text:
\section{Introduction}

	Let $k=\bar k$ be an algebraically closed field of characteristic $p>0$. Let
	$\mathcal O=W(k)$ be the ring of Witt vectors over $k$, and denote by $K$ the field of fractions of $\mathcal O$. In the representation theory of a finite group $G$ over either of these rings, permutation modules, $p$-permutation modules and endo-permutation modules play a pivotal role. However, to even define permutation modules, one needs to know a group basis of the group ring, a piece of information which is lost when passing to isomorphic or Morita equivalent algebras. Recovering the information lost when forgetting the group basis, or at least quantifying the loss, is a fundamental problem in modular representation theory. This is, for instance, the problem once faces when trying to bridge the gap between Donovan's and Puig's respective conjectures. There are scant results in this direction, apart from Weiss' seminal theorem \cite{WeissRigidity}, which gives a criterion for a lattice to be $p$-permutation requiring only limited knowledge of a group basis (and in some cases none at all). In the present article we study the following property of lattices over an $\mathcal O$-order $\Lambda$ such that $A=K\otimes_\mathcal O \Lambda$ is a separable $K$-algebra, which of course includes lattices over a finite group algebra $\mathcal O G$:
	\begin{defi}
		A $\Lambda$-lattice $L$ is called \emph{rigid} if ${\operatorname{Ext}}^1_\Lambda(L,L)=0$.
	\end{defi}
	First and foremost we should point out that permutation lattices over a finite group algebra $\mathcal O G$ are rigid in this sense. The notion of rigid modules over finite-dimensional {$k$-algebras} is widely known and well-studied (see for example \cite{DonaldDeformations, DadeAlgRigid}), but unfortunately permutation $kG$-modules and their ilk usually do not have that property. To see why permutation \emph{lattices} do, we can use a well-known alternative characterisation of rigidity in terms of endomorphism rings, which works for arbitrary $\Lambda$-lattices $L$. Consider the following long exact sequence obtained by applying $\operatorname{Hom}_\Lambda(L,-)$ to the short exact sequence ${0\rightarrow L\rightarrow L \rightarrow L/pL\rightarrow 0}$:
	\begin{equation}
		0 \longrightarrow \operatorname{End}_{\Lambda}(L) \stackrel{p}{\longrightarrow} \operatorname{End}_{\Lambda}(L)\longrightarrow\operatorname{End}_{\Lambda}(L/pL) \longrightarrow {\operatorname{Ext}}^1_\Lambda(L,L) \stackrel{p}{\longrightarrow} {\operatorname{Ext}}^1_\Lambda(L,L).
	\end{equation}
	As $A$ is separable, we must have $K\otimes_\mathcal O {\operatorname{Ext}}^1_{\Lambda}(L,L)\cong {\operatorname{Ext}}^1_{A} (K\otimes_\mathcal O L, K\otimes_\mathcal O L)=0$, meaning that ${\operatorname{Ext}}^1_{\Lambda}(L,L)$ is a finitely generated torsion $\mathcal O$-module. By the Nakayama lemma ${\operatorname{Ext}}^1_{\Lambda}(L,L)$ is zero if and only if multiplication by $p$ is injective on it, which, by exactness, happens if and only if the reduction map
	\begin{equation}
	\operatorname{End}_\Lambda(L)\longrightarrow \operatorname{End}_{\Lambda}(L/pL)
	\end{equation}
	is surjective.
	  Now if $R[H\backslash G]$ is a permutation module over an arbitrary commutative ring $R$, then its endomorphisms can be identified with sums $\sum_{h\in HgH} h$ over double cosets $HgH \in H\backslash G /H$. That clearly implies the (well-known) fact that the reduction map $\operatorname{End}_{\mathcal O G}(\mathcal O[H\backslash G]) \longrightarrow \operatorname{End}_{kG}(k[H\backslash G])$ is surjective, which by the above implies that $\mathcal O[H \backslash G]$ is rigid.

	After this short digression on permutation lattices let us state our first main result, which is that rigid lattices enjoy the same discreteness property as rigid modules over finite-dimensional algebras.
	\begin{namedtheorem}[Theorem A]\label{thm rigid main}
		For every $n\in \mathbb N$ there are at most finitely many isomorphism classes of rigid $\Lambda$-lattices of $\mathcal O$-rank at most $n$.
	\end{namedtheorem}
 An easy corollary of this is that only finitely many isomorphism classes of $\mathcal O G$-lattices of any given character can be images of permutation lattices under Morita or stable equivalences originating from another group or block algebra.
	Perhaps unsurprisingly, this result is proved using geometric methods. The basic idea of using a variety of modules (or a variety of complexes, as the case may be) has been successfully applied to modules and complexes over finite-dimensional algebras in many different contexts \cite{DadeAlgRigid, DadeGreenCorrespondent, DonaldDeformations, ZimmermannGeometryChainComplexes, RickardRigidity, RouquierAutomorphFrench}. The idea is typically that one gets a homomorphism from the tangent space of such a variety in a point $M$ into ${\operatorname{Ext}}^1(M,M)$, whose kernel is the tangent space of the subvariety of points $N$~isomorphic~to~$M$.

	The way we obtain a variety parametrising all lattices with a given $K$-span is quite different from how one proceeds for finite-dimensional $k$-algebras. We essentially start with a fixed lattice, and conjugate an affording representation by a generic matrix. That is how we obtain our analogue of a variety of modules, a \emph{smooth family of $\Lambda$-lattices} (see Definition~\ref{defi smooth family}).
	  Despite working over $\mathcal O$, these smooth families of lattices are actually parametrised by varieties over $k$.
	  This is possible due to the theory of Witt vectors. The two main ingredients of Theorem~A are the fact that each  smooth family of lattices contains at most one rigid lattice, up to isomorphism (see Theorem~\ref{thm smooth family open subset}), and the fact that lattices in a given finite-dimensional $A$-module can be appropriately partitioned into finitely many smooth families (see Theorem~\ref{thm lattice alg}).

	Our second main result is an immediate consequence of Theorem~A, but we state it as a theorem nevertheless, since it was the main motivation for writing this paper. The study of Picard groups of blocks of finite group algebras over $\mathcal O$ was initiated in \cite{BKL}. While \cite{BKL} primarily studies the group of Morita self-equivalences induced by bimodules of endo-permutation source, this is where the possibility of Picard groups of blocks always being finite was first raised. The theorem we obtain is much more general, and provides further evidence for the speculation put forward in \cite{EiselePicard} that the Lie algebra of ${\operatorname{Pic}}_{\mathcal O}(\Lambda)$ (which is shown to be an algebraic group in that paper) should be related to the first Hochschild cohomology of $\Lambda$ in some way or form.

	\begin{namedtheorem}[Theorem B]\label{thm pic trivial}
		Assume that $\operatorname{HH}^1(\Lambda)=0$. Then ${\operatorname{Pic}}_{\mathcal O}(\Lambda)$ is a finite group.
	\end{namedtheorem}
	Here ${\operatorname{Pic}}_{\mathcal O}(\Lambda)$ denotes the group of $\mathcal O$-linear self-equivalences of the module category~of~$\Lambda$. This group is commensurable with the outer automorphism group $\operatorname{Out}_\mathcal O(\Lambda)$, as well as $\operatorname{Outcent}(\Lambda)$ and $\operatorname{Picent}(\Lambda)$, and we could just as well have stated Theorem~B with ${\operatorname{Pic}}_{\mathcal O}(\Lambda)$ replaced by any of these groups.
	The theorem of course immediately implies the following:
	\begin{corollary}\label{cor pic finite}
		Let $G$ be a finite group, and let $\mathcal O Gb$ be a block. Then ${\operatorname{Pic}}_\mathcal O(\mathcal O Gb)$ is finite.
	\end{corollary}
	There is also a more elementary formulation of this fact which is reminiscent of the second Zassenhaus conjecture.
	\begin{corollary}
		For a finite group $G$ there are only finitely many $\mathcal U(\mathcal O G)$-conjugacy classes of group bases of $\mathcal O G$.
	\end{corollary}

	Corollary~\ref{cor pic finite} shows that block algebras over $\mathcal O$ are in some sense extremely rigid, and it turns their Picard groups into a very interesting finite invariant. This is particularly important since K\"ulshammer \cite{KuelshammerDonovan} showed that Picard groups play an important role in both Donovan's conjecture and the classification of blocks of a given defect group. This theme is explored in \cite{EatonDefect16, EEL}. In the way of actual computations, \cite{NebeHertweck} determines the Picard group of the principal block of $A_6$ for $p=3$, \cite{EatonLiveseyPicard} determines Picard groups of almost all blocks of abelian $2$-defect of rank three, and this will certainly not mark the end of the story. Unfortunately, we have very little to offer to aid the calculation of Picard groups. Still, the following might be useful. Note that $\mathcal{T}(\mathcal O Gb)$ denotes the subgroup of ${\operatorname{Pic}}_{\mathcal O}(\mathcal O Gb)$ of equivalences induced by $p$-permutation bimodules, which is determined in \cite{BKL}.

	\begin{prop}\label{prop det pic}
		Let $G$ be a finite group, let $\mathcal O G b$ be a block and let $P\leq G$ be one of its defect groups.
		\begin{enumerate}
		\item An element of ${\operatorname{Pic}}_{\mathcal O}(\mathcal O G b)$ lies in $\mathcal T(\mathcal O Gb)$ if and only if it sends the $p$-permutation lattice $\mathcal O[P\backslash G]\cdot b$ to a $p$-permutation lattice.
		\item If $\mathcal O[P\backslash G]\cdot b$ is, up to isomorphism, the only rigid lattice in $K[P \backslash G]\cdot b$ with endomorphism ring isomorphic to $\operatorname{End}_{\mathcal O G}(\mathcal O[P\backslash G]\cdot b)$, then
		$\operatorname{Picent}(\mathcal O Gb)\subseteq \mathcal T(\mathcal O Gb)$.
		\end{enumerate}
	\end{prop}
	The same is true if we set $b=1$ and let $P$ be a Sylow $p$-subgroup of $G$. It is also sufficient to prove that each indecomposable summand $L$ of $\mathcal O[P\backslash G]\cdot b$ is the unique rigid lattice in $K\otimes_\mathcal O L$ with endomorphism ring $\operatorname{End}_{\mathcal O Gb}(L)$. It is however unclear whether one can show such uniqueness in any interesting examples, even though the construction explained in \S\ref{section lattice var} is in principle constructive. Nevertheless,
	Proposition~\ref{prop det pic} highlights the importance of understanding and classifying rigid lattices with a given character.

	One last thing to note is that while we focus on applications to block algebras in this article, there are other types of $\mathcal O$-orders with vanishing first Hochschild cohomology to which Theorem~B applies. Iwahori-Hecke algebras defined over $\mathcal O$, for instance, should have this property by \cite[Theorem 5.2]{GeckRouquier}.

	\begin{namedtheorem}[Notation and conventions] By $\nu_p:\ K \longrightarrow \mathbb Z\cup\{\infty\}$ we denote the $p$-adic valuation
		on $K$. Modules are right modules by default. All varieties are reduced, and by a ``point'' we mean a closed point. $\Lambda$ will always denote an $\mathcal O$-order in a separable $K$-algebra. We will assume that the reader is familiar with the theory of Witt vectors as laid out in \cite[\S5--\S6]{SerreLocalFields}.
	\end{namedtheorem}

	\section{Prerequisites}

	Recall that a commutative ring $R$ is called a \emph{strict $p$-ring} if $R$ is complete and Hausdorff with respect to the topology induced by the filtration $R\supseteq pR \supseteq p^2 R \supseteq \ldots$, the element $p$ is not a zero-divisor in $R$ and $\bar R=R/pR$ is a perfect ring of characteristic $p$. The condition that $R$ be complete and Hausdorff is equivalent to the natural homomorphism $R \longrightarrow \varprojlim R/p^i R$ being an isomorphism. The theory of Witt vectors shows (see \cite[\S5 Proposition~10 and \S6 Theorem~8]{SerreLocalFields}) that a strict $p$-ring $R$ with residue ring $\bar R$ is unique up to unique isomorphism (or, to be more precise, the pair $(R, R/pR\rightarrow\bar R)$ is). In particular, there is a unique ring homomorphism $\varphi:\ R \longrightarrow W(\bar R)$, where $W(\bar R)$ denotes the ring of Witt vectors over $\bar R$, making the diagram
	\begin{equation}
		\xymatrix{
			R \ar@{->>}[dr]\ar[r]^\varphi&  W(\bar R)\ar@{->>}[d]\\
			&\bar R
		}
	\end{equation}
	commute, the arrows going down being the natural homomorphism from $R$ into $\bar R$ and the projection onto the first Witt vector component, respectively.

	\begin{notation}
	Let $R$ be a strict $p$-ring with perfect residue ring $\bar R$.
	For $l\in \mathbb N$ we let
	\begin{equation}
	\rho_l:\ R \longrightarrow \bar R^l
	\end{equation}
	denote the map which sends $r\in R$ to the first $l$ components of the Witt vector $\varphi(r)\in W(\bar R)$.
	We say that $r$ \emph{reduces to} $\rho_l(r)$.
	\end{notation}
	We will extend the map $\rho_l$ to vectors and matrices over $R$. There is also no need to explicitly record the ring $R$ in our notation, due to the uniqueness of the isomorphism between a strict $p$-ring and the corresponding ring of Witt vectors explained above.

	A strict $p$-ring with a perfect residue \emph{field} is a complete discrete valuation ring, and if this residue field is moreover algebraically closed we may view any set of truncated Witt vectors over it as affine space. This applies to our ring $\mathcal O$. It is clear that if $f\in \mathcal O[X_1,\ldots,X_n]$ is a polynomial and $r\in \mathbb N$, then the condition ``$\nu_p(f(\widehat x_1,\ldots,\widehat x_n))\geq r$'' for $\widehat x_1,\ldots,\widehat x_n\in \mathcal O^n$ is equivalent to certain polynomials in the entries of $\rho_r(\widehat x_1),\ldots,\rho_r(\widehat x_n)$  vanishing. That is, ``$\nu_p(f(\widehat x_1,\ldots,\widehat x_n))\geq r$'' is essentially a ``closed condition'' on Witt vectors. The point of the remainder of this section is to understand the implications of this simple observation.

	\begin{defi}\label{defi generic val}
		Let $f\in \mathcal O[X_1, \ldots, X_n]$ be a polynomial, and let $l\in\mathbb N$.
		\begin{enumerate}
		\item We define $\nu_p(f)$ as the minimal $p$-valuation of a coefficient of $f$.
		\item For a point $\mathbf x = (x_{1,0},\ldots,x_{1,l-1},\ldots,x_{n,0},\ldots,x_{n,l-1})\in \mathbb A^{n\cdot l}(k)$ we define
		 \begin{equation}
		 	\nu_{p, \mathbf x}(f)=\min \{ \nu_p(f(\widehat x_1,\ldots, \widehat x_n)) \ |\ \widehat{\mathbf x}=(\widehat x_1,\ldots, \widehat x_n) \in \mathcal O^n \textrm{ such that } \rho_l(\mathbf{\widehat x})=\mathbf x \}
		 \end{equation}
		 	We call $\nu_{p, \mathbf x}(f)$ the \emph{generic valuation} of $f$ at $\mathbf{x}$.
		 \end{enumerate}
	\end{defi}
 We may extend this generic valuation to $K(X_1\ldots,X_n)$ in the obvious way. One should note though that if $f$ is a rational function rather than a polynomial, then $\nu_p(f(\widehat{\mathbf x}))$ could be either bigger or smaller than $\nu_{p, \mathbf x}(f)$, depending on the choice of $\widehat{\mathbf x}\in \mathcal O^n$ reducing to $\mathbf x$.

	\begin{prop}\label{prop generic val first properties} Assume the situation of Definition~\ref{defi generic val}.
	\begin{enumerate}
		\item For any $\widehat{\mathbf x}=(\widehat x_1, \ldots, \widehat x_n)\in \mathcal O^n$ reducing to $\mathbf x$
		we have
		\begin{equation}
		\nu_{p,\mathbf x}(f) = \nu_p(f(\widehat x_1 + p^l\cdot Z_1, \ldots, \widehat x_n+p^l\cdot Z_n))
		\end{equation}
		where $Z_1,\ldots, Z_n$ are indeterminates.
	\item If an $\widehat{\mathbf x} \in \mathcal O^n$  reduces to ${\mathbf x}$, then there is a $\widehat{\mathbf z}\in W(\bar{\mathbb F}_p)^n$ such that
	\begin{equation}
	\nu_p(f(\widehat {\mathbf x} + p^l \cdot \widehat {\mathbf z}))=\nu_{p, \mathbf x}(f).
	\end{equation}
	\end{enumerate}
	\end{prop}
	\begin{proof}
		\begin{enumerate}
		\item Note that the right hand side cannot be bigger than the left hand side. On the other hand,
		if $\nu_p(f(\widehat x_1 + p^l Z_1, \ldots, \widehat x_n+p^lZ_n))=m\in \mathbb Z_{\geq 0}$, then
		$p^{-m}f(\widehat x_1 + p^l Z_1, \ldots, \widehat x_n+p^lZ_n)$ reduces to a non-zero polynomial in
		$k[Z_1,\ldots, Z_n]$, and we can certainly find values for the $Z_i$ for which the polynomial does not vanish. This shows that the right hand side cannot be smaller than the left hand side either.
	\item By the first part we know that $p^{-\nu_{p, \mathbf x}(f)}f(\widehat x_1 + p^l Z_1, \ldots, \widehat x_n+p^lZ_n)$ reduces to a non-zero polynomial $g\in k[Z_1,\ldots, Z_n]$, and
$\nu_p(f(\widehat {\mathbf x} + p^l \cdot \widehat {\mathbf z}))=\nu_{p, \mathbf x}(f)$ if and only if $g(\mathbf z)\neq 0$, where $\mathbf z$ is the reduction of $\widehat{\mathbf z}$ modulo $p$. Hence, what we are looking for is a $\mathbf z \in \bar {\mathbb F}_p^n$ which avoids the vanishing set of a polynomial defined over $k$. Since $\mathbb A^n(\bar {\mathbb F}_p)$ is Zariski-dense in $\mathbb A^n(k)$ such a $\mathbf z$ exists trivially. \qedhere
		\end{enumerate}
	\end{proof}

	\begin{corollary}\label{cor gen val ext}
	Let $\mathcal E$ be the ring of Witt vectors over an algebraically closed field $k'$ containing $k$,
	and let $f \in \mathcal O[X_1,\ldots,X_n]$ be a polynomial. For $\mathbf x \in \mathbb A^{n\cdot l}(k)$ the value of
	$\nu_{p, \mathbf x}(f)$ is independent of whether we consider $f$ as a polynomial over $\mathcal O$ or over $\mathcal E$.
	\end{corollary}
	\begin{proof}
		This follows from the first part of Proposition~\ref{prop generic val first properties}.
	\end{proof}

	\begin{corollary}
	In the situation of Proposition~\ref{prop generic val first properties} let $f_1,\ldots,f_d\in \mathcal O[X_1,\ldots, X_n]$ ($d\in \mathbb N$) be non-zero polynomials. Then, for any $\widehat{\mathbf x}\in \mathcal O^n$ reducing to $\mathbf x$, we have \begin{equation}
		\nu_{p,\mathbf x}(f_1\cdots f_d)=\nu_p(f_1(\widehat{\mathbf x})\cdots f_d(\widehat{\mathbf x}))
	\end{equation}
	if and only if
	\begin{equation}
	 \nu_{p,\mathbf x}(f_i)=\nu_p(f_i(\widehat{\mathbf x})) \textrm{ for all $1\leq i\leq d$}.
	\end{equation}
	\end{corollary}
	\begin{proof}
		By the first part of Proposition~\ref{prop generic val first properties} we have
		$\nu_{p,\mathbf x}(f_1\cdots f_d) = \sum_i \nu_{p, \mathbf x}(f_i)$. Moreover, the valuation of a polynomial at an $\widehat{\mathbf x}$ is always greater than or equal to the generic valuation of the polynomial at $\mathbf x$, but never smaller. The claim follows.
	\end{proof}

	The above shows that if we have finitely many non-zero polynomials $f_1,\ldots,f_d$ ($d\in \mathbb N$), then there is always an $\widehat{\mathbf x}$ reducing to $\mathbf x$ such that $\nu_{p}(f_i(\widehat{\mathbf x}))=\nu_{p, \mathbf x}(f_i)$ for all $i$ at once. One last thing we need to understand is what the set of all points $\mathbf x \in \mathbb A^{n\cdot l}(k)$ with $\nu_{p,\mathbf x}(f)\geq r$ (for some given polynomial $f$ and $r\in \mathbb Z_{\geq 0}$) looks like geometrically.
	 Proposition~\ref{prop gen val zariski closed} below answers this, and is essentially a consequence of the fact that such a set is the complement of the image of an open set under the projection onto the first $l$ components of a Witt vector, which will be a closed set.
	\begin{prop}\label{prop gen val zariski closed}
		Given $f \in \mathcal O[X_1,\ldots,X_n]$, $l\in \mathbb N$ and $r\in \mathbb Z_{\geq 0}$, there is a closed subvariety ${\mathcal V_{l,\nu_{p,-}(f)\geq r}\subseteq \mathbb A^{n\cdot l}}$ defined over $k$ such that for any algebraically closed $k'\supseteq k$ we have
		\begin{equation}
		\mathcal V_{l,\nu_{p,-}(f)\geq r}(k')=\{ \mathbf x \in \mathbb A^{n\cdot l}(k') \ | \ \nu_{p,\mathbf x}(f) \geq r \}
		\end{equation}
	\end{prop}
	\begin{proof}
		Fix an algebraically closed $k'\supseteq k$, and set $\mathcal E=W(k')$.
		Let
		 $\mathbf x \in \mathbb A^{n\cdot l}(k')$, and let $\widehat{\mathbf x}\in \mathcal E^n$ be the element reducing to it such that the first $l$ components of the Witt vector $\widehat x_i$ are given by the
		 $x_{i,j}$ for $0\leq j \leq l-1$, and all other components are zero (note that we use two indices to refer to the $n\cdot l$ entries of $\mathbf x$, which is more natural since we want to think of $\mathbf x$ as an element of $k^{n\times l}$, rather than $k^{n\cdot l}$).
		As $\mathcal E$ is the ring of Witt vectors over $k'$ and $f$ is defined over $k$, we get polynomials ${f_i\in k[X_{1,0},\ldots, X_{1,l-1}, \ldots, X_{n,0},\ldots, X_{n,l-1}, Z_{1,j},\ldots, Z_{n,j} \ | \ 0\leq j \leq i]}$ (where $i\in \mathbb Z_{\geq 0}$)
		such that $f(\widehat {\mathbf x}+p^l\cdot \widehat {\mathbf z})$ (for arbitrary $\widehat {\mathbf z}\in \mathcal E^n$) is given by the Witt vector whose $i$-th component is
		the evaluation of $f_i$ at $\mathbf x$ and $\rho_{i+1}(\widehat {z}_1), \ldots, \rho_{i+1}(\widehat {z}_n)$.
		The polynomials $f_i$ do not depend on $k'$.

		 Now $\nu_{p,\mathbf x}(f) \geq r$ if and only if, for all
		$0\leq i < r$, $\mathbf x$ is a zero of all coefficients of $f_i$ as a polynomial in $ k[X_{1,0},\ldots, X_{1,l-1}, \ldots, X_{n,0},\ldots, X_{n,l-1}][Z_{1,j},\ldots, Z_{n,j} \ | \ 0\leq j \leq i]$. Hence we can define $\mathcal V_{l,\nu_{p,-}(f)\geq r}$ as the zero locus of these coefficients, which are elements of $ k[X_{1,0},\ldots, X_{1,l-1}, \ldots, X_{n,0},\ldots, X_{n,l-1}]$.
	\end{proof}

	\section{Smooth families of lattices and rigidity}

	In this section we will prove the main ingredient going into Theorem~A, which is Theorem~\ref{thm smooth family open subset} below. To do this, we first need to introduce the structure we use to endow parametric families of lattices with the structure of a variety over $k$. We call this structure a \emph{smooth family of $\Lambda$-lattices}. Its definition is  quite straight-forward, and it is not difficult to show that we can parametrise all $\Lambda$-lattices of a given index in a fixed lattice by finitely many such families (see \S\ref{section lattice var}). Note that lattices in a smooth family as defined below have isomorphic $K$-span, and therefore the same character in case $\Lambda$ is a block or a group algebra.

	\begin{defi}\label{defi smooth family}
		A \emph{smooth family of $\Lambda$-lattices} $L(-)$ is a pair $(\Delta, \mathcal Z \cap \mathcal U)$, where
		\begin{enumerate}
			\item $\Delta: \Lambda \longrightarrow K[X_1, \ldots, X_n]^{m\times m}$ (for certain $m,n\in \mathbb N$) is  a representation,
			\item $\mathcal Z \cap \mathcal U$ is the intersection of  an irreducible closed subvariety $\mathcal Z \subseteq \mathbb A^{n\cdot l}=\mathbb A^l\times \ldots \times \mathbb A^l$, and open subvariety $\mathcal U \subseteq  \mathbb A^{n\cdot l}=\mathbb A^l\times \ldots \times \mathbb A^l$ (for some $l\in \mathbb N$), both defined over $k$,
		\end{enumerate}
		such that the following hold:
		\begin{enumerate}
		\item ${\rm sing}(\mathcal Z)\cap \mathcal U = \emptyset$.
		\item Let $\mathcal E=W(k')$ for some algebraically closed field $k'\supseteq k$.
		If $\widehat{\mathbf x} \in \mathcal E^n$ reduces to a point $\mathbf x = \rho_l(\mathbf{\widehat x}) \in \mathcal Z(k') \cap \mathcal U(k')$
		then
		\begin{equation}
			\Delta_{\widehat {\mathbf x}}:\ \Lambda \longrightarrow \mathcal E^{m\times m}:\ \lambda \mapsto \Delta(\lambda)|_{(X_1,\ldots, X_n)=(\widehat x_1, \ldots, \widehat x_n)}
		\end{equation}
		is well-defined, and the isomorphism type of the corresponding $\mathcal E\otimes_{\mathcal O}\Lambda$-lattice only depends on $\mathbf x$.
		\end{enumerate}
	\end{defi}

	A few remarks are in order. Firstly, the elements of $\mathbb A^{n\cdot l}(k')$ represent the first $l$ Witt vector components of elements in $\mathcal E^n$. To reflect this fact we identify the coordinate ring of $\mathbb A^{n\cdot l}$ with $k[X_{i,j}\ | \ 1\leq i \leq n,\ 0\leq j \leq l-1]$. A second thing to note is that the reason we are considering extensions $\mathcal E$ of $\mathcal O$ is that we need those to specialise at ``generic points'' in the proof of Theorem~\ref{thm smooth family open subset} below. In applications we actually only require the property that specialisation at points in $\mathcal Z(k)\cap \mathcal U(k)$ is well-defined. The last thing we should note is that the requirement that $\mathcal Z\cap \mathcal U$ be smooth and irreducible only serves to get uniqueness in Corollary~\ref{cor rigid unique} and make the proof of Theorem~\ref{thm smooth family open subset} slightly nicer. As such, its inclusion in the definition is really a matter of preference.

	\begin{notation}
		In the situation of Definition~\ref{defi smooth family} we denote the $\mathcal E\otimes_\mathcal O\Lambda$-lattice  corresponding to a point $\mathbf x \in \mathcal Z(k')\cap \mathcal U(k')$ by $L({\mathbf x})$. Of course $L({\mathbf x})$ is only defined up to isomorphism.
		We say that $L(-)=(\Delta, \mathcal Z\cap \mathcal U)$ \emph{contains} (the isomorphism class of) $L({\mathbf x})$. We also refer to the (common) $\mathcal O$-rank of $\Lambda$-lattices contained in $L(-)$ as the ``$\mathcal O$-rank of $L(-)$''.
	\end{notation}

	\begin{thm}\label{thm smooth family open subset}
		If $L(-)=(\Delta, \mathcal Z\cap \mathcal U)$ is a smooth family of $\Lambda$-lattices, then the subset of ${\mathcal U(k) \cap \mathcal Z(k)}$ such that the corresponding $\Lambda$-lattices are isomorphic to some fixed rigid $\Lambda$-lattice contains a Zariski-open subset.
	\end{thm}
	\begin{proof}
		Assume that we have an $\mathbf x \in \mathcal Z(k)\cap \mathcal U(k)$ such that $L({\mathbf x})$ is rigid.
		Let $\mathfrak m$ be the ideal of elements of $k[\mathcal Z]$ vanishing at $\mathbf x$. As $\mathcal Z$ is smooth at $\mathbf x$, the completion of the local ring $k[\mathcal Z]_{\mathfrak m}$ is isomorphic to
		$k[[T_1,\ldots, T_{\dim \mathcal Z}]]$. Hence we get a map ${\varphi:\ k[X_{1,0},\ldots,X_{1,l-1},\ldots, X_{n,0},\ldots, X_{n,l-1}]\longrightarrow k[[T_1,\ldots, T_r]]}$, where $r=\dim (\mathcal Z)+n$, with image contained in $k[[T_1,\ldots,T_{\dim\mathcal Z}]]$,  which factors through $k[\mathcal Z]$, and whose composition with the evaluation map $k[[T_1,\ldots,T_r]]\twoheadrightarrow k$ is the same as evaluation at $\mathbf x$ (the $n$ spare variables will come in handy later on).
		Define ${\mathbf w = (\varphi(X_{i,j}) \ |\ 1\leq i\leq n, 0 \leq j\leq l-1)\in k[[T_1,\ldots,T_r]]^{n\cdot l}}$.
		Then $\mathbf w$ lies in $\mathcal Z(\overline{k((T_1,\ldots, T_r))})$, and $\mathbf w(0,\ldots,0)=\mathbf x$. Here $\mathbf w(0,\ldots,0)$ denotes the evaluation of $\mathbf w$ at $(T_1,\ldots,T_r)=(0,\ldots,0)$.

		Define
		\begin{equation}
			R=\bigcup_{i=0}^\infty \mathcal O[[T_1^{1/p^{i}},\ldots,T_r^{1/p^{i}}]] \quad\textrm{and}\quad \mathcal E_0=\varprojlim R/p^i R.
		\end{equation}
		That is, $\mathcal E_0$ is the $p$-adic completion of $R$. The residue ring $\mathcal E_0/p\mathcal E=R/pR$ can be identified with
		\begin{equation}
			  \bar R=\bigcup_{i=0}^\infty k[[T_1^{1/p^{i}},\ldots,T_r^{1/p^{i}}]],
		\end{equation}
		which is a perfect ring of characteristic $p$.  In particular $\mathcal E_0$  is a strict $p$-ring. Hence $\mathcal E_0$ is isomorphic to $W(\bar R)$ by a unique isomorphism preserving the surjection onto $\bar R$. Now $W(\bar R)$ embeds into  $\mathcal E=	 W(k')$, where
		${k'=\overline{k((T_1,\ldots,T_r))}}$. Long story short, we get an embedding of $\mathcal O$-algebras $R \hookrightarrow \mathcal E = W(k')$ factoring through an embedding of rings of Witt vectors $\mathcal E_0\hookrightarrow \mathcal E$.

		Now choose, for each $1\leq i\leq n$, a $\widehat w_i'\in R$ such that $\rho_l(\widehat w'_i)=(w_{i,0},\ldots, w_{i,l-1})$.
		The reason we can do this is that $\rho_l:\ \mathcal E_0 \longrightarrow \bar R ^l$ factors through $\mathcal E_0/p^l\mathcal E_0$, and $R$ surjects onto $R/p^lR \cong \mathcal E_0/p^l\mathcal E_0$, the latter two being canonically isomorphic since $\mathcal E_0$ is the completion of $R$. Thus we get a $\widehat{\mathbf w}'=(\widehat w'_1,\ldots,\widehat w'_n)\in R^n$ such that
		$\rho_l(\mathbf{\widehat w'})=\mathbf w$. Since $R$ is defined as a union of rings, we actually get that
		\begin{equation}
		\mathbf{\widehat w'}\in \mathcal O[[S_1,\ldots,S_r]]^n\textrm{ where $S_1=T_1^{1/p^{e}},\ldots, S_r=T_r^{1/p^{e}}$ for some $e\in \mathbb N$.}
		\end{equation}
		Since $\mathbf w$ only involves the indeterminates $T_1,\ldots,T_{\dim \mathcal Z}$, we can actually assume that ${\mathbf{\widehat w'}\in \mathcal O[[S_1,\ldots,S_{\dim \mathcal Z}]]^n}$. We then define
		$\widehat w_i = \widehat w_i' + p^l\cdot S_{\dim \mathcal Z+i}$ for $1\leq i \leq n$. The upshot is
		that $\mathbf{\widehat w}=(\widehat w_1,\ldots,\widehat w_n)\in \mathcal O[[S_1,\ldots,S_r]]^n$ satisfies $\rho_l(\mathbf{\widehat w})=\mathbf w$ and the individual entries $\widehat w_i$ are algebraically independent over $K$. Moreover, $\rho_l(\mathbf{\widehat w}(0,\ldots,0))=\mathbf x$, since $\rho_l$ commutes with evaluation at $(0,\ldots,0)$ by the universal property of Witt vectors. Note that our construction of $\mathbf{\widehat w}$ also ensures that $\nu_{p,\mathbf w}(g)= \nu_p(g(\widehat{\mathbf w}))$ for all $g\in \mathcal O[X_1,\ldots,X_n]$ (where we view $\widehat{\mathbf w}$ as an element of $\mathcal E^n$), since $\mathbf{\widehat w}$ is polynomial in the spare variables $T_{\dim \mathcal Z+i}$, and therefore
		\begin{equation}
		\nu_{p,\mathbf w}(g)=\nu_p(g(\widehat{w}'_1 + p^l\cdot T_{\dim\mathcal Z+1}, \ldots, \widehat{w}'_n + p^l\cdot T_{\dim\mathcal Z+n}))=\nu_p(g(\widehat{\mathbf w})),
		\end{equation}
		which is seen by
		using Proposition~\ref{prop generic val first properties} and the fact that $\nu_p$ is independent of whether we regard the $T_{\dim\mathcal Z+i}$ as indeterminates or as elements of the valuation ring $\mathcal E$.

		 Using $\widehat{\mathbf w}$ we can now define an $\mathcal O$-algebra homomorphism
		 \begin{equation}
		 \widehat \varphi:\ \mathcal O[X_1,\ldots, X_n]\longrightarrow \mathcal O[[S_1,\ldots, S_r]]: X_i \mapsto \widehat w_i
		 \end{equation}
		 ``lifting'' $\varphi$. Our assumptions ensure  that $\widehat \varphi$ is injective, which will be important later.
		 Moreover, the element $\widehat{\mathbf{w}}$, by construction, gives us representations $\Delta_{\widehat{\mathbf w}}$
		 and  $\Delta_{\widehat{\mathbf w}(0,\ldots,0)}$, affording $L({\mathbf w})$ and $L({\mathbf x})$, respectively.
		 A priori, the image of $\Delta_{\widehat{\mathbf w}}$ is only contained in $m\times m$-matrices over
		 $\mathcal E$. However, since the images of $\Delta_{\widehat{\mathbf w}}$ are obtained from the images of $\Delta$ (which live in $K[X_1,\ldots,X_n]^{m\times m}$) by substituting $X_i=\widehat w_i$, we actually get that the image of $\Delta_{\mathbf{\widehat w}}$ is also contained in $m\times m$-matrices over $K\cdot \mathcal O[[S_1,\ldots, S_r]]$. Hence the images of $\Delta_{\mathbf{\widehat w}}$ actually have entries in $\mathcal E \cap (K\cdot \mathcal O [[S_1,\ldots, S_r]])=\mathcal O [[S_1,\ldots,S_r]]$.

		 It now follows that
		 \begin{equation}\label{eqn expansion delta}
		 	\Delta_{\widehat{\mathbf w}}(\lambda) = \Delta_{\widehat{\mathbf w}(0,\ldots,0)}(\lambda)+\sum_{\mathbf i\in \mathbb Z_{\geq 0}^r\setminus \{(0,\ldots,0)\}} \Gamma_{\mathbf i}(\lambda) \cdot S_1^{i_1}\cdots S_r^{i_r}\quad \textrm{for all $\lambda \in \Lambda$,}
		 \end{equation}
		 where each $\Gamma_{\mathbf i}$ is a map from $\Lambda$ into $\mathcal O^{m\times m}$. Assume that at least one of the $\Gamma_{\mathbf i}$ is not the zero map, and  let $\mathbf j\in \mathbb Z_{\geq 0}^r$ be degree-lexicographically minimal such that $\Gamma_{\mathbf j}$ is non-zero. Then
		 \begin{equation}
		 	\Gamma_{\mathbf j}(\lambda\cdot \gamma)= \Delta_{\widehat{\mathbf w}(0,\ldots,0)}(\lambda)\cdot \Gamma_{\mathbf j}(\gamma) +
		 	\Gamma_{\mathbf j}(\lambda)\cdot  \Delta_{\widehat{\mathbf w}(0,\ldots,0)}(\gamma)
		 \end{equation}
		 for any $\lambda,\gamma \in \Lambda$. That is, $\Gamma_{\mathbf j}$ defines an element of
		 ${\operatorname{Ext}}^1_{\Lambda}(L({\mathbf x}), L({\mathbf x}))$, which we assumed was zero. Hence there is a $\mathbf B \in \mathcal O^{m\times m}$ such that, for any $\lambda \in \Lambda$, $\Gamma_{\mathbf j}(\lambda)= \Delta_{\widehat{\mathbf w}(0,\ldots,0)}(\lambda) \cdot \mathbf B - \mathbf B \cdot \Delta_{\widehat{\mathbf w}(0,\ldots,0)}(\lambda)$. Conjugating $\Delta_{\widehat{\mathbf w}}$ by $\operatorname{id} + \mathbf B\cdot S_1^{j_1}\cdots S_r^{j_r}$ yields a representation with an expansion as in  \eqref{eqn expansion delta} where the degree-lexicographically smallest $\mathbf i$ with $\Gamma_{\mathbf i}\neq 0$ is strictly bigger than $\mathbf j$. Iterating this process gives a sequence of conjugating elements in $\operatorname{GL} _m(\mathcal O[[S_1,\ldots S_r]])$ that converge with respect to the topology induced by the realisation of $\mathcal O[[S_1,\ldots, S_r]]$ as $\varprojlim \mathcal O[S_1,\ldots, S_r]/(S_1,\ldots,S_r)^i$. The limit of these elements will conjugate $\Delta_{\widehat{\mathbf w}}$ to $\Delta_{\widehat{\mathbf w}(0,\ldots,0)}$.

		 We now know that $\mathcal E\otimes_{\mathcal O} L({\mathbf x})$ is isomorphic to $L({\mathbf w})$, as these are the modules afforded by the two representations we just showed are conjugate. In elementary terms, this means that the linear system of equations
		 \begin{equation}\label{eqn hom lats}
		 	\Delta_{\widehat{\mathbf{w}}}(\lambda) \cdot \mathbf M - \mathbf M\cdot \Delta_{\widehat{\mathbf{w}}(0,\ldots, 0)}(\lambda) \quad \textrm{(for $\lambda$ in a basis of $\Lambda$)}
		 \end{equation}
		 has a solution $\mathbf M\in \operatorname{GL}_m(\mathcal E)$. However, the coefficients of the system of equations
		 \eqref{eqn hom lats} lie in the subring of $\mathcal E$ generated by elements of the discrete valuation ring $R_{pR}$ which have  preimages in $K(X_1,\ldots, X_n)$ under the map $\widehat \varphi$ from earlier (extended to fields of fractions). Hence the $\mathcal E$-lattice of solutions to  \eqref{eqn hom lats} has a basis consisting of matrices $\mathbf M_1,\ldots,\mathbf M_d$ (with $d=\operatorname{rank}_\mathcal E \operatorname{Hom}(L({\mathbf w}), \mathcal E\otimes_{\mathcal O}L({\mathbf x}))$) with entries in the aforementioned subring.  Now the fact that
		 $L_{\mathbf w}$ and  $\mathcal E\otimes_{\mathcal O}L_{\mathbf x}$ are isomorphic is equivalent to the assertion that the polynomial
		 $$
		 	\det(\mathbf M_1 \cdot Z_1 + \ldots + \mathbf M_d \cdot Z_d) \in \mathcal E[Z_1, \ldots, Z_d]
		 $$
		 has $p$-valuation zero (i.e. some coefficient has $p$-valuation zero). But then one can easily find $\mathbf z\in \mathcal O^d$ (or even $W(\bar{\mathbb F}_p)^d$) such that the $p$-valuation of $\det(\mathbf M_1\cdot z_1+\ldots+\mathbf M_d\cdot z_d)$ is zero. Then $\mathbf M=\mathbf M_1\cdot z_1+\ldots+ \mathbf M_d\cdot z_d$ is an element of $\operatorname{GL}_m(\mathcal E)$ with entries that have preimages in $K(X_1,\ldots, X_n)$ such that $\mathbf M^{-1} \cdot \Delta_{\widehat{\mathbf w}}(\lambda) \cdot \mathbf M = \Delta_{\widehat{\mathbf w}(0,\ldots,0)}(\lambda)$ for all $\lambda \in \Lambda$. Now recall that $\Delta_{\widehat{\mathbf w}}(\lambda)$ is obtained from $\Delta(\lambda)$ by entry-wise application of $\widehat \varphi$ (again, extended to fields of fractions). Let $\mathbf M'$ be a preimage under $\widehat \varphi$ of $\mathbf M$, that is, $\mathbf M'$ has entries in $K(X_1,\ldots,X_n)$. Then $\mathbf M'^{-1}\cdot \Delta(\lambda) \cdot \mathbf  M'$ must be a preimage under $\widehat \varphi$ of $\Delta_{\widehat{\mathbf w}(0,\ldots,0)}(\lambda)$ (for any $\lambda \in \Lambda$). But since $\widehat \varphi$ is injective by construction, we get
		 \begin{equation}\label{eqn conj}
		 	\mathbf M'^{-1}\cdot \Delta(\lambda) \cdot \mathbf M' = \Delta_{\widehat{\mathbf{w}}(0,\ldots,0)}(\lambda) \quad \textrm{for all $\lambda \in \Lambda$,}
		 \end{equation}
		 which is now an equation entirely in $K(X_1,\ldots, X_n)={\rm frac}(\mathcal O[X_1,\ldots, X_n])$.

		 Now let $\mathbf y$ be another point in $\mathcal U(k)\cap \mathcal Z(k)$, and let $\widehat{\mathbf y}\in \mathcal O^n$ be an element such that $\rho_l(\widehat{\mathbf y})=\mathbf y$. Since $\Delta_{\widehat{\mathbf y}}$ is obtained from $\Delta$ by substituting $X_i=\widehat y_i$, equation~\eqref{eqn conj} implies that $L({\mathbf y})\cong L({\mathbf x})$ provided $\mathbf M'|_{(X_1,\ldots,X_n)=\widehat {\mathbf y}}\in \operatorname{GL}_m(\mathcal O)$. Note that all $g\in \mathcal O[X_1,\ldots,X_n]$ which occur as numerators or denominators of either $\mathbf M'$ or $\mathbf M'^{-1}$ satisfy $\nu_{p,\mathbf w}(g)=\nu_p(g(\widehat{\mathbf w}))$, and substituting $(X_1,\ldots,X_n)=\widehat{\mathbf w}$ in $\mathbf M'$ and $\mathbf M'^{-1}$ gives back $\mathbf M$ and $\mathbf M^{-1}$ by definition. By Proposition~\ref{prop gen val zariski closed} there are closed subvarieties $\mathcal V_{l,\nu_{p,-}(g)\geq\nu_{p,\mathbf w}(g)}$ and $\mathcal V_{l,\nu_{p,-}(g)\geq\nu_{p,\mathbf w}(g)+1}$ of $\mathbb A^{n\cdot l}$ defined over $k$ such that $\mathbf w$ lies in $\mathcal V_{l,\nu_{p,-}(g)\geq\nu_{p,\mathbf w}(g)}(k')$ but not in $\mathcal V_{l,\nu_{p,-}(g)\geq\nu_{p,\mathbf w}(g)+1}(k')$. Since $\mathbf w$ was chosen as a generic point for $\mathcal Z$, any subvariety of $\mathbb A^{n\cdot l}$ defined over $k$  contains $\mathbf w$ if and only if it contains $\mathcal Z$. It follows that $(\mathcal V_{l,\nu_{p,-}(g)\geq\nu_{p,\mathbf w}(g)} \setminus \mathcal V_{l,\nu_{p,-}(g)\geq\nu_{p,\mathbf w}(g)+1})\cap \mathcal Z$ is an open subvariety of $\mathcal Z$, whose $k$-rational points are by definition those $\mathbf y$ for which $\nu_{p, \mathbf y}(g)=\nu_{p, \mathbf w}(g)$.

		  We conclude that there is an open subvariety $\mathcal V$ of $\mathcal U$  such that for any $\mathbf y\in \mathcal V(k)$ there is a $\widehat {\mathbf y}$ for which $\mathbf M'|_{(X_1,\ldots,X_n)=\widehat {\mathbf y}}$
		 and $\mathbf M'^{-1}|_{(X_1,\ldots,X_n)=\widehat {\mathbf y}}$ lie in $\mathcal O^{m\times m}$ (in fact, the valuation of each entry is the same as that of the corresponding one of $\mathbf M$). Hence $L({\mathbf y})\cong L({\mathbf x})$ for all $\mathbf y \in \mathcal V(k)$, which completes the proof.
	\end{proof}

	\begin{corollary}\label{cor rigid unique}
		A smooth family of $\Lambda$-lattices contains at most one isomorphism class of rigid lattices.
	\end{corollary}
	\begin{proof}
		We assume $\mathcal Z$ to be irreducible, which means that any two non-empty Zariski-open subsets have non-trivial intersection. In particular, if the family contains two rigid lattices, then their respective sets of points parametrising lattices  isomorphic to them have non-trivial intersection. This implies that any two rigid lattices in the family must be isomorphic.
	\end{proof}

	\section{Varieties of lattices}\label{section lattice var}

	In this section we will show how to parametrise all $\Lambda$-lattices in a given finite-dimensional $K\otimes_\mathcal O\Lambda$-module $V$, up to isomorphism. The idea is to start with a fixed lattice, and then to conjugate an affording representation by an upper-triangular basis matrix with ``generic entries'' above the diagonal. The condition that the specialisation of the resulting ``generic representation'' $\Delta$ at some $\widehat{\mathbf x}$ be integral is a closed condition on the Witt vector components of $\widehat{\mathbf x}$, which, after a minimal amount of work, gives sets $\mathcal Z \cap \mathcal U$ such that each $(\Delta, \mathcal Z\cap \mathcal U)$ defines a smooth family of $\Lambda$-lattices.

	\begin{lemma}\label{lemma sublattice fams}
		Let $L$ be a $\Lambda$-lattice of rank $m\in \mathbb N$ and let $l$ be some non-negative integer. Then there are finitely many smooth families of
		$\Lambda$-lattices $L_1(-),\ldots,L_d(-)$ ($d\in\mathbb N$) such that each $\Lambda$-sublattice $L'\leq L$ for which the quotient $L/L'$ has length $l$ as an $\mathcal O$-module is contained in one of the $L_i(-)$.
	\end{lemma}
	\begin{proof}
	Let $\Delta_L:\ \Lambda \longrightarrow \mathcal O^{m\times m}$ be a representation affording $L$, and let us fix $v_1,\ldots, v_m\in \mathbb Z_{\geq 0}$ such that $v_1+\ldots+v_m=l$. Let us also fix an algebraically closed $k'\supseteq k$ and set $\mathcal E=W(k')$ (this is just to formally verify the conditions of a smooth family of $\Lambda$-lattices). Consider the upper-diagonal matrix
	\begin{equation}
	\mathbf B=
	\left( \begin{array}{ccccc}
	p^{v_1} & X_{1} & X_{2} & \cdots & X_{m-1} \\
	0 & p^{v_2} & X_{m}& \cdots& X_{2m-3} \\
	0 & 0 & p^{v_3} & \cdots & X_{3m-6} \\
	\vdots & \vdots  & \vdots & \ddots & \vdots \\
	0 & 0 & 0 & \cdots & p^{v_m}\
	\end{array}\right) \in K[X_1,\ldots, X_{m(m-1)}]^{m\times m}
	\end{equation}
	We will define finitely many smooth families of $\Lambda$-lattices such that every $\Lambda$-sublattice of $\mathcal O^m$ (considered as a $\Lambda$-lattice via $\Delta_L$) which has an $\mathcal O$-basis  consisting of the rows of $\mathbf B|_{(X_1,\ldots,X_{m(m-1)})=\widehat{\mathbf x}}$ for some $\widehat{\mathbf x}\in\mathcal O^{m(m-1)}$ is contained in one of these families. That will actually prove the lemma, since every $\mathcal O$-sublattice of $\mathcal O^m$ with quotient of $\mathcal O$-length $l$ has a basis given by the rows of a matrix of the same form as $\mathbf B$ for some $v_1,\ldots, v_m\in \mathbb Z_{\geq 0}$ with $v_1+\ldots+v_m=l$.

	First define
	\begin{equation}
	\Delta:\ \Lambda\longrightarrow K[X_1,\ldots,X_{m(m-1)}]^{m\times m}:\ \lambda\mapsto \mathbf B \cdot \Delta_L(\lambda)\cdot \mathbf B^{-1}
	\end{equation}
	 Note that $p^l\cdot \Delta$ takes values in $\mathcal O[X_1,\ldots,X_{m(m-1)}]^{m\times m}$, and for a given $\lambda\in \Lambda$ and $1\leq i,j\leq m$ we have that
	$p^l \cdot \Delta(\lambda)_{i,j}$ is an element  $f\in {\mathcal O[X_1,\ldots,X_{m(m-1)}]}$.
	From the theory of Witt vectors we get polynomials $f_0,\ldots, f_{l-1}\in {k[X_{i',j'} \ | \ 1\leq i' \leq m(m-1), 0 \leq j' \leq l-1]}$ such that
	\begin{equation}
	\rho_l (f(\widehat{\mathbf x}))=\left(f_0(\rho_l(\widehat{\mathbf x})), \ldots, f_{l-1}(\rho_l(\widehat{\mathbf x}))\right)
	\end{equation}
	for all $\widehat{\mathbf x} \in \mathcal E^{m(m-1)}$.
	The intersection of the vanishing sets of the polynomials $f_0,\ldots, f_{l-1}$ defines a closed subvariety
	$\mathcal X_{i,j,\lambda} \subseteq \mathbb A^{m(m-1)\cdot l}$ defined over $k$.
	By definition, the representation $\Delta_{\widehat{\mathbf x}}$ (for  $\widehat{\mathbf x} \in \mathcal E^{m(m-1)}$) takes values in $\mathcal E^{m\times m}$ if and only if $\widehat{\mathbf x}$ reduces to a point of $\mathcal X(k')$, where
	\begin{equation}
		\mathcal X= \bigcap_{i,j=1}^m \bigcap_{\lambda} \mathcal X_{i,j,\lambda}\quad \textrm{ ($\lambda$ running over a basis of $\Lambda$)}
	\end{equation}
	Moreover, one checks that the row space of the matrix $\mathbf B$ specialised at any element of $\mathcal E^{m(m-1)}$ contains $p^l\cdot \mathcal E^m$, which implies that if $\widehat{\mathbf x}, \widehat{\mathbf y}\in \mathcal E^{m(m-1)}$ reduce to the same point in $\mathcal X(k')$, then the respective row spaces of the matrices $\mathbf B|_{(X_1,\ldots,X_{m(m-1)})=\widehat{\mathbf x}}$ and $\mathbf B|_{(X_1,\ldots,X_{m(m-1)})=\widehat{\mathbf y}}$ are equal, which implies that the representations $\Delta_{\mathbf{\widehat x}}$ and $\Delta_{\mathbf{\widehat y}}$ are conjugate.

	The only remaining problem is the fact that $\mathcal X$ is in general neither smooth nor irreducible. However, we can decompose $\mathcal X$ as a union of finitely many irreducible components, remove the singular loci from each irreducible component, then decompose the singular loci into irreducible components, and so on and so forth. We ultimately obtain finitely many $\mathcal Z_1,\ldots, \mathcal Z_d \subseteq \mathbb A^{m(m-1)\cdot l}$ closed
	and $\mathcal U_1,\ldots, \mathcal U_d \subseteq \mathbb A^{m(m-1)\cdot l}$ open ($d\in \mathbb N$) such that
	\begin{equation}
		\mathcal X = \bigcup_{i=1}^d \mathcal Z_i \cap \mathcal U_i
	\end{equation}
	By construction, the $L_i(-)=(\Delta, \mathcal Z_i\cap \mathcal U_i)$ are smooth families of $\Lambda$-lattices such that each $\Lambda$-sublattice of $L$ with a basis of the same shape as $\mathbf B$ is contained in one of the $L_i(-)$, as required.
	\end{proof}

	\begin{thm}\label{thm lattice alg}
		Let $V$ be a finite-dimensional $K\otimes_\mathcal O \Lambda$-module. Then there are finitely many smooth families of $\Lambda$-lattices
		$M_1(-),\ldots, M_d(-)$ such that each full $\Lambda$-lattice $L\leq V$ is contained in one of the $M_i(-)$.
	\end{thm}
	\begin{proof}
		Fix some full $\Lambda$-lattice $L_0\leq V$.
		It is well-known that every full $\Lambda$-lattice in $V$ is isomorphic to a $\Lambda$-lattice $L$ such that $L_0\cdot (\Gamma:\Lambda)^2 \leq L \leq L_0$, where $(\Gamma : \Lambda)$ denotes the biggest two-sided $\Gamma$-ideal contained in $\Lambda$,  $\Gamma$ being a maximal order containing $\Lambda$. Therefore there is an upper bound $n\in \mathbb N $ on the composition length of $L_0/L$ as an $\mathcal O$-module which depends only on $V$ and $\Lambda$. Now we can just apply Lemma~\ref{lemma sublattice fams} to $L_0$ for all $0\leq l\leq n$, such as to obtain finitely many smooth families of $\Lambda$-lattices containing all $\Lambda$-sublattices of~$L$, up to isomorphism.
	\end{proof}

	\section{Proofs of the main theorems and applications}

	It should be fairly clear by now how Corollary~\ref{cor rigid unique} and Theorem~\ref{thm lattice alg} imply Theorem~A, and Theorem~B is an immediate consequence of that. We still include proofs for completeness' sake.

	\begin{proof}[Proof of Theorem~A]
		As $K\otimes_{\mathcal O} \Lambda$ is assumed to be separable, there are only finitely many isomorphism classes of $K\otimes_\mathcal O\Lambda$-modules $V$ of dimension $\leq n$.
		By Theorem~\ref{thm lattice alg} there are, for each such $V$, finitely many smooth families of $\Lambda$-lattices such that each full $\Lambda$-lattice in $V$ is contained in one of these families. Hence every $\Lambda$-lattice of rank $\leq n$ is contained in one of finitely many smooth families of $\Lambda$-lattices, and by Corollary~\ref{cor rigid unique} each such family can contain at most one isomorphism class of rigid lattices.
	\end{proof}

	\begin{proof}[Proof of Theorem~B]
		We can assume without loss of generality that $\Lambda$ is basic. Then every element of ${\operatorname{Pic}}_{\mathcal O}(\Lambda)$ is represented by a $\Lambda$-$\Lambda$-bimodule $\Lambda{_\alpha}$, where $\alpha \in \operatorname{Aut}_{\mathcal O}(\Lambda)$.
		Define $\Lambda^{\rm e} = \Lambda^{\rm op}\otimes_{\mathcal O} \Lambda$. Then $\Lambda^{\rm e}$ is again an $\mathcal O$-order in a separable $K$-algebra, and we can view elements of ${\operatorname{Pic}}_{\mathcal O}(\Lambda)$ as
		$\Lambda^{\rm e}$-lattices. Note that if $\alpha\in \operatorname{Aut}_\mathcal O(\Lambda)$, then $\operatorname{id}\otimes \alpha\in \operatorname{Aut}_{\mathcal O}(\Lambda^{\rm e})$, and $\Lambda_{\alpha}$ (that is, the $\Lambda$-$\Lambda$-bimodule $\Lambda$ twisted by $\alpha$ on the right) is the same as $\Lambda_{\operatorname{id}\otimes \alpha}$ (that is, the right $\Lambda^{\rm e}$-module $\Lambda$ twisted by $\operatorname{id}\otimes \alpha$). Hence
		${\operatorname{Ext}}^1_{\Lambda^{\rm e}}(\Lambda_\alpha, \Lambda_\alpha)\cong {\operatorname{Ext}}^1_{\Lambda^{\rm e}}(\Lambda, \Lambda)=\operatorname{HH}^1(\Lambda)$, and the latter is zero by assumption. That is, the elements of ${\operatorname{Pic}}_\mathcal O(\Lambda)$ are rigid $\Lambda^{\rm e}$-lattices of $\mathcal O$-rank equal to the $\mathcal O$-rank of $\Lambda$. By Theorem~A there are only finitely many such $\Lambda^{\rm e}$-lattices, up to isomorphism.
	\end{proof}
	Proposition~\ref{prop det pic} is actually just the combination of Theorem~B and Weiss' criterion.
	\begin{proof}[Proof of Proposition~\ref{prop det pic}]
		Consider $(\mathcal O G b)^{\rm e}$ as a block of $\mathcal O (G^{\rm op}\times G)$.
		Let $M$ be an {$(\mathcal O G b)^{\rm e}$-module} representing an element of $\operatorname{Pic}(\mathcal O Gb)$. By Weiss' criterion (see \cite{WeissRigidity}, and \cite{WeissArbitraryCoeffs} for a version allowing $\mathcal O$ as a coefficient ring) our $M$ has trivial source if and only if the lattice of $P$-fixed points taken on the left
		\begin{equation}
		{^{P}M} \cong {^P(\mathcal O Gb\otimes _{\mathcal O G b} M)} \cong {(^P\mathcal O Gb)\otimes _{\mathcal O G b} M}   \cong \mathcal O [P\backslash G]\cdot b\otimes_{\mathcal O Gb} M
		\end{equation}
		has trivial source as an $\mathcal O Gb$-module, that is, is $p$-permutation.  That proves the first part.

		Now if $M$ represents an element of $\operatorname{Picent}(\mathcal O Gb)$, then
		$\mathcal O[P\backslash G]\cdot b\otimes_{\mathcal O Gb}M$ has the same character as $\mathcal O[P\backslash G]\cdot b$, and isomorphic endomorphism ring. Moreover, we clearly have
		\begin{equation}
		{\operatorname{Ext}}^1_{\mathcal O G b}(\mathcal O[P\backslash G]\cdot b\otimes_{\mathcal O Gb}M, \mathcal O[P\backslash G]\cdot b\otimes_{\mathcal O Gb}M)\cong {\operatorname{Ext}}^1_{\mathcal O G b}(\mathcal O[P\backslash G]\cdot b, \mathcal O[P\backslash G]\cdot b)=0.
		\end{equation}
		Hence, by our uniqueness assumption, we have
		$\mathcal O[P\backslash G]\cdot b\otimes_{\mathcal O Gb}M \cong \mathcal O[P\backslash G]\cdot b$ as $\mathcal O Gb$-modules, which, by the above, implies that $M$ lies in $\mathcal T(\mathcal O G b)$.
	\end{proof}

	To finish, let us briefly mention the following nice consequence of Theorems~A~and~B, even though it is implied by \cite[Theorem (38.6)]{Thevenaz}, a theorem due to Puig.
	\begin{prop}
		Let $\{\mathcal O G_ib_i\}_{i\in I}$ (for some index set $I$) be a family of block algebras defined over $\mathcal O$ with fixed defect group $P$, each of which  Morita equivalent to some fixed $\mathcal O$-algebra $A$ by means of some fixed $\mathcal O G_ib_i$-$A$-bimodule $M_i$. If there is a bound, independent of $i$,  on the dimension of $L\otimes_{\mathcal O G_ib_i}M_i$ for indecomposable $p$-permutation $\mathcal O G_ib_i$-modules $L$, then the $\mathcal O G_ib_i$ split into finitely many equivalence classes with respect to splendid Morita equivalence.
	\end{prop}
	\begin{proof}
		Note that by assumption $P$ is a subgroup of $G_i$ for every $i\in I$. Equivalently, one could also assume that there is a fixed embedding $P\hookrightarrow G_i$ for each $i$, but we are going to take the former point of view.
		By Theorem~A there are only finitely many rigid $A$-lattices of rank smaller than the given bound on images of indecomposable $p$-permutation modules. Hence $I$ splits up into finitely many sets $I_1,\ldots, I_d$ such that
		$I=\bigcup_{j=1}^d I_j$ and, for every $i\in I_j$, the $A$-module $\mathcal O[P\backslash G_i]\cdot b_i\otimes_{\mathcal O G_ib_i} M_i$ has the same indecomposable summands as some fixed $A$-module $L_j$. It follows that if $i,i'\in I_j$, then $M_i\otimes_A M_{i'}^\vee$ maps the indecomposable summands of $\mathcal O[P\backslash G_i]\cdot b_i$ to the indecomposable summands of $\mathcal O[P\backslash G_{i'}]\cdot b_{i'}$. One can show using the same argument as in the proof of Proposition~\ref{prop det pic} that  $M_i\otimes_{A}M_{i'}^\vee$ is a $p$-permutation $\mathcal O(G_i^{\rm op}\times G_{i'})$-module. By \cite[7.5.1]{PuigBook} (or, independently, by results of \cite{ScottUnpublished}) this implies that the $M_i\otimes_A M_{i'}^\vee$ for $i,i'$ in a fixed $I_j$ are splendid up to restriction along an automorphism of $P$, of which there are only a finite number. That is, each $I_j$ splits into finitely many subsets such that the blocks parametrised by any one of the subsets are pair-wise source algebra equivalent by means of the $M_i\otimes_{A}M_{i'}^\vee$.
	\end{proof}

	\paragraph*{Acknowledgements} I would like to thank Radha Kessar and Markus Linckelmann for pointing the finiteness problem for Picard groups out to me, and for helpful comments on a first version of this preprint.

redpajama_set_name: RedPajamaArXiv


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DODOMA, 28 May 2021 – Tanzanian authorities have signaled their intent to reverse an earlier decision committing to withdraw from the African Court of Human and People's Rights. The administration of late President John Pombe Magufuli announced plans to...
Tanzania: Finalizing Plans to Ratify AfCFTA
DAR ES SALAAM, 27 May 2021 – Tanzanian authorities have pledged to ratify the Africa Continental Free Trade Area (AfCFTA) between June and October 2021. The country's cabinet is expected to endorse AfCFTA prior to its presentation for debate, adoption, and...
Africa: Celebrations Marking Africa Day
HARARE, 25 May 2021 – Celebrations are underway in several African countries notably in Ethiopia to mark Africa Day. The day commemorates the founding in 1963 of the erstwhile Organization of African Unity (OAU) renamed the African Union (AU). Zimbabwe's President...
Africa: EU to Build Vaccine Hubs
ROME, 22 May 2021 – The European Union (EU) says it intends to invest up to a billion euros, approximately 1.2 billion dollars, building COVID-19 vaccine manufacturing hubs in Africa. The commitment was given Friday by Ursula von der Leyen, the president of the...
Africa: Destroy Expired COVID-19 Vaccines – WHO
GENEVA, 19 May 2021 – The World Health Organization (WHO) has reversed itself, calling now for expired COVID-19 vaccines to be destroyed. The world body's earlier guidance was for countries to continue administering the vaccines up to 36 months after the expiry date...

redpajama_set_name: RedPajamaCommonCrawl


Effective immediately, Dafran is suspended from the active roster of our Overwatch team.
Upon further VOD review of Dafran's behavior, especially this past night, we have found his actions to be completely unacceptable, and in clear violation of Blizzard's rules around player conduct, in addition to our own expectations of player/teammate conduct. Blizzard has also reviewed the case and has suspended Dafran's account from Competitive Play for the duration of Season 5. In addition, Dafran has been disqualified from competing in the Overwatch Contenders Season Zero, as well as Season One. Having finished our own detailed review, we agree with and fully accept their decision.
While Dafran will no longer be playing with our team, we will be assisting him in finding the help, counseling, and guidance that he needs to improve his mental and emotional well-being. He is a dear friend, and we will be as supportive as possible during this time off from competitive play, so that he can come back stronger and better than ever. In the meantime, we will be finding a substitute/replacement for the remainder of the Contenders Season Zero.
As fans of the game and its competitive scene, we appreciate Blizzard's quick and righteous decision. We apologize to our fans, the Overwatch community, and any/all persons and players impacted by Dafran's actions.


Pebble Records opened as a physical record store in 2014 after trading successfully online as a specialist indie music webstore. The shop has broaden it's range to cover all styles of music including new releases and an extensive back catalogue of classic rock and pop, jazz, indie, metal, psych, prog rock, blues, reggae, folk, world, electronica and dance, soundtracks and much more. We sell new vinyl, compact discs, tapes and flexi discs! We also stock hi-fi seperates including Rega and Pro-ject with more coming soon.
From Record Shops" worth £8.99 for free.


1. An amount paid or required, often as an installment payment, for an insurance policy.
a. An amount paid to obtain a loan.
b. An interest payment, usually on a regular basis, on a loan.
3. A sum of money or bonus paid in addition to a regular price, salary, or other amount: Many people are willing to pay a premium to live near the ocean.
a. The amount at which something, such as a security or a currency, is valued above its initial or nominal price.
b. The amount at which a securities option is bought or sold.
a. Something offered free or at a reduced price as an inducement to buy something else: offered the backpack as a premium for students opening new bank accounts.
b. A prize or award.
6. An unusual or high value: Employers put a premium on honesty and hard work.
7. Archaic Payment for training in a trade or profession.
Of superior quality or value: premium gasoline.
More valuable than usual, as from scarcity: Fresh water was at a premium after the reservoir was contaminated.


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About - Beth Krensky
page-template,page-template-full_width,page-template-full_width-php,page,page-id-40461,ajax_fade,page_not_loaded,,select-theme-ver-1.8.1,smooth_scroll,wpb-js-composer js-comp-ver-4.4.4,vc_responsive
Beth Krensky is a professor of art education and the Area Head of Art Teaching at the University of Utah. She is an artist, activist and educator. She received her formal art training from the Boston Museum School. She has exhibited widely throughout the United States and internationally. She is a founding member of the international artist collective, the Artnauts. Her work is intended to provoke reflection about what is happening in our world as well as to create a vision of what is possible.
She is also a scholar in the area of youth-created art and social change. She received a master's degree with a focus on critical pedagogy and art education from the Harvard Graduate School of Education and a Ph.D. in Education from the University of Colorado at Boulder. She co-founded and spent a decade as Artistic Director for the award-winning youth arts/service/action organization, Project YES (Youth Envisioning Social change). Her co-authored book, Engaging Classrooms and Communities through Art: A Guide to Designing and Implementing Community-Based Art Education, was published by AltaMira Press in 2009.
"As an academic, I'm known largely for my scholarly works, writings, and also for my work in the field — specifically, going into diverse communities and working with children, making art, promoting dialogue and healing. What's fueling my classroom, scholarship and social actions, however, is a very personal and highly material-based studio practice: one in which I internalize the outside forces and inner beliefs that guide me, and transform them into objects. Professors are also practitioners; artists are also activists. I make art because I am compelled to: a reason that I've always felt was the best motivation for creating new forms.
Change—affecting it personally, and socially—is very important to me. Often, I look to ancient sources as my guide for this, in part because they hearken from a time more comfortable with notions of Transformation. I've used wood gleaned from olive groves in Bethlehem to create staffs; I've compounded anointing oil based on the exact components (cassia, myrrh, cinnamon) as laid down in the book of Exodus, and offered it up in hand-blown glass dispensers to exhibition visitors—allowing them to anoint themselves as they see fit. I've made reliquaries of bronze (a material thought to have powerful "memorializing" qualities), and copper (for its "conduit"-like properties between the spirit and material worlds). Modern inventions like the sewing machine, and familiar office-tools like rolling casters, make their way into my sculptures as well. But at their root, these are forms that embody the sort of ineffable sense of 'shift' and internal re-orientation that I seek.
Many of the pieces are small enough to carry on one's person, and its only in close view that you'll truly get as sense of their presence: the rust on the metal tip of a staff; the hollow crab shells inside a hand-shaped reliquary; the touch of oil-rubbed wood. The tactile is a Formal response to the notion of the personal; materials are latent with message. Hence, I've spent a lot of time researching just what it is they might be trying to tell us; I've learned about early Jewish practices previously kept secret, of mysticism and magic—to the point where my studio practice can almost seem to border on the anthropological. That's an admixture with which I feel comfortable. In many ways I'm a visual archeologist, unearthing fragments of the past, digging up metaphor and parable that I then site into a personal narrative. The only way I can advocate change for Others is to begin with my own practice, put my own self and beliefs on the line. Viewers often respond with intense catharsis to these works, and I welcome that. Through the gentle power of shared emotion, darkness is exposed, and healing is brought to light."
2002 Ph.D. Educational Foundations, Policy and Practice, The University of Colorado, Boulder, Colorado; Ph.D. Dissertation: An Act of Hope: Developing Social Responsibility through Issue-Oriented Community-Based Art Education
1991 Ed.M. Multicultural Arts Education and Critical Pedagogy, Harvard Graduate School of Education, Cambridge, Massachusetts
Center for Advanced Visual Studies, Massachusetts Institute of Technology, Cambridge, Massachusetts
1988 B.F.A. Sculpture, School of the Museum of Fine Arts/Tufts University, Boston, Massachusetts
2016- present Professor, Art Teaching Area Head, Department of Art and Art History, College of Fine Arts, University of Utah, Salt Lake City, Utah
2010- 2016 Associate Professor, Art Teaching Area Head, Department of Art and Art History, College of Fine Arts, University of Utah, Salt Lake City, Utah
2003-2010 Assistant Professor, Art Teaching Area Head, Department of Art and Art History, College of Fine Arts, University of Utah, Salt Lake City, Utah
1995-2002 Instructor, INVST Community Studies Program, The University of Colorado, Boulder, Colorado
RECENT AWARDS/HONORS/GRANTS/FELLOWSHIPS
Presidential Scholar, University of Utah
Fellow, Jewish Art Salon, New York City, New York
Community-Engaged Learning Microgrant, Lowell Bennion Community Service Center, University of Utah ($1,000)
Faculty Fellow, Office of Undergraduate Studies, University of Utah
Faculty Fellow, University Research Committee, University of Utah
SOLO/TWO-PERSON EXHIBITIONS
Portable Sanctuaries, CCA Christensen Gallery, Granary Art Center, Ephraim, Utah, February-April
On Sin and Redemption: On Line Exhibition (with Andi Arnovitz), Sh'ma: A Journal of Jewish Ideas, August, 2013
Beth Krensky: Portable Sanctuaries, Salt Lake Arts Council Finch Lane Gallery, Salt Lake City, Utah, May-June
Beth Krensky: Portable Sanctuary #3, 3///3: Three Walls on Wednesdays, Athens, Greece, March
We Make the Road by Walking (with Sama Alshaibi), La Galería De La Misión De Senecú, Ysleta Independent School District, El Paso, Texas
Motherhood and Revolution (with Sama Alshaibi), California Institute of the Arts, Valencia, California
We Make the Road by Walking (with Sama Alshaibi), Dinnerware Artspace, Tucson, Arizona
We Make the Road by Walking (with Sama Alshaibi), Mizel Museum, Denver, Colorado
Beth Krensky: Objects for Ritual and Transformation, Salt Lake Arts Council Finch Lane Gallery, Salt Lake City, Utah
Beth Krensky: New Work, Michael Gray Gallery, Longmont, Colorado
Beth Krensky and Melody Morrow Reeves, Granary Gallery, Longmont, Colorado
Would You Rather Fly or Be Invisible?, Arts and Humanities Assembly of Boulder County, Boulder, Colorado
My Sister/Myself, Gutman Library, Harvard University, Cambridge, Massachusetts
RECENT GROUP EXHIBITIONS (SELECTED)
3rd Jerusalem Biennale (Curator: Ori Z. Soltes), Jerusalem, Israel, October-November
Far From the Front Lines, Evanston Art Center, Evanston, Illinois, September
Meta Journey
The Historical Museum, Sarajevo, Bosnia and Herzegovina, February
Art Gallery Bihac (UMJETNIČKA GALERIJA BIHAĆ), Bihac, Bosnia and Herzegovina, February
Souvenir: A travelling exhibit of work by members of the Rocky Mountain Chapter of the Guild of Book Workers, in collaboration with Marnie Powers-Torrey
Harold B. Lee Library, Brigham Young University, Provo, UT, Jan.-March
Denver Public Library, Denver, CO, April-June
University of Wyoming, Laramie, WY, July-Sept.
Globalocation
University of Texas-Austin, Austin, Texas, March-June
Marriott Library, University of Utah, Salt Lake City, Utah, Jan.-March
Globalocation, RedLine, Denver, Colorado, November-December
Rally 'Round the Flag of Justice (Curator: Linda Weintraub), RedLine, Denver, Colorado, November-December
Souvenir, in collaboration with Marnie Powers-Torrey, Phoenix Public Library, Oct.-Dec
Social Justice: It Happens to One, It Happens to All, St. Mary's College Museum of Art, Moraga, California, September-December
Greatmore Studios, Capetown, South Africa, July
Harare Academy of Inspiration, Khayelitsha, Capetown, South Africa, July
On the Edge of Your Seat: Chairs for the 21st Century, The Center for Art in Wood (Jurors: Nora Atkinson, Lloyd Herman Curator of Craft, Smithsonian American Art Museum; Jasper Brinton, Principal, Brinton Design; Benjamin Colman, Associate Curator of American Art, Detroit Institute of Arts; Susie J. Silbert, Curator of Modern and Contemporary Glass, Corning Museum of Glass), Philadelphia, PA May-July
The Painted Veil (Curator: Jorge Rojas), Rio Gallery, Salt Lake City, Utah, May-July
Art & Resistance, Exhibition Gallery, Dar Al-Kalima University College of Arts and Culture, Bethlehem, Palestine, May
Soul Drift: Visualizations of a Cultural Phenomenon, Winter Festival Sarajevo, Museum of Natural History Sarajevo, Sarajevo, Bosnia/Herzegovena, February
Where Do We Come From? What Are We? Where Are We Going?, Winter Festival Sarajevo, Museum of Natural History Sarajevo, Sarajevo, Bosnia/Herzegovena, February
Evento Cultural Proyecto Comunidad, Barrio Macatoa, Granada, Colombia, November
Evento Cultural Proyecto Comunidad, Barrio Sabana, Granada, Colombia, November
Colegio Artistico Normal Superior, Acacias, Colombia, October
Evento Cultural Proyecto Comunidad, Barrio la Madrid, Villaviciencio, Colombia
Biblioteca Sede Barcelona, Universidad de los Llanos, Villaviciencio, Colombia
Winter Festival Sarajevo, Museum of Natural History Sarajevo, Sarajevo, Bosnia/Herzegovena, February
Biodiversity/Extinction (Curators: Elizabeth Corr, Natural Resources Defense Council; Dr. Paula J. Ehrlich, E.O. Wilson Biodiversity Foundation), New York Hall of Science, New York City, New York, October-February 2016
Visualizing Peace, Galleriji Alu, Academy of Fine Arts, Sarajevo, Bosnia, November
Diaspora of Being, International Center of Bethlehem, Bethlehem, Palestine, August
Re: Visioning the West, University Art Gallery Banner Project, New Mexico State University, Las Cruces, New Mexico, May-August
Collective Experience, Rio Gallery, Salt Lake City, Utah, March-April
Identity, Place and Memory (Arte Norteamericano: Identidad, Lugar y Memoria), Bogota Arte Contemporaneo Gallery, Bogota, Colombia, March
Identity (Identidad En Circulacion)
Teatro La Voragine, Sala de Exposiciones, Villavicencio, Meta, Colombia
Universidad de Los Llanos – Sede San Antonio, Villavicencio, Meta, Colombia
Escuela Normal Superior de Acacias, Villavicencio, Meta, Colombia
Escuela Normal Superior de Acacias, Acacias, Meta, Colombia
Casa de la Cultura, Hall de Exposiciones, Granada, Meta, Colombia
Parque Central Rafael Reyes, Restrepo, Meta, Colombia
Universidad Nacional de Colombia, Hall de Exposicions, Leticia, Colombia
Fundacion AMAZARTE, Leticia, Amazonas, Colombia
Alcaldia, Sala Exposiciones, Puerto Narino, Amazonas, Colombia
Gobernacion del Guaviare, Sala Exposiciones – Centro Cultural, San Jose del
Guaviare, Guaviare, Colombia
Casa de la Cultura, Yopal, Casanare, Colombia
Biblioteca Publica, Mitu, Vaupes, Colombia
Casa de la Cultura, Inirida, Museo Comunitario de Guainia, Puerto Inirida, Guaninia, Colombia
Casa de la Cultura, Saravena, Vichada, Colombia
Biblioteca Publica, Tame, Arauca, Colombia
Casa de la Cultura, sala de Exposiciones, Arauca, Arauca, Colombia
Where Do We Come From? What Are We? Where Are We Going?, Galerija Atelje Kunik, Center for Peacebuilding, Sanski Most, Bosnia and Herzogovina, November
Memory Remains, Galeria Aguafuerte, Mexico City, Mexico, October
Religious and Spiritual Art of Utah (Honorable Mention), Springville Museum of Art, Springville, Utah, November-January
Movement and Meaning: The Power of Pilgrimage (Featured Artist), B. F. Larsen Gallery, Brigham Young University, Provo, Utah, October-November
New Narratives: Recent Work by U of U Art Faculty, Utah Museum of Fine Arts, Salt Lake City, Utah, October-January
Readymade@100 (Curator: Mark Cameron Boyd, Corcoran College of Art and Design), American University Museum at the Katzen Art Center, Washington, DC, September-October
Art & Poetics
Mahmoud Darwish Museum, Ramallah, Palestinian Territories, August
Dar al-Kalima University College of Arts and Culture, Bethlehem, Palestine, August-September
Art as Catalyst, Center for Global Justice, San Miguel de Allende, Mexico, July-August
30 Years at Finch Lane: A Retrospective, Finch Lane Gallery, Salt Lake City, Utah, June-August
Windows to the Divine, ArtHaus, Denver, Colorado, May-June
University of the Andes, Bogota, Colombia
Thirteen venues in the Colombian Amazon, including:
Sala de Exposiciones de CORCUNVI / en la Casa de la Cultura, Meta
Villavicencio Biblioteca Pública / Colegio Publico Agropecuario de Puerto Lopéz, Meta Puerto Lopez
Biblioteca Pública / Colegio Publico Normal Superior Acacias, Meta Acacias
Casa de la Cultura / Biblioteca Pública, Meta Granada
Casa de la Cultura, Biblioteca Pública / Sena, Vaupes Mitú
Universidad Nacional de Colombia / Area Cultural Banco de la Republica, Amazonas Leticia
Sala de Exposiciones de la Gobernación, San José del Guaviare
Alcadía o Biblioteca pública municipal, Arauca
Colegio Público, Arauca Tame Casa de la Cultura / Biblioteca Pública, Arauca Saravena
Asociacion plenilunio, Casa de la Cultura, Biblioteca, Casanare Yopal
Museo Comunitario / Casa de la Cultura, Guainia Puerto Inirida
Only Light, Rio Gallery, Salt Lake City, Utah, January-February
Home/Land, Dar al-Kalima College, Bethlehem, Palestine, August
Portas Abertas/Open Doors, (Curator: Claudia Giannetti), Fórum Eugénio de Almeida, Évora, Portugal, July 11-Oct 6
A Better World, Eighth International Conference on the Arts in Society, Eotvos Lorand University, Golyavar Conference Center, Budapest, Hungary, June 24-26
A Better Society (Public Billboard at Metro Stops), Budapest, Hungary, June 15-30
ELTE University: V. district , University square 1-3.
Citylight:V. district Deák square 9 bus endstation
V. district Széchenyi square 2 tram station
V. district Deák square, 47-49 tram endstation
V. district, Astoria , 9 bus station
V.district Jászai Mari square, 4-6 tram station
V. district, Fővám square, 47-49 tram station
Diptych America: Private Spaces/Public Places, Central Academy of Fine Arts (CAFA), Beijing, China, April
Place/Lugar
This exhibition, which was sponsored by the Colombian Ministry of Culture, will travel to the following venues in Colombian Amazon (along the border with Venezuela and Brazil):
Biblioteca Publica Muncipal, Taraira, Departamento de Vaupes, Colombia
April 24-30
Museo comunitario del Guaini, Puerto Inirida, Departamento de Guainia, Colombia, June 15-30
Casa de la Cultura, Mitu, Departamento de Vaupes, Colombia, August 30 – September
Biblioteca Publica Muncipal, Arauca, Departamento de Arauca, Colombia
October 15 – October 30
Biblioteca Publica Muncipal, Vichada, Departamento de Santa Rosalia, Colombia
November 18 – November 29
The Herb & Dorothy Show, Nox Contemporary, Salt Lake City, Utah, November-
Death Is…, Public Projection
KGNU Radio Station Façade, Denver, Colorado, November
Museo de la Autonomia Universitaria, Patio Principal
Centro Historico, Mexico City, October-November
Mixta, Universidad de los Llanos, Villavicencio, Colombia
Memory Remains, Museo de la Autonomia Universitaria, Mexico City, Mexico, October-November
Amulets, Nazars & Evil Eyes: Artists Looking Forward, Queens College Art Center, Flushing, NY, May-June
This exhibition, which was sponsored by the Colombian Ministry of Culture, traveled to the following 15 towns in Colombia that are located along the Amazon River. In each town and state, the exhibition was taken to other locations of tribal groups along tributaries of the Amazon River located near the towns. These other places do not have names. The art was exhibited in the tribal communal house, which is considered a sacred place to conduct tribal affairs.
Universidad de los Llanos, Villavicendo, Colombia
Universidad de los Llanos, San Martin, Meta, Colombia
Universidad de los Llanos, Cabuyaro, Meta, Colombia
Parque principal, Villavicencio, Meta, Colombia
Corcunvi, Villavicencio, Meta, Colombia
Parque principal, Granada, Meta, Colombia
Parque Concha Acusstica, San Martin de los Llanos, Meta, Colombia
Parque principal, Acacias, Meta, Colombia
Parque principal, Cabuyaro, Meta, Colombia
Parque principal, Barranca de Upia, Meta, Colombia
Parque principal, San Jose del Guaviare, Guaviare, Colombia
Parque principal, Aguazul, Casanare, Colombia
Parque principal, Tame, Arauca, Colombia
Taller de Arte, Saravena, Arauca, Colombia
Piazoleta Alcaldia, Arauca, Arauca, Colombia
The Ecological Imperative, Museo Contemporáneo, Valdivia, Chile, catalogue essay by Suzi Gablik, January-March
ART & DIS(PLACE)MENT, Diyar Consortium, Higher Education and Research Building, Mureir Mountain, Bethlehem, Palestine
The Faculty Show: Recent Work by the U of U Art Faculty, Utah Museum of Fine Arts, January-May
Memory Remains, Casa de las Ciencias, Universidad Autónoma del Estado de Morelos,
Cuernavaca, Morelos, Mexico, October-November
Reaction and Healing: The Tenth Anniversary of 9-11, Pennsylvania State University-Lehigh Valley, Center Valley, Pennsylvania, September-October
Ernesto Pujol-Walking Ground, Contemporary Museum Honolulu, Honolulu, Hawaii, May-August
Galeria De Artes/CAUA, Universidade Federal Do Amazonas,
Universidad Nacional de Colombia, Bogota, Colombia
Casa de la Cultura de la Gobernacion, Departamento de Guania, Región Amazonica, Inírida, Guainía
Biblioteca del Banco de la Republica, Leticia, Colombia
4,000 Year Road Trip: Gathering Sparks, Mizel Museum, Denver, Colorado, permanent exhibition
ART & DIS(PLACE)MENT
Al Kahf Gallery, International Center of Bethlehem, Bethlehem, Palestine
Dar al-Kalima College, Bethlehem, Palestine
AZAG/GAZA: Mirroring War, Al Kahf Gallery, Bethlehem, Palestine
La Casa de la Cultura de la Gobernacion, Inirida, Colombia
Fondo Mixto de Guainia, Departmento de Guaninia, Region Amazonica, Guainia, Colombia (16 Indigenous villages along the Rio Guainia)
Horror, Terror & Death
The Museum of Edgar Allan Poe, Richmond, Virginia, September
Galería Antonio Ramírez, Escuela Nacional de Artes Plasticas, Xochimilco, Mexico City
Casa de la Cultura Oaxaqueña, Oaxaca, Mexico
Academia de San Carlos, Escuela Nacional de Artes Plasticas, Universidad Nacional Autonoma de Mexico, Mexico City, Mexico
Proof of Purchase, Sams∅n, Boston, Massachusetts
Galeria Tinoco y Palacios, Oaxaca, Mexico
Goguette Gallery, Mexico City, Mexico
Art, Ecology & the Environment
Universidad Nacional de Colombia Sede Amazonia, Leticia, Colombia
Departamento de Cundinamarca, Bogotá, Colombia
Centro Cultural Santa Teresita, Bogota, Colombia
Escuela de Artes Academia Guerrero, Bogota, Colombia
Departamento del Meta, Ciudad Villavicencio, Lugar Universidad del Llano y Corporación Universitaría del Meta
Departamento del Guaviare, Gobernación del Departamento y Espacios nocturnos, San Jose, Colombia
Departamento del Valle, Comunidad Indigena Nasa Yuwe, Grupo Etnico Tradicional
Signs, Signifiers and the Signified
Alcaldía de Funza, Central Cultural de Funza, Cundinamarca, Colombia
Universidad de Estudios Adminitrativos y de Negocios, Bogota, Colombia
Universidad Nacional de Bogota, Bogota, Colombia
Centro Cultural de Santa Teresita, Bogota, Colombia
University de Llanos, Villavicencio, Colombia
Universidad de Meta, Villavicencio, Colombia
Camara de Comercio, Villavicencio, Colombia
Museo de la Universidad de San Carlos, Guatemala City, Guatemala
LART23TRES Contemporary Art Gallery, Mexico City, Mexico
SP(L)ACES Gallery, Mexico City, Mexico
Don Apolinio, San Adrés Cholula, Mexico
Death/La Muerte: Art from the United States
Galeria ESCOOL, Colonia Roma Norte, Mexico
Don Apolonio Presenta, Cholula, Puebla, Mexico
Climate Change & the Environment: Art of Portrait and Text
Anderson Ranch Arts Center, Snowmass, Colorado
Garden Gallery, Zayed University, Abu Dhabi, United Arab Emirates
Orchard Street Shul Project, John Slade Ely House Center for Contemporary Art, New Haven, Connecticut
Ecology & the Environment
Sala de Escuela Superior de Arte el Paranimfo, University of San Carlos, Guatemala City, Guatemala
University of Guatemala, Guatemala City, Guatemala
Directions Gallery, Colorado State University, Fort Collins, Colorado
Museum of Natural History, Boulder, Colorado
Death: Art from the United States, Galeria Tinoco y Palacios, Oaxaca, Mexico
Windows to the Divine, The Madden Museum, Denver, Colorado
Parallel Worlds, ScalaMata Gallery, Venice, Italy
Words for Life, Muriel Sibell Woole Gallery, Boulder, Colorado
Signs of Globalization
Salón de Exposiciones de la Facultad de Arquitectura, University of San Carlos, Guatemala City, Guatemala
Sala Iquitos, Universidad Nacional del Perú Estado Loreto, Iquitos, Peru
Sala de Exposiciones Amazonas, Universidad del Estado del Amazonas UEA, Tabatinga, Brazil
Sala de Exposiciones Orellana, Biblioteca del Banco de la Republica, Leticia, Colombia
Sala Jania-Iko (Casa de Imágenes en Uitoto), Universidad Nacional de Colombia Sede Amazonia, Leticia, Colombia
Polifacetica: Etnografia Grafica
Encuentro de Fotografía de Medellín, Medellin, Colombia
Sala 2, Galeria Tinoco y Palacios, Oaxaca, Mexico
French Building Alianza Francesa, Socrates 156 Esquina Homero, Colonia Los Morales, Polanco, Mexico
Art Now, Utah Museum of Fine Arts, Salt Lake City, Utah
AZAG/GAZA: Mirroring War
Casa Borda, Taxco, Mexico
Galeria Zapata, Oaxaca, Mexico
Post Tractatus: Art in Amazonia,
Museo de Arte Contemporaneo, Universidad Nacional de Colombia, Sede Bogotá, Bogotá, Colombia
copyright (c) 2015 Beth Krensky
unless otherwise noted, all images and video by Josh Blumental


text:
Demonstration of integrated longitudinal and lateral control for the operation of automated vehicles in platoons
Rajesh Rajamani, Han Shue Tan, Boon Kait Law, Wei Bin Zhang
This paper presents the design and experimental implementation of an integrated longitudinal and lateral control system for the operation of automated vehicles in platoons. The challenges handled in the design of the longitudinal control system include nonlinear vehicle dynamics, string-stable operation with very small inter-vehicle spacing, operation at all speeds from a complete stop to high-speed cruising, and the execution of longitudinal split and join maneuvers in the presence of communication constraints. The challenges handled in the design of the lateral control system include high-speed operation using a purely 'look down' sensor system and lane changing without transitional lateral position measurements. The paper also describes the design of an on-board supervisor that utilizes inter-vehicle communication and coordinates the operation of the lateral and longitudinal controllers in order to execute entry and exit maneuvers. Extensive experimental results are included in the paper from the NAHSC1 demonstration of automated highways conducted in August 1997 at San Diego, CA. The demonstration included a closely spaced eight-car platoon operating continuously over several weeks, with passenger rides being given to visitors. The maneuvers demonstrated included starting the automated vehicles from complete rest, accelerating to cruising speed, automated lane keeping, allowing any vehicle to exit from the platoon with an automated lane change, allowing new vehicles to join the platoon and bringing the platoon to a complete stop at the end of the highway.
IEEE Transactions on Control Systems Technology
https://doi.org/10.1109/87.852914
Published - Jul 1 2000
Fingerprint Dive into the research topics of 'Demonstration of integrated longitudinal and lateral control for the operation of automated vehicles in platoons'. Together they form a unique fingerprint.
Longitudinal control Engineering & Materials Science
Integrated control Engineering & Materials Science
Control systems Engineering & Materials Science
Position measurement Engineering & Materials Science
Intelligent vehicle highway systems Engineering & Materials Science
Supervisory personnel Engineering & Materials Science
Rajamani, R., Tan, H. S., Law, B. K., & Zhang, W. B. (2000). Demonstration of integrated longitudinal and lateral control for the operation of automated vehicles in platoons. IEEE Transactions on Control Systems Technology, 8(4), 695-708. https://doi.org/10.1109/87.852914
Demonstration of integrated longitudinal and lateral control for the operation of automated vehicles in platoons. / Rajamani, Rajesh; Tan, Han Shue; Law, Boon Kait; Zhang, Wei Bin.
In: IEEE Transactions on Control Systems Technology, Vol. 8, No. 4, 01.07.2000, p. 695-708.
Rajamani, R, Tan, HS, Law, BK & Zhang, WB 2000, 'Demonstration of integrated longitudinal and lateral control for the operation of automated vehicles in platoons', IEEE Transactions on Control Systems Technology, vol. 8, no. 4, pp. 695-708. https://doi.org/10.1109/87.852914
Rajamani R, Tan HS, Law BK, Zhang WB. Demonstration of integrated longitudinal and lateral control for the operation of automated vehicles in platoons. IEEE Transactions on Control Systems Technology. 2000 Jul 1;8(4):695-708. https://doi.org/10.1109/87.852914
Rajamani, Rajesh ; Tan, Han Shue ; Law, Boon Kait ; Zhang, Wei Bin. / Demonstration of integrated longitudinal and lateral control for the operation of automated vehicles in platoons. In: IEEE Transactions on Control Systems Technology. 2000 ; Vol. 8, No. 4. pp. 695-708.
@article{9c1e311ebbba4e11abfa8d12f4ea2d31,
title = "Demonstration of integrated longitudinal and lateral control for the operation of automated vehicles in platoons",
abstract = "This paper presents the design and experimental implementation of an integrated longitudinal and lateral control system for the operation of automated vehicles in platoons. The challenges handled in the design of the longitudinal control system include nonlinear vehicle dynamics, string-stable operation with very small inter-vehicle spacing, operation at all speeds from a complete stop to high-speed cruising, and the execution of longitudinal split and join maneuvers in the presence of communication constraints. The challenges handled in the design of the lateral control system include high-speed operation using a purely 'look down' sensor system and lane changing without transitional lateral position measurements. The paper also describes the design of an on-board supervisor that utilizes inter-vehicle communication and coordinates the operation of the lateral and longitudinal controllers in order to execute entry and exit maneuvers. Extensive experimental results are included in the paper from the NAHSC1 demonstration of automated highways conducted in August 1997 at San Diego, CA. The demonstration included a closely spaced eight-car platoon operating continuously over several weeks, with passenger rides being given to visitors. The maneuvers demonstrated included starting the automated vehicles from complete rest, accelerating to cruising speed, automated lane keeping, allowing any vehicle to exit from the platoon with an automated lane change, allowing new vehicles to join the platoon and bringing the platoon to a complete stop at the end of the highway.",
author = "Rajesh Rajamani and Tan, {Han Shue} and Law, {Boon Kait} and Zhang, {Wei Bin}",
doi = "10.1109/87.852914",
journal = "IEEE Transactions on Control Systems Technology",
T1 - Demonstration of integrated longitudinal and lateral control for the operation of automated vehicles in platoons
AU - Rajamani, Rajesh
AU - Tan, Han Shue
AU - Law, Boon Kait
AU - Zhang, Wei Bin
N2 - This paper presents the design and experimental implementation of an integrated longitudinal and lateral control system for the operation of automated vehicles in platoons. The challenges handled in the design of the longitudinal control system include nonlinear vehicle dynamics, string-stable operation with very small inter-vehicle spacing, operation at all speeds from a complete stop to high-speed cruising, and the execution of longitudinal split and join maneuvers in the presence of communication constraints. The challenges handled in the design of the lateral control system include high-speed operation using a purely 'look down' sensor system and lane changing without transitional lateral position measurements. The paper also describes the design of an on-board supervisor that utilizes inter-vehicle communication and coordinates the operation of the lateral and longitudinal controllers in order to execute entry and exit maneuvers. Extensive experimental results are included in the paper from the NAHSC1 demonstration of automated highways conducted in August 1997 at San Diego, CA. The demonstration included a closely spaced eight-car platoon operating continuously over several weeks, with passenger rides being given to visitors. The maneuvers demonstrated included starting the automated vehicles from complete rest, accelerating to cruising speed, automated lane keeping, allowing any vehicle to exit from the platoon with an automated lane change, allowing new vehicles to join the platoon and bringing the platoon to a complete stop at the end of the highway.
AB - This paper presents the design and experimental implementation of an integrated longitudinal and lateral control system for the operation of automated vehicles in platoons. The challenges handled in the design of the longitudinal control system include nonlinear vehicle dynamics, string-stable operation with very small inter-vehicle spacing, operation at all speeds from a complete stop to high-speed cruising, and the execution of longitudinal split and join maneuvers in the presence of communication constraints. The challenges handled in the design of the lateral control system include high-speed operation using a purely 'look down' sensor system and lane changing without transitional lateral position measurements. The paper also describes the design of an on-board supervisor that utilizes inter-vehicle communication and coordinates the operation of the lateral and longitudinal controllers in order to execute entry and exit maneuvers. Extensive experimental results are included in the paper from the NAHSC1 demonstration of automated highways conducted in August 1997 at San Diego, CA. The demonstration included a closely spaced eight-car platoon operating continuously over several weeks, with passenger rides being given to visitors. The maneuvers demonstrated included starting the automated vehicles from complete rest, accelerating to cruising speed, automated lane keeping, allowing any vehicle to exit from the platoon with an automated lane change, allowing new vehicles to join the platoon and bringing the platoon to a complete stop at the end of the highway.
U2 - 10.1109/87.852914
DO - 10.1109/87.852914
JO - IEEE Transactions on Control Systems Technology
JF - IEEE Transactions on Control Systems Technology


text:
Herald.ie› Entertainment› Film›
Saoirse Ronan speaks up about 'wage gap' at Brooklyn premiere
Kirsty Blake Knox – 23 October 2015 07:50 AM
Saoire Ronan,Ali Hewson and husband Bono pictured tonight at The Irish premiere screening of the film Brooklyn at The Savoy Cinema Dublin, Pic Brian McEvoy No Repro fee for one use
ACTRESS Saoirse Ronan has praised Jennifer Lawrence for writing her "wage gap" essay and for criticising double standards in the film industry.
Saoirse was at the Irish premiere of Brooklyn in Dublin's Savoy cinema last night and spoke of the frustration she feels with regard to the lack of strong female roles.
"I hope the likes of Jennifer Lawrence speaking out forces things to change," she said.
"There are so many roles where women are simply supporting. They are the crutch, the sister, the assistant or the girlfriend, and that's as far as the definition goes. That needs to change."
Saoirse said she was shocked recently when a close friend auditioned for a role in a movie where the female character was described as "the leading man's type".
"It said 'just his type', that was the character's description, that was her definition," she said. "That has to change, and in order to change we need to have more female directors, producers and writers."
Set in the 1950s, Brooklyn tells the story of a young Irish woman who moves to the US. The film has generated plenty of Oscar buzz, and Saoirse hopes it will shift industry attitudes on female roles.
Saoire Ronan,Ali Hewson and husband Bono pictured tonight Saoirse Ronan at The Irish premiere screening of the film Brooklyn at The Savoy Cinema Dublin, Pic Brian McEvoy No Repro fee for one use
"I hope we'll see more films about women and women's interactions. I hope we'll see films where women help each other rather than being catty of bitchy," she said.
The premiere attracted plenty of big names including Bono, Ali Hewson and Colm Toibin as well as Caroline Downey, Eva Birthistle and Jenn Murray.


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text:
package net.sf.mmm.ui.toolkit.impl.swing.model;

import javax.swing.AbstractListModel;

import net.sf.mmm.ui.toolkit.api.event.UIListModelEvent;
import net.sf.mmm.ui.toolkit.api.event.UIListModelListener;
import net.sf.mmm.ui.toolkit.api.model.data.UiListMvcModel;
import net.sf.mmm.util.event.api.ChangeType;

/**
 * This class adapts a {@link net.sf.mmm.ui.toolkit.api.model.data.UiListMvcModel} to a
 * swing {@link javax.swing.ListModel}.
 *
 * @param <E> is the templated type of the elements that can be selected.
 *
 * @author Joerg Hohwiller (hohwille at users.sourceforge.net)
 * @since 1.0.0
 */
public class ListModelAdapter<E> extends AbstractListModel implements UIListModelListener {

  /** uid for serialization */
  private static final long serialVersionUID = -6062581165914420738L;

  /** the original list model */
  protected final UiListMvcModel<E> model;

  /**
   * The constructor.
   *
   * @param listModel is the list model to adapt to swing.
   */
  public ListModelAdapter(UiListMvcModel<E> listModel) {

    super();
    this.model = listModel;
    this.model.addListener(this);
  }

  /**
   * {@inheritDoc}
   */
  public Object getElementAt(int index) {

    return this.model.getElement(index);
  }

  /**
   * {@inheritDoc}
   */
  public int getSize() {

    return this.model.getElementCount();
  }

  /**
   * {@inheritDoc}
   */
  public void listModelChanged(UIListModelEvent event) {

    if (event.getType() == ChangeType.ADD) {
      fireIntervalAdded(this, event.getStartIndex(), event.getEndIndex());
    } else if (event.getType() == ChangeType.REMOVE) {
      fireIntervalRemoved(this, event.getStartIndex(), event.getEndIndex());
      System.out.println(event.getStartIndex() + "-" + event.getEndIndex());
    } else if (event.getType() == ChangeType.UPDATE) {
      fireContentsChanged(this, event.getStartIndex(), event.getEndIndex());
    }
  }

  /**
   * This method gets the list-model that is adapted.
   *
   * @return the actual list-model.
   */
  public UiListMvcModel<E> getModel() {

    return this.model;
  }

  /**
   * This method disposes this model adapter. It is called when the adapter is
   * NOT used anymore and can be eaten by the garbarge-collector. The
   * implementation of this method can tidy-up (e.g. remove registered
   * listeners).
   */
  public void dispose() {

    this.model.removeListener(this);
  }

}


text:
Q: Having trouble rendering 2 images side-by-side the original images being altered I'm trying to create a program that can open two jpegs with the same dimensions, and create a new jpeg that has those two images placed side-by-side with no change to wither image.
The code that creates the new image is this:
public static Picture sumPicturesHorizontally(Picture left, Picture right) {

    BufferedImage newImage = new BufferedImage(
            left.getWidth() + right.getWidth(),
            left.getHeight(),
            left.getImage().getType());

    Graphics2D g = (Graphics2D) newImage.getGraphics();
    g.drawImage(left.getImage(), 0, 0, null);
    g.drawImage(right.getImage(), left.getWidth(), 0, null);

    Picture p = new Picture(newImage);
    return p;
}

(Picture just my a wrapper class that contains a BufferedImage and that image's dimensions)
For the most part, this works.  However, the image that's output isn't just the left and right images side-by-side, it seems that something is modifying the border between the left and right images.
I have a yellow square, and a red square; my source images are 1 constant color.  The sum image, however, has some artifacts around the new border that aren't in the original.  Here's a zoomed-in screenshot of what I'm talking about:

If I zoom to the same level on my source images, I don't see the same artifacts on the edges. I'd like it to be just yellow to the edge, then just red.  There should'nt be different shades of yellow and red.  Is there some config to make it behave this way, or a different method of rendering?
Also, I don't normally work with graphics, is there a name for what this edge blending is called?  Thanks for the help!

A: your code is OK and have no problem
and your image without zoom also OK and have no artifacts
the only problem in the "zoom" level which you do
when you make the Zoom you will need to get additional pixel not in your original image
this pixel is calculated by algorithm , this algorithm used the neighbor pixels around the edge between the two color (the red and the yellow pixel) to calculate the new pixel in the zoomed image

see this image : it make a zoom for specific part , you can see like squares ,but in the original image you can see this is tiny pixel

Note: this is why when you buy a camera you need to ask about it's resolution , because when the resolution increase the image quality increase even if you make a zoom to this level.

I used your code and this is the output :
1-
this is the two image without any zoom and have no artifact
2-
this is the two image after make a zoom in , here i can see the artifact

A: Works fine for me on Windows 7 using JDK7. Post your SSCCE so we can see exactly how you tested this.
This is the SSCCE I used:
import java.awt.*;
import java.awt.image.*;
import javax.swing.*;

public class MergeImageSSCCE extends JPanel
{
    public MergeImageSSCCE()
    {
        setLayout( new FlowLayout() );
        int size = 100;

        Image left = createImage(size, Color.YELLOW);
        Image right = createImage(size, Color.RED);
        Image merged = merge(left, right);

        add( new JLabel( new ImageIcon(left) ) );
        add( new JLabel( new ImageIcon(right) ) );
        add( new JLabel( new ImageIcon(merged) ) );
    }

    public static Image createImage(int size, Color color)
    {
        BufferedImage image = new BufferedImage(size, size, BufferedImage.TYPE_INT_RGB);
        Graphics2D g = image.createGraphics();
        g.setColor( color );
        g.fillRect(0, 0, size, size);
        g.dispose();

        return image;
    }

    public static Image merge(Image left, Image right)
    {
        BufferedImage merged = new BufferedImage(
            left.getWidth(null) + right.getWidth(null), left.getHeight(null), BufferedImage.TYPE_INT_RGB);
        Graphics2D g = (Graphics2D) merged.getGraphics();
        g.drawImage(left, 0, 0, null);
        g.drawImage(right, left.getWidth(null), 0, null);

        return merged;
    }

    private static void createAndShowUI()
    {
        JFrame frame = new JFrame("MergeImageSSCCE");
        frame.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
        frame.add( new MergeImageSSCCE() );
        frame.setSize(250, 250);
        frame.setLocationByPlatform( true );
        frame.setVisible( true );
    }

    public static void main(String[] args)
    {
        EventQueue.invokeLater(new Runnable()
        {
            public void run()
            {
                createAndShowUI();
            }
        });
    }
}

A: I found this in the internet:
System.Drawing.Image thumbnail = new Bitmap(width, height, PixelFormat.Format32bppArgb);
Graphics graphic = Graphics.FromImage(thumbnail);

graphic.CompositingQuality = CompositingQuality.HighQuality;
graphic.InterpolationMode = InterpolationMode.HighQualityBicubic;
graphic.PixelOffsetMode = PixelOffsetMode.Half;

redpajama_set_name: RedPajamaStackExchange


text:
Maciej Stachura (ur. 10 stycznia 1968) – polski rugbysta, trener rugby.

Jest wychowankiem Ogniwa Sopot, był zawodnikiem formacji ataku. Grał w Ogniwie od 1988 roku, następnie przeszedł do Arki Gdynia. W latach 2003–2017 był trenerem Arki. W tym czasie zespół czterokrotnie zdobył mistrzostwo Polski (2004, 2005, 2011, 2015), trzy razy wicemistrzostwo (2007, 2009, 2013), a czterokrotnie zakończył sezon na trzecim miejscu (2008, 2010, 2012, 2014).

Przypisy

Reprezentanci Polski w rugby union
Rugbyści Arki Gdynia
Urodzeni w 1968

redpajama_set_name: RedPajamaWikipedia