document: an @xmath0-hypergeometric system @xmath1 is a @xmath4-module determined by an integral matrix @xmath0 and a complex parameter vector @xmath5 . 
   these systems are also known as _ gkz - systems _ , as they were introduced in the late 1980 s by gelfand , graev , kapranov , and zelevinsky @xcite . 
   their solutions occur naturally in mathematics and physics , including the study of roots of polynomials , picard 
   fuchs equations for the variation of hodge structure of calabi - yau toric hypersurfaces , and generating functions for intersection numbers on moduli spaces of curves , see @xcite . 
   the ( holonomic ) rank of @xmath1 coincides with the dimension of its solution space at a nonsingular point . in this article 
   , we provide a combinatorial formula for the rank of @xmath1 in terms of certain lattice translates determined by @xmath0 and @xmath6 . for a fixed matrix @xmath0 , 
   this computation yields a geometric stratification of the parameter space @xmath7 that refines its stratification by the rank of @xmath1 . 
   let @xmath8 $ ] be an integer @xmath9-matrix with integral column span @xmath10 . 
   assume further that @xmath0 is _ pointed _ , meaning that the origin is the only linear subspace of the cone @xmath11 . 
   a subset @xmath12 of the column set of @xmath0 is called a _ face _ of @xmath0 , denoted @xmath13 , if @xmath14 is a face of the cone @xmath15 . 
   let @xmath16 be variables and @xmath17 their associated partial differentiation operators . 
   in the polynomial ring @xmath18 $ ] , let @xmath19 denote the toric ideal associated to @xmath0 , and let @xmath20 be its quotient ring . 
   note that @xmath21 is isomorphic to the semigroup ring of @xmath0 , which is @xmath22 : = \bigoplus_{a\in { { \ensuremath{\mathbb{n}}}}a } { { \ensuremath{\mathbb{c}}}}\cdot t^a \end{aligned}\ ] ] with multiplication given by semigroup addition of exponents . 
   the weyl algebra @xmath23=\delta_{ij } , [ x_i , x_j]=0=[\del_i,\del_j]\>\ ] ] is the ring of @xmath24-linear differential operators on @xmath25 $ ] . 
   [ def - gkz ] the _ @xmath0-hypergeometric system _ with parameter @xmath26 is the left @xmath4-module @xmath27 where @xmath28 are _ euler operators _ associated to @xmath0 . 
   the _ rank _ of a left @xmath4-module @xmath29 is @xmath30}m.\ ] ] the rank of a holonomic @xmath4-module is finite and equal to the dimension of its solution space of germs of holomorphic functions at a generic nonsingular point @xcite . 
   in @xcite , gelfand , kapranov , and zelevinsky showed that when @xmath21 is cohen  macaulay and standard @xmath31-graded , the @xmath0-hypergeometric system @xmath1 is holonomic of rank @xmath32 for all parameters @xmath6 , where @xmath32 is @xmath33 times the euclidean volume of the convex hull of @xmath0 and the origin . 
   adolphson established further that @xmath1 is holonomic for all choices of @xmath0 and @xmath6 and that the holonomic rank of @xmath1 is generically given by @xmath32 @xcite . 
   however , an example found by sturmfels and takayama showed that equality need not hold in general @xcite ( see also @xcite ) . 
   at the same time , cattani , dandrea , and dickenstein produced an infinite family of such examples through a complete investigation of the rank of @xmath1 when @xmath2 defines a projective monomial curve @xcite . 
   the relationship between @xmath32 and the rank of @xmath1 was made precise by matusevich , miller , and walther , who used euler 
   koszul homology to study the holonomic 
   _ generalized @xmath0-hypergeometric system _ @xmath34 associated to a toric module @xmath29 ( see definition  [ def : toric ] ) . 
   the euler  koszul homology @xmath35 of @xmath29 with respect to @xmath6 is the homology of a twisted koszul complex on @xmath36 given by the sequence @xmath37 . 
   this includes the @xmath0-hypergeometric system @xmath38 as the special case that @xmath39 . as in this special case , and for the purposes of this article , suppose that the generic rank of @xmath40 is @xmath32 . 
   the matrix @xmath0 induces a natural @xmath41-grading on @xmath42 ; the _ quasidegree set _ of a finitely generated @xmath41-graded @xmath42-module @xmath43 is defined to be the zariski closure in @xmath7 of the set of vectors @xmath44 for which the graded piece @xmath45 is nonzero . in @xcite , an explicit description of the _ exceptional arrangement _ 
   @xmath46 associated to @xmath29 is given in terms of the quasidegrees of certain @xmath47 modules involving @xmath29 ( see ) . 
   this description shows that @xmath48 is a subspace arrangement in @xmath7 given by translates of linear subspaces that are generated by the faces of the cone @xmath15 , and that @xmath48 is empty exactly when @xmath49 is a maximal cohen 
   macaulay @xmath21-module . 
   it is also shown in @xcite that the rank of @xmath35 is upper semi - continuous as a function of @xmath6 . 
   thus the exceptional arrangement @xmath48 is the nested union over @xmath50 of the zariski closed sets @xmath51 in particular , the rank of @xmath34 induces a stratification of @xmath48 , which we call its _ 
   rank stratification_.      the present article is a study of the rank stratification of @xmath48 when @xmath52 $ ] is @xmath41-graded such that the degree set @xmath53 of @xmath29 is a nontrivial @xmath54monoid . in particular , @xmath29 is weakly toric ( see definition  [ def - wtoric ] ) . 
   if @xmath55 , then @xmath29 is the semigroup ring @xmath21 from and @xmath56 is the @xmath0-hypergeometric system at @xmath6 . 
   the module @xmath29 could also be a localization of @xmath21 along a subset of faces of @xmath0 . as @xmath29 will be fixed throughout this article 
   , we will often not indicate dependence on @xmath29 in the notation . 
   examination of the long exact sequence in euler  koszul homology induced by the short exact sequence of weakly toric modules @xmath57 \rightarrow q \rightarrow 0\end{aligned}\ ] ] reveals that the _ rank jump _ of @xmath29 at @xmath6 , @xmath58 we define the _ ranking arrangement _ @xmath59 of @xmath29 to be the quasidegrees of @xmath60 . 
   vanishing properties of euler 
   koszul homology imply that the exceptional arrangement @xmath48 is contained in the ranking arrangement @xmath59 . 
   we show in theorem  [ thm - str comparison ] that @xmath59 is the union of @xmath48 and an explicit collection of hyperplanes .    for a fixed @xmath61 , we then proceed to compute @xmath62 . in section  [ sec : r - t mods ] , we combinatorially construct a _ finitely generated _ 
   @xmath41-graded _ ranking toric module _ 
   @xmath63 with @xmath64 . since @xmath62 is determined by the euler 
   koszul homology of @xmath60 by , we see that @xmath63 contains the information essential to computing the rank jump @xmath62 . to outline the construction of the module @xmath63 , 
   let @xmath65 be the polyhedral complex of faces of @xmath0 determined by the components of the ranking arrangement @xmath59 that contain @xmath6 . 
   we call the collection of integral points @xmath66 the _ ranking lattices _ @xmath67 of @xmath29 at @xmath6 . 
   this set is a union of translates of lattices generated by faces of @xmath0 , where the vectors in these lattice translates of @xmath68 in @xmath67 are precisely the degrees of @xmath60 which cause @xmath69 to lie in the ranking arrangement . 
   since it contains full lattice translates , @xmath67 can not be the degree set of a finitely generated @xmath21-module . 
   thus , to complete the construction of the degree set @xmath70 of @xmath63 , we intersect @xmath67 with an appropriate half space ( see definition  [ def - cp ] ) . to give a flavor of our approach for @xmath71 , this is equivalent to intersecting @xmath72 with @xmath73 $ ] . by setting up the proper module structure , @xmath74 gives the @xmath41-graded degree set of the desired toric module @xmath63 with @xmath75 . 
   after translating the computation of the rank jump @xmath62 to @xmath63 , we obtain a generalization of the formula given by okuyama in the case @xmath76 @xcite . 
   [ thm - compute jump ] the rank jump @xmath62 of @xmath29 at @xmath6 can be computed from the combinatorics of the ranking lattices @xmath67 of @xmath29 at @xmath6 .    in particular , the rank of the hypergeometric system is the same at parameters which share the same ranking lattices . 
   the proof of theorem  [ thm - compute jump ] can be found in section  [ subsec : rank jumps ] as a special case of our main result , theorem  [ thm - main ] . 
   let @xmath77 and @xmath78 be subspace arrangements in @xmath7 . 
   we say that a stratification @xmath79 of @xmath77 _ respects _ @xmath78 if for each irreducible component @xmath80 of @xmath78 and each stratum @xmath81 , either @xmath82 or @xmath83 . 
   a _ ranking slab _ of @xmath29 is a stratum in the coarsest stratification of @xmath48 that respects the arrangements @xmath59 and the negatives of the quasidegrees of each of the @xmath47 modules that determine @xmath48 ( see definition  [ def - ranking slab ] ) . 
   proposition  [ prop - zc e ] states that the parameters @xmath84 belong to the same ranking slab of @xmath29 exactly when their ranking lattices coincide , that is , @xmath85 . combining this with theorem  [ thm - compute jump ] 
   , we see that the rank of @xmath40 is constant on each ranking slab . 
   [ cor - strat vs equiv ] the function @xmath86 is constant on each ranking slab . 
   in particular , the stratification of the exceptional arrangement @xmath48 by ranking slabs refines its rank stratification . 
   hence , like @xmath48 , each set @xmath87 is a union of translated linear subspaces of @xmath7 which are generated by faces of @xmath15 . in order for the ranking slab stratification of @xmath48 to refine its rank stratification 
   , it must respect each of the arrangements appearing in its definition ; this can be seen from examples  [ example - hidden ] ,  [ example - nonconstant slab ] ,  and  [ plane - line-4diml ] . in particular , as @xmath88 , the exceptional arrangement @xmath48 does not generally contain enough information to determine its rank stratification . 
   when @xmath39 , the ranking lattices @xmath67 are directly related to the combinatorial sets @xmath89 defined by saito , which determine the isomorphism classes of @xmath1 . in @xcite , various @xmath90-functions arising from an analysis of the symmetry algebra of @xmath0-hypergeometric systems are used to link these isomorphism classes to the sets @xmath89 . 
   we conclude this paper with a shorter proof , replacing the use of @xmath90-functions with euler  koszul homology . 
   the following is a brief outline of this article . in section  [ sec : defns ] , we summarize definitions and results on weakly toric modules and euler  koszul homology , following @xcite . 
   section  [ sec : ekstr ] is a study of the structure of the euler 
   koszul complex of maximal cohen 
   macaulay toric face modules . 
   the relationship between the exceptional and ranking arrangements of @xmath29 is made precise in section  [ sec : exceptional arrangement ] . in section  [ sec : r - t mods ] , we define the class of ranking toric modules , which play a pivotal role in calculating the rank jump @xmath62 . section  [ sec : comb of rank ] contains our main theorem , theorem  [ thm - main ] , which results in the computation of @xmath62 for a fixed parameter @xmath6 . 
   we close with a discussion on the isomorphism classes of @xmath0-hypergeometric systems in section  [ sec : isom classes ] . 
   i am grateful to my advisor uli walther for many inspiring conversations and thoughtful suggestions throughout the duration of this work . 
   i would also like to thank laura felicia matusevich for asking the question  is rank constant on a slab ? " as well as helpful remarks on this text and conversations that led directly to the proof of theorem  [ thm - isom classes ] . 
   finally , my thanks to ezra miller for many valuable comments for improving this paper , including the addition of definition  [ def - cellular definition ] , and to the referee for detailed notes that clarified this article . 
   in this section , we summarize definitions found in the literature and set notation . 
   most important are the definitions of a weakly toric module @xcite and euler  koszul homology @xcite . 
   let @xmath91 denote the columns of @xmath0 . 
   for a face @xmath13 , let @xmath92 denote the complement of a face @xmath12 in the column set of @xmath0 . 
   if @xmath12 is any subset of the columns of @xmath0 , the _ codimension _ of @xmath12 is @xmath93 , the codimension of the @xmath94-vector space generated by @xmath12 . 
   the _ dimension _ of @xmath12 is @xmath95 . 
   a face @xmath12 of @xmath0 is a _ facet _ of @xmath0 if @xmath96 . 
   recall that the _ primitive integral support function _ of a facet @xmath13 is the unique linear functional @xmath97 such that    @xmath98    @xmath99 for all @xmath100 , and    @xmath101 exactly when @xmath102 . 
   the _ volume _ of a face @xmath12 , denoted @xmath103 , is the integer @xmath104 times the euclidean volume in @xmath105 of the convex hull of @xmath12 and the origin . 
   [ def : semigroup ring d ] let @xmath106 be the semigroup generated by the face @xmath12 and , as in , @xmath107\ ] ] is the corresponding semigroup ring , called a _ face ring _ of @xmath0 . 
   let @xmath108 and @xmath109 . 
   define @xmath110\ ] ] to be the polynomial ring in @xmath111 and @xmath112 = \delta_{ij},[x_i , x_j ] = 0 = [ \del_i,\del_j]\>\ ] ] to be the _ weyl algebra _ associated to @xmath12 . 
   note that @xmath113    [ def : monoid ] let @xmath114 be variables . for a face @xmath13 , we say that a subset @xmath115 is an _ @xmath116module _ if @xmath117 . 
   further , we call an @xmath116module @xmath118 an _ @xmath116monoid _ if it is closed under addition , that is , for all @xmath119 , @xmath120 . given an @xmath116module @xmath118 , 
   define the @xmath121-module @xmath122 as a @xmath24-vector space with @xmath121-action given by @xmath123 . 
   further , @xmath124 is equipped with a multiplicative structure given by @xmath125 for @xmath119 and extended @xmath24-linearly . by definition 
   , @xmath116 is an @xmath116monoid and @xmath126 as rings . 
   define a @xmath41-grading on @xmath127 by setting @xmath128 then @xmath124 is naturally a @xmath41-graded @xmath121-module by setting @xmath129 . 
   the _ saturation _ of @xmath12 in @xmath68 is the semigroup @xmath130 . 
   the _ saturation _ , or _ normalization _ , of @xmath121 is the semigroup ring of the saturation of @xmath12 in @xmath68 , which is given by @xmath131 as a @xmath41-graded @xmath121-module . 
   note that @xmath132 is a cohen 
   macaulay @xmath121-module @xcite . 
   if @xmath43 is a @xmath41-graded @xmath42-module and @xmath133 , the _ degree set _ of @xmath43 , denoted @xmath134 , is the set of all @xmath133 such that @xmath135 . 
   let @xmath136 denote the @xmath41-graded module with @xmath137-graded piece @xmath138 . 
   we now recall the definitions of toric and weakly toric modules and their quasidegree sets , which can be found in ( * ? ? ? 
   * definition  4.5 ) and ( * ? ? ? 
   * section  5 ) , respectively . 
   [ def : toric ] a @xmath41-graded @xmath42-module is _ toric _ if it has a filtration @xmath139 such that for each @xmath140 , @xmath141 is a @xmath41-graded translate of @xmath142 for some face @xmath143 . 
   notice that toric modules are necessarily finitely generated @xmath42-modules . 
   [ def - qdeg ] if @xmath43 is a finitely generated @xmath41-graded @xmath42-module , a vector @xmath144 is a _ quasidegree _ of @xmath43 , written @xmath145 , if @xmath146 lies in the zariski closure of @xmath134 under the natural embedding @xmath147 . 
   notice that if @xmath43 is toric , then @xmath148 is a finite subspace arrangement in @xmath7 , consisting of translated subspaces generated by faces of @xmath0 , see ( * ? ? ? 
   * lemma  2.5 ) .    a partially ordered set @xmath149 is _ filtered _ if for each @xmath150 there exists @xmath151 with @xmath152 and @xmath153 . 
   [ def - wtoric ] we say that a @xmath41-graded @xmath42-module @xmath29 is _ weakly toric _ if there is a filtered partially ordered set @xmath149 and a @xmath41-graded direct limit @xmath154 where @xmath155 is a toric @xmath42-module for each @xmath151 . 
   we then define the _ quasidegree set _ of @xmath29 to be @xmath156 where each @xmath157 is defined by definition  [ def - qdeg ] . 
   [ ex - m wt ] if @xmath158 is an @xmath54module , then @xmath159 is weakly toric because it is a direct limit over @xmath160 of @xmath161 under the natural @xmath0-homogeneous inclusion @xmath162\cong { { \ensuremath{\mathbb{c}}}}[{{\ensuremath{\mathbb{z}}}}^d]$ ] .    consider the matrix @xmath163 $ ] with face @xmath164 $ ] . 
   the module @xmath165 $ ] is weakly toric with quasidegree set @xmath166\right ) = { { \ensuremath{\mathbb{c}}}}f\ ] ] because it is a filtered direct limit over @xmath167 of @xmath168 . 
   similarly , the module @xmath169 $ ] is weakly toric with @xmath170\right ) = { { \ensuremath{\mathbb{c}}}}^2 $ ] . 
   the quotient @xmath169/s_a$ ] is also weakly toric . 
   its quasidegree set consists of the point @xmath171 $ ] and the union of lines in @xmath172 , where @xmath173 are the coordinates of @xmath174 . 
   we now recall the definition of the euler 
   koszul complex of a weakly toric module @xmath29 with respect to a parameter @xmath26 . for @xmath175 , each _ euler operator _ 
   @xmath176 determines a @xmath41-graded @xmath4-linear endomorphism of @xmath36 , defined on a homogeneous @xmath177 by @xmath178 and extended @xmath24-linearly . 
   this sequence @xmath179 of commuting endomorphisms determines a koszul complex @xmath180 on the left @xmath4-module @xmath36 , called the _ 
   euler  koszul complex _ of @xmath29 with parameter @xmath6 . 
   @xmath181 euler 
   koszul homology module _ of @xmath29 is @xmath182 . 
   our object of study will be the _ 
   generalized @xmath0-hypergeometric system _ @xmath34 associated to @xmath29 . 
   the euler 
   koszul complex defines an exact functor from the category of weakly toric modules with degree - preserving morphisms to the category of bounded complexes of @xmath41-graded left @xmath4-modules with degree - preserving morphisms , so short exact sequences of weakly toric modules yield long exact sequences of euler  koszul homology . 
   notice also that euler 
   koszul homology behaves well under @xmath41-graded translations : for @xmath183 , @xmath184 we close this section by recording two important vanishing results for euler  koszul homology . 
   [ prop - hh qis 0 ] for a weakly toric module @xmath29 , the following are equivalent :    1 . 
   @xmath185 for all @xmath186 , 2 . 
   @xmath187 , 3 . 
   @xmath188    see ( * ? ? ? 
   * theorem  5.4 ) . 
   [ thm - cm hh rel ] let @xmath29 be a weakly toric module . 
   then @xmath185 for all @xmath189 and for all @xmath26 if and only if @xmath29 is a maximal cohen 
   macaulay @xmath21-module . 
   see ( * ? ? ? 
   * theorem  6.6 ) for the toric case . 
   the extension to the weakly toric case can be found in @xcite . 
   theorem  [ thm - cm hh rel ] provides a criterion for higher vanishing of euler 
   koszul homology via maximal cohen  macaulay @xmath21-modules . in this section , 
   we provide a description of the euler  koszul homology modules of maximal cohen  macaulay @xmath121-modules for a face @xmath13 and 
   use it to understand the images of maps between such modules .    throughout this section , 
   @xmath43 is a toric @xmath121-module for a face @xmath13 . 
   recall the definitions of toric , @xmath121 , @xmath190 , and @xmath191 from definition  [ def : semigroup ring d ] , and let @xmath192    [ not - hhf ] let @xmath193 be the lexicographically first subset of @xmath194 of cardinality @xmath195 such that @xmath196 is a set of linearly independent euler operators on @xmath197 . 
   the existence of @xmath193 follows from the fact that the matrix @xmath0 has full rank . 
   we use @xmath198 to denote the euler 
   koszul complex on @xmath199 given by the operators @xmath200 , and set @xmath201    using the standard basis of @xmath10 , let @xmath202 and let @xmath203 denote a complex with trivial differentials . 
   we show now that when @xmath204 , @xmath205 is quasi - isomorphic to a complex involving @xmath198 and @xmath203 . 
   [ prop - str cm ek complex ] let @xmath13 and @xmath43 be a toric @xmath121-module . 
   if @xmath206 , then there is a quasi - isomorphism of complexes @xmath207 \otimes_{{\ensuremath{\mathbb{c}}}}\kk_\bullet^f(n,\beta ) \otimes_{{\ensuremath{\mathbb{z}}}}\left(\textstyle\bigwedge^\bullet{{\ensuremath{\mathbb{z}}}}f^\bot\right).\end{aligned}\ ] ] in particular , if @xmath43 is maximal cohen  macaulay as an @xmath121-module , there is a decomposition @xmath208 \otimes_{{\ensuremath{\mathbb{c}}}}\hh_0^f(n,\beta ) \otimes_{{\ensuremath{\mathbb{z}}}}\left(\textstyle\bigwedge^\bullet{{\ensuremath{\mathbb{z}}}}f^\bot\right).\end{aligned}\ ] ]    under the hypotheses of proposition  [ prop - str cm ek complex ] , @xmath209 \otimes_{{\ensuremath{\mathbb{c}}}}\hh_0^f(n,\beta^f)^{\binom{\codim(f)}{i}},\ ] ] for @xmath189 , as shown in @xcite . in particular , @xmath210 we show in proposition  [ prop - str / rk image ] that surjections of maximal cohen 
   macaulay toric modules for nested faces yield induced maps on euler  koszul homology that respect the decompositions of . 
   the additional information stored in @xmath203 of shows how images of collections of such surjections overlap , which will be vital to our calculation of @xmath62 in section  [ sec : comb of rank ] . 
   [ pf - gf ] fix a matrix @xmath211 such that the entries of each row of @xmath212 not corresponding to @xmath193 are zero and the rows of @xmath0 that do correspond to @xmath193 are identical in @xmath0 and @xmath213 . setting @xmath214 , @xmath21 and @xmath215 
   are isomorphic rings , and the matrix @xmath216 gives a bijection of their degree sets , sending @xmath54 to @xmath217 . 
   this identification makes @xmath43 a @xmath218-graded @xmath219-module , where @xmath220 , and there is a quasi - isomorphism of complexes @xmath221    let @xmath222 and @xmath223 , and recall that @xmath224 . by the definition of @xmath216 , @xmath225 for @xmath226 because @xmath227 . 
   let @xmath228 and @xmath229 denote the weyl algebra and the polynomial ring @xmath230 $ ] with an @xmath231-grading . since @xmath43 is an @xmath219-module , @xmath232 , and so there is an isomorphism @xmath233 \otimes_{{\ensuremath{\mathbb{c}}}}d_{f ' } \otimes_{r_{f ' } } n$ ] . 
   hence the action of each element in @xmath234 on @xmath235 is 0 .    if @xmath236 denotes the standard basis of @xmath237 , then the set @xmath238 generates @xmath239 by choice of @xmath216 . applying the isomorphism @xmath240 in the reverse direction , we obtain 
   finally , if @xmath43 is maximal cohen  macaulay as an @xmath121-module , @xmath241 for all @xmath189 by theorem  [ thm - cm hh rel ] . 
   [ remark - diffl in cm ek complex ] let @xmath242 and @xmath243 respectively denote the differentials of the euler  koszul complexes @xmath244 and @xmath198 . under the hypotheses of proposition  [ prop - str cm ek complex ] , if @xmath245 and @xmath246 \otimes_{{\ensuremath{\mathbb{c}}}}\kk^f_i(n,\beta ) \otimes_{{\ensuremath{\mathbb{z}}}}\left(\textstyle\bigwedge^j{{\ensuremath{\mathbb{z}}}}f^\bot\right)$ ] 
   , then @xmath247 is an element of @xmath248 \otimes_{{\ensuremath{\mathbb{c}}}}\kk^f_{i-1}(n,\beta ) \otimes_{{\ensuremath{\mathbb{c}}}}\left(\textstyle\bigwedge^j{{\ensuremath{\mathbb{z}}}}f^\bot\right ) \subseteq \kk^{a'}_{q-1}(n,\beta).$ ]    [ example - ek cm comp ] consider the matrix @xmath249.\ ] ] set @xmath250 \in{{\ensuremath{\mathbb{c}}}}^3 $ ] , and let @xmath251 denote the standard basis vectors in @xmath252 . 
   notice that every face ring of @xmath0 is cohen  macaulay because the semigroup generated by each face of @xmath0 is saturated . for the face @xmath253 
   , we choose @xmath254 to be the identity matrix . 
   the proof of proposition  [ prop - str cm ek complex ] shows that there is an isomorphism of complexes @xmath255\cdot e_i \ar[r]^{0\cdot } & { { \ensuremath{\mathbb{c}}}}[x]}\right),$ ] so the euler 
   koszul homology of @xmath256 at @xmath6 is @xmath257 \cdot e_i\right).\ ] ] for the face @xmath258 $ ] of @xmath0 , @xmath259 is again already in the desired form , so take @xmath216 to be the identity matrix and write @xmath25\<\del_1,\del_2\>$ ] in place of @xmath260 \otimes_{{\ensuremath{\mathbb{c}}}}d_f \otimes_{r_f } s_f$ ] 
   . then proposition  [ prop - str cm ek complex ] implies that @xmath261\<\del_1,\del_2\ > & \text{if $ q=0$,}\\          { { \ensuremath{\mathbb{c}}}}[x]\<\del_1,\del_2\>\cdot e_3 & \text{if $ q=1$,}\\          0 & \text{otherwise . }      \end{cases } \intertext{for $ g_1=[a_1]$ and $ g_{g_1}$ as the identity matrix , } \hh_q(s_{g_1},\beta ) & =      \begin{cases }          { { \ensuremath{\mathbb{c}}}}[x]\<\del_1\ > & \text{if $ q=0$,}\\          { { \ensuremath{\mathbb{c}}}}[x]\<\del_1\>\cdot e_2 \oplus { { \ensuremath{\mathbb{c}}}}[x]\<\del_1\>\cdot e_3 & \text{if $ q=1$,}\\          { { \ensuremath{\mathbb{c}}}}[x]\<\del_1\>\cdot e_2\wedge e_3 & \text{if $ q=2$,}\\          0 & \text{otherwise . }      \end{cases } \intertext{for the face $ g_2 = [ a_2]$ , setting   $ g_{g_2 } = \left[\begin{smallmatrix }          1&\phantom{-}0&0\\          1&-1&0\\          0&\phantom{-}0&1 \end{smallmatrix}\right]$ yields the decomposition } \hh_q(s_{g_2},\beta ) & =      \begin{cases }          { { \ensuremath{\mathbb{c}}}}[x]\<\del_2\ > & \text{if $ q=0$,}\\          { { \ensuremath{\mathbb{c}}}}[x]\<\del_2\>\cdot ( e_1-e_2 ) \oplus { { \ensuremath{\mathbb{c}}}}[x]\<\del_1\>\cdot e_3 & \text{if $ q=1$,}\\          { { \ensuremath{\mathbb{c}}}}[x]\<\del_2\>\cdot ( e_1-e_2)\wedge e_3 & \text{if $ q=2$,}\\          0 & \text{otherwise . 
   }      \end{cases}\end{aligned}\ ] ]    [ lemma - ek morph ] let @xmath262 be faces of @xmath0 , @xmath43 be a toric @xmath121-module , and @xmath263 be a toric @xmath264-module . 
   regard @xmath43 and @xmath263 as toric @xmath21-modules via the natural maps @xmath265 . 
   let @xmath266 be a morphism of @xmath21-modules . 
   then there is a commutative diagram @xmath267^{\kk_\bullet(\pi,\beta ) } \ar[d ] & \kk_\bullet(l,\beta ) \ar[d ] \\ { { \ensuremath{\mathbb{c}}}}[x_{f^c } ] \otimes_{{\ensuremath{\mathbb{c}}}}\kk^{f}_\bullet(n,\beta ) \otimes_{{\ensuremath{\mathbb{z}}}}\left ( \textstyle\bigwedge^\bullet { { \ensuremath{\mathbb{z}}}}f^\bot \right ) \ar[r ] & { { \ensuremath{\mathbb{c}}}}[x_{g^c } ] \otimes_{{\ensuremath{\mathbb{c}}}}\kk^{g}_\bullet(l,\beta ) \otimes_{{\ensuremath{\mathbb{z}}}}\left ( \textstyle\bigwedge^\bullet { { \ensuremath{\mathbb{z}}}}g^\bot \right ) } \end{aligned}\ ] ] with vertical maps as in .    by choice of @xmath193 and @xmath268 in notation 
   [ not - hhf ] and the corresponding @xmath216 and @xmath269 , the diagram @xmath270^{\kk_\bullet(\pi,\beta ) } \ar[d ] & & \kk_\bullet(l,\beta ) \ar[d ] \\ 
   \kk^{g_f a}_\bullet(n,\beta ) \ar[r ] &   \kk^{g_f a}_\bullet(l,\beta ) \ar[r ] & \kk^{g_g a}_\bullet(l,\beta ) } \ ] ] commutes . 
   hence the result follows from the proof of proposition  [ prop - str cm ek complex ] . 
   [ prop - str / rk image ] let @xmath262 be faces of @xmath0 , @xmath43 be a maximal cohen  macaulay toric @xmath121-module , and @xmath263 be a maximal cohen 
   macaulay toric @xmath264-module . 
   regard @xmath43 and @xmath263 as toric @xmath21-modules via the natural maps @xmath265 . 
   let @xmath271 be a surjection of @xmath21-modules . 
   if @xmath272 , then @xmath273      \otimes_{{\ensuremath{\mathbb{c}}}}\hh_0^g(l,\beta ) \otimes_{{\ensuremath{\mathbb{c}}}}\left(\textstyle\bigwedge^\bullet{{\ensuremath{\mathbb{z}}}}f^\bot\right)\ ] ] as a submodule of @xmath274 \otimes_{{\ensuremath{\mathbb{c}}}}\hh_0^g(l,\beta ) \otimes_{{\ensuremath{\mathbb{c}}}}\left(\textstyle\bigwedge^\bullet{{\ensuremath{\mathbb{z}}}}g^\bot\right)$ ] . 
   ( continuation of example  [ example - ek cm comp ] ) [ example - ek cm comp i m ] the surjection of face rings given by @xmath275 induces the following image in euler 
   koszul homology : @xmath276\<\del_1\ > & \text{if $ q=0$,}\\      { { \ensuremath{\mathbb{c}}}}[x]\<\del_1\>\cdot e_3 & \text{if $ q=1$,}\\      0 & \text{otherwise . }      \end{cases}\ ] ]    with @xmath224 , the image of @xmath277 is isomorphic to the image of @xmath278 . by proposition  [ prop - str cm ek complex ] , 
   there are decompositions @xmath279 \otimes_{{\ensuremath{\mathbb{c}}}}\hh^f_0(n,\beta ) \otimes_{{\ensuremath{\mathbb{z}}}}\left(\textstyle\bigwedge^\bullet { { \ensuremath{\mathbb{z}}}}f^\bot \right ) \intertext{and } \hh^{a'}_\bullet(l,\beta ) & =      { { \ensuremath{\mathbb{c}}}}[x_{f^c } ] \otimes_{{\ensuremath{\mathbb{c}}}}\hh^f_\bullet(l,\beta ) \otimes_{{\ensuremath{\mathbb{z}}}}\left(\textstyle\bigwedge^\bullet { { \ensuremath{\mathbb{z}}}}f^\bot \right),\end{aligned}\ ] ] so it is enough to find the image of @xmath280 as a submodule of @xmath281 . 
   the result now follows because the sequence @xmath282^{\hh_0^f(\pi,\beta ) } & &      \hh_0^f(l,\beta ) \ar[r ] & 0}\ ] ] is exact , @xmath283 \otimes_{{\ensuremath{\mathbb{c}}}}\hh^g_0(l,\beta)$ ] , and @xmath284 for @xmath189 . 
   ( continuation of example  [ example - ek cm comp i m ] ) [ example - ek cm comp i m intersect ] let @xmath285 for @xmath286 . 
   then @xmath287 & \text{if $ q=0$,}\\          { { \ensuremath{\mathbb{c}}}}[x]\cdot e_2 \oplus { { \ensuremath{\mathbb{c}}}}[x]\cdot e_3 & \text{if $ q=1$,}\\          { { \ensuremath{\mathbb{c}}}}[x]\cdot e_2\wedge e_3 & \text{if $ q=2$,}\\          0 & \text{otherwise , }      \end{cases } \intertext{and } \image 
   \hh_q(\pi_2,\beta ) & =      \begin{cases }          { { \ensuremath{\mathbb{c}}}}[x ] & \text{if $ q=0$,}\\          { { \ensuremath{\mathbb{c}}}}[x]\cdot ( e_2-e_1 ) \oplus { { \ensuremath{\mathbb{c}}}}[x]\cdot e_3 & \text{if $ q=1$,}\\          { { \ensuremath{\mathbb{c}}}}[x]\cdot ( e_2-e_1 ) \wedge e_3 & \text{if $ q=2$,}\\          0 & \text{otherwise . }      \end{cases}\end{aligned}\ ] ] the intersection of the images of euler 
   koszul homology at @xmath6 applied to @xmath288 and @xmath289 is @xmath290\cap[\image \hh_q(\pi_2,\beta ) ] =      \begin{cases }      { { \ensuremath{\mathbb{c}}}}[x ] & \text{if $ q=0$,}\\      { { \ensuremath{\mathbb{c}}}}[x]\cdot e_3 & \text{if $ q=1$,}\\      0 & \text{otherwise }      \end{cases}\ ] ] because @xmath291 . 
   we close this section with an observation that is vital to our rank jumps computations . for faces 
   @xmath292 , set @xmath293 . 
   let @xmath294 be maximal cohen  macaulay toric @xmath142-modules , @xmath263 be a maximal cohen 
   macaulay toric @xmath264-module , and @xmath295 be @xmath21-module surjections . 
   suppose that @xmath272 . 
   using the equality @xmath296^\bot$ ] , proposition  [ prop - str / rk image ] implies that @xmath297 \otimes_{{\ensuremath{\mathbb{c}}}}\hh_0^g(l,\beta ) \otimes_{{\ensuremath{\mathbb{c}}}}\left(\textstyle\bigwedge^i { { \ensuremath{\mathbb{z}}}}[f\cup g]^\bot\right ) , \end{array}\ ] ] which has rank @xmath298 
   let @xmath299 be a nonempty @xmath54monoid ( see definition  [ def : monoid ] ) , so that the nontrivial module @xmath300 $ ] is weakly toric ( see example  [ ex - m wt ] ) and @xmath301 since @xmath302 is an @xmath54-monoid , the generic rank of @xmath40 is @xmath32 . 
   rank jump _ of @xmath29 at @xmath6 is the nonnegative integer @xmath303 of parameters with nonzero rank jump . 
   by @xcite and @xcite , the exceptional arrangement can be described in terms of certain @xmath47 modules involving @xmath29 , namely @xmath304 where @xmath305 . 
   it follows that @xmath48 is a union of translates of linear subspaces spanned by the faces of @xmath0 , see ( * ? ? ? 
   * corollary  9.3 ) . 
   we begin our study of @xmath62 with the short exact sequence @xmath306 \rightarrow q \rightarrow 0.\end{aligned}\ ] ] while @xmath60 is not a noetherian @xmath21-module , it is a filtered limit of noetherian @xmath41-graded @xmath21-modules and is therefore weakly toric ( see definition  [ def - wtoric ] ) . thus the _ ranking arrangement _ of @xmath29 @xmath307 is an infinite union of translates of linear subspaces of @xmath7 spanned by proper faces of @xmath0 . since @xmath308 $ ] is a maximal cohen 
   macaulay @xmath21-module , theorem  [ thm - cm hh rel ] implies that @xmath309,\beta)=0 $ ] for all @xmath189 . 
   moreover , by ( * ? ? ? 
   * theorem  4.2 ) , @xmath310,\beta)=\vol(a)$ ] for all @xmath6 . 
   examination of the long exact sequence in euler  koszul homology from reveals that @xmath311 this implies that for @xmath61 , @xmath312 is nonzero . 
   therefore there is an inclusion @xmath313 . 
   we make this relationship precise in theorem  [ thm - str comparison ] . 
   [ lemma - finite inside ] let @xmath133 . 
   the number of irreducible components of @xmath314 which intersect @xmath315 is finite . 
   view @xmath21 and its shifted saturation @xmath316 as graded submodules of @xmath308 $ ] . to see that the intersection @xmath317 involves only a finite number of irreducible components of @xmath314 , 
   it is enough to show that the arrangement given by the quasidegrees of the module @xmath318 has finitely many irreducible components . 
   this follows since @xmath316 is toric . 
   recall from that @xmath53 . for 
   @xmath183 and @xmath13 , let @xmath319 . 
   [ lemma - mm05 ] let @xmath52 $ ] be a weakly toric module , @xmath183 , @xmath13 be a face of codimension at least two , and @xmath320 be an interior vector of @xmath116 . if @xmath12 is maximal among faces of @xmath0 not in @xmath321 , then for all sufficiently large positive integers @xmath322 , the vector @xmath323 is an exceptional degree of @xmath29 . 
   this is ( * ? ? ? 
   * lemma  14 ) when @xmath324 and @xmath0 is homogeneous . 
   ( the matrix @xmath0 is called _ homogeneous _ when the vector @xmath325 is in the @xmath326-row span of @xmath0 . ) the same argument yields this generalization by @xmath41-graded local duality , see ( * ? ? ? 
   * section  3.5 ) . 
   [ thm - str comparison ] let @xmath52 $ ] be a weakly toric module . 
   the ranking arrangement @xmath59 contains the exceptional arrangement @xmath48 and @xmath327 where @xmath328 is pure of codimension 1 . 
   we must show that @xmath48 contains all irreducible components of @xmath59 of codimension at least two . to this end , let @xmath329 be such that @xmath330 is an irreducible component with @xmath331 . 
   then there are submodules @xmath332 and @xmath333 such that @xmath334 and @xmath335 . in fact 
   , there is a @xmath336 with @xmath337 and @xmath338 . 
   we may further choose @xmath90 so that @xmath12 is maximal among faces of @xmath0 that are not in the set @xmath321 . to see this 
   , first note that @xmath339 for all @xmath340 . indeed , for 
   if @xmath341 , then there are @xmath342 and @xmath343 with @xmath344 , a contradiction . 
   since @xmath338 is an irreducible component of @xmath345 , it suffices to show that @xmath90 can be chosen so that each facet @xmath346 of @xmath0 is in @xmath321 . 
   first , if @xmath347 , then by lemma  [ lemma - finite inside ] , there are at most a finite number of translates of @xmath348 that define components of @xmath345 and intersect @xmath349 ; write these as @xmath350 . if necessary , replace @xmath90 by a vector @xmath351 such that @xmath352 to assume that @xmath346 is in @xmath321 . 
   note that after such a replacement , it is still true that @xmath353 by the previous paragraph , so @xmath354 . next , suppose that @xmath355 . 
   if @xmath356 , then @xmath357 , an impossibility because @xmath358 defines an irreducible component of @xmath345 . 
   thus it must be that @xmath359 . in this case , @xmath360 , so @xmath346 is in @xmath321 . 
   hence every facet @xmath346 of @xmath0 is in @xmath321 , and the claim on the choice of @xmath90 is established . 
   let @xmath361 be an interior vector of @xmath116 . 
   lemma [ lemma - mm05 ] implies that for all sufficiently large integers @xmath322 , the vector @xmath362 . 
   therefore @xmath363 . 
   [ not - beta cmpt ] for @xmath26 , the _ @xmath6-components _ @xmath364 of the ranking arrangement of @xmath29 are the union of the irreducible components of @xmath59 which contain @xmath6 . 
   since @xmath0 has a finite number of faces , @xmath364 has finitely many irreducible components .    by ( 
   * porism  9.5 ) , the exceptional arrangement @xmath365 of the @xmath0-hypergeometric system @xmath366 has codimension at least two . in the following example 
   we show that there may be components of @xmath365 which are embedded in codimension 1 components of the ranking arrangement @xmath314 . 
   in particular , the zariski closure of @xmath367 may not agree with @xmath365 . 
   [ example - hidden ] let @xmath368\ ] ] with @xmath369 , and label the faces @xmath370 $ ] and @xmath371 $ ] . with @xmath372 $ ] , the exceptional arrangement of @xmath324 is properly contained in a hyperplane component of the ranking arrangement : @xmath373 we will discuss the rank jumps of @xmath1 in examples  [ example - hidden2 ] and  [ example - hidden3 ] .    one goal of section  [ sec : comb of rank ] is to understand the structure of the sets @xmath374 . 
   we will achieve this by stratifying @xmath48 by ranking slabs ( see definition  [ def - ranking slab ] ) . 
   another description of ranking slabs ( via translates of certain lattices contained in the @xmath6-components @xmath364 ) will be given in proposition  [ prop - zc e ] . 
   let @xmath77 and @xmath78 be subspace arrangements in @xmath7 . 
   we say that a stratification @xmath79 of @xmath77 _ respects _ @xmath78 if for each irreducible component @xmath80 of @xmath78 and each stratum @xmath81 , either @xmath82 or @xmath83 . 
   [ def - ranking slab ] a _ ranking slab _ of @xmath29 is a stratum in the coarsest stratification of @xmath48 that respects each of the following arrangements : @xmath59 and @xmath375 for @xmath376 . 
   since each of the arrangements used in definition  [ def - ranking slab ] is determined by the quasidegrees of a weakly toric module , the closure of each ranking slab of @xmath29 is the translate of a linear subspace of @xmath7 that is generated by a face of @xmath0 . 
   corollary  [ cor - strat vs equiv ] states that @xmath86 is constant on each ranking slab , so each @xmath87 with @xmath186 is a union of translates of linear subspaces of @xmath7 that are spanned by faces of @xmath0 . 
   it then follows that the stratification of @xmath48 by ranking slabs refines its rank stratification . while this is generally a strict refinement , examples  [ example - hidden3 ]  and  [ example-4diml hidden ] 
   show that the two stratifications may coincide for parameters close enough to the cone @xmath15 . 
   we wish to emphasize that the rank jump @xmath62 is not simply determined by holes within the semigroup @xmath54 , as can be seen in example  [ ex - first example ] . 
   [ def : slab ] a _ slab _ is a set of parameters in @xmath7 that lie on a unique irreducible component of the exceptional arrangement @xmath48 @xcite . 
   we will show by example that rank need not be constant on a slab . in example  [ example-4diml hidden ] , this failure results from  embedded " components of @xmath48 , while in example  [ example - nonconstant slab ] , it is due to the hyperplanes of @xmath59 that strictly refine the arrangement stratification of @xmath48 . 
   together with example  [ plane - line-4diml ] , these examples show that each of the arrangements listed in definition  [ def - ranking slab ] is necessary to determine such a geometric refinement of the rank stratification of @xmath48 . 
   as in section  [ sec : exceptional arrangement ] , let @xmath299 be a nonempty @xmath54monoid ( see definition  [ def : monoid ] ) , so that @xmath300 $ ] is a nontrivial weakly toric module ( see example  [ ex - m wt ] ) . for a fixed @xmath61 , 
   we know from that @xmath60 can be used to compute the rank jump @xmath62 . however 
   , this module contains a large amount of excess information that does not play a role in @xmath377 . to isolate the graded pieces of @xmath60 that impact @xmath62 
   , we will define weakly toric modules @xmath378 so that    @xmath379{12 cm }   \begin{enumerate } \item \label{prop1 } $ m \subseteq s^{\beta } \subseteq          t^{\beta } \subseteq s_a[\del_a^{-1}],$ \item \label{prop2 } 
   $ \cr_a(m,\beta ) =          \qdeg\left(\dfrac{t^{\beta}}{s^{\beta}}\right),$ \item \label{prop3 } $ \beta \notin          \qdeg\left(\dfrac{s_a[\del_a^{-1}]}{t^{\beta}}\right)$ , \item \label{prop4 } $ \beta \notin          \qdeg\left(\dfrac{s^{\beta}}{m}\right)$ , and \item \label{prop5 } $ \pp^{\beta } =          \deg\left(\dfrac{t^{\beta}}{s^{\beta}}\right)$          is a union of translates of $ \wt{{{\ensuremath{\mathbb{n}}}}f}$ for various $ f \preceq a$. \end{enumerate } \end{minipage}\end{aligned}\ ] ]    in proposition  [ prop - qis ] , we show that properties  - of allow us to replace @xmath60 with @xmath380 when calculating @xmath62 . to use this module to actually compute @xmath62 
   , we will encounter other toric modules with structure similar to @xmath63 , which are called _ ranking toric modules_. property   allows @xmath63 ( and similarly , any ranking toric module ) to be decomposed into _ simple ranking toric modules_. these modules are constructed so that their euler  koszul homology modules have easily computable ranks . at the end of section  [ subsec : simple rtm ] , we outline more specifically how simple ranking toric modules will play a role in our computation of @xmath62 . 
   we now construct the class of _ ranking toric modules _ , which includes the module @xmath63 coming from . 
   these modules will be constructed via their degree sets , which are unions of @xmath381modules for various @xmath13 . 
   we begin by isolating the translated lattices contained in @xmath382 that lie in the @xmath6-components @xmath383 of the ranking arrangement of @xmath29 ( see notation  [ not - beta cmpt ] ) . 
   the union of these translated lattices will be denoted by @xmath67 . 
   [ def - ep ]     1 . 
   let @xmath65 be the set of faces of @xmath0 corresponding to the @xmath6-components @xmath364 of the ranking arrangement of @xmath29 . 
   this set @xmath384 is a polyhedral cell complex , and @xmath364 is the union @xmath385 2 .   for each @xmath386 
   , let @xmath387 lemma  [ lemma - mm05 ] and theorem  [ thm - str comparison ] together imply that @xmath388 is nonempty exactly when there is a containment @xmath389 . 
   3 .   since @xmath302 is an @xmath54monoid , @xmath390 
   is closed under addition , so @xmath388 is @xmath68-stable . 
   thus there is a finite set of @xmath68-orbit representatives @xmath391 such that @xmath392 is partitioned into @xmath68-orbits as a disjoint union over @xmath391 . 
   notice that @xmath393 $ ] . 
   the @xmath68-orbits @xmath394 in are the translated lattices that we will use to construct ranking toric modules . 
   each is determined by the pair @xmath395 . 
   we denote the collection of such pairs by @xmath396 5 .   for a subset @xmath397 , 
   let @xmath398    [ not - omitj ] many of the objects we define in this section are dependent upon a subset @xmath399 , and this dependence is indicated by the subscript @xmath400 . 
   whenever we omit this subscript , it is understood that @xmath401 .    by the upcoming proposition  [ prop - zc e ] , two parameters @xmath84 belong to the same ranking slab exactly when @xmath85 . 
   this is what will be used to show that the rank jump @xmath62 ) is constant on ranking slabs . 
   [ lemma - zc e addition ] the zariski closure of the ranking lattices @xmath67 of @xmath29 at @xmath6 coincides with the @xmath6-components @xmath364 of the ranking arrangement . 
   it is clear from the definitions that @xmath402 . 
   for the reverse containment , if @xmath403 , then there exists a vector @xmath404 such that @xmath405 . 
   this implies that @xmath406 is empty , so the claim now follows from the definition of quasidegree sets in definitions  [ def - qdeg ]  and  [ def - wtoric ] . 
   [ prop - zc e ] the parameters @xmath84 belong to the same ranking slab if and only if the ranking lattices of @xmath29 at @xmath6 and @xmath407 coincide , that is , if @xmath408    this is a consequence of lemma  [ lemma - zc e addition ] , lemma  [ lemma - mm05 ] , and theorem  [ thm - str comparison ] .    in light of proposition  [ prop - zc e ] , 
   use equality of ranking lattices to extend the ranking slab stratification of @xmath48 to the parameter space @xmath7 . 
   one might try making the sets @xmath409 in definition  [ def - ep ] the degree sets of ranking toric modules . 
   however , while the natural map @xmath410 given by faces @xmath262 induces a vector space map @xmath411 , this induced map is not a morphism of @xmath121-modules because it sends units to zero . to overcome this 
   , we introduce the lattice points in a certain polyhedron , denoted by @xmath412 , and intersect it with @xmath413 to produce the degree set of a ranking toric module . 
   [ def - cp ]     1 . 
   recall the primitive integral support functions @xmath414 from the beginning of section  [ sec : defns ] . in order to construct a ranking toric module from @xmath409 , ( and achieve the various quasidegree sets proposed in ) , set @xmath415 for @xmath71 , notice that @xmath416 is simply the integral points in the cone @xmath417 after translation by @xmath6 . 
   2 .   for a pair @xmath418 , 
   let @xmath419 . 
   \nonumber      \intertext{the degree sets of ranking toric modules are of the form }      \label{17 }      \pp_j^{\beta } & \phantom{:}=   \bigcup_{(f , b)\in j } \pp_{f , b}^{\beta }       \ = \ \cc_a(\beta ) \cap \bbe_j^{\beta }      \intertext{for $ j\subseteq\cj(\beta)$. the largest of these is }      \label{eqn - p }      \pp^{\beta } \ & : = \ \pp_{\cj(\beta)}^{\beta } \       = \ 
   \cc_a(\beta ) \cap \bbe^{\beta } ,       \end{aligned}\ ] ] the degree set appearing in . 
   ( continuation of example  [ example - hidden ] ) [ example - hidden2 ] with @xmath420 \in\ee_a(s_a)$ ] and @xmath421 $ ] , the sets of definitions  [ def - ep ]  and  [ def - ts and r - t ] are @xmath422 \sqcup [ b + { { \ensuremath{\mathbb{z}}}}g ] , \\   \text{and } \phantom{xx }   \pp^{\beta } & = & [ \beta + { { \ensuremath{\mathbb{n}}}}f ] \sqcup [ b + { { \ensuremath{\mathbb{n}}}}g ] . \end{aligned}\ ] ]    having defined the degree sets of ranking toric modules in , we now construct the modules themselves . along the way 
   , we meet the modules that satisfy the requirements of . 
   [ def - ts and r - t ]     1 . 
   each ranking toric module will be a quotient of the module @xmath423 ( for some @xmath26 ) , where @xmath424      \phantom{xx } \text { and } \phantom{xx }      t^{\beta}\ = \ { { \ensuremath{\mathbb{c}}}}\{\tt^{\beta}\}.\ ] ] notice that if @xmath425 , then @xmath426 . 
   the simplest case occurs when @xmath55 and @xmath427 , so that @xmath428 . 
   2 .   for @xmath429 , 
   let @xmath430 we show in proposition  [ prop - basics r - toric ] that @xmath423 and @xmath431 are indeed weakly toric modules ( see definition  [ def - wtoric ] ) . 
   when @xmath401 , these modules satisfy the properties . by notation  [ not - omitj ] , @xmath432 . 
   3 .   for a subset @xmath429 , 
   the quotient @xmath433 has degree set @xmath434 , as recorded in proposition  [ prop - basics r - toric ] . in proposition  [ prop - qis ] , we show that @xmath60 in can be replaced by @xmath435 when computing @xmath62 . 
   if a toric module @xmath43 is isomorphic to @xmath436 for a pair @xmath437 and a subset @xmath429 , we say that @xmath43 is a _ ranking toric module _ determined by @xmath400 . 
   [ prop - basics r - toric ] let @xmath397 . 
   there are containments of weakly toric modules : @xmath438 . 
   \end{aligned}\ ] ] in particular , @xmath439 is a ranking toric module with degree set @xmath440 .    by construction , we have the containment @xmath441 . since the intersection of @xmath67 and @xmath302 is empty , @xmath442 is empty as well . 
   hence @xmath443 . 
   the other containments in are obvious . 
   it is clear from the definitions that the degree sets of all modules in question are closed under addition with elements of @xmath54 , so they are all weakly toric modules . for the second statement , since @xmath444 , @xmath436 is a finitely generated @xmath21-module and therefore @xmath436 is toric .    by lemma  [ lemma - zc e addition ] and the definition of @xmath445 , 
   the arrangement @xmath446 coincides with the @xmath6-components @xmath364 of @xmath29 at @xmath6 . 
   further , the construction of @xmath63 in definition  [ def - ts and r - t ] is such that @xmath63 can replace @xmath60 in when calculating @xmath62 . 
   [ prop - qis ] the euler  koszul complexes @xmath447 and @xmath448 are quasi - isomorphic . in particular , @xmath449    consider the short exact sequences @xmath450}{m } \rightarrow \dfrac{s_a[\del_a^{-1}]}{t^{\beta } } \rightarrow 0 \quad \text{and } \quad 0 \rightarrow \dfrac{s^{\beta}}{m } \rightarrow \dfrac{t^{\beta}}{m } \rightarrow \dfrac{t^{\beta}}{s^{\beta } } \rightarrow 0.\end{aligned}\ ] ] the definition of @xmath412 ensures that @xmath6 is not a quasidegree of either @xmath451 or @xmath308/t^{\beta}$ ] . 
   thus we obtain the result from long exact sequences in euler  koszul homology and proposition  [ prop - hh qis 0 ] . 
   [ def - e  k char ] the _ @xmath452 partial euler 
   koszul characteristic _ of a weakly toric module @xmath43 is the ( nonnegative ) integer @xmath453    the main result of this article , theorem  [ thm - main ] , states that the partial euler  koszul characteristics of ranking toric modules are determined by the combinatorics of the ranking lattices @xmath67 of @xmath29 at @xmath6 . 
   lemma  [ lemma - chi p ] describes @xmath62 as the @xmath454 partial euler 
   koszul characteristic of @xmath63 , so our results regarding the combinatorics of rank jumps are a consequence of theorem  [ thm - main ] . 
   [ lemma - chi p ] the @xmath455 partial euler  koszul characteristic of every nontrivial ranking toric module for a pair @xmath437 is 0 . 
   in particular , @xmath456 . 
   this follows from ( * ? ? ? 
   * theorem  4.2 ) and . 
   the polyhedral structure of the degree sets of ranking toric modules plays an important role in our computation of their partial euler  koszul characteristics . 
   we will use that each face @xmath457 determines an @xmath121-module that is the quotient of a ranking toric module . 
   modules of this type are called _ simple ranking toric modules_.    [ def - rtoric simple ]     1 .   for a subset @xmath429 , the ranking toric module 
   @xmath436 is _ simple _ if there is a unique @xmath458 such that all pairs in @xmath400 are of the form @xmath395 . 
   2 .   for each @xmath459 and @xmath429 , denote by @xmath460 the simple ranking toric module determined by the set @xmath461 . 
   the degree set of this module is denoted by @xmath462 . 
   3 .   call the parameter @xmath6 _ simple _ for @xmath29 if @xmath63 is a simple ranking toric module , or equivalently , if there is an @xmath459 such that @xmath463 ( see notation  [ not - omitj ] ) . 
   we show in theorem  [ thm - rank simple r - toric ] that for @xmath13 , each simple ranking toric module @xmath460 is a maximal cohen 
   macaulay toric @xmath121-module . 
   thus the results of section  [ sec : ekstr ] can be applied to compute the rank of their euler  koszul homology modules . 
   notice that by setting @xmath464 it follows from definition  [ def - rtoric simple ] that @xmath465 . 
   in particular , when @xmath401 , @xmath466 . 
   [ prop - simple r - t ] for @xmath459 and @xmath429 , the simple ranking toric module @xmath460 of definition  [ def - rtoric simple ] admits an @xmath121-module structure that is compatible with its @xmath21-module structure .    by construction , @xmath462 is closed under addition with elements of @xmath116 . 
   [ def - triangle ] we define a partial order @xmath467 on @xmath397 by @xmath468 if and only if @xmath469 for pairs @xmath470 . we let @xmath471 denote the subset of @xmath400 consisting of maximal elements with respect to @xmath467 .    for @xmath429 
   , @xmath471 is the smallest subset of @xmath400 that determines a direct sum of simple ranking toric modules into which @xmath436 embeds . 
   the calculation of the partial euler  koszul characteristics of a ranking toric module @xmath436 will be achieved by homologically replacing it by an acyclic complex @xmath472 composed of simple ranking toric modules . 
   we then examine the spectral sequences determined by the double complex @xmath473 to obtain a formula for the partial euler 
   koszul characteristics of @xmath436 . 
   we now define an equivalence relation on the union of the various @xmath68-orbit representatives of . 
   we show in proposition  [ prop - p decomp ] that for @xmath429 , the ranking toric module @xmath436 splits as direct sum over the equivalence classes of this relation . 
   thus by additivity of rank , can be expressed as a sum involving simpler ranking toric modules . 
   [ def - tilde ]     1 . 
   let @xmath474 be the collection of all @xmath68-orbit representatives from . 
   let @xmath475 be the equivalence relation on the elements of @xmath476 that is generated by the relations @xmath477 if there exist @xmath478 such that @xmath479 3 
   .   let @xmath480 denote the set of equivalence classes of @xmath475 . 
   4 .   for @xmath481 and @xmath429 , 
   let @xmath482 hence for @xmath429 , there is a partition of @xmath440 over @xmath480 :   @xmath483 . 
   [ prop - p decomp ] for @xmath429 , there is a decomposition @xmath484 .    for distinct @xmath485 , the sets @xmath486 and @xmath487 are disjoint by definition of @xmath475 . 
   thus there is a decomposition @xmath488 .    as a special case of proposition  [ prop - p decomp ] 
   , the simple ranking toric module @xmath489 can be expressed as the direct sum @xmath490 ( see ) .    for @xmath401 , 
   let the _ rank jump from @xmath491 of @xmath29 at @xmath6 _ 
   be @xmath492    [ cor - j decomp ] the rank jump @xmath62 can be expressed as the sum @xmath493 . 
   this follows from , proposition  [ prop - p decomp ] , and the additivity of @xmath62 .    as stated in corollary  [ cor - j decomp ] , computing @xmath62 
   is reduced to finding @xmath494 for each @xmath481 . 
   when working with examples , it is typically useful to consider each @xmath494 individually . 
   in contrast , as we continue with the theory , it is more efficient for our notation to study @xmath62 directly . in sections  [ subsec : simple case ] and 
   [ subsec : non - simple case ] , replacing @xmath384 , @xmath63 , and @xmath62 by their corresponding @xmath491 counterparts calculates @xmath494 . 
   we retain the notation of section  [ sec : r - t mods ] . 
   this section contains our main result , theorem  [ thm - main ] , which states that for any subset @xmath429 , the partial euler  koszul characteristics of the ranking toric module @xmath436 are determined by the combinatorics of @xmath409 ( refer to definitions  [ def - e  k char ] ,  [ def - ts and r - t ] , and  [ def - ep ] ) . as a special case of this result , we will have computed @xmath456 in terms of the ranking lattices @xmath67 , resulting in a proof of theorem  [ thm - compute jump ] . 
   we begin by examining the partial euler 
   koszul characteristics of simple ranking toric modules @xmath495 from definition  [ def - rtoric simple ] . 
   we will compute the partial euler 
   koszul characteristics of a ranking toric module @xmath436 by homologically approximating it by a cellular resolution ( see definition  [ def - cellular definition ] ) built from simple ranking toric modules . 
   the next theorem shows that simple ranking toric modules are maximal cohen  macaulay toric face modules , which will be useful in the general case . 
   this allows us to compute the rank jump @xmath62 of @xmath29 at @xmath6 when @xmath6 is simple for @xmath29 , as in ( * ? ? ? 
   * theorem  2.5 ) . 
   [ thm - rank simple r - toric ] fix @xmath26 , @xmath459 , and @xmath429 . 
   then the simple ranking toric module @xmath460 is a maximal cohen 
   macaulay toric @xmath121-module . 
   further , there is a decomposition @xmath496 \otimes_{{\ensuremath{\mathbb{c}}}}\hh^f_0(p_{f , j}^{\beta},\beta )       \otimes_{{\ensuremath{\mathbb{z}}}}\left(\textstyle\bigwedge^\bullet { { \ensuremath{\mathbb{z}}}}f^\bot\right ) , \end{aligned}\ ] ] and for all @xmath497 , @xmath498    fix a @xmath68-orbit representative @xmath499 , chosen so that @xmath500 . 
   this implies that @xmath501 is a short exact sequence of toric modules . since @xmath502 the definition of @xmath412 ensures that @xmath503 . 
   proposition  [ prop - hh qis 0 ] and imply that induces the isomorphism @xmath504 . 
   as @xmath505 , proposition  [ prop - str cm ek complex ] gives the decomposition , in light of proposition  [ prop - p decomp ] . by ( 
   * ; * ? ? ? 
   * lemma  3.3 ) , @xmath506 . 
   now the additivity of rank and combine to complete the claim . 
   [ cor - simple ] if @xmath61 is simple for @xmath29 , then the rank jump of @xmath29 at @xmath6 is @xmath507 \cdot \vol(f).\ ] ]    since @xmath6 is simple for @xmath29 , @xmath463 for some @xmath13 . 
   hence apply theorem  [ thm - rank simple r - toric ] to corollary  [ cor - j decomp ] , noting that @xmath508 . 
   ( continuation of examples  [ example - hidden ] and  [ example - hidden2 ] ) [ example - hidden3 ] with @xmath509 $ ] , we have the set @xmath510 . 
   both @xmath511 and @xmath512 are simple ranking toric modules . by corollary  [ cor - simple ] , @xmath513\cdot 1 \ = 
   \ 1   \quad \text{and } \quad j_{\widehat{b}}(\beta )   =   1\cdot [ 1 - 1]\cdot 1 \ = \ 0.\end{aligned}\ ] ] it now follows from proposition  [ prop - p decomp ] that @xmath514 . 
   a similar calculation shows that @xmath515 for any @xmath516 . 
   [ example-4diml hidden ] let @xmath517\ ] ] with @xmath518 , and consider the faces @xmath370 $ ] and @xmath371 $ ] . 
   note that the semigroup @xmath54 of example  [ example - hidden ] embeds into the @xmath54 here . with @xmath519 $ ] , the exceptional arrangement of @xmath21 is @xmath520 , where @xmath521 thus the ranking slab stratification of @xmath365 is strictly finer than its arrangement stratification . 
   further , this finer stratification coincides with the rank stratification of @xmath365 inside the cone @xmath15 . for @xmath522 , @xmath523 , while @xmath524 for @xmath525 $ ] . 
   calculations similar to those of example  [ example - hidden3 ] show that @xmath526$. }      \end{cases}\ ] ] in particular , the rank of the @xmath0-hypergeometric system @xmath366 is not constant on the slab @xmath527\subseteq \ee_a(s_a)$ ] ( see definition  [ def : slab ] ) . 
   [ example-2 lines c4 ] let @xmath528 for @xmath529\ ] ] and consider the saturated faces @xmath530 $ ] and @xmath531 $ ] . 
   here @xmath532 , @xmath533 , and @xmath534 . 
   computations in macaulay 2 @xcite with reveal that @xmath535\cup[\beta'+{{\ensuremath{\mathbb{c}}}}f_2],\ ] ] where @xmath536 $ ] . with @xmath537 $ ] and @xmath538 , @xmath539\setminus\beta'$ , } \\ 
   \beta+{{\ensuremath{\mathbb{c}}}}f_2 &          \text{if $ \beta\in[\beta'+{{\ensuremath{\mathbb{c}}}}f_2]\setminus\beta'$ , }      \end{cases}\ ] ] @xmath540\cup [ \beta+{{\ensuremath{\mathbb{n}}}}f_1]\cup       [ \beta+{{\ensuremath{\mathbb{n}}}}f_2 ] & \text{if $ \beta=\beta'$}\\      [ \beta+b+{{\ensuremath{\mathbb{n}}}}f_1]\cup \beta+{{\ensuremath{\mathbb{n}}}}f_1 &               \text{if $ \beta\in\wt{{{\ensuremath{\mathbb{n}}}}a}\cap[\beta'+{{\ensuremath{\mathbb{c}}}}f_1]\setminus\beta'$ , } \\ 
   \beta+{{\ensuremath{\mathbb{n}}}}f_2 & \text{if $ \beta\in\wt{{{\ensuremath{\mathbb{n}}}}a}\cap[\beta'+{{\ensuremath{\mathbb{c}}}}f_2]\setminus\beta'$ , }      \end{cases}\ ] ] and @xmath541\setminus\beta'$ , } \\    \ 
   { \widehat{\beta } \ }   & \text{if $ \beta\in\wt{{{\ensuremath{\mathbb{n}}}}a}\cap[\beta'+{{\ensuremath{\mathbb{c}}}}f_2]\setminus\beta'$. }      \end{cases}\ ] ]    by corollary  [ cor - simple ] , for @xmath542\setminus\beta'$ ] , @xmath543\cdot\vol(f_2)\ = \          [ 3 - 1]\cdot 1\ = \ 2,\ ] ] while for @xmath544\setminus\beta'$ ] , @xmath545 , and @xmath546\cdot\vol(f_1)\ = \          2 \cdot [ 3 - 1]\cdot 1\ = \ 
   4.\ ] ] to compute the rank jump of @xmath21 at @xmath407 , we must move to the general case . 
   we will see in example  [ example-2 lines c4 again ] that @xmath547 , which arises as the sum of the generic rank jumps along irreducible components of @xmath548 that is then corrected by error terms that arise from a spectral sequence calculation .      we are now prepared to compute the partial euler 
   koszul characteristics of ranking toric modules . 
   the proof of our main theorem , theorem  [ thm - main ] , will be given at the end of this section , after a sequence of lemmas . 
   the definitions of @xmath549 , @xmath409 , and @xmath436 can be found in definitions  [ def - ep ]  and  [ def - ts and r - t ] , respectively . 
   [ thm - main ] for @xmath429 , the partial euler 
   koszul characteristics of the ranking toric module @xmath436 are determined by the combinatorics of @xmath409 . 
   we compute the partial euler 
   koszul characteristics of the ranking toric module @xmath436 will be achieved by homologically approximating @xmath436 by simple ranking toric modules ; note that the ranks of the euler 
   koszul homology modules of simple ranking toric modules have been computed in theorem  [ thm - rank simple r - toric ] . 
   [ def - cellular definition ] let @xmath550 be an oriented cell complex ( e.g. cw , simplicial , polyhedral ) . 
   then @xmath550 has the cochain complex @xmath551 where @xmath552 is multiplication by some integer coeff@xmath553 . 
   let @xmath554 be the category with the nonempty faces of @xmath550 as objects and morphisms @xmath555 fix an abelian category @xmath556 and suppose there is a covariant functor @xmath557 . 
   let @xmath558 for each @xmath559 . 
   a sequence of morphisms in @xmath556 @xmath560 is _ cellular _ and _ supported on _ 
   @xmath550 if the @xmath561 component of @xmath562 is coeff@xmath553 @xmath563 . 
   since @xmath556 is abelian , a cellular sequence is necessarily a complex .    in a manner analogous to definition  [ def - cellular definition ] , a cellular complex supported on @xmath550 can also be obtained from the chain complex @xmath564 of @xmath550 and a contravariant functor @xmath565 . 
   further , we could replace @xmath564 in this construction with the _ reduced _ chain or cochain complexes of @xmath550 . 
   we say that a complex in @xmath556 is _ cellular _ if it can be constructed from the underlying topological data of a cell complex and a functor @xmath557 . 
   when @xmath550 is a simplicial or polyhedral cell complex , coeff@xmath553 of definition  [ def - cellular definition ] is simply @xmath566 if the orientation of @xmath567 induces the orientation of @xmath568 and @xmath569 if it does not .    the generality in which we define cellular complexes is alluded to in the introduction of @xcite and appears as ( * 
   * definition  3.2 ) . 
   an introduction to these complexes , in the polyhedral case , can be found in ( * ? ? ? 
   * chapter  4 ) . 
   recall from definition  [ def - triangle ] that @xmath471 was defined so that it yields the smallest set of faces of @xmath0 that determines a direct sum of simple ranking toric modules into which @xmath436 embeds . 
   [ not - cpx i ] we wish to take intersections of faces in the set @xmath471 . in order to keep track of which 
   faces were involved in each intersection , set @xmath570 with @xmath571 , let @xmath572 be the standard @xmath573-simplex with vertices corresponding to the elements of @xmath574 . to the @xmath575-face of @xmath550 spanned by the vertices corresponding to the elements in @xmath576 , 
   assign the ranking toric module @xmath577 . 
   choosing the natural maps @xmath578 for @xmath579 induces a cellular complex supported on @xmath550 , @xmath580 with @xmath581    [ lemma - i acyclic ] the cohomology of the cellular complex @xmath472 of is concentrated in cohomological degree 0 and is isomorphic to @xmath436 . 
   given @xmath582 , let @xmath583 be the faces @xmath584 such that @xmath585 . 
   the degree @xmath44 part of @xmath472 computes the cohomology of the @xmath586-subsimplex of @xmath550 given by the vertices with labels corresponding to @xmath583 ; in particular , it is acyclic with @xmath587-cohomology @xmath588 .    by construction , @xmath436 is a @xmath41-graded monomial module over the saturated semigroup ring @xmath589 , and it can be translated by some @xmath361 so that @xmath590 . 
   after translation by @xmath44 , is similar to an _ irreducible resolution _ 
   , as defined in ( * ? ? ? 
   * ; * ? ? ? 
   * definition  2.1 ) . 
   we continue to view @xmath436 as an @xmath21-module , so we use maximal cohen  macaulay toric face modules instead of irreducible quotients of @xmath589 . 
   consider the @xmath41-graded double complex @xmath591 with @xmath592 let @xmath593 and @xmath594 denote the horizontal and vertical differentials of @xmath591 , respectively . by the exactness of , 
   taking homology of @xmath591 with respect to @xmath593 yields @xmath595 we now apply the decomposition of these homologies given in theorem  [ thm - rank simple r - toric ] to obtain a new description of the euler 
   koszul homology of the ranking toric module @xmath436 . 
   [ lemma - right ss ] the vertical spectral sequence obtained from the double complex @xmath596 \otimes_{{\ensuremath{\mathbb{c}}}}\kk^{f_s}_i(p_{f_s , j}^{\beta},\beta ) \otimes_{{\ensuremath{\mathbb{z}}}}\left ( \textstyle\bigwedge^j { { \ensuremath{\mathbb{z}}}}f_s^\bot \right)\end{aligned}\ ] ] ( with differentials as in lemma  [ lemma - ek morph ] ) has abutment @xmath597    by theorem  [ thm - rank simple r - toric ] and lemma  [ lemma - ek morph ] , the vertical differentials @xmath594 of @xmath598 are compatible with the quasi - isomorphism @xmath599 \otimes_{{\ensuremath{\mathbb{c}}}}\kk^{f_s}_\bullet(p_{f_s , j}^{\beta},\beta ) \otimes_{{\ensuremath{\mathbb{z}}}}\left ( \textstyle\bigwedge^\bullet { { \ensuremath{\mathbb{z}}}}f_s^\bot \right).\end{aligned}\ ] ] since @xmath600 and @xmath601 converge to the same abutment , the result follows from . 
   note that the first page of the spectral sequence in lemma  [ lemma - right ss ] is @xmath602 \otimes_{{\ensuremath{\mathbb{c}}}}\hh^{f_s}_0(p_{f_s , j}^{\beta},\beta ) \otimes_{{\ensuremath{\mathbb{z}}}}\left ( \textstyle\bigwedge^q { { \ensuremath{\mathbb{z}}}}f_s^\bot \right).\end{aligned}\ ] ]    for @xmath603 , let @xmath604 denote the differential of @xmath605 , and let @xmath606 respectively denote the vertical and horizontal differentials of @xmath607 . 
   if @xmath245 with @xmath608 , then by remark  [ remark - diffl in cm ek complex ] , the element @xmath609 \otimes_{{\ensuremath{\mathbb{c}}}}\kk_i^{f_s}(p_{f_s , j}^{\beta},\beta ) \otimes_{{\ensuremath{\mathbb{c}}}}\left ( \textstyle\bigwedge^q { { \ensuremath{\mathbb{z}}}}f_s^\bot \right )      \subseteq \ { ' e}_0^{p ,- q}\ ] ] has vertical differential @xmath610 we will use that is an element of @xmath611\otimes \kk_{i-1}^{f_s } ( p_{f_s , j}^{\beta},\beta ) \otimes_{{\ensuremath{\mathbb{c}}}}\left ( \textstyle\bigwedge^j { { \ensuremath{\mathbb{z}}}}f_s^\bot \right ) \subseteq { ' e}_0^{p ,- q+1}\ ] ] to show that @xmath612 degenerates quickly . 
   this is the main technical result of this article . 
   [ lemma - ss terminates ] the spectral sequence @xmath613 of lemma  [ lemma - right ss ] degenerates at the second page .    for @xmath614 , 
   let @xmath615 denote the image of @xmath616 in @xmath617 , if it exists . 
   let @xmath618 denote the differential of @xmath619 , so @xmath620 .    to see that @xmath621 , consider an element @xmath622 with @xmath623 . 
   then there is an element @xmath624 such that @xmath625 , which is used to define @xmath626 ( recall that is independent of the choice of @xmath627 . ) 
   we write @xmath628 as an element of . 
   note that @xmath629 \otimes_{{\ensuremath{\mathbb{c}}}}\kk_{i-1}^{f_s}(p_{f_s , j}^{\beta},\beta )   \otimes_{{\ensuremath{\mathbb{z}}}}\left ( \textstyle\bigwedge^j { { \ensuremath{\mathbb{z}}}}f_s^\bot \right ) ,   \nonumber   \intertext{so $ \alpha^s_{ij}$ is in the kernel of $ { } _ v\delta$ for all $ s , i , j$.   by \eqref{eq : nice 1st page } , $ \alpha^{s}_{ij}$ is in the image of $ { } _ v\delta$   whenever $ i>0$. hence without changing $ \oalpha$ , we may assume that for all $ s\in\cg^p_j$ ,   $ \alpha^{s}_{ij}=0 $ when $ i>0 $ , so that } \alpha & \in \bigoplus_{s\in\cg^p_j } { { \ensuremath{\mathbb{c}}}}[x_{f_s^c } ] \otimes_{{\ensuremath{\mathbb{c}}}}\kk_0^{f_s}(p_{f_s , j}^{\beta},\beta ) \otimes_{{\ensuremath{\mathbb{z}}}}\left ( \textstyle\bigwedge^q { { \ensuremath{\mathbb{z}}}}f_s^\bot \right ) . 
   \nonumber \intertext{as the differential $ { } _ h\delta$ is induced by \eqref{9 } , } { } _ h\delta(\alpha ) & \in \bigoplus_{s\in\cg^{p+1}_j } { { \ensuremath{\mathbb{c}}}}[x_{f_s^c } ] \otimes_{{\ensuremath{\mathbb{c}}}}\kk_0^{f_s}(p_{f_s , j}^{\beta},\beta ) \otimes_{{\ensuremath{\mathbb{z}}}}\left ( \textstyle\bigwedge^q { { \ensuremath{\mathbb{z}}}}f_s^\bot \right ) . 
   \nonumber \intertext{by hypothesis on $ \alpha$ and \eqref{31 } , there is an element } \label{34 } \eta & \in \bigoplus_{s\in\cg^{p+1}_j } { { \ensuremath{\mathbb{c}}}}[x_{f_s^c } ] \otimes_{{\ensuremath{\mathbb{c}}}}\kk_1^{f_s}(p_{f_s , j}^{\beta},\beta ) \otimes_{{\ensuremath{\mathbb{z}}}}\left ( \textstyle\bigwedge^q { { \ensuremath{\mathbb{z}}}}f_s^\bot \right ) \intertext{such that $ { } _ v\delta(\eta)={}_h\delta(\alpha)$. set $ \zeta={}_h\delta(\eta)$ and note that $ \delta_2(\oalpha)=\ol{\zeta}$. using again that the differential $ { } _ h\delta$ is induced by \eqref{9 } ,   applied now to \eqref{34 } , we see that } \zeta & \in \bigoplus_{s\in\cg^{p+2}_j }      { { \ensuremath{\mathbb{c}}}}[x_{f_s^c } ] \otimes_{{\ensuremath{\mathbb{c}}}}\kk_1^{f_s}(p_{f_s , j}^{\beta},\beta ) \otimes_{{\ensuremath{\mathbb{z}}}}\left ( \textstyle\bigwedge^q { { \ensuremath{\mathbb{z}}}}f_s^\bot \right ) . 
   \nonumber\end{aligned}\ ] ] since @xmath630 , @xmath631 is in the kernel of @xmath632 . 
   hence implies that @xmath633 vanishes . 
   [ lemma - formula ] for @xmath429 , the @xmath452 partial euler 
   koszul characteristic of the ranking toric module @xmath436 is given by @xmath634    for @xmath635 , let @xmath636 by lemma [ lemma - ss terminates ] , @xmath637 . from the abutment , we see that @xmath638 for @xmath639 . 
   since @xmath640 by lemma  [ lemma - chi p ] , also @xmath641 . 
   thus the @xmath452 partial euler  koszul characteristic of @xmath436 can be expressed as @xmath642 now follows from the definition of @xmath643 and the quasi - isomorphism , as the isomorphic first pages of the spectral sequences there are and . 
   we will compute the ranks of @xmath644 and the image of @xmath645 from in subsequent lemmas . 
   the first is an immediate consequence of theorem  [ thm - rank simple r - toric ] . 
   [ lemma - rank - hqip ] if @xmath646 , then @xmath647    by definition of @xmath648 and additivity of rank , @xmath649 . now apply theorem  [ thm - rank simple r - toric ] . 
   the rank of the image of @xmath645 is determined combinatorially because the spectral sequence rows @xmath650 are cellular complexes . 
   [ lemma-e cell rows ] the complexes @xmath650 are cellular with support @xmath651 of notation  [ not - cpx i ] .    in notation 
   [ not - cpx i ] , we constructed the cellular complex @xmath472 from a labeling of the simplex @xmath572 . if we assign in this construction @xmath652 \otimes_{{\ensuremath{\mathbb{c}}}}\hh_q^f(p_f^{\beta},\beta )   \otimes_{{\ensuremath{\mathbb{z}}}}\left ( \textstyle\bigwedge^q { { \ensuremath{\mathbb{z}}}}f^\bot \right)\ ] ] in place of @xmath460 and use the induced maps , we obtain the cellular complex @xmath650 with differential @xmath653 , see . 
   the existence and compatibility of the differentials follows from lemma  [ lemma - ek morph ] . 
   [ lemma - rank image ] the rank of the image of @xmath645 is determined by the combinatorics of @xmath409 .    by lemma  [ lemma-e cell rows ] , 
   the image of @xmath645 is determined by the @xmath575-coboundaries of @xmath550 and the corresponding labels of @xmath550 , which come from @xmath409 . 
   if @xmath400 is simple for @xmath29 , the result follows from theorem  [ thm - rank simple r - toric ] , so suppose that @xmath436 is not simple for @xmath29 . 
   by lemma  [ lemma - i acyclic ] , @xmath436 is the @xmath587-cohomology of the acyclic cellular complex @xmath472 . 
   thus the abutment of the spectral sequences arising from the double complex @xmath473 is @xmath654 . by lemma  [ lemma - right ss ] , 
   the vertical spectral sequence obtained from the double complex @xmath655 of has the same abutment . 
   since this spectral sequence degenerates on the second page by lemma  [ lemma - ss terminates ] , lemma  [ lemma - formula ] yields the formula , and by lemmas  [ lemma - rank - hqip ] and  [ lemma - rank image ] , the summands of are dependent only on the combinatorics of @xmath409 . 
   recall formula : @xmath656 lemma  [ lemma - rank - hqip ] computes the first summand , but lemma  [ lemma - rank image ] does not explicitly state the rank of the image of @xmath645 . 
   a method to do this is provided by proposition  [ prop : alg ] . 
   [ def : circuit ] for @xmath657 , a subset @xmath658 corresponds to a subcomplex @xmath659 of the simplex @xmath660 , as described in notation  [ not - cpx i ] . 
   if @xmath661 and there is a minimal generator of @xmath662 of the form @xmath663 $ ] , where all coefficients @xmath664 are nonzero , then we say that @xmath665 is a _ circuit _ for @xmath666 .    for @xmath657 , 
   let @xmath667 denote the set of circuits for @xmath666 , and set @xmath668 for @xmath669 , @xmath670 , and @xmath671 , set @xmath672    [ prop : alg ] let @xmath673 be the ranking toric module in . 
   the rank of the image of @xmath645 from is equal to @xmath674 further , can be computed by combining theorem  [ thm - rank simple r - toric ] and induction on the dimension of @xmath675 .    before providing the proof of proposition  [ prop : alg ] , we state two lemmas .    [ lemma : easier comp ] for @xmath676 , @xmath677 where @xmath673 is the ranking toric module given by . 
   the rank of the image of @xmath678 is @xmath679 by proposition  [ prop - str / rk image ] . 
   view the image of @xmath680 as a quotient of @xmath681 . 
   if @xmath44 is one of its nonzero multigraded components , then it also appears in the degree set of another summand of @xmath682 . 
   the collection of such degrees is exactly @xmath683 . 
   [ lemma : harder comp ] for @xmath657 , @xmath684.\end{aligned}\ ] ]    to see this , notice first that for @xmath685 , @xmath686 is generated by the images coming from circuits for @xmath666 . by proposition  [ prop - str / rk image ] , 
   given a fixed circuit @xmath665 for @xmath666 , the rank of @xmath687 can be computed as the rank of @xmath688 times the @xmath31-rank of @xmath689 =   \textstyle\bigwedge^q { { \ensuremath{\mathbb{z}}}}f(\lambda)^\bot . 
   \end{aligned}\ ] ] by the same reasoning used to obtain , the rank of @xmath690 equals the rank of . 
   the @xmath31-rank of is a binomial coefficient in the codimension in @xmath7 of the span of the vectors in @xmath691 , so the rank of is @xmath692 further , for a collection of circuits @xmath671 , @xmath693 gives the rank of the intersection over @xmath694 of the images of type , so the inclusion - exclusion principle yields . 
   recall from that the domain of @xmath645 is the direct sum @xmath695 \otimes_{{\ensuremath{\mathbb{c}}}}\hh^{f_s}_0(p_{f_s , j}^{\beta},\beta ) \otimes_{{\ensuremath{\mathbb{z}}}}\left ( \textstyle\bigwedge^q { { \ensuremath{\mathbb{z}}}}f_s^\bot \right).\end{aligned}\ ] ] for @xmath696 , let @xmath697 denote the restriction of @xmath645 to the summands in @xmath698 . 
   order the elements of @xmath699 , so that @xmath700 . 
   \nonumber\end{aligned}\ ] ] lemmas  [ lemma : easier comp ] and  [ lemma : harder comp ] respectively computed the summands of , resulting in . 
   thus it remains to show the second statement . 
   if a ranking toric module has dimension 0 , then it is necessarily a simple ranking toric module , so theorem  [ thm - rank simple r - toric ] computes the rank of its euler 
   koszul homology modules . thus by induction on dimension 
   , we can compute the summand in corresponding to @xmath669 if the dimension of @xmath701 is strictly less than the dimension of @xmath681 . 
   if it is the case that the dimension of @xmath701 equals the dimension of @xmath681 , notice first that each pair @xmath702 has @xmath703 . 
   this implies that @xmath701 is a direct sum ( as in proposition  [ prop - p decomp ] ) of the simple ranking toric module @xmath704 and a lower - dimensional ranking toric module . 
   therefore induction together with theorem  [ thm - rank simple r - toric ] still completes the computation . 
   finally , the same argument applies to computing the rank of @xmath705 for @xmath671 , since @xmath706 .      by lemma  [ lemma - chi p ] , our results on the partial euler 
   koszul characteristics of ranking toric modules reveal the combinatorial nature of rank jumps of the generalized @xmath0-hypergeometric system @xmath34 .    by and 
   lemma  [ lemma - chi p ] , @xmath456 , so the result is an immediate consequence of theorem  [ thm - main ] and proposition  [ prop : alg ] . 
   [ example-2 cmpts ] if @xmath26 is such that @xmath707 , the proof of theorem  [ thm - main ] and section  [ subsec : compute ] show that the rank jump of @xmath29 at @xmath6 is @xmath708 \cdot \vol(f_i )      \right ) + |b_g^{\beta}| \cdot c^{\beta } \cdot\vol(g),\end{aligned}\ ] ] where @xmath709 and the constant @xmath710 is given by @xmath711    ( continuation of example  [ example-2 lines c4 ] ) [ example-2 lines c4 again ] with @xmath712 $ ] , the set @xmath713 , and @xmath714 by proposition  [ prop - p decomp ] . by , 
   @xmath715 \cdot \vol(f_i )          \right ) \ + \ 
   |\widehat{\beta ' } \cap b_g^{\beta}| \cdot c^{\beta } \cdot\vol(g)\\      & = & 2 + 2 + 1\cdot(-2)\cdot 1 \ = \ 2 , \end{aligned}\ ] ] and @xmath716 by corollary  [ cor - simple ] . 
   thus proposition  [ prop - p decomp ] implies that the rank jump of the @xmath0-hypergeometric system @xmath717 is @xmath718 . 
   when @xmath76 and @xmath719 for some @xmath481 , ( * ? ? 
   * theorem  2.6 ) implies that the rank jump @xmath62 of @xmath29 at @xmath6 corresponds to the reduced homology of the lattice @xmath384 . 
   the formula given by okuyama involves this homology and the volumes of the 1-dimensional faces of @xmath0 in @xmath384 . for higher - dimensional cases , 
   the cellular structure of the complex @xmath720 of notation  [ not - cpx i ] shows that , in general , more information than the reduced homology of @xmath384 is needed to compute @xmath62 , or even a single @xmath494 . 
   recall from definition  [ def - ranking slab ] that a ranking slab of @xmath29 is a stratum in the coarsest stratification of @xmath48 that respects a specified collection of subspace arrangements . 
   we are now prepared to prove corollary  [ cor - strat vs equiv ] , which states that the ranking slab stratification of @xmath48 refines its rank stratification . from this 
   it follows that each @xmath374 is a union of ranking slabs , making each a union of translated linear subspaces of @xmath7 . 
   if @xmath84 belong to the same ranking slab , then the ranking lattices @xmath721 coincide by proposition  [ prop - zc e ] . by theorem  [ thm - compute jump ] , 
   the rank jumps @xmath62 and @xmath722 coincide as well . 
   [ cor - eea union linear translates ] for all integers @xmath186 , @xmath87 is a union of translates of linear subspaces that are generated by faces of @xmath0 . 
   this is an immediate consequence of theorem  [ thm - str comparison ] and corollary  [ cor - strat vs equiv ] . 
   the following is the second example promised at the end of section  [ sec : exceptional arrangement ] , showing that the rank of @xmath34 need not be constant on a slab ( see definition  [ def : slab ] ) . 
   further , this example shows that neither the arrangement stratification of @xmath48 nor its refinement given by the @xmath47 modules in determine its rank stratification .    [ example - nonconstant slab ] consider the matrix @xmath723\ ] ] with @xmath724 , and label the faces @xmath725,$ ] @xmath726 $ ] , and @xmath727 $ ] . with @xmath728 $ ] and @xmath729 ,      \left[\begin{smallmatrix } 3 \\ 2 \\ 3 \end{smallmatrix}\right ] ,      \left[\begin{smallmatrix } 3 \\ 5 \\ 3 \end{smallmatrix}\right ] ,      \left[\begin{smallmatrix } 5 \\ 3 \\ 3 \end{smallmatrix}\right ] ,      \left[\begin{smallmatrix } 3 \\ 3 \\ 5 \end{smallmatrix}\right ] ,      \left[\begin{smallmatrix } 5 \\ 5 \\ 6 \end{smallmatrix}\right ]      \right\}$ ] , the exceptional arrangement of @xmath21 is @xmath730 for @xmath731 , @xmath732\setminus 
   \beta'$,}\\      \bigcup_{i=1}^3 [ \beta ' + \cc f_i ]      & \text{if $ \beta = \beta'$,}\\      \end{cases}\ ] ] so by the proof of theorem  [ thm - main ] , the rank jump of @xmath29 at @xmath61 is @xmath733\setminus\beta'$,}\\      2 & \text{otherwise . } 
   \end{cases}\ ] ] here the arrangement stratification of @xmath365 agrees with the one given by the @xmath47 modules that determine it , but @xmath86 is not constant on the slab @xmath734\subseteq\ee_a(s_a)$ ] .    to show that all of the arrangements in the definition of ranking slabs ( definition  [ def - ranking slab ] ) are necessary to obtain a refinement of the rank stratification of @xmath48 
   , we include the following example . here 
   , @xmath62 changes where components of @xmath365 that correspond to different @xmath47 modules intersect . 
   [ plane - line-4diml ] let @xmath735,\ ] ] @xmath736 $ ] , @xmath737 $ ] , and @xmath738 $ ] . here 
   @xmath739 , and the exceptional arrangement of @xmath21 is @xmath740 \cup [ \beta ' + { { \ensuremath{\mathbb{c}}}}g],\ ] ] where @xmath741 by corollary  [ cor - simple ] and example  [ example-2 cmpts ] , @xmath742\setminus\beta'$ , } \\      4 & \text{if $ \beta\in[\beta'+{{\ensuremath{\mathbb{c}}}}g]\setminus\beta'$. }      \end{cases}\ ] ]    we include a final example to show that @xmath62 is not determined simply by @xmath743 , the holes in the semigroup @xmath54 . 
   [ ex - first example ] the matrix @xmath744\ ] ] has volume 16 . 
   the exceptional arrangement of @xmath324 is the union of four lines and a point : @xmath745       + { { \ensuremath{\mathbb{c}}}}f      \right )      \cup      \left (      \left[\begin{smallmatrix }      1\\ 1\\ 0      \end{smallmatrix}\right ]       + { { \ensuremath{\mathbb{c}}}}g      \right )      \cup      \left (      \left[\begin{smallmatrix }      1\\ 0\\ 0      \end{smallmatrix}\right ]       + { { \ensuremath{\mathbb{c}}}}[a_5 ]      \right )      \cup      \left (      \left[\begin{smallmatrix }      0\\ 1\\ 0      \end{smallmatrix}\right ]       + { { \ensuremath{\mathbb{c}}}}[a_6 ]      \right )      \cup      \left\ {      \left[\begin{smallmatrix }      2\\ 2\\ 1      \end{smallmatrix}\right ]      \right\},\ ] ] where @xmath746 $ ] and @xmath747 $ ] . 
   the generic rank jumps along each component are as follows : @xmath748       + { { \ensuremath{\mathbb{c}}}}f       \mapsto   3 , \quad            \left[\begin{smallmatrix }      1\\ 0\\ 0      \end{smallmatrix}\right ]       + { { \ensuremath{\mathbb{c}}}}[a_5 ]       \mapsto   1 , \quad           \left[\begin{smallmatrix }      1\\ 1\\ 0      \end{smallmatrix}\right ]       + { { \ensuremath{\mathbb{c}}}}g       \mapsto   3 , \quad          \left[\begin{smallmatrix }      0\\ 1\\ 0      \end{smallmatrix}\right ]       + { { \ensuremath{\mathbb{c}}}}[a_6 ]       \mapsto   1 , \quad          \text{and }      \left[\begin{smallmatrix }      2\\ 2\\ 1      \end{smallmatrix}\right ]       \mapsto   2 .   \end{aligned}\ ] ] these generic rank jumps are achieved everywhere except at the points @xmath749 ,      \left[\begin{smallmatrix }      1\\ 0\\ 0      \end{smallmatrix}\right ] ,      \left[\begin{smallmatrix }      0\\ 1\\ 0      \end{smallmatrix}\right ] , \text { and }      \left[\begin{smallmatrix }      \phantom{-}0\\ \phantom{-}0\\ -1      \end{smallmatrix}\right].\ ] ] the point that may be unexpected in this collection is @xmath750 $ ] . 
   both of the components of @xmath365 that contain @xmath90 have generic rank jumps of 1 ; however , @xmath751 . 
   this is because the ranking arrangement of @xmath21 has three components that contain @xmath90 : @xmath752     \right )         \cup      \left (     b + { { \ensuremath{\mathbb{c}}}}[a_6 ]     \right )     \cup      \left (     b + { { \ensuremath{\mathbb{c}}}}[a_1\ a_2\ a_3\ a_4 ]     \right ) . 
   \end{aligned}\ ] ] if the plane @xmath753)$ ] were not in @xmath754 , then the rank jump of @xmath21 at @xmath90 would only be 1 by . 
   thus the hyperplane in , although unrelated to the holes in @xmath54 , accounts for the higher value of @xmath755 . 
   the rank jumps at the other parameters are @xmath756 \right ) = j\left ( s_a ,       \left[\begin{smallmatrix }      0\\ 1\\ 0      \end{smallmatrix}\right ] \right ) = 3 \quad \text { and } \quad j\left ( s_a ,       \left[\begin{smallmatrix }      1\\ 1\\ 0      \end{smallmatrix}\right ] \right ) = 5.\ ] ]    the algebraic upper semi - continuity of the rank of @xmath34 implies that most of the codimension 1 components of the ranking arrangement @xmath59 do not increase the rank of @xmath34 . 
   it would interesting to know if the set of such hyperplanes can be identified . 
   when @xmath324 , the results of section  [ sec : comb of rank ] apply to the @xmath0-hypergeometric system @xmath366 . for a face @xmath567 of @xmath0 , @xmath757 is a finite set . 
   it is shown in @xcite and @xcite that @xmath1 and @xmath758 are isomorphic as @xmath4-modules precisely when @xmath759 for all faces @xmath567 of @xmath0 . 
   we will now use euler  koszul homology to give a simple proof of one direction of this equivalence ; first we exhibit a complementary relationship between @xmath89 and @xmath760 ( see ) . 
   [ obs - etau ] it is shown in theorem  [ thm - compute jump ] that as @xmath567 runs through the faces of @xmath0 , the sets @xmath761 determine the rank jump of @xmath1 at @xmath6 . 
   notice that @xmath762 is the complement of @xmath89 in the group @xmath763 .            if @xmath770 is not an isomorphism , then @xmath771 by lemma  [ lemma - nisom ] . by the definition of quasidegrees 
   , there exist vectors @xmath772 and @xmath773 for some face @xmath567 such that @xmath774 , @xmath775 , and @xmath776 for all @xmath140 . 
   hence @xmath777 , so @xmath778 . 
   further , the condition @xmath779 for all @xmath140 implies that @xmath780 . 
   for the  if " direction , suppose that @xmath783 for all faces @xmath567 of @xmath0 . as stated in ( 
   * ; * ? ? ? 
   * proposition  2.2 ) , @xmath784 implies by definition that @xmath785 , so @xmath764 for some @xmath786 . 
   there are unique vectors @xmath787 with disjoint support such that @xmath788 . in light of lemma  [ lemma - netau ] 
   , we may assume that both @xmath789 and @xmath790 are nonzero . 
   we claim that at least one of @xmath791 or @xmath792 defines an isomorphism from @xmath1 to @xmath758 . 
   if @xmath793 or @xmath794 defines an isomorphism , then the  only - if " direction and lemma  [ lemma - netau ] imply that @xmath791 or @xmath792 , respectively , give the desired isomorphism . 
   we are left to consider the case when @xmath795 does not define an isomorphism from either domain . by lemma  [ lemma - nisom ] , this is equivalent to @xmath796 from the right side of , we see that the nonempty face @xmath797 is such that @xmath798 , so @xmath799 . however , the shift @xmath800 in the left side of implies that @xmath801 . 
   thus @xmath802 , which is a contradiction . 
   it is not yet understood how the holomorphic solution space of @xmath1 varies as a function of @xmath6 ; different functions of @xmath6 suggest alternative behaviors . 
   walther showed in @xcite that the reducibility of the monodromy of @xmath1 varies with @xmath6 in a lattice - like fashion . when the convex hull of @xmath0 and the origin is a simplex , saito used the sets @xmath89 to construct a basis of holomorphic solutions of @xmath1 with a common domain of convergence @xcite . 
   thus theorem  [ thm - compute jump ] and the complementary relationship in observation  [ obs - etau ] between the @xmath89 and the ranking lattices of @xmath21 at @xmath6 suggest that the ranking slabs give the coarsest stratification over which there could be a constructible sheaf of solutions for the hypergeometric system . 
   victor  v.  batyrev and duco  van  straten , _ generalized hypergeometric functions and rational curves on calabi  yau complete intersections in toric varieties _ , ( english summary ) comm . math . phys . 
   * 168 * ( 1995 ) , no .  3 , 493533 .            i.  m.  gelfand , a.  v.  zelevinski , and m.  m.  kapranov , _ hypergeometric functions and toric varieties _ , funktsional 
   i prilozhen . 
   * 23 * ( 1989 ) , no .  2 , 1226 . 
   correction in ibid , * 27 * ( 1993 ) , no .  4 ,  91 .                                  mutsumi  saito and william  n.  traves , _ differential algebras on semigroup algebras _ , _ symbolic computation : solving equations in algebra , geometry , and engineering _ ( south hadley , ma , 2000 ) , 207226 , contemp . math 
   , * 286 * , amer . 
   soc . , providence , ri , 2001 . 
   bernd  sturmfels , _ solving algebraic equations in terms of @xmath0-hypergeometric series _ ( english summary ) , formal power series and algebraic combinatorics ( minneapolis , mn , 1996 ) , discrete math . * 210 * ( 2000 ) , no .  1 - 3 , 171181 . 
   bernd  sturmfels and nobuki  takayama , _ grbner bases and hypergeometric functions _ , grbner bases and applications ( linz , 1998 ) , 246258 , london math . 
   lecture note ser . , * 251 * , cambridge univ . press , cambridge , 1998 .
summary: the holonomic rank of the @xmath0-hypergeometric system @xmath1 is the degree of the toric ideal @xmath2 for generic parameters ; in general , this is only a lower bound . to the semigroup ring of @xmath0 
   we attach the ranking arrangement and use this algebraic invariant and the exceptional arrangement of nongeneric parameters to construct a combinatorial formula for the rank jump of @xmath1 . as consequences , we obtain a refinement of the stratification of the exceptional arrangement by the rank of @xmath1 and show that the zariski closure of each of its strata is a union of translates of linear subspaces of the parameter space . 
   these results hold for generalized @xmath0-hypergeometric systems as well , where the semigroup ring of @xmath0 is replaced by a nontrivial weakly toric module @xmath3 $ ] . 
   we also provide a direct proof of the main result in @xcite regarding the isomorphism classes of @xmath1 .