Let $(R, \mathfrak m)$ denote an $n$-dimensional Gorenstein ring. For an ideal $I \subset R$ of height $c$ we are interested in the endomorphism ring $B = \Hom_R(H^c_I(R), H^c_I(R)).$ It turns out that $B$ is a commutative ring. In the case of $(R,\mathfrak m)$ a regular local ring containing a field $B$ is a Cohen-Macaulay ring. Its properties are related to the highest Lyubeznik number $l = \dim_k \Ext_R^d(k,H^c_I(R)).$ In particular $R \simeq B$ if and only if $l = 1.$ Moreover, we show that the natural homomorphism $\Ext_R^d(k, H^c_I(R)) \to k$ is non-zero.
In this article, we study certain local cohomology modules over $F$-pure rings. We give sufficient conditions for the vanishing of some Lyubeznik numbers, derive a formula for computing these invariants when the $F$-pure ring is standard graded and, by its means, we provide some new examples of Lyubeznik tables. We study associated primes of certain Ext-modules, showing that they are all compatible ideals. Finally, we focus on properties that Lyubeznik numbers detect over a globally $F$-split projective variety.
We give an explicit recipe for determining iterated local cohomology groups with support in ideals of minors of a generic matrix in characteristic zero, expressing them as direct sums of indecomposable D-modules. For non-square matrices these indecomposables are simple, but this is no longer true for square matrices where the relevant indecomposables arise from the pole order filtration associated with the determinant hypersurface. Specializing our results to a single iteration, we determine the Lyubeznik numbers for all generic determinantal rings, thus answering a question of Hochster.
We establish a "second vanishing theorem" for local cohomology modules over regular rings of unramified mixed characteristic, which relates the connectedness of the spectrum of a ring with the vanishing of local cohomology. Applying this, and new results on the mixed characteristic Lyubeznik numbers, we further study connectedness properties of the spectra of a certain class of rings.
We study the structure of local cohomology with support in Pfaffian varieties as a module over the Weyl algebra D_X of differential operators on the space X of n x n complex skew-symmetric matrices. The simple composition factors of these modules are known by the work of Raicu-Weyman in 2016, and when n is odd, the general theory implies that the local cohomology modules are semi-simple. When n is even, we show that the local cohomology is a direct sum of indecomposable modules coming from the pole order filtration of the Pfaffian hypersurface. We then determine the Lyubeznik numbers for Pfaff…
In this note I give a description of Lyubeznik's local cohomology invariants for a certain natural class of local rings, namely the ones which have the same local cohomology vanishing as one expects from an isolated singularity. This strengthens the results of Bondu and myself in math.AG/0406265 while at the same time somewhat simplifying the proofs. Through examples I further point out the bad behavior of these invariants under reduction to positive characteristic.
We discuss for an affine variety $Y$ embedded in affine space $X$ two sets of integers attached to $Y\subseteq X$ via local and de Rham cohomology spectral sequences. We give topological interpretations, study them in small dimension, and investigate to what extent one can attach them to projective varieties.
We investigate the relationship between connectedness properties of spectra and the Lyubeznik numbers, numerical invariants defined via local cohomology. We prove that for complete equidimensional local rings, the Lyubeznik numbers characterize when connectedness dimension equals one. More generally, these invariants determine a bound on connectedness dimension. Additionally, our methods imply that the Lyubeznik number with indices (1,2) of the local ring at the vertex of the affine cone over a projective variety is independent of the choice of its embedding into projective space.
We investigate the Lyubeznik numbers, and the injective dimension of local cohomology modules, of finitely generated $\mathbb{Z}$-algebras. We prove that the mixed characteristic Lyubeznik numbers and the standard ones agree locally for almost all reductions to positive characteristic. Additionally, we address an open question of Lyubeznik that asks whether the injective dimension of a local cohomology module over a regular ring is bounded above by the dimension of its support. Although we show that the answer is affirmative for several families of $\mathbb{Z}$-algebras, we also exhibit an exa…
We study Bass numbers of local cohomology modules supported on squarefree monomial ideals paying special attention to Lyubeznik numbers. We build a dictionary between local cohomology modules and minimal free resolutions that allow us to interpret Lyubeznik numbers as the obstruction to the acyclicity of the linear strands of the Alexander dual ideals. The methods we develop also help us to give a bound for the injective dimension of the local cohomology modules in terms of the dimension of the small support.
Given a local ring containing a field, we define and investigate a family of invariants that includes the Lyubeznik numbers, but that captures finer information. These "generalized Lyubeznik numbers" are defined as lengths of certain iterated local cohomology modules in a category of D-modules, and in order to define them, we develop the theory of a functor Lyubeznik utilized in proving that his original invariants are well defined. In particular, this functor gives an equivalence of categories with a category of D-modules. These new invariants are indicators of F-regularity and F-rationality …
The Frobenius depth denoted by F-depth defined by Hartshorne-Speiser in 1977 and later by Lyubeznik in 2006, in a different way, for rings of positive characteristic. The first aim of the present paper is to compare the F-depth with formal grade and reprove some results of Lyubeznik using formal local cohomology. Then the endomorphism rings of local cohomology modules will be considered. As an application, we reprove the results due to Huneke-Koh in positive characteristic and Lyubeznik in characteristic zero on the annihilators of local cohomology modules.
School of Mathematics, Institute for Advanced Study（高等研究院数学学院）；Department of Mathematics, Princeton University（普林斯顿大学数学系）；Department of Mathematics, Purdue University（普渡大学数学系）
Let $R$ be a commutative Noetherian ring that is a smooth $\mathbb Z$-algebra. For each ideal $I$ of $R$ and integer $k$, we prove that the local cohomology module $H^k_I(R)$ has finitely many associated prime ideals. This settles a crucial outstanding case of a conjecture of Lyubeznik asserting this finiteness for local cohomology modules of all regular rings.
This paper presents an algorithm for calculation of the Lyubeznik numbers of a local ring which is a homomorphic image of a regular local ring $R$ of prime characteristic. The methods used employ Lyubeznik's $F$-modules over $R$, particularly his $F$-finite $F$-modules, and also the modules of generalized fractions of Sharp and Zakeri. It is shown that many modules of generalized fractions over $R$ have natural structures as $F$-modules; these lead to $F$-module structures on certain local cohomology modules over $R$, which are exploited, in conjunction with $F$-module structures on injective …
We give a description of certain local cohomology invariants (introduced by Lyubeznik) for a class of mildly singular varieties in positive characteristic. The novelty of our approuch lies in using the new Riemann-Hilbert type correspondenc of Emerton and Kisin to express these invariants in terms of 'etale cohomology with ZZ/pZZ coefficients. Thus we are able to obtain results in positive characteristic whereas earlier approaches were restricted (due to the use of de Rham theory) to the analytic case.
Lyubeznik conjectured that local cohomology modules of regular rings have finitely many associated primes. We examine this conjecture for polynomial rings over the integers, and record some equational identities that arise from studying special cases.
For a polynomial ring $R=k[x_1,...,x_n]$, we present a method to compute the characteristic cycle of the localization $R_f$ for any nonzero polynomial $f\in R$ that avoids a direct computation of $R_f$ as a $D$-module. Based on this approach, we develop an algorithm for computing the characteristic cycle of the local cohomology modules $H^r_I(R)$ for any ideal $I\subseteq R$ using the Čech complex. The algorithm, in particular, is useful for answering questions regarding vanishing of local cohomology modules and computing Lyubeznik numbers. These applications are illustrated by examples of com…
This paper applies G. Lyubeznik's notion of $F$-finite modules to describe in a very down-to-earth manner certain annihilator submodules of some top local cohomology modules over Gorenstein rings. As a consequence we obtain an explicit description of the test ideal of Gorenstein rings in terms of ideals in a regular ring.
Let $(R,\mathfrak m)$ denote an $n$-dimensional complete local Gorenstein ring. For an ideal $I$ of $R$ let $H^i_I(R), i \in \mathbb Z,$ denote the local cohomology modules of $R$ with respect to $I.$ If $H^i_I(R) = 0$ for all $i \not= c = \height I,$ then the endomorphism ring of $H^c_I(R)$ is isomorphic to $R$ (cf. \cite{HSt} and \cite{HS}). Here we prove that this is true if and only if $H^i_I(R) = 0, i = n, n -1$ provided $c \geq 2$ and $R/I$ has an isolated singularity resp. if $I$ is set-theoretically a complete intersection in codimension at most one. Moreover, there is a vanishing resu…
We study a question raised by Eisenbud, Mustata, and Stillman regarding the injectivity of natural maps from Ext modules to local cohomology modules. We obtain some positive answers to this question which extend earlier results of Lyubeznik. In the process, we also prove a vanishing theorem for local cohomology modules which connects theorems previously known in the case of positive characteristic and in the case of monomial ideals.
Let $(R,\mathfrak{m})$ denote an $n$-dimensional Gorenstein ring. For an ideal $I \subset R$ with $\grade I = c$ we define new numerical invariants $τ_{i,j}(I)$ as the socle dimensions of $H^i_{\mathfrak{m}}(H^{n-j}_I(R))$. In case of a regular local ring containing a field these numbers coincide with the Lyubeznik numbers $λ_{i,j}(R/I)$. We use $τ_{d,d}(I), d = \dim R/I,$ to characterize the surjectivity of the natural homomorphism $f : \hat{R} \to \Hom_{\hat{R}}(H^c_{I\hat{R}}(\hat{R}),H^c_{I\hat{R}}(\hat{R}))$. As a technical tool we study several natural homomorphisms. Moreover we prove a …
Let $R$ be a polynomial ring in finitely many variables over the integers, and fix an ideal $I$ of $R$. We prove that for all but finitely prime integers $p$, the Bockstein homomorphisms on local cohomology, $H^k_I(R/pR)\to H^{k+1}_I(R/pR)$, are zero. This provides strong evidence for Lyubeznik's conjecture which states that the modules $H^k_I(R)$ have a finite number of associated prime ideals.
Let $K$ be an algebraically closed field of characteristic zero and let $R = K[X_1,\ldots,X_n]$. Let $I$ be an ideal in $R$. Let $A_n(K)$ be the $n^{th}$ Weyl algebra over $K$. By a result of Lyubeznik, the local cohomology modules $H^i_I(R)$ are holonomic $A_n(K)$-modules for each $i \geq 0$. In this paper we compute the Euler characteristic of De-Rahm cohomology of $H^{\htt P}_P(R)$ for certain classes of prime ideals $P$ in $R$.
We study the behaviour of the finiteness of the set of associated primes of local cohomology modules, more generally of Lyubeznik functors, under various ring extensions. At first, we review the results for flat and faithfully flat extensions and we present new applications of them. Then, we focus how the finiteness property of the set of associated primes of local cohomology modules and Lyubeznik functors is transferred from extended ring to the base ring of pure and cyclically pure ring extensions. We show that finiteness property can be transferred from a ring to its pure local subring and …
Given a local domain $(R,m)$ of prime characteristic that is a homomorphic image of a Gorenstein ring, Huneke and Lyubeznik proved that there exists a module-finite extension domain $S$ such that the induced map on local cohomology modules $H^i_m(R)\to H^i_m(S)$ is zero for each $i<\dim R$. We prove that the extension $S$ may be chosen to be generically Galois, and analyze the Galois groups that arise.
Let $M$ denote a finitely generated module over a Noetherian ring $R$. For an ideal $I \subset R$ there is a study of the endomorphisms of the local cohomology module $H^g_I(M), g = \operatorname{grade} (I,M),$ and related results. Another subject is the study of left derived functors of the $I$-adic completion $Λ^I_i(H^g_I(M))$, motivated by a characterization of Gorenstein rings given in the book by Simon and the author. This provides another Cohen-Macaulay criterion. The results are illustrated by several examples. There is also an extension to the case of homomorphisms of two different loc…
The aim of this paper is to extend the main result of C. Huneke and G. Lyubeznik in [Adv. Math. 210 (2007), 498--504] to the class of rings that are images of Cohen-Macaulay local rings. Namely, let $R$ be a local Noetherian domain of positive characteristic that is an image of a Cohen-Macaulay local ring. We prove that all local cohomology of $R$ (below the dimension) maps to zero in a finite extension of the ring. As a direct consequence we obtain that the absolute integral closure of $R$ is a big Cohen-Macaulay algebra. Since every excellent local ring is an image of a Cohen-Macaulay local …
Let $K$ be a field of characteristic zero, $R = K[X_1,...,X_n]$ and let $I$ be an ideal in $R$. Let $A_n(K) = K<X_1,...,X_n, \partial_1,..., \partial_n>$ be the $n^{th}$ Weyl algebra over $K$. By a result due to Lyubeznik the local cohomology modules $H^i_I(R)$ are holonomic $A_n(K)$-modules for each $i \geq 0$. In this article we compute the Koszul homology modules $H_*(\partial_1,...,\partial_n ; H^*_I(R))$ for certain classes of ideals.
Let $R=k[x_1,\dots,x_n]$ be a ring of polynomials over a field $k$ of characteristic $p>0$. There is an algorithm due to Lyubeznik for deciding the vanishing of local cohomology modules $H^i_I(R)$ where $I\subset R$ is an ideal. This algorithm has not been implemented because its complexity grows very rapidly with the growth of $p$ which makes it impractical. In this paper we produce a modification of this algorithm that consumes a modest amount of memory.
Let I be an ideal of a Complete Cohen-Macaulay local ring R of dimension n. We wil show that the natural homomorphism Rto HomR(HcI(KR), HcI(KR)) is an isomorphism provided that I is a cohomologically compltete intersection ideal of grade c where KR (resp. HiI(.)) denote the canonical module (resp. i-th local cohomology with respect to the ideal I) of R. The same result is true for the Matlis dual of HcI(KR).
Let $I$ denote an ideal of a local ring $(R,\mathfrak{m})$ of dimension $n$. Let $M$ denote a finitely generated $R$-module. We study the endomorphism ring of the local cohomology module $H^c_I(M), c = \grade (I,M)$. In particular there is a natural homomorphism $\Hom_{\hat{R}^I}(\hat{M}^I, \hat{M}^I)\to \Hom_{R}(H^c_{I}(M),H^c_{I}(M))$, where $\hat{\cdot}^I$ denotes the $I$-adic completion functor. We prove sufficient conditions such that it becomes an isomorphism. Moreover, we study a homomorphism of two such endomorphism rings of local cohomology modules for two ideals $J \subset I$ with th…
Let R be a local complete ring. For an R-module M the canonical ring map R\to End_R(M) is in general neither injective nor surjective; we show that it is bijective for every local cohomology module M := H^h_I(R) if H^l_I(R) = 0 for every l\neq h(:= height(I)) (I an ideal of R); furthermore the same holds for the Matlis dual of such a module. As an application we prove new criteria for an ideal to be a set-theoretic complete intersection.
We explain how some results of M. Nori (on motives) and F. Ivorra (on perverse motives) can be used to define "motivic" versions of Lyubeznik numbers, a set of numerical invariants for local rings. We also discuss on additional structures that might be put on some local cohomology modules (besides those of D- or F-module). This note is mainly expository, it is an expanded version of my talk at the FACARD workshop in Barcelona, on January 2019.
Let $R$ be a standard graded polynomial ring that is finitely generated over a field, and let $I$ be a homogenous prime ideal of $R$. Bhatt, Blickle, Lyubeznik, Singh, and Zhang examined the local cohomology of $R/I^t$, as $t$ grows arbitrarily large. Such rings are known as thickenings of $R/I$. We consider $R = \mathbb{F}[X]$ where $\mathbb{F}$ is a field of characteristic 0, $X$ is a $2 \times m$ matrix, and $I$ is the ideal generated by size two minors. We give concrete constructions for the local cohomology modules of thickenings of $R/I$. Bizarrely, these local cohomology modules can be …
Let $I$ be an ideal of a local ring $(R,\mathfrak m)$ with $d = \dim R.$ For the local cohomology module $H^i_I(R)$ it is a well-known fact that it vanishes for $i > d$ and is an Artinian $R$-module for $i = d.$ In the case that the Hartshorne-Lichtenbaum Vanishing Theorem fails, that is $H^d_I(R) \not= 0,$ we explore its fine structure. In particular, we investigate its endomorphism ring and related connectedness properties. In the case $R$ is complete we prove - as a technical tool - that $H^d_I(R) \simeq H^d_{\mathfrak m}(R/J)$ for a certain ideal $J \subset R.$ Thus, properties of $H^d_I(R…
We develop splitting techniques to study Lyubeznik numbers of cover ideals of graphs which allow us to describe them for large families of graphs including forests, cycles, wheels and cactus graphs. More generally we are able to compute all the Bass numbers and the shape of the injective resolution of local cohomology modules by considering the connected components of the corresponding subgraphs. Indeed our method gives us a very simple criterion for the vanishing of these local cohomology modules in terms of the connected components.
Lyubeznik's conjecture, (\cite{Ly1}, Remark 3.7) asserts the finiteness of the set ssociated primes of local cohomology modules for regular rings. But, in the case of ramified regular local ring, it is open. Recently, in Theorem 1.2 of \cite{Nu1}, it is proved that in any Noetherian regular local ring $S$ and for a fixed ideal $J\subset S$, associated primes of local cohomology $H^i_J(S)$ for $i\geq 0$ is finite, if it does not contain $p$. In this paper, we use this result to construct examples of local cohomology modules for ramified regular local ring so that they have finitely many associa…
We establish a criterion for the strong $F$-regularity of a (non-Gorenstein) Cohen-Macaulay reduced complete local ring of dimension at least $2$, containing a perfect field of prime characteristic $p$. We also describe an explicit generating morphism (in the sense of Lyubeznik) for the top local cohomology module with support in certain ideals arising from an $n\times (n-1)$ matrix $X$ of indeterminates. For $p\geq 5$, these results led us to derive a simple, new proof of the well-known fact that the generic determinantal ring defined by the maximal minors of $X$ is strongly $F$-regular.
One classical topic in the study of local cohomology is whether the non-vanishing of a specific local cohomology module is equivalent to the vanishing of its annihilator; this has been studied by several authors, including Huneke, Koh, Lyubeznik and Lynch. Motivated by questions raised by Lynch and Zhang, the goal of this paper is to provide some new results about this topic, which provide some partial positive answers to these questions. The main technical tool we exploit is the structure of local cohomology as module over rings of differential operators.
Let $(R,\mathfrak{m},k)$ denote a local ring. For $I$ and $J$ ideals of $R$, for all integer $i$, let $H^i_{I,J}(-)$ denote the $i$-th local cohomology functor with respect to $(I,J)$. Here we give a generalized version of Local Duality Theorem for local cohomology defined by a pair of ideals. Also, for $M$ be a finitely generated $R$-module, we study the behavior of the endomorphism rings $H^t_{I,J}(M)$ and $D(H^t_{I,J}(M))$ where $t$ is the smallest integer such that the local cohomology with respect to a pair of ideals is non-zero and $D(-):= {\rm Hom}_R(-,E_R(k))$ is the Matlis dual functo…
Local and category-theoretical entropies associated with an endomorphism of finite length (i.e., with zero-dimensional closed fiber) of a commutative Noetherian local ring are compared. Local entropy is shown to be less than or equal to category-theoretical entropy. The two entropies are shown to be equal when the ring is regular, and also for the Frobenius endomorphism of a complete local ring of positive characteristic. Furthermore, given a flat morphism of Cohen-Macaulay local rings endowed with compatible endomorphisms of finite length, it is shown that local entropy is "additive". Finally…
Let $A$ be a complete local ring with a coefficient field $k$ of characteristic zero, and let $Y$ be its spectrum. The de Rham homology and cohomology of $Y$ have been defined by R. Hartshorne using a choice of surjection $R \rightarrow A$ where $R$ is a complete regular local $k$-algebra: the resulting objects are independent of the chosen surjection. We prove that the Hodge-de Rham spectral sequences abutting to the de Rham homology and cohomology of $Y$, beginning with their $E_2$-terms, are independent of the chosen surjection (up to a degree shift in the homology case) and consist of fini…
In this paper, we will consider derived equivalences for differential graded endomorphism algebras by Keller's approaches. First we construct derived equivalences of differential graded algebras which are endomorphism algebras of the objects from a triangle in the homotopy category of differential graded algebras. We also obtain derived equivalences of differential graded endomorphism algebras from a standard derived equivalence of finite dimensional algebras. Moreover, under some conditions, the cohomology rings of these differential graded endomorphism algebras are also derived equivalent. T…
Let $S$ be the coordinate ring of the space of $n\times n$ complex skew-symmetric matrices. This ring has an action of the group $\textrm{GL}_n(\mathbb{C})$ induced by the action on the space of matrices. For every invariant ideal $I\subseteq S$, we provide an explicit description of the modules $\textrm{Ext}^{\bullet}_S(S/I,S)$ in terms of irreducible representations. This allows us to give formulas for the regularity of basic invariant ideals and (symbolic) powers of ideals of Pfaffians, as well as to characterize when these ideals have a linear free resolution. In addition, given an inclusi…
Let $K$ be a field of characteristic zero, $R = K[X_1,...,X_n]$. Let $A_n(K) = K<X_1,...,X_n, \partial_1, ..., \partial_n>$ be the $n^{th}$ Weyl algebra over $K$. We consider the case when $R$ and $A_n(K)$ is graded by giving $°X_i = ω_i $ and $°\partial_i = -ω_i$ for $i =1,...,n$ (here $ω_i$ are positive integers). Set $ω= \sum_{k=1}^{n}ω_k$. Let $I$ be a graded ideal in $R$. By a result due to Lyubeznik the local cohomology modules $H^i_I(R)$ are holonomic $A_n(K)$-modules for each $i \geq 0$. In this article we prove that the De Rahm cohomology modules $H^*(\bP ; H^*_I(R))$ is concentrated …
In this paper we show that the Frobenius test exponent for parameter ideals of a local ring of prime characteristic is always bigger than or equal to its Hartshorne-Speiser-Lyubeznik number. Our argument is based on an isomorphism of Nagel and Schenzel on local cohomology that we will provide an elementary proof.
We investigate how to compare Hochschild cohomology of algebras related by a Morita context. Interpreting a Morita context as a ring with distinguished idempotent, the key ingredient for such a comparison is shown to be the grade of the Morita defect, the quotient of the ring modulo the ideal generated by the idempotent. Along the way, we show that the grade of the stable endomorphism ring as a module over the endomorphism ring controls vanishing of higher groups of selfextensions, and explain the relation to various forms of the Generalized Nakayama Conjecture for Noetherian algebras. As appl…
We consider the ring S=C[x_ij] of polynomial functions on the vector space C^(m x n) of complex m x n matrices. We let GL= GL_m x GL_n and consider its action via row and column operations on C^(m x n) (and the induced action on S). For every GL-invariant ideal I in S and every j>=0, we describe the decomposition of the modules Ext^j_S(S/I,S) into irreducible GL-representations. For any inclusion I into J of GL-invariant ideals we determine the kernels and cokernels of the induced maps Ext^j_S(S/I,S) -> Ext^j_S(S/J,S). As a consequence of our work, we give a formula for the regularity of the p…
In this article, we prove that if $R\to S$ is a homomorphism of Noetherian rings that splits, then for every $i\geq 0$ and ideal $I\subset R$, $\Ass_R H^i_I(R)$ is finite when $\Ass_S H^i_{IS}(S)$ is finite. In addition, if $S$ is a Cohen-Macaulay ring that is finitely generated as an $R$-module, such that all the Bass numbers of $H^i_{IS}(S)$, as an $S$-module, are finite, then all the Bass numbers of $H^i_{I}(R)$, as an $R$-module, are finite. Moreover, we show these results for a larger class a functors introduced by Lyubeznik. As a consequence, we exhibit a Gorenstein $F$-regular UFD of po…
Let $K$ be a field of characteristic zero, $R = K[X_1,\ldots,X_n]$. Let $A_n(K) = K<X_1,\ldots,X_n, \partial_1, \ldots, \partial_n>$ be the $n^{th}$ Weyl algebra over $K$. We consider the case when $R$ and $A_n(K)$ is graded by giving $°X_i = ω_i $ and $°\partial_i = -ω_i$ for $i =1,\ldots,n$ (here $ω_i$ are positive integers). Set $ω= \sum_{k=1}^{n}ω_k$. Let $I$ be a graded ideal in $R$. By a result due to Lyubeznik the local cohomology modules $H^i_I(R)$ are holonomic $A_n(K)$-modules for each $i \geq 0$. In this article we compute the de Rham cohomology modules $H^j(\mathbb{\partial}; H^1_{…
This paper continues the investigation of quasilength, of content of local cohomology with respect to generators of the support ideal, and of robust algebras begun in joint work of Hochster and Huneke. We settle several questions raised by Hochster and Huneke. In particular, we give a family of examples of top local cohomology modules both in equal characteristic 0 and in positive prime characteristic that are nonzero but have content 0. We use the notion of a robust forcing algebra (the condition turns out to be strictly stronger than the notion of a solid forcing algebra in, for example, equ…
