\(H\)-vectors of simplicial complexes with Serre's conditions

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Abstract: We study

h

-vectors of simplicial complexes which satisfy Serre's condition (

S_{r}

). We say that a simplicial complex

D e l t a

satisfies Serre's condition (

S_{r}

) if

i l d e H_{i} (l k_{D} e l t a (F); K) = 0

for all faces

F i n D e l t a

and for all

i < m i n r - 1, d i m l k_{D} e l t a (F)

, where

l k_{D} e l t a (F)

is the link of

D e l t a

with respect to

F

and where

i l d e H_{i} (D e l t a; K)

is the reduced homology groups of

D e l t a

over a field

K

. The main result of this paper is that if

D e l t a

satisfies Serre's condition (

S_{r}

) then (i)

h_{k} (D e l t a)

is non-negative for

k = 0, 1, ..., r

and (ii)

s u m_{k g e q r} h_{k} (D e l t a)

is non-negative.

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