(G(y)\).

3.3 Shellings↩︎

The concept of a shelling, which we now recall, can be useful in computing the cohomology groups of the topological realization of a simplicial complex. We will use this in Example 12.

Let \(\mathcal{K}\) be an abstract simplicial complex. A shelling (cf. [17]) of \(\mathcal{K}\) is a sequence \((f_1,\ldots,f_k)\) whose entries \(f_i\) are the maximal faces of \(\mathcal{K}\), and that satisfies the following property for each \(1\leq j<k\): each maximal face of \[\label{eqn:shelling} \bigcup_{i=1}^{j} \Delta(f_i) \cap \Delta(f_{j+1})\tag{12}\] has dimension \((\dim f_{j+1})-1\).

Not every simplicial complex is shellable, but for those that are, it is easy to understand their topology. Given a shelling \((f_1,\ldots,f_k)\), and \(1\leq j<k\), we say that \(f_{j+1}\) is a homology face if the complex from 12 is the boundary of \(\Delta(f_{j+1})\), that is, contains all proper subfaces of \(f_{j+1}\).

4 The localization formula↩︎

Let \(\mathcal{K}\) be an abstract simplicial complex on vertex set \(\mathcal{V}\). As in §1.2, given \(\mathbf{c}\in\mathbb{Z}^n\), denote by \(\mathbf{a},\mathbf{b}\in\mathbb{Z}^\mathcal{V}_{\geq0}\) the unique weights with disjoint supports \(a,b\subseteq \mathcal{V}\) satisfying \(\mathbf{c}=\mathbf{a}-\mathbf{b}\).

5 Weight bounds↩︎

5.1 General bounds↩︎

We now use the Koszul-Tate-Taylor resolution to establish some general bounds on the negative part of the weight \(\mathbf{c}\) implying vanishing of \(T^i_\mathbf{c}(\mathcal{K})\).

5.2 Koszul case↩︎

We now show how to improve on the bounds of §5.1 in a special case. Let \(S\) be a finitely generated \(\mathbb{Z}\)-graded \(\Bbbk\)-algebra, generated in degree \(1\) with homogeneous maximal ideal \(\mathfrak{m}\). Recall that \(S\) is Koszul if for all \(i\geq 0\), \(\mathop{\mathrm{Ext}}^i_S(S/\mathfrak{m},S/\mathfrak{m})\) is concentrated in weight \(i\). See [18] for alternate characterizations of this property.

6 The non-splitting complex↩︎

6.1 Splittings↩︎

Fix a monomial ideal \(I\subseteq \Bbbk[x_1,\ldots,x_n]\) and a weight \(\mathbf{c}=\mathbf{a}-\mathbf{b}\in \mathbb{Z}^n\). Fix a Koszul-Tate-Taylor resolution \(P\to S=R/I\). The complex \(W_\bullet^\mathbf{c}\) of Theorem 7 is quite large in general. In cases of special \(\mathbf{b}\), we would like to replace it by a smaller complex \(\widetilde{W}_\bullet^\mathbf{c}\) that will be more amenable to computations.

Since primitive generators belong to \(W_\bullet^\mathbf{c}\) only if \(\mathbf{b}\in\{0,1\}^\mathcal{V}\), we will assume throughout this section that \(\mathbf{b}\in\{0,1\}^\mathcal{V}\).

For \(i\geq 0\), let \(\widetilde{W}_i^\mathbf{c}\subseteq W_i^\mathbf{c}\) be the subspace generated by primitive generators that do not admit splittings. There is a natural projection \(\widetilde{\pi} \colon W_\bullet^\mathbf{c}\to \widetilde{W}_\bullet^\mathbf{c}\). We set \[\widetilde{\partial}_i^\mathbf{c}=\widetilde{\pi}_{i-1}\circ\partial_i^\mathbf{c}\colon \widetilde{W}_i^\mathbf{c}\to \widetilde{W}_{i-1}^\mathbf{c}.\]

We will show:

We will call \(\widetilde{W}_\bullet^\mathbf{c}\) the non-splitting complex. We will also show the following:

The proofs of both the lemma and the proposition are delayed until §6.3.

6.2 A homological algebra intermezzo↩︎

To prove Proposition 1 we take a brief detour into homological algebra. Let \((A_\bullet,d_A)\) and \((B_\bullet,d_B)\) be complexes of \(\Bbbk\)-vector spaces (or abelian groups), and \(f \colon A_\bullet\to B_\bullet\) be a map of complexes. We let \(\mathop{\mathrm{Cone}}(f)_\bullet\) be the mapping cone of \(f\). We thus have, for every \(i\in\mathbb{Z}\), that \(\mathop{\mathrm{Cone}}(f)_i= A_{i-1} \oplus B_i\) with differential given by \[d_{\mathop{\mathrm{Cone}}(f)} \colon (a,b) \mapsto (-d_A a, f a + d_B b).\] Of course there is a short exact sequence of complexes \[0 \to B_\bullet \to \mathop{\mathrm{Cone}}(f)_\bullet \to A_\bullet[-1] \to 0\] with maps \(b \mapsto (0,b)\) and \((a,b) \mapsto -b\). Note that only in this subsection and the very next subsection, we will use \(a,b\) to refer to elements of the complexes \(A,B\), as opposed to referring to the support of the weights \(\mathbf{a},\mathbf{b}\).

We also recall that \(\mathop{\mathrm{coker}}(f)_\bullet\) has a natural differential making \[A_\bullet \overset{f}\to B_\bullet \overset{g} \to \mathop{\mathrm{coker}}(f)_\bullet \to 0\] into an exact sequence of complexes. Moreover, although the projection \(\mathop{\mathrm{Cone}}(f)_\bullet\to B_\bullet\) sending \((a,b)\in A_{i-1}\oplus B_i\) to \(b\) is in general not a map of complexes, it is straightforward to check that composing with \(g\) does give a map of complexes \(h:\mathop{\mathrm{Cone}}(f)_\bullet \to \mathop{\mathrm{coker}}(f)_\bullet\).

6.3 Proof of Lemma 1 and Proposition 1↩︎

To prove Lemma 1 and Proposition 1, we will realize the complex \(W_\bullet^\mathbf{c}\) as a mapping cone. Indeed, for each \(i\in \mathbb{Z}\), let \(B_i\) be the subspace of \(W_i^\mathbf{c}\) generated by primitive generators. Since \(\partial^\mathbf{c}\) takes \(B_i\) to \(B_{i-1}\), we obtain that \(B_\bullet\) is a subcomplex of \(W_\bullet^\mathbf{c}\).

We similarly let \(A_{i-1}\) be the subspace of \(W_i^\mathbf{c}\) generated by generators that are not primitive. This also gives a complex \(A_\bullet\), with differential \(d_A\) induced by \(-\partial^\mathbf{c}\). The map \(f_i:A_i\to B_i\) obtained by composing \(\partial^\mathbf{c}\) with the projection from \(W_i^\mathbf{c}\) to \(B_i^\mathbf{c}\) gives a map of complexes \(f:A_\bullet\to B_\bullet\). Indeed, for \(a\in A_i\), we have \[0=\partial^\mathbf{c}\circ\partial^\mathbf{c}(a)=\partial^\mathbf{c}(-d_A(a)+f(a))=-f(d_A(a))+d_B(f(a)).\]

We now observe that \(W_\bullet^\mathbf{c}\) is precisely the mapping cone of \(f:A_\bullet\to B_\bullet\). Moreover, the cokernel of \(f_i\) is \(\widetilde{W}_i^\mathbf{c}\), and \(\widetilde{\partial}^\mathbf{c}\) agrees with the natural differential on \(\mathop{\mathrm{coker}}f\). Lemma 1 follows.

For Proposition 1, we will apply Lemma 2. Fix \(i\geq 2\), and assume that \(W_{i-1}^\mathbf{c}\) is generated by primitive and quasiprimitive generators, and \(\mathbf{b}\in\{0,1\}^\mathcal{V}\). Since \(\mathbf{b}\in\{0,1\}^\mathcal{V}\) the image of any quasiprimitive generator \(y\in A_{i-2}\) under \(f_{i-2}\) is always non-zero, and in fact, it is precisely a primitive generator \(f_{i-2}(y)\). Thus, we see that if every primitive string in \(W_{i-2}^\mathbf{c}\) has at most one splitting, \(f_{i-2}\) must be injective.

In a similar fashion, under the assumption that \(W_{i}^\mathbf{c}\) is generated by primitive and quasiprimitive strings, we see that the kernel of \(f_{i-1}\) is generated by differences of splittings of primitive generators in \(W_{i-1}^\mathbf{c}\). Under the hypotheses of Proposition 1, it thus follows from Lemma 2 that the map on homology induced by \(\widetilde{\pi}\) is an isomorphism. 0◻

6.4 Applicable cases↩︎

When can we use the non-splitting complex introduced in §6.1 to compute cotangent cohomology? The following proposition gives sufficient conditions:

7 Weight bounds for \(T^3\) and \(T^4\)↩︎

In this section, we will further improve on the bounds from §5 on the weights in which \(T^3\) and \(T^4\) are supported in the special case of flag complexes. We first consider the case of \(T^3\).

We now move on to \(T^4\).

We will apply the above lemma to show the following.

8 Topological interpretations↩︎

8.1 Preliminaries↩︎

Consider an arbitrary simplicial complex \(\mathcal{K}\). Fix some subset \(b\) of \(\mathcal{V}(\mathcal{K})\). We introduce and recall some notation connected to \(b\).

Recall from §1.2 that we defined a subcomplex of \(\mathcal{K}\) \[\mathcal{K}_b=\left\{f\in \mathcal{K}\;\Big|\;f\cap \big(b \cup \mathcal{V}(\bigcap_{v\in b}\mathop{\mathrm{lk}}(v,\mathcal{K}))\big)=\emptyset\right\}.\] We will also need the related complex: \[\mathcal{K}_b'=\left\{\coprod_{v\in b} f_v\;\Big|\;f_v\subseteq \mathcal{V}(\mathcal{K})\smallsetminus\big(b \cup \mathcal{V}(\mathop{\mathrm{lk}}(v,\mathcal{K}))\big)\;\forall v\in b,\;\textrm{and}\;\bigcup_{v\in b}f_v\in \mathcal{K}\right\}.\] Here \(\coprod\) denotes the disjoint union. This is a simplicial complex whose vertex set is contained in the disjoint union of \(\#b\) copies of \(\mathcal{V}(\mathcal{K})\). There is a natural simplicial map \(\phi_b:\mathcal{K}_b'\to \mathcal{K}_b\) via \(\coprod_{v\in b} f_v\mapsto \bigcup_{v\in b}f_v\). In particular, for any vertex \(w\) of \(\mathcal{K}_b'\), we obtain a corresponding vertex \(\phi_b(w)\) of \(\mathcal{K}_b\). For any vertex \(w\) of \(\mathcal{K}_b'\), we will denote by \(\nu(w)\) the element of \(b\) corresponding to the copy of \(\mathcal{V}(\mathcal{K})\) to which \(w\) belongs. Thus, any vertex \(w\) of \(\mathcal{K}_b'\) is determined by \(\phi_b(w)\) and \(\nu(w)\).

Suppose now that \(\mathcal{K}\) is a flag complex, and \(b\subseteq \mathcal{V}(\mathcal{K})\) with \(|b|\geq 2\). We let \(\mathcal{E}_b\) be the set of all edges of \(G(I_\mathcal{K})\) contained in \(b\), or equivalently, the set of all two-element subsets of \(b\) not contained in \(\mathcal{K}\).

We also need some notation related to simplicial homology. See e.g. [23] for details. Let \(X\) be a finite simplicial complex. Recall that for \(k\in\mathbb{Z}_{\geq 0}\), an oriented \(k\)-simplex of \(X\) is a sequence \((u_0,\ldots,u_k)\) with the set \(\{u_0,\ldots,u_k\}\) a face of \(X\). A \(k\)-chain with coefficients in \(\Bbbk\) is a formal (finite) \(\Bbbk\)-linear combination of oriented \(k\)-simplices, where we make the identification that for any permutation \(\tau\) of \(\{0,\ldots,k\}\), \[(u_{\tau(0)},\ldots,u_{\tau(k)})=\mathop{\mathrm{sign}}(\tau) (u_0,\ldots,u_k).\]

We let \({C}_\bullet(X)\) denote the complex of simplicial chains in \(X\) with coefficients in \(\Bbbk\); the differential is defined via \[(u_0,\ldots,u_k)\mapsto \sum_{j=0}^k(-1)^j (u_0,\ldots,u_{j-1},u_{j+1},\ldots,u_k).\] Let \(\widetilde{C}_\bullet(X)\) denote the augmented version of this complex, that is, \(C(X)_{-1}\cong \Bbbk\) with the obvious map \(C(X)_0\to C(X)_{-1}\) taking any oriented \(0\)-simplex to \(1\in\Bbbk\). Likewise, for a subcomplex \(X'\) of \(X\), we let \({C}_\bullet(X,X')\) denote the complex of relative simplicial chains for \(X'\) in \(X\) with coefficients in \(\Bbbk\), that is, the quotient of \(C_\bullet(X)\) by \(C_\bullet(X')\).

8.2 Type \((1)\)↩︎

We now prove Theorem 2. Since \(\mathbf{b}\) is of type \((1)\), the only generators in \(W_\bullet^{-\mathbf{b}}\) are primitive. Let \[\begin{align} Y=\{f\in\mathcal{K}\;|\;f\cup b\notin \mathcal{K}\}=\mathcal{K}\smallsetminus(\Delta(b)*\mathop{\mathrm{lk}}(b,\mathcal{K})). \end{align}\] Following [6], for \(j\geq 0\) set \[Y^{(j)}=\{(f_0,\ldots,f_j)\in Y^{j+1}\;|\;f_0\cup\ldots\cup f_j\in Y\}.\] From loc. cit., \(Y\) determines a cochain complex \(K^\bullet(Y)\): \(K^j(Y)\) is the set of alternating maps \(Y^{(j)}\to \Bbbk\), and the differential is the obvious Koszul-like differential.

There is a bijection between strings \(s\) representing elements of \(W_{j+1}^{-\mathbf{b}}\) and elements of \(Y^{(j)}\) sending \(s=\{g_0\ldots g_j\}\) with \(\overline{s}\in W_{j+1}^{-\mathbf{b}}\) to \((\mathbf{wt}(g_0),\ldots,\mathbf{wt}(g_j))\). It follows that the complex \(\mathop{\mathrm{Hom}}(W_\bullet^{-\mathbf{b}},\Bbbk)\) in cohomological degrees \(\geq 1\) is isomorphic to the complex \(K^\bullet(Y)[-1]\).

Following the notation of [6], \(\langle Y\rangle\) is the complement of \((\Delta(b)*\mathop{\mathrm{lk}}(b,\mathcal{K}))\) in the topological realization of \(\mathcal{K}\). For \(i\geq 2\) we then have \[T^i_{-\mathbf{b}}(\mathcal{K})\cong H^i(\mathop{\mathrm{Hom}}(W_\bullet^{-\mathbf{b}},\Bbbk) \cong H^{i-1}(K^\bullet(Y))\cong H^{i-1}(\langle Y \rangle ,\Bbbk)\] with the first isomorphism following from Theorem 7, the second from the above discussion, and the third isomorphism following from [6]. For \(i=1\), one also has \(T^1_{-\mathbf{b}}(\mathcal{K})\cong \widetilde{H}^{0}(\langle Y \rangle ,\Bbbk)\) by [6] (in the notation of loc. cit. \(\langle N_{a-b}\rangle=\langle Y \rangle\) and \(\langle \widetilde{N}_{a-b}\rangle=\emptyset\)).

The claim of the theorem now follows by observing that (the topological realization of) \(\mathcal{K}_b\) is a deformation retract of \(\langle Y\rangle\). Indeed, in this instance \[\mathcal{K}_b=\{f\in \mathcal{K}\;|\;f\cap \mathcal{V}(\Delta(b)*\mathop{\mathrm{lk}}(b,\mathcal{K}))=\emptyset\}\] and for any simplex \(f\in \mathcal{K}\), we may retract the complement \(f^\circ\) of \(\Delta(v)*\mathop{\mathrm{lk}}(v,\mathcal{K})\) in the topological realization of \(f\) to the topological realization of \(f\smallsetminus\mathcal{V}(\Delta(v)*\mathop{\mathrm{lk}}(v,\mathcal{K}))\). This gives a retraction of \(\langle Y\rangle\) to \(\mathcal{K}_b\).

0◻

8.3 Type \((1,1)\)↩︎

We now assume that \(\mathcal{K}\) is a flag complex and we prove Theorem 3. We first note by Proposition 1 that we may apply Proposition 1 to conclude that \(W_\bullet^{-\mathbf{b}}\) and \(\widetilde{W}_\bullet^{-\mathbf{b}}\) are quasiisomorphic. Thus, by Theorem 7 it will suffice to show that \(H_i(\widetilde{W}_\bullet^{-\mathbf{b}})\) is \(0\) (if \(b\in\mathcal{K})\) or isomorphic to \(\widetilde{H}_{i-2}(\mathcal{K}_b,\Bbbk)\) (if \(b\notin\mathcal{K}\)).

Next, for any primitive generator \(y\in W_j^{-\mathbf{b}}\), let \(G(y)\) be the subgraph of \(G(I_\mathcal{K})\) as in §3.2 and Remark 1. Then every edge of \(G(y)\) must be adjacent to an element of \(b\), every element of \(b\) must be a vertex of \(G(y)\), and the vertices of \(G(y)\) outside of \(b\) must form a simplex in \(\mathcal{K}\). Moreover, again by Remark 1, any primitive generator \(y\) is non-split if and only if \(b\) is an edge of \(G(y)\).

If \(b\in \mathcal{K}\), then from the previous paragraph it follows that \(\widetilde{W}_\bullet^{-\mathbf{b}}=0\) and we are done. We thus instead assume that \(b\notin\mathcal{K}\). By the previous paragraph, there is an isomorphism \(\widetilde{C}_\bullet(\mathcal{K}_b',\Bbbk)[-2]\cong \widetilde{W}_\bullet^{-\mathbf{b}}\) sending the oriented simplex \((u_2,\ldots,u_j)\) to \(\{f_2\ldots f_jf_{j+1}\}\) where \[f_{j+1}=x^{\mathbf{b}} \quad \text{ and } \quad f_k=x_{\phi_b(u_k)}x_{\nu(u_k)},\qquad \text{for } k=2,\ldots,j.\] Note that this is an isomorphism because \(u_k\) is uniquely determined by the pair of vertices \((\phi_b(u_k),\nu(u_k))\). By Lemma 4 we then have for \(i\geq 1\) that \[H_i(\widetilde{W}_\bullet^{-\mathbf{b}}) \cong \widetilde{H}_{i-2}(\mathcal{K}_b',\Bbbk)\cong \widetilde{H}_{i-2}(\mathcal{K}_b,\Bbbk)\] as desired. 0◻

8.4 Type \((1,1,1)\)↩︎

Again assuming \(\mathcal{K}\) is a flag complex, we now prove Theorem 4. By Proposition 1 we may apply Proposition 1 to conclude that \(H_i(W_\bullet^{-\mathbf{b}})\cong H_i(\widetilde{W}_\bullet^{-\mathbf{b}})\) for \(i=3,4\). Thus, by Theorem 7 it will suffice to show that \(H_i(\widetilde{W}_\bullet^{-\mathbf{b}})\) has the desired form for \(i=3,4\).

As in §8.3, for any primitive generator \(y\in W_j^{-\mathbf{b}}\), let \(G(y)\) be the subgraph of \(G(I_\mathcal{K})\) as in §3.2 and Remark 1. Then every edge of \(G(y)\) must be adjacent to an element of \(b\), every element of \(b\) must be a vertex of \(G(y)\), and the vertices of \(G(y)\) outside of \(b\) must form a simplex in \(\mathcal{K}\). Again by Remark 1, any primitive generator \(y\) admits a splitting unless the restriction of \(G(y)\) to \(b\) is connected. It immediately follows that if \(G(I_\mathcal{K})\) restricted to \(b\) contains fewer than two edges, i.e. it is disconnected, then \(\widetilde{W}_\bullet^{-\mathbf{b}}=0\), and hence \(T^i_{-\mathbf{b}}(\mathcal{K})=0\). But this is equivalent to \(\mathcal{K}\) having more than one edge that is a subset of \(b\). Therefore, for the rest of the proof, we may thus assume that \(G(I_\mathcal{K})\) restricted to \(b\) contains at least two edges.

We now describe more precisely when generators \(y\in W_j^{-\mathbf{b}}\) admit a splitting. By the discussion above, we can restrict our attention to generators with \(G(y)\) containing at least two edges between the elements of \(b\). By Remark 1, if \(G(y)\) contains three edges between the elements of \(b\), then it has no splitting. On the other hand, if \(G(y)\) contains precisely two edges \(e_1,e_2\) between the elements of \(b\), then \(y\) admits a splitting if and only if the complement of \(b\) in the set of vertices of \(G(y)\) belongs to \(\mathop{\mathrm{lk}}(w_{\{e_1,e_2\}},\mathcal{K})\), where \(w_{\{e_1,e_2\}}\in b\) is the unique vertex belonging to both \(e_1\) and \(e_2\). Indeed, a partition \(E_1,E_2\) as in Remark 1 cannot have both \(e_1,e_2\) in e.g. \(E_1\), since then any edge in \(E_2\) has a vertex in common with \(E_1\) (since it has a vertex in \(b\), see Remark 1). But then without loss of generality, we have \(e_1\in E_1\), \(e_2\in E_2\), and so the union of \(w_{\{e_1,e_2\}}\) with the complement of \(b\) in the set of vertices of \(G(y)\) must be a face in \(\mathcal{K}\).

We now define a surjective map of complexes \[\label{eqn:psi} \psi:C_\bullet(\Delta(\mathcal{E}_b)*\mathcal{K}_b')[-1]\to \widetilde{W}_\bullet^{-\mathbf{b}}.\tag{14}\] Indeed, for a vertex \(u\) of \(\Delta(\mathcal{E}_b)*\mathcal{K}_b'\), set \[g(u)= \begin{cases} x_vx_w & u=\{v,w\}\in \mathcal{E}_b\\ x_{\phi_b(u_j)}x_{\nu(u)} & u\in \mathcal{V}(\mathcal{K}_b')\\ \end{cases}.\] We then let \(\psi\) be the map sending an oriented simplex \(\sigma=(u_1,\ldots,u_j)\in C_{j-1}(\Delta(\mathcal{E}_b)*\mathcal{K}_b')\) to \((-1)^j\) times the class of the primitive generator \(y=\{g(u_1)\cdots g(u_j)\}\) in \(\widetilde{W}_j^{-\mathbf{b}}\) if \(y\in W_j^{-\mathbf{b}}\), and to \(0\) if \(y\notin W_j^{-\mathbf{b}}\).

We now describe the kernel of \(\psi\). There are two reasons that the image of an oriented simplex \(\sigma=(u_1,\ldots,u_j)\) might vanish: the resulting generator \(y=\{g(u_1)\cdots g(u_j)\}\) might not belong to \(W_j^{-\mathbf{b}}\), or it might have a splitting. By the above discussion on \(G(y)\), we know that one of these will certainly happen if \(\sigma\) is a simplex for the subcomplex \(\Delta(\mathcal{E}_b)^{(0)}*\mathcal{K}_b'\), where \(\Delta(\mathcal{E}_b)^{(0)}\) is the \(0\)-skeleton of \(\Delta(\mathcal{E}_b)\). On the other hand, if \(\sigma\) is not a simplex for the subcomplex \(\Delta(\mathcal{E}_b)^{(0)}*\mathcal{K}_b'\), then it is guaranteed that the generator \(y\) belongs to \(W_j^{-\mathbf{b}}\), although it might still have a splitting.

Consider then an oriented simplex \(\sigma=(u_1,\ldots,u_j)\) not supported on the complex \(\Delta(\mathcal{E}_b)^{(0)}*\mathcal{K}_b'\). By our above characterization of the generators admitting splittings, we see that \(\sigma\) lies in the kernel of \(\psi\) if and only if \(\sigma\) is a chain supported on a subcomplex of the form \[\Delta(f)*\phi^{-1}_b(\mathop{\mathrm{lk}}(w_f,\mathcal{K})\cap \mathcal{K}_b)\subseteq \Delta(\mathcal{E}_b)*\mathcal{K}_b',\] where \(f\) is an edge of \(\Delta(\mathcal{E}_b)\) and \(w_f\in b\) is as above the unique vertex belonging to the two edges corresponding to the vertices of \(f\).

Letting \(\widehat \mathcal{K}_b'\) be the union of these complexes \(\Delta(f)*\phi^{-1}_b(\mathop{\mathrm{lk}}(w_f,\mathcal{K})\cap\mathcal{K}_b)\) with \(\Delta(\mathcal{E}_b)^{(0)}*\mathcal{K}_b'\), we thus obtain that \(\psi\) induces an isomorphism of complexes \[C_\bullet(\Delta(\mathcal{E}_b)*\mathcal{K}_b',\widehat\mathcal{K}_b')[-1]\cong \widetilde{W}_\bullet^{-\mathbf{b}}.\] Since \(\Delta(\mathcal{E}_b)*\mathcal{K}_b'\) is contractible, from the long exact sequence of homology we obtain isomorphisms \[H_{i-2}(\widehat \mathcal{K}_b',\Bbbk) \cong H_i(\widetilde{W}_\bullet^{-\mathbf{b}}).\] Finally, the map \(\phi_b\) induces a surjective simplicial map \(\widehat\phi_b:\widehat \mathcal{K}_b'\to \widehat \mathcal{K}_b\). As in Lemma 4, this again induces a homotopy equivalence of \(\widehat \mathcal{K}_b'\) and \(\widehat \mathcal{K}_b\) since the preimage of any simplex under \(\widehat\phi_b\) is again a simplex. The claim of Theorem 4 now follows. 0◻

The complex \(\widehat\mathcal{K}_b\) is not as nice to work with as \(\mathcal{K}_b\) since it is not a subcomplex of \(\mathcal{K}\). However, we have the following result. Recall that for any topological space, \[\widetilde{H}^{-1}(X,\Bbbk):=\begin{cases} 0 & X\neq \emptyset;\\ \Bbbk& X=\emptyset. \end{cases}\]

We first prove the following lemma:

We return to the proof of the corollary.

8.5 Type \((2,1,1)\)↩︎

For the case \(\mathbf{b}\) of type \((2,1,1)\), we need to slightly modify the complex \(\mathcal{K}_b\). Let \(\mathcal{K}\) be a simplicial complex on \(\mathcal{V}\), \(b\subseteq \mathcal{V}\), and \(u\in \mathcal{V}\). We set \[\mathcal{K}_{u,b}=\left\{f\in \mathcal{K}\;\Big|\;f\cap \left(\{u\} \cup \mathcal{V}(\bigcap_{v\in b}\mathop{\mathrm{lk}}(v,\mathcal{K}))\right)=\emptyset\right\}.\]

8.6 Type \((1,1,1,1)\)↩︎

For this case, we need some extra notation. Fix \(\mathbf{b}\in\mathbb{Z}_{\geq 0}^\mathcal{V}\) of type \((1,1,1,1)\). We will fix an order on the elements of \(b\). This induces an order on the edges of the subgraph of \(G(I_\mathcal{K})\) obtained by restricting to the vertex set \(b\) (e.g. by taking lex order). For \(i=3,4,5,6\), let \(\Gamma_i(b)\) be the set of all connected graphs with vertex set \(b\) and \(i\) edges, none of which belong to \(\mathcal{K}\). These are thus subgraphs of \(G(I_\mathcal{K})\). The potential combinatorial types of elements of \(\Gamma_i(b)\) for \(i=3,4,5,6\) are pictured in Figure 11. We may view each element of \(\Gamma_i(b)\) as a face of \(\Delta(\mathcal{E}_b)\). Given \(\gamma\in\Gamma_i(b)\), we let \(e_1(\gamma),\ldots,e_i(\gamma)\) be the edges of \(\gamma\), written in order from least to greatest, and we let \(\gamma_j\) be the graph obtained by deleting the edge \(j\).

If \(\gamma'\in \Gamma_{i-1}(b)\) is a subgraph of \(\gamma\in \Gamma_i(b)\), we may consider the intersection of the partition \(\mathfrak{S}\) with \(\gamma'\) which we will denote by \(\mathfrak{S}_{|\gamma'}\). Note that this might fail to be a splitting of \(\gamma'\).

Graphs \(\gamma\) in \(\Gamma_3(b)\) have one, two, or three splittings. For \(\gamma\) a path of length three, the splitting with \(\cap \mathfrak{S}\) maximal is non-standard. The other splitting, and all three splittings for the other combinatorial type in \(\Gamma_3(v)\), are standard. Graphs in \(\Gamma_4(b)\) have zero, one, or two splittings. A splitting of \(\gamma\in \Gamma_4(b)\) is standard if for all subgraphs \(\gamma'\in \Gamma_3(b)\) of \(\gamma\), either \(\mathfrak{S}_{|\gamma'}\) is standard or fails to be a splitting.⁹

We now define a subcomplex \(\widehat\mathcal{K}_b^\mathrm{std}\) of \(\Delta(\mathcal{E}_b)*\mathcal{K}_b\). Indeed, let \(\widehat\mathcal{K}_b^\mathrm{std}\) consist of the union of the complexes: \[\begin{align} \left(\Delta(\mathcal{E}_b)\smallsetminus\bigcup_{i=3}^6 \Gamma_i(b)\right)*\mathcal{K}_b;\;\textrm{and}\\ \Delta(\gamma)*\left(\mathop{\mathrm{lk}}(\cap \mathfrak{S},\mathcal{K})\cap \mathcal{K}_b\right) \end{align}\] as \(\gamma,\mathfrak{S}\) range over all \(\gamma\in \bigcup_{i=3,4} \Gamma_i(b)\) and standard splittings \(\mathfrak{S}\) of \(\gamma\).

Consider the map \(\eta:K_1\to K_0\) arising from the sum of \(\eta_1,\ldots,\eta_4\) in the following diagram: \[\begin{tikzcd}[ampersand replacement=\&,column sep=tiny] K_1:\& \begin{array}{c} \displaystyle\bigoplus_{\substack{\gamma\in \Gamma_4(b)\\ \mathfrak{S}\;\textrm{non-std.}}} \Bbbk\cdot (\gamma,\mathfrak{S}) \end{array} \& \oplus \&\begin{array}{c} \displaystyle\bigoplus_{\substack{\gamma\in \Gamma_3(b)\\ \mathfrak{S}\;\textrm{non-std.}}} C_0(\mathop{\mathrm{lk}}(\cap \mathfrak{S},\mathcal{K})\cap \mathcal{K}_b)\cdot (\gamma,\mathfrak{S}) \end{array}\\ K_0: \&{\displaystyle\bigoplus_{\substack{\gamma\in\Gamma_3(b)\\ \mathfrak{S}\;\textrm{non-std.}}}\Bbbk\cdot (\gamma,\mathfrak{S})} \&\oplus \& H_2(\widehat\mathcal{K}_b^\mathrm{std},\Bbbk) \arrow["\eta_1",from=1-2, to=2-2] \arrow["\eta_3"{pos=0.7},from=1-2, to=2-4] \arrow["\eta_2",from=1-4, to=2-2] \arrow["\eta_4",from=1-4, to=2-4] \end{tikzcd}\] where the maps \(\eta_i\) are defined as follows. We have \[\eta_1((\gamma,\mathfrak{S}))=\sum_j (-1)^j\cdot \mathop{\mathrm{sign}}(\mathfrak{S})\cdot \mathop{\mathrm{sign}}(\mathfrak{S}_{|\gamma_j})\cdot(\gamma_j,\mathfrak{S}_{|\gamma_j})\] where we treat \((\gamma_j,\mathfrak{S}_{|\gamma_j})\) as zero if \(\gamma_j\notin \Gamma_3(b)\) or \(\mathfrak{S}_{|\gamma_j}\) is not a non-standard splitting. Likewise, \[\eta_2((v)\cdot (\gamma,\mathfrak{S}))=(\gamma,\mathfrak{S}).\]

Under \(\eta_3\), \((\gamma,\mathfrak{S})\) maps to the class of the boundary of the simplicial chain \(\mathop{\mathrm{sign}}(\mathfrak{S})\cdot (e_1(\gamma),e_2(\gamma),e_3(\gamma),e_4(\gamma))\). Likewise, under \(\eta_4\), \((v)\cdot (\gamma,\mathfrak{S})\) maps to the class of the boundary of the simplicial chain \(\mathop{\mathrm{sign}}(\mathfrak{S})\cdot (e_1(\gamma),e_2(\gamma),e_3(\gamma),v)\).

We delay the proof until the end of the section, first providing an example illustrating the theorem.

We also obtain the following result guaranteeing vanishing of \(T^4_{-\mathbf{b}}(\mathcal{K})\).

9 \(T^3\) vanishing and the associahedron↩︎

9.1 Criteria for \(T^3\) vanishing↩︎

Putting our topological results together, we obtain the following sufficient criterion for \(T^3\) vanishing.

We now prove Corollary 1 from §1.4:

9.2 Triangulations of \(S^1\) with \(T^3=0\).↩︎

A triangulation of \(S^1\) with \(n\) vertices is simply the boundary of an \(n\)-gon.

9.3 The associahedron↩︎

Fix \(n\geq 3\). Consider an \(n\)-gon \(\mathcal{R}_n\), with vertices labeled \(1,\ldots,n\) in counterclockwise order. Let \(\mathcal{V}\) be the set of \({n\choose 2}-n\) diagonals in this polygon. For \(1\leq i<j\leq n\) we will refer to the diagonal with endpoints \(i,j\) simply as \(ij\). Two diagonals are said to cross if they intersect in their relative interiors. We denote by \(\mathcal{A}_n\) the simplicial complex on the vertex set \(\mathcal{V}\) consisting of sets of diagonals that do not pairwise cross; it follows immediately from the definition that \(\mathcal{A}_n\) is a flag complex. Maximal faces of \(\mathcal{A}_n\) thus correspond to triangulations of \(\mathcal{R}_n\). The complex \(\mathcal{A}_3\) is the empty set, \(\mathcal{A}_4\) consists of two isolated vertices, \(\mathcal{A}_5\) is (coincidentally) the boundary of a pentagon, and \(\mathcal{A}_6\) is complex (i) from Figure 14 in §9.4. In general \(\mathcal{A}_n\) is an \((n-4)\)-dimensional sphere, and is the boundary of the polytope dual to the associahedron on \(n-1\) elements, see e.g. [26].

We now prove Corollary 1.

9.4 Triangulations of \(S^2\) with \(T^3=0\).↩︎

We now prove Corollary 1. By Proposition 1 and Theorem 9, we only need to consider those triangulations of \(S^2\) where each vertex has valency at most \(5\). These are pictured in Figure 14; see [8]. There is a vertex at infinity not pictured.

The triangulation (i) is just the complex \(\mathcal{A}_6\), so has \(T^3=0\) by Corollary 1. Starting with any of the triangulations (a)–(g), one may obtain (i) via a sequence of edge subdivisions, so the \(T^3\) vanishing for (i) implies \(T^3\) vanishing for these complexes as well by [25] and semicontinuity of cotangent cohomology (cf. [2]).

It remains to see that \(T^3\) is non-zero for (h), (j), and (k). For each of these complexes \(\mathcal{K}\), we consider \(T^3_{-\mathbf{b}}(\mathcal{K})\) with \(\mathbf{b}\) of type \((1,1)\) and \(b\) given by the two red vertices in Figure 14. In each case, these two vertices have disjoint links, so \(\mathcal{K}_b\) is homotopy equivalent to \(S^2\) with \(2\) points removed. It follows by Theorem 3 that \(T^3_{-\mathbf{b}}(\mathcal{K})\cong H^1(\mathcal{K}_b,\Bbbk)\cong \Bbbk\), so in particular \(T^3(\mathcal{K})\neq 0\). Note that Theorem 3 applies on the nose in cases (j) and (k), as these are flag complexes. Case (h) is not a flag complex, but by Remark 1 we may still apply the conclusion of Theorem 3.

We know by Remark 1 that \(T^4\neq 0\) for the triangulation (i). We will see below in Example 14 that for (G), \(T^4=0\). The vanishing of \(T^4\) for the triangulations (a)–(f) then again follows by [25] and semicontinuity of cotangent cohomology (cf. [2]). 0◻

9.5 The dual quotient bundle↩︎

We now give a sample geometric application of the vanishing of higher cotangent cohomology. For any \(n\in \mathbb{Z}\), \(n\geq 3\), let \(G(2,n)\) be the Grassmannian parametrizing \(2\)-dimensional subspaces of \(\Bbbk^n\). The quotient bundle \(\mathcal{Q}\) is the rank \(n-2\) vector bundle whose fiber at \(x\in G(2,n)\) is \(\Bbbk^n/L_x\), where \(L_x\subseteq \Bbbk^n\) is the linear space corresponding to the point \(x\). We thus have a surjection of vector bundles \[G(2,n)\times \Bbbk^n \to \mathcal{Q}.\] Let \(\mathcal{Q}^*\) be the dual vector bundle, i.e.  the dual quotient bundle. The above surjection dualizes to an inclusion \[\mathcal{Q}^*\hookrightarrow G(2,n)\times (\Bbbk^n)^*.\]

The Grassmannian \(G(2,n)\) has a natural embedding in \(\mathbb{P}^{ {n\choose 2}-1}\) sending a point \(x\) to the \(2\times 2\) minors of any matrix whose rowspan is \(L_x\); this is the Plücker embedding. This induces an embedding of \(\mathcal{Q}^*\) in \(\mathbb{P}^{ {n\choose 2}-1}\times \mathbb{A}^n\). Taking the projectivization¹⁰ \(X\) of \(\mathcal{Q}^*\), we obtain an embedding of \(X\) in \(\mathbb{P}^{ {n\choose 2}-1}\times \mathbb{P}^{n-1}\).

In [27], Turo and the first author describe a certain simplicial complex \(\mathcal{K}_n\) for \(n\geq 5\) obtained by an edge subdivision of \(\mathcal{A}_n*\mathcal{A}_2\). The complex \(\mathcal{K}_n\) is related to the dual quotient bundle \(\mathcal{Q}^*\) by the following:

We may now extend [27] from the case of hypersurfaces to codimension-two complete intersections:

References↩︎

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1. i.e. radical↩︎

2. We use the term weight here instead of degree to differentiate from the (co)homological degrees that will occur elsewhere in the paper.↩︎

3. Recall that \(\widetilde{H}^{-1}(\mathcal{K}_b,\Bbbk)\) vanishes unless \(\mathcal{K}_b=\emptyset\), in which case it is \(\Bbbk\).↩︎

4. This means it is free as a commutative graded algebra after forgetting the differential.↩︎

5. The terminology “arborescent” comes from [9], in which rooted trees play an important role in constructing the resolution. We avoid the use of such trees here, but keep the name.↩︎

6. We continue using homological conventions, so \(d\) decreases degree by \(1\).↩︎

7. In other words, \(P_\bullet\) is the colimit of the \(P(i)\) in the category of commutative graded algebras.↩︎

8. Note that the string \(``\{\}"\) representing \(1\in P_0=R\) is primitive.↩︎

9. Graphs in \(\Gamma_5(b)\) and \(\Gamma_6(b)\) never have splittings.↩︎

10. We use the convention that the projectivization of a vector bundle means taking all lines in each fiber (as opposed to codimension-one quotients).↩︎